Using ANSYS for Finite Element Analysis, Volume II: Dynamic, Probabilistic Design and Heat Transfer Analysis [2] 1947083228, 9781947083226

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Using ANSYS for Finite Element Analysis, Volume II: Dynamic, Probabilistic Design and Heat Transfer Analysis [2]
 1947083228, 9781947083226

Table of contents :
Cover
Contents
List of Figures
Preface
Chapter 1: Dynamic Analysis
Chapter 2: Composite Materials
Chapter 3: Probabilistic Design Analysis
Chapter 4: APDL Programming
Chapter 5: Design Optimization
Bibliography
About the Authors
Index
Adpage
Backcover

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THE CONTENT

Dynamic, Probabilistic Design and Heat Transfer Analysis, Volume II Wael A. Altabey • Mohammad Noori • Libin Wang Over the past two decades, the use of finite element method as a design tool has grown rapidly. Easy to use commercial software, such as ANSYS, have become common tools in the hands of students as well as practicing engineers. The objective of this book is to demonstrate the use of one of the most commonly used Finite Element Analysis software, ANSYS, for linear static, dynamic, and thermal analysis through a series of tutorials and examples. Some of the topics covered in these tutorials include development of beam, frames, and Grid Equations; 2-D elasticity problems; dynamic analysis; composites, and heat transfer problems. These simple, yet, fundamental tutorials are expected to assist the users with the better understanding of finite element modeling, how to control modeling errors, and the use of the FEM in designing complex load bearing components and structures. These tutorials would supplement a course in basic finite element or can be used by practicing engineers who may not have the advanced training in finite element analysis. Wael A. Altabey is an assistant professor in the department of mechanical engineering, faculty of engineering, Alexandria University, Alexandria, Egypt and has been a postdoctoral researcher at the

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International Institute for Urban Systems Engineering, Southeast

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For further information, a free trial, or to order, contact:  [email protected]

Mohammad Noori is a professor of mechanical engineering at California Polytechnic State University in San Luis Obispo, California, USA, and a fellow of the American Society of Mechanical Engineers. Dr Noori has over 34 years of experience as a scholar and educator. He has also been a distinguished visiting professor at the International Institute for Urban Systems Engineering, Southeast University, Nanjing, China. Libin Wang is a professor and the dean of the school of civil engineering at Nanjing Forestry University, in Nanjing, China. He has been an educator and scholar, for over 20 years, and has taught the subject of finite element analysis both at the undergraduate and graduate level.

ISBN: 978-1-94708-322-6

Using ANSYS for Finite Element Analysis, Volume II

• Manufacturing Engineering • Mechanical & Chemical Engineering • Materials Science & Engineering • Civil & Environmental Engineering • Advanced Energy Technologies

Using ANSYS for Finite Element Analysis

ALTABEY • NOORI • WANG

EBOOKS FOR THE ENGINEERING LIBRARY

SUSTAINABLE STRUCTURAL SYSTEMS COLLECTION Mohammad Noori, Editor

Using ANSYS for Finite Element Analysis Dynamic, Probabilistic Design and Heat Transfer Analysis Volume II

Wael A. Altabey Mohammad Noori Libin Wang

Using ANSYS for Finite Element Analysis

Using ANSYS for Finite Element Analysis Dynamic, Probabilistic Design and Heat Transfer Analysis Volume II

Wael A. Altabey, Mohammad Noori, and Libin Wang

MOMENTUM PRESS, LLC, NEW YORK

Using ANSYS for Finite Element Analysis: Dynamic, Probabilistic Design and Heat Transfer Analysis, Volume II Copyright © Momentum Press®, LLC, 2018. All rights reserved. No part of this publication may be reproduced, stored in a retrieval system, or transmitted in any form or by any means—­ electronic, mechanical, photocopy, recording, or any other—except for brief quotations, not to exceed 400 words, without the prior permission of the publisher. First published by Momentum Press®, LLC 222 East 46th Street, New York, NY 10017 www.momentumpress.net ISBN-13: 978-1-94708-322-6 (print) ISBN-13: 978-1-94708-323-3 (e-book) Momentum Press Sustainable Structural Systems Collection Collection ISSN: 2376-5119 (print) Collection ISSN: 2376-5127 (electronic) Cover and interior design by Exeter Premedia Services Private Ltd., Chennai, India 10 9 8 7 6 5 4 3 2 1 Printed in the United States of America

Abstract Finite Element Method (FEM) is a well-established technique for analyzing the behavior and the response of structures or mechanical components under static, dynamic, or thermal loads. Over the past two decades the use of finite element analysis as a design tool has grown rapidly. Easy to use commercial software have become common tools in the hands of students as well as practicing engineers. The objective of this two volume book is to demonstrate the use of one of the most commonly used Finite Element Analysis software, ANSYS, for linear static, dynamic, and thermal analysis through a series of tutorials and examples. Some of the topics and concepts covered in these tutorials include development of beam, frames, and grid equations; 2-D elasticity problems; dynamic analysis; and heat transfer problems. We are hoping these simple, yet, fundamental tutorials will assist the users with the better understanding of finite element modeling, how to control modeling errors, the safe use of the FEM in support of designing complex load bearing components and structures. There are many good textbooks currently used for teaching the fundamentals of finite element methods. There are also detailed users manuals available for commercial software (ANSYS). However, those sources are useful for advanced students and users. Therefore, there was a need to develop a tutorial that would supplement a course in basic finite element or can be used by practicing engineers who may not have the advanced training in finite element analysis. That is the gap addressed by this book.

Keywords ANSYS, composite materials, Dynamics, Failure analysis, Fatigue loads, FEM, optimization, statistics

Contents List of Figures

ix

Preface 

xi

1  Dynamic Analysis  1.1 Tutorial 1: Harmonic Analysis of Structure  1.2 Tutorial 2: Modal Analysis of Structure 2  Composite Materials

1 1 11 19

2.1  Composites—A Basic Introduction

19

2.2  Modeling Composites Using ANSYS

26

2.3 Tutorial 3: Simply Supported Laminated Plate Under Pressure

38

3  Probabilistic Design Analysis

69

3.1  Probabilistic Design

69

3.2  Probability Distributions

75

3.3 Choosing a Distribution for a Random Variable

91

3.4  Probabilistic Design Techniques

96

3.5 Postprocessing Probabilistic Analysis Results

98

3.6 Tutorial 4: Probabilistic Design Analysis of Circular Plate Bending 4  APDL Programming

107 145

4.1  Create the Analysis File

145

4.2 Tutorial 5: Stress Analysis of Bicycle Wrench

148

4.3 Tutorial 6: Heat Loss from a Cylindrical Cooling Fin

155

viii  •  Contents

5  Design Optimization

167

5.1  Optimum Design

167

5.2  Design Optimization Using ANSYS

172

5.3  Tutorial 7: Design Optimization Tutorial

193

Bibliography

217

About the Authors

219

Index

221

List of Figures Figure 2.1. Illustrating the combined effect on modulus of the addition of fibers to a resin matrix.

21

Figure 2.2. Illustrates the tensile load applied to a composite body.22 Figure 2.3. Illustrates the compression load applied to a composite body.

22

Figure 2.4. Illustrates the shear load applied to a composite body.23 Figure 2.5. Illustrates the loading due to flexure on a composite body.23 Figure 2.6.  Tensile strength of common structural materials.

24

Figure 2.7.  Tensile modulus of common structural materials.

25

Figure 2.8. Specific tensile strength of common structural materials.25 Figure 2.9. Specific tensile modulus of common structural materials.26 Figure 2.10.  Layered model showing dropped layer.

29

Figure 2.11.  Sandwich construction.

32

Figure 2.12.  Layered shell with nodes at midplane.

32

Figure 2.13.  Layered shell with nodes at bottom surface.

33

Figure 2.14.  Example of an element display.

36

Figure 2.15. Sample LAYPLOT display for [45/−45/−45/45] sequence.37 Figure 3.1. The flow of information during a probabilistic design analysis.75 Figure 3.2.  The normal probability density function (pdf).

77

x  •   List of Figures

Figure 3.3.  The normal cumulative distribution function (cdf).

78

Figure 3.4.  The uniform probability density function.

83

Figure 3.5.  The uniform cumulative distribution function.

84

Figure 3.6. The lognormal probability density function for four values of s.86 Figure 3.7. The lognormal cumulative distribution function for four values of s.87 Figure 3.8. The Weibull probability density function for four values of g.89 Figure 3.9. The Weibull cumulative distribution function for four values of g.90 Figure 3.10. The graph of X1 and X2 illustrating bad sample distribution.97 Figure 3.11. The graph of X1 and X2 illustrating good sample distribution.98 Figure 3.12. The cumulative distribution function of the random property X.

100

Figure 5.1.  Optimization tree listing the optimization methods.

171

Figure 5.2.  Optimization data flow.

175

Preface Finite element method (FEM) is a well-established technique for analyzing the behavior of mechanical and structural components of systems. In recent years, the use of finite element analysis (FEA) as a design tool has grown rapidly. Easy-to-use commercial software have become common tools in the hands of students as well as practicing engineers. In the first volume of this tutorial, we demonstrated the use of ANSYS for Static Analysis of solid structures. In this volume we introduce the ­following applications: • • • • •

Dynamic Analysis Composite Materials Probabilistic Design Heat Transfer Design Optimization Problems.

The main objective of this book is to serve as a practical tutorial to help the readers gain insight into appropriate use of finite element modeling, understand how to control modeling errors, benefit from hands-on exercise at the computer workstation, and understand the safe use of the FEM in support of designing complex load-bearing components and structures. There are many good textbooks already in existence that cover the theory of FEMs. Similarly, there are detailed user’s manuals available for commercial software (ANSYS). But, those are useful for advanced students and users. Therefore, there was a need to develop a computer session manual in line with the flow of a course and utilizing a software platform, ANSYS, that is available in most engineering schools. Students will be able to acquire the required level of understanding and skill in modeling, analysis, validation, and report generation for various design problems. This book could also be very helpful for the students of senior design (Mechanical System Design) and FEA for Large Deformation Problems.

xii  •  Preface

In addition, it could be used for computer sessions of short courses on stress analysis techniques and FEA offered by Mechanical Engineering departments. After giving a brief introduction to the finite element analysis and modeling, various guided examples have been included in this book. Several new tutorials have been developed and others adapted from ­ ­different sources including ANSYS manuals in 2-volumes, ANSYS workshops, and Internet resources. Tutorials have been arranged in each ­volume according to the flow of the course and covered topics, such as solid modeling using 2D and 3D primitives available in ANSYS, dynamic analysis (harmonic and modal analysis), and thermal analysis.

Chapter 1

Dynamic Analysis Structural analysis is mainly concerned with finding out the behavior of a physical structure when subjected to force. This action can be in the form of load due to the weight of things such as people, furniture, wind, snow, and so on, or some other kind of excitation such as an earthquake, shaking of the ground due to a blast nearby, and so on. In essence all these loads are dynamic, including the self-weight of the structure because at some point in time these loads were not there. The distinction is made between the dynamic and the static analysis on the basis of whether the applied action has enough acceleration in comparison to the structure’s natural frequency. If a load is applied sufficiently slowly, the inertia forces (Newton’s first law of motion) can be ignored and the analysis can be simplified as static analysis. Structural dynamics, therefore, is a type of structural analysis that covers the behavior of structures subjected to dynamic (actions having high acceleration) loading. Dynamic loads include people, wind, waves, traffic, earthquakes, and blasts. Any structure can be subjected to dynamic loading. Dynamic analysis can be used to find dynamic displacements, time history, and modal analysis. A dynamic analysis is also related to the inertia forces developed by a structure when it is excited by means of dynamic loads applied suddenly (e.g., wind blasts, explosion, and earthquake). Dynamic analysis for simple structures can be carried out manually, but for complex structures finite element analysis can be used to calculate the mode shapes and frequencies.

1.1 Tutorial 1: Harmonic Analysis of Structure In this tutorial, the harmonic analysis of a cantilever beam will be addressed. Harmonic analysis is used to determine the response of a

2  •  Using ANSYS for Finite Element Analysis

structure to harmonically time-varying loads. This tutorial was created using ANSYS 7.0. The purpose of this tutorial is to explain the steps required to perform harmonic analysis on the cantilever beam shown as follows.

0.01 m 0.01 m

1.0 m

Modulus of elasticity (E) = 206800(106) N/m2 Density = 7830 kg/m3

We will now conduct a harmonic forced response test by applying a cyclic load (harmonic) at the end of the beam. The frequency of the load will be varied from 1– 100 Hz. The following figure depicts the beam with the application of the load.

Cyclic load Magnitude: 100 N Frequency range: 1–100 Hz.

ANSYS provides three methods for conducting a harmonic analysis. These three methods are the Full, Reduced, and Modal Superposition methods. This example demonstrates the Full method because it is simple and easy to use as compared to the other two methods. However, this method makes use of the full stiffness and mass matrices and thus is the slower and costlier option.

1.1.1  Step-by-Step ANSYS Solution 1.1.1.1  Preprocessing: Defining the Problem Student should be able to make simple cantilever model himself or otherwise use the following command list:

Dynamic Analysis   •  3

1.1.1.2  The Command Log File /TITLE, Dynamic Analysis /FILNAME,Dynamic,0 /PREP7 K,1,0,0 K,2,1,0 L,1,2 ET,1,BEAM3 R,1,0.0001,8.33e-10,0.01 MP,EX,1,2.068e11 MP,PRXY,1,0.33 MP,DENS,1,7830 LESIZE,ALL,,,10 LMESH,1 FINISH

! This sets the jobname to “Dynamic”

1.1.1.3  Solution: Assigning Loads and Solving 1. Define analysis type (Harmonic) Solution > Analysis Type > New Analysis > Harmonic ANTYPE,3 2. Set options for analysis type Select Solution > Analysis Type > Analysis Options. The following window will appear. As shown, select the Full Solution method, the Real + imaginary DOF (degrees of freedom) printout format and do not use lumped mass approx. Click “OK”

The following window will appear. Use the default settings (shown as follows).

4  •  Using ANSYS for Finite Element Analysis

3. Apply constraints Select Solution > Define Loads > Apply > Structural > Displacement > On Nodes The following window will appear once you select the node at x=0 (Note small changes in the window compared to the static examples):

Constrain all DOF as shown in the above window. 4. Apply Loads Select Solution > Define Loads > Apply > Structural > Force/Moment > On Nodes Select the node at x=1 (far right). The following window will appear. Fill it in as shown to apply a load with a real value of 100 and an imaginary value of 0 in the positive “y” direction. Note: By specifying a real and imaginary value of the load we are providing information on magnitude and phase of the load. In this case the magnitude of the load is 100 N and its phase is 0. Phase information is important when you have two or more cyclic loads

Dynamic Analysis   •  5

being applied to the structure as these loads could be in or out of phase. For harmonic analysis, all loads applied to a structure must have the SAME FREQUENCY.

5. Set the frequency range Select Solution > Load Step Opts > Time/Frequency > Freq and Substps... As shown in the f o l l o w i n g window, specify a frequency range of 0–100 Hz, 100 substeps, and stepped b.c. By doing this we will be subjecting the beam to loads at 1 Hz, 2 Hz, 3 Hz, ..... 100 Hz. We will specify a stepped boundary condition (KBC) as this will ensure that the same amplitude (100 N) will be applied for each of the frequencies. The ramped option, on the other hand, would ramp up the amplitude where at 1 Hz the amplitude would be 1 N and at 100 Hz the amplitude would be 100 N.

You should now have the following in the ANSYS Graphics window

6  •  Using ANSYS for Finite Element Analysis

6. Solve the system Solution > Solve > Current LS SOLVE 1.1.1.4  Postprocessing: Viewing the Results We want to observe the response at x=1 (where the load was applied) as a function of frequency. We cannot do this with General PostProcessing (POST 1); rather we must use TimeHist Post-Processing (POST26). POST26 is used to observe certain variables as a function of either time or frequency. 1. Open the TimeHist Processing (POST26) Menu Select TimeHist Postpro from the ANSYS Main Menu. 2. Define variables Here we have to define variables that we want to see plotted. By default, Variable 1 is assigned either Time or Frequency. In our case it is assigned Frequency. We want to see the displacement UY at the node at x=1, which is node #2. (To get a list of nodes and their attributes, select Utility Menu > List > nodes.) Select Time Hist Postpro > Variable Viewer...

Dynamic Analysis   •  7

And the following window should pop up:

Select Add (the green “+” sign in the upper left corner) from this window and the following window should appear:

8  •  Using ANSYS for Finite Element Analysis

We are interested in the Nodal Solution > DOF Solution > Y-Component of displacement. Click OK. Graphically select node 2 when prompted and click OK. The “Time History Variables” window should now look as follows:

3. List Stored Variables In the “Time History Variables” window click the “List” ­button, three buttons to the left of “Add.” The following window will appear listing the data:

Dynamic Analysis   •  9

4. Plot UY vs. frequency In the “Time History Variables” window click the “Plot” button, two buttons to the left of “Add.” The following graph should be plotted in the main ANSYS window.

Note that we get peaks at frequencies of approximately 8.3 and 51 Hz. This corresponds with the predicted frequencies of 8.311 and 51.94 Hz. To get a better view of the response, view the log scale of UY. Select Utility Menu > PlotCtrls > Style >Graphs > Modify Axis The following window will appear:

10  •  Using ANSYS for Finite Element Analysis

As marked by an “A” in the above window, change the Y-axis scale to “Logarithmic.” Select Utility Menu > Plot > Replot You should now see the following:

This is the response at node 2 for the cyclic load applied at this node from 0–100 Hz.

Dynamic Analysis   •  11

For ANSYS version lower than 7.0, the “Variable Viewer” window is not available. Use the “Define Variables” and “Store Data” functions under TimeHist Postpro. See the help file for instructions.

1.2 Tutorial 2: Modal Analysis of Structure In this tutorial, the modal analysis of a cantilever beam will be addressed. Modal analysis—used to calculate the natural frequencies and mode shapes of a structure. Different mode extraction methods are available. This tutorial was created using ANSYS 7.0. The purpose of this t­utorial is to outline the steps required to do a simple modal analysis of the ­cantilever beam shown as follows.

0.01 m

1.0 m

0.01 m

Modulus of elasticity (E) = 206800(106) N/m2 Density = 7830 kg/m3

1.2.1  Step-by-Step ANSYS Solution 1.2.1.1  Preprocessing: Defining the Problem The simple cantilever beam is same as used in the previous tutorial. 1.2.1.2  Solution: Assigning Loads and Solving 1. Define analysis type Solution >Analysis Type > NewAnalysis > Modal ANTYPE,2 2. Set options for analysis type: Select Solution > Analysis Type > Analysis Options..

12  •  Using ANSYS for Finite Element Analysis

The following window will appear:

As shown, select the Subspace method and enter 5 in the “No. of modes to extract.” Check the box beside “Expand mode shapes” and enter 5 in the “No. of modes to expand.” Click “OK.” Note that the default mode extraction method chosen is the Reduced Method. This is the fastest method as it reduces the system matrices to only consider the Master Degrees of Freedom (MDOFs) (see the following figure). The Subspace Method extracts modes for all DOFs. It is therefore more exact but it also takes longer to ­compute (especially when the geometry is complex). The following window will then appear:

Dynamic Analysis   •  13

For a better understanding of these options see the Commands manual. For this problem, we will use the default options so click on OK. 1. Apply constraints Solution > Define Loads > Apply > Structural > Displacement > On Keypoints Fix Keypoint 1 (i.e., all DOFs constrained). 2. Solve the system Solution > Solve > Current LS SOLVE 1.2.1.3  Postprocessing: Viewing the Results 1. Verify extracted modes against theoretical predictions Select General Postproc > Results Summary... The following window will appear:

14  •  Using ANSYS for Finite Element Analysis

The following table compares the mode frequencies (in Hz) predicted by theory and ANSYS. Mode 1 2 3 4 5

Theory    8.311  51.94 145.68 285.69 472.22

ANSYS    8.300  52.01 145.64 285.51 472.54

Percent error 0.1 0.2 0.0 0.0 0.1

Note: To obtain accurate higher mode frequencies, this mesh would have to be refined even more (i.e., instead of ten elements, we would have to model the cantilever using 15 or more ­elements depending on the highest mode frequency of interest). 2. View mode shapes Select General Postproc > Read Results > First Set This selects the results for the first mode shape. Select General Postproc > Plot Results > Deformed shape. Select “Def + undef edge” The first mode shape will now appear in the graphics window. To view the next mode shape, select General Postproc > Read Results > Next Set As above choose General Postproc > Plot Results > Deformed shape. Select “Def + undef edge.” The first four mode shapes should look like the following:

Dynamic Analysis   •  15

3. Animate mode shapes Select Utility Menu (Menu at the top) > Plot Ctrls > Animate > Mode Shape The following window will appear:

Keep the default setting and click “OK.” The animated mode shapes for Mode 3 is shown as follows. Mode 3

16  •  Using ANSYS for Finite Element Analysis

1.2.1.4  Using the Reduced Method for Modal Analysis This method employs the use of MDOFs. These are DOFs that govern the dynamic characteristics of a structure. For example, the MDOFs for the bending modes of cantilever beam are: Master degrees of freedom (MDOFs)

For this option, a detailed understanding of the dynamic behavior of a structure is required. However, going this route means a smaller (reduced) stiffness matrix, and thus faster calculations. The steps for using this option are quite simple. Instead of specifying the Subspace method, select the Reduced method and specify 5 modes for extraction. Complete the window as shown in the following:

Note: For this example both the number of modes and frequency range was specified. ANSYS then extracts the minimum number of modes between the two. Select Solution > Master DOF > User Selected > Define When prompted, select all nodes except the left-most node (fixed).

Dynamic Analysis   •  17

The following window will appear:

Select UY as the 1st degree of freedom (shown above). The same constraints are used as above. The following table compares the mode frequencies ( in Hz) predicted by theory and ANSYS (Reduced). Mode 1 2 3 4 5

Theory    8.311  51.94 145.68 285.69 472.22

ANSYS    8.300  52.01 145.66 285.71 473.66

Percent error 0.1 0.1 0.0 0.0 0.3

As you can see, the error does not change significantly. However, for more complex structures, larger errors would be expected using the reduced method.

Chapter 2

Composite Materials 2.1 Composites—A Basic Introduction Composite materials have been used in structures for a long time. In recent times, composite parts have been used extensively in aircraft structures, automobiles, sporting goods, and many consumer products. Composite materials are those containing more than one bonded material, each with different structural properties. The main advantage of composite materials is the potential for a high ratio of stiffness to weight. Composites used for typical engineering applications are advanced fiber or laminated composites, such as fiberglass, glass epoxy, graphite epoxy, and boron epoxy. To fully appreciate the role and application of composite materials to a structure, an understanding is required of the component materials themselves and of the ways in which they can be processed; this article therefore looks at basic composite theory and properties of materials used. In its most basic form a composite material is one that is composed of at least two elements working together to produce material properties that are different to the properties of those elements on their own. In practice, most composites consist of a bulk material (the “matrix”), and a reinforcement of some kind, added primarily to increase the strength and stiffness of the matrix. This reinforcement is usually in fiber form. Today, the most common man-made composites can be divided into three main groups: • Polymer matrix composites (PMCs)—These are the most common and will be discussed here. Also known as FRP—fiber-reinforced polymers (or plastics)—these materials use a polymer-based resin as the matrix, and a variety of fibers such as glass, carbon, and ­aramid as the reinforcement. • Metal matrix composites (MMCs)—Increasingly found in the automotive industry, these materials use a metal such as aluminum as the matrix, and reinforce it with fibers such as silicon carbide.

20  •  Using ANSYS for Finite Element Analysis

• Ceramic matrix composites (CMCs)—Used in very high temperature environments, these materials use a ceramic as the matrix and reinforce it with short fibers, or whiskers such as those made from silicon carbide and boron nitride. 2.1.1  Polymer Matrix Composites Resin systems such as epoxies and polyesters have limited use for the manufacture of structures on their own, since their mechanical properties are not very high when compared to, for example, most metals. However, they have desirable properties, most notably their ability to be easily formed into complex shapes. Materials such as glass, aramid, and boron have extremely high tensile and compressive strength but in “solid form” these properties are not readily apparent. This is due to the fact that when stressed, random surface flaws will cause each material to crack and fail well below its theoretical “breaking point.” To overcome this problem, the material is produced in fiber form, so that, although the same number of random flaws will occur, they will be restricted to a small number of fibers with the remainder exhibiting the material’s theoretical strength. Therefore a bundle of fibers will reflect more accurately the optimum performance of the material. However, fibers alone can only exhibit tensile properties along the fiber’s length, in the same way as fibers in a rope. It is when the resin systems are combined with reinforcing fibers such as glass, carbon, and aramid that exceptional properties can be obtained. The resin matrix spreads the load applied to the composite between each of the individual fibers and also protects the fibers from damage caused by abrasion and impact. High strengths and stiffnesses, ease of molding complex shapes, high environmental resistance all coupled with low densities, make the resultant composite superior to metals for many applications. Since PMCs combine a resin system and reinforcing fibers, the properties of the resulting composite material will combine something of the properties of the resin on its own with that of the fibers on their own, as surmised in Figure 2.1. Overall, the properties of the composite are determined by: • The properties of the fiber. • The properties of the resin. • The ratio of fiber to resin in the composite (fiber volume fraction (FVF)). • The geometry and orientation of the fibers in the composite.

Composite Materials   •  21

Tensile stress

Fiber

FRP composite Resin Strain

Figure 2.1.  Illustrating the combined effect on modulus of the addition of fibers to a resin matrix.

The ratio of the fiber to resin derives largely from the manufacturing process used to combine resin with fiber. However, it is also influenced by the type of resin system used, and the form in which the fibers are incorporated. In general, since the mechanical properties of fibers are much higher than those of resins, the higher the fiber volume fraction the higher will be the mechanical properties of the resultant composite. In practice there are limits to this, since the fibers need to be fully coated in resin to be effective, and there will be an optimum packing of the generally circular cross-section fibers. In addition, the manufacturing process used to combine fiber with resin leads to varying amounts of imperfections and air inclusions. Typically, with a common hand lay-up process as widely used in the boat-building industry, a limit for FVF is approximately 30–40 percent. With the higher quality, more sophisticated and precise processes used in the aerospace industry, FVFs approaching 70 percent can be successfully obtained. The geometry of the fibers in a composite is also important since fibers have their highest mechanical properties along their lengths, rather than across their widths. This leads to the highly anisotropic properties of composites, where, unlike metals, the mechanical properties of the composite are likely to be very different when tested in different directions. This means that it is very important when considering the use of composites to understand at the design stage, both the magnitude and the direction of the applied loads. When correctly accounted for, these anisotropic properties can be very advantageous since it is only necessary to put material where loads will be applied, and thus redundant material is avoided. It is also important to note that with metals the material supplier largely determines the properties of the materials, and the person who

22  •  Using ANSYS for Finite Element Analysis

fabricates the materials into a finished structure can do almost nothing to change those “in-built” properties. However, a composite material is formed at the same time, as the structure is itself being fabricated. This means that the person who is making the structure is creating the properties of the resultant composite material, and so the manufacturing processes they use have an unusually critical part to play in determining the performance of the resultant structure. 2.1.1.1  Loading There are four main direct loads that any material in a structure has to withstand: tension, compression, shear, and flexure. Tension: Figure 2.2 shows a tensile load applied to a composite. The response of a composite to tensile loads is very dependent on the tensile stiffness and strength properties of the reinforcement fibers, since these are far higher than the resin system on its own. Compression: Figure 2.3 shows a composite under a compressive load. Here, the adhesive and stiffness properties of the resin system are crucial, as it is the role of the resin to maintain the fibers as straight columns and to prevent them from buckling. Shear: Figure 2.4 shows a composite experiencing a shear load. This load is trying to slide adjacent layers of fibers over each other. Under shear loads the resin plays the major role, transferring the stresses across the composite. For the composite to perform well under shear loads the resin element must not only exhibit good mechanical properties but must also have high adhesion to the reinforcement fiber. The interlaminar shear strength (ILSS) of

Figure 2.2.  Illustrates the tensile load applied to a composite body.

Figure 2.3.  Illustrates the compression load applied to a composite body.

Composite Materials   •  23

Figure 2.4.  Illustrates the shear load applied to a composite body.

Figure 2.5.  Illustrates the loading due to flexure on a composite body.

a composite is often used to indicate this property in a multi-layer composite (“laminate”). Flexure: Flexural loads are really a combination of tensile, compression, and shear loads. When loaded as shown in Figure 2 . 5 , the upper face is put into compression, the lower face into tension, and the central portion of the laminate experiences shear. 2.1.1.2  Comparison With Other Structural Materials Owing to the factors described previously, there is a very large range of mechanical properties that can be achieved with composite materials. Even when considering one fiber type on its own, the composite properties can vary by a factor of 10 with the range of fiber contents and orientations that are commonly achieved. The comparisons that follow therefore show a range of mechanical properties for the composite materials. The lowest properties for each material are associated with simple manufacturing processes and material forms (e.g., spray lay-up glass fiber), and the higher properties are associated with higher technology manufacture (e.g., autoclave molding of unidirectional glass fiber prepreg), such as would be found in the aerospace industry. For the other materials shown, a range of strength and stiffness (modulus) figures are

24  •  Using ANSYS for Finite Element Analysis

also given to indicate the spread of properties associated with different alloys, for example. Figures 2.6 and 2.7 clearly show the range of properties that different composite materials can display. These properties can best be summed up as high strengths and stiffnesses combined with low densities. It is these properties that give rise to the characteristic high strength and stiffness to weight ratios that make composite structures ideal for so many applications. This is particularly true of applications, which involve movement, such as cars, trains, and aircraft, since lighter structures in such applications play a significant part in making these applications more efficient. The strength and stiffness to weight ratio of composite materials can best be illustrated by the following graphs that plot “specific” properties. These are simply the result of dividing the mechanical properties of a material by its density. Generally, the properties at the higher end of the ranges illustrated in the previous graphs (Figures 2.6 and 2.7) are produced from the highest density variant of the material. The spread of specific properties shown in the following graphs (Figures 2.8 and 2.9) takes this into account.

2,800

Tensile modulus (MPa)

2,400 2,000 1,600 1,200 800

Figure 2.6.  Tensile strength of common structural materials.

IM carbon composites

HS carbon composites

Aramid composites

S-Glass composites

E-Glass composites

Steels

Titanium

Al.Alloys

0

Woods

400

IM carbon composites

HS carbon composites

Aramid composites

S-Glass composites

E-Glass composites

Steels

Titanium

Al.Alloys

Woods

Specific tensile strength

IM carbon composites

HS carbon composites

Aramid composites

S-Glass composites

E-Glass composites

Steels

Titanium

Al.Alloys

Woods

Tensile modulus (GPa)

Composite Materials   •  25

210

180

150

120

90

60

30

0

Figure 2.7.  Tensile modulus of common structural materials.

2,000

1,800

1,600

1,400

1,200

1,000

800

600

400

200

0

Figure 2.8.  Specific tensile strength of common structural materials.

26  •  Using ANSYS for Finite Element Analysis 120 110 Specific tensile modulus

100 90 80 70 60 50 40 30 20 10 IM carbon composites

HS carbon composites

Aramid composites

S-Glass composites

E-Glass composites

Steels

Titanium

Al.Alloys

Woods

0

Figure 2.9.  Specific tensile modulus of common structural materials.

2.2 Modeling Composites Using ANSYS ANSYS allows you to model composite materials with specialized elements called layered elements. Once you build your model using these elements, you can do any structural analysis (including nonlinearities such as large deflection and stress stiffening). Composites are somewhat more difficult to model than an isotropic material such as iron or steel. You need to take special care in defining the properties and orientations of the various layers since each layer may have different orthotropic material properties. In this section, we will concentrate on the following aspects of building a composite model: • • • •

Choosing the proper element type. Defining the layered configuration. Specifying failure criteria. Following modeling and postprocessing guidelines.

2.2.1  Choosing the Proper Element Type The following element types are available to model layered composite materials: SHELL99, SHELL91, SHELL181, SOLID46, and

Composite Materials   •  27

SOLID191. Which element you choose depends on the application, the type of results that need to be calculated, and so on. Check the individual element descriptions to determine if a specific element can be used in your ANSYS product. All layered elements allow failure criterion calculations. 2.2.1.1  SHELL99—Linear Layered Structural Shell Element SHELL99 is an 8-node, 3-D shell element with six degrees of freedom at each node. It is designed to model thin to moderately thick plate and shell structures with a side-to-thickness ratio of roughly 10 or greater. For structures with smaller ratios, you may consider using SOLID46. The SHELL99 element allows a total of 250 uniform-thickness layers. Alternately, the element allows 125 layers with thicknesses that may vary bilinearly over the area of the layer. If more than 250 layers are required, you can input your own material matrix. It also has an option to offset the nodes to the top or bottom surface.

2.2.1.2  SHELL91—Nonlinear Layered Structural Shell Element SHELL91 is similar to SHELL99 except that it allows only up to 100 layers and does not allow you to input a material property matrix. However, SHELL91 supports plasticity, large-strain behavior, and a special sandwich option, whereas SHELL99 does not. SHELL91 is also more robust for large deflection behavior.

2.2.1.3  SHELL181—Finite Strain Shell SHELL181 is a 4-node 3-D shell element with six degrees of freedom at each node. The element has full nonlinear capabilities including large strain and allows 255 layers. The layer information is input using the section commands rather than real constants. Failure criteria is available using the FC commands.

2.2.1.4  SOLID46—3-D Layered Structural Solid Element SOLID46 is a layered version of the 8-node, 3-D solid element, SOLID45, with three degrees of freedom per node (UX, UY, UZ). It is designed to model thick-layered shells or layered solids and allows up to

28  •  Using ANSYS for Finite Element Analysis

250 ­uniform-thickness layers per element. Alternately, the element allows 125 layers with thicknesses that may vary bilinearly over the area of the layer. An advantage with this element type is that you can stack several elements to model more than 250 layers to allow through-the-thickness deformation slope discontinuities. The user-input constitutive matrix option is also available. SOLID46 adjusts the material properties in the transverse direction permitting constant stresses in the transverse direction. In comparison to the 8-node shells, SOLID46 is a lower order element and finer meshes may be required for shell applications to provide the same accuracy as SHELL91 or SHELL99. 2.2.1.5  SOLID191—Layered Structural Solid Element SOLID191 is a layered version of the 20-node 3-D solid element SOLID95, with three degrees of freedom per node (UX, UY, UZ). It is designed to model thick-layered shells or layered solids and allows up to 100 layers per element. As with SOLID46, SOLID191 can be stacked to model through-the-thickness discontinuities. SOLID191 has an option to adjust the material properties in the transverse direction permitting constant stresses in the transverse direction. In spite of its name, the element does not support nonlinear materials or large deflections. In addition to the layered elements mentioned earlier, other composite element capabilities exist in ANSYS, but will not be considered further in the chapter: • SOLID95, the 20-node structural solid element, with KEYOPT (1) = 1 functions similarly to a single-layered SOLID191 including the use of an orientation angle and failure criterion. It allows nonlinear materials and large deflections. • SHELL63, the 4-node shell element, can be used for rough, approximate studies of sandwich shell models. A typical application would be a polymer between two metal plates, where the bending stiffness of the polymer would be small relative to the bending stiffness of the metal plates. The bending stiffness can be adjusted by the real constant RMI to represent the bending stiffness due to the metal plates, and distances from the middle surface to extreme fibers (real constants CTOP, CBOT) can be used to obtain output stress estimates on the outer surfaces of the sandwich shell. It is not used as frequently as SHELL91, SHELL99, or SHELL181, and will not be considered for anything other than sandwich structures in this section.

Composite Materials   •  29

• SOLID65, the 3-D reinforced concrete solid element, models an isotropic medium with optional reinforcing in three different user-defined orientations. • BEAM188 and BEAM189, the 3-D finite strain beam elements, can have their sections built up with multiple materials. 2.2.2  Defining the Layered Configuration The most important characteristic of a composite material is its layered configuration. Each layer may be made of a different orthotropic material and may have its principal directions oriented differently. For laminated composites, the fiber directions determine layer orientation. Two methods are available to define the layered configuration: • By specifying individual layer properties. • By defining constitutive matrices that relate generalized forces and moments to generalized strains and curvatures (available only for SOLID46 and SHELL99). 2.2.2.1  Specifying Individual Layer Properties With this method, the layer configuration is defined layer-by-layer from bottom to top. The bottom layer is designated as layer 1, and additional layers are stacked from bottom to top in the positive Z (normal) direction of the element coordinate system. You need to define only half of the layers if stacking symmetry exists. At times, a physical layer will extend over only part of the model. In order to model continuous layers, these dropped layers may be modeled with zero thickness. Figure 2.10 shows a model with four layers, the second of which is dropped over part of the model. For each layer, the following properties are specified in the element real constant table [R, RMORE, RMODIF] Main Menu> Preprocessor> Real Constants Accessed with REAL attributes.

4 3 2 1

Figure 2.10.  Layered model showing dropped layer.

Layer 2 4 is dropped 3 1

30  •  Using ANSYS for Finite Element Analysis

• Material properties (via a material reference number MAT). • Layer orientation angle commands (THETA). • Layer thickness (TK). Layered sections may also be defined through the Section Tool. Prep>Sections>Shell—Add/edit For each layer, the following are specified in the section definition through the section commands; or through the Section Tool (SECTYPE, SECDATA) (accessed with the SECNUM attributes). • • • •

Material properties (via a material reference number MAT) Layer orientation angle commands (THETA) Layer thickness (TK) Number of integration points per layer (NUMPT).

Material Properties as with any other element, the MP command: Main Menu> Preprocessor> Material Props> Material Models> Structural> Linear> Elastic> Isotropic or Orthotropic Is used to define the linear material properties, and the TB command is used to define the nonlinear material data tables (plasticity is only available for elements SOLID191 and SHELL91). The only difference is that the material attribute number for each layer of an element is specified in the element’s real constant table. For the layered elements, the MAT command: Main Menu> Preprocessor> Meshing> Mesh Attributes> Default Attribs Attribute is only used for the DAMP and REFT arguments of the MP command. The linear material properties for each layer may be either isotropic or orthotropic (see Linear Material Properties in the ANSYS Elements Reference). Typical fiber-reinforced composites contain ­orthotropic materials and these properties are most often supplied in the major Poisson’s ratio form (see the ANSYS, Inc. Theory Reference). Material property directions are parallel to the layer coordinate system, which is defined by the element coordinate system and the layer orientation angle (described as follows). Layer orientation angle—This defines the orientation of the layer coordinate system with respect to the element coordinate system. It is the angle (in degrees) between X-axes of the two systems. By default, the layer coordinate system is parallel to the element coordinate system. All elements have a default coordinate system that you can change using the ESYS element attribute [ESYS] Main Menu> Preprocessor> Meshing> Mesh Attributes> Default Attribs

Composite Materials   •  31

You may also write your own subroutines to define the element and layer coordinate systems (USERAN and USANLY); see the Guide to ANSYS User Programmable Features for details. Layer thickness—If the layer thickness is constant, you only need to specify TK(I), the thickness at node I. Otherwise, the thicknesses at the four corner nodes must be input. Dropped layers may be represented with zero thickness. Number of integration points per layer—This allows you to determine in how much detail the program should compute the results. For very thin layers, when used with many other layers, one point would be appropriate. But for laminates with few layers, more would be needed. The default is three points. This feature applies only to sections defined through the section commands. Currently, the graphical user interface (GUI) only allows layer real constant input of up to 100 layers. If more layers are needed for SHELL99 or SOLID46, the R and RMORE commands must be used.

2.2.2.2  Defining the Constitutive Matrices This is an alternative to specifying the individual layer properties and is available as an option KEYOPT (2) for SOLID46 and SHELL99. The matrices, which represent the force–moment and strain–curvature relationships for the element, must be calculated outside the ANSYS program as outlined in the ANSYS, Inc. Theory Reference. They can be included as part of the solution printout with KEYOPT(10). The main advantages of the matrix approach are: • It allows you to incorporate an aggregate composite material behavior. • A thermal load vector may be supplied. • The matrices may represent an unlimited number of layers. The terms of the matrices are defined as real constants. Mass effects are incorporated by specifying an average density (real constants AVDENS) for the element. If the matrix approach is used, detailed results in each layer cannot be obtained since individual layer information is not input.

2.2.2.3  Sandwich and Multiple-Layered Structures Sandwich structures have two thin faceplates and a thick, but relatively weak, core. Figure 2.11 illustrates sandwich construction.

32  •  Using ANSYS for Finite Element Analysis Faceplate Core (at least 1/2 of local thickness) Faceplate

Figure 2.11.  Sandwich construction.

You can model sandwich structures with SHELL63, SHELL91, or SHELL181. SHELL63 has one layer but permits sandwich modeling through the use of real constants. You can modify the effective bending moment of inertia and the distance from the middle surface to the extreme fibers to account for the weak core. KEYOPT(9) = 1 of SHELL91 specifies the sandwich option. The core is assumed to carry all of the transverse shear; the faceplates carry none. Conversely, the faceplates are assumed to carry all (or almost all) of the bending load. Only SHELL91 has this sandwich option. SHELL181 models the transverse shear deflection using as energy equivalence method that makes the need for a sandwich option unnecessary.

2.2.2.4  Node Offset For SHELL181 using sections defined through the section commands, nodes can be offset during the definition of the section, using the ­SECOFFSET command. For SHELL91, and SHELL99 the node offset option (KEYOPT(11)) locates the element nodes at the bottom, middle, or top surface of the shell. The following figures illustrate how you can conveniently model ply drop off in shell elements that are adjacent to each other. In Figure 2.12, the nodes are located at the middle surfaces (KEYOPT(11) = 0) and these surfaces are aligned. In Figure 2.13, the

Nodes located at the midplane with KEYOPT(11) = 0

Figure 2.12.  Layered shell with nodes at midplane.

Composite Materials   •  33 Ply 3 Ply 2 Ply 1

Nodes located on bottom surface with KEYOPT(11) = 1

Figure 2.13.  Layered shell with nodes at bottom surface.

nodes are located at the bottom surfaces (KEYOPT(11) = 1) and these surfaces are aligned.

2.2.3  Specifying Failure Criteria Failure criteria are used to learn if a layer has failed due to the applied loads. You can choose from three predefined failure criteria or specify up to six failure criteria of your own (user-written criteria). The three predefined criteria are: • Maximum Strain Failure Criterion, which allows nine failure strains. • Maximum Stress Failure Criterion, which allows nine failure stresses. • Tsai-Wu Failure Criterion, which allows nine failure stresses and three additional coupling coefficients. You have a choice of two methods of calculating this criterion. The methods are defined in the ANSYS, Inc. Theory Reference. The failure strains, stresses, and coupling coefficients may be temperature-dependent. See the ANSYS Elements Reference for details about the data required for each criterion. To specify a failure criterion, use either the family of TB commands or the family of FC commands. The TB commands are TB, TBTEMP, and TBDATA. Main Menu> Preprocessor> Material Props> Failure Criteria A typical sequence of commands to specify a failure criterion using these commands is shown as follows. TB,FAIL,1,2 ! Data table for failure criterion, material 1, ! no. of temperatures = 2

34  •  Using ANSYS for Finite Element Analysis

TBTEMP,,CRIT ! Failure criterion key TBDATA,2,1 ! Maximum Stress Failure Criterion (Const. 2 = 1) TBTEMP,100! Temperature for subsequent failure properties TBDATA,10,1500,,400,,10000 ! X, Y, and Z failure tensile stresses (Z value! set to a large number) TBDATA,16,200,10000,10000 ! XY, YZ, and XZ failure shear stresses TBLIST TBTEMP,200! Second temperature TBDATA,... The FC → commands → are FC, FCDELE, and FCLIST commands: Main Menu> Preprocessor> Material Props> Material Models> Structural> Nonlinear> Inelastic> Non-Metal Plasticity> Failure Criteria Main Menu> General Postproc> Failure Criteria A typical sequence of commands to specify a failure criterion using these commands is shown as follows. FC,1,TEMP,, 100, 200 ! Temperatures FC,1,S,XTEN, 1500, 1200 ! Maximum stress components FC,1,S,YTEN, 400, 500 FC,1,S,ZTEN,10000, 8000 FC,1,S,XY , 200, 200 FC,1,S,YZ ,10000, 8000 FC,1,S,XZ ,10000, 8000 FCLIST, ,100 ! List status of Failure Criteria at 100.0 degrees FCLIST, ,150 ! List status of Failure Criteria at 150.0 degrees FCLIST, ,200 ! List status of Failure Criteria at 200.0 degrees PRNSOL,S,FAIL ! Use Failure Criteria The TB commands (TB, TBTEMP, and TBDATA) can be used only for SHELL91, SHELL99, SOLID46, or SOLID191, but the FC and FCLIST commands can be used for any 2-D or 3-D structural solid element or any 3-D structural shell element. See the ANSYS Commands Reference for a discussion of the TB, TBTEMP, TBDATA, TBLIST, FC, FCDELE, and FCLIST commands. Some notes about specifying failure criteria: • The criteria are orthotropic, so you must input the failure stress or failure strain values for all directions. (The exception is that compressive values default to tensile values.)

Composite Materials   •  35

• If you don’t want the failure stress or strain to be checked in a particular direction, specify a large number in that direction (as shown in the previous example). User-written failure criteria may be specified via user subroutines USRFC1 through USRFC6. These subroutines should be linked with the ANSYS program beforehand; see the ANSYS Advanced Analysis Techniques Guide for a brief description of user-programmable features. 2.2.4  A  dditional Modeling and Postprocessing Guidelines Some additional guidelines for modeling and postprocessing of composite elements are presented in the following list. 1. Composites exhibit several types of coupling effects, such as coupling between bending and twisting, coupling between extension and bending, and so on. This is due to stacking of layers of differing material properties. As a result, if the layer stacking sequence is not symmetric, you may not be able to use model symmetry even if the geometry and loading are symmetric, because the displacements and stresses may not be symmetric. 2. Interlaminar shear stresses are usually important at the free edges of a model. For relatively accurate interlaminar shear stresses at these locations, the element size at the boundaries of the model should be approximately equal to the total laminate thickness. For shells, increasing the number of layers per actual material layer does not necessarily improve the accuracy of interlaminar shear stresses. With elements SOLID46, SOLID95, and SOLID191, however, stacking elements in the thickness direction should result in more accurate interlaminar stresses through the thickness. Interlaminar transverse shear stresses in shell elements are based on the assumption that no shear is carried at the top and bottom surfaces of the element. These interlaminar shear stresses are only computed in the interior and are not valid along the shell element boundaries. Use of shell-to-solid submodeling is recommended to accurately compute all of the free-edge interlaminar stresses. 3. Since a large amount of input data is required for composites, you should verify the data before proceeding with the solution. Several commands are available for this purpose: • ELIST (Utility Menu> List> Elements) lists the nodes and attributes of all selected elements.

36  •  Using ANSYS for Finite Element Analysis

• EPLOT (Utility Menu> Plot> Elements) displays all selected elements. Using the /ESHAPE,1 command (Utility Menu> PlotCtrls> Style> Size and Shape) before EPLOT causes shell elements to be displayed as solids with the layer thicknesses obtained from real constants or section definition (see Figure 2.14. This example uses element SHELL99 with /ESHAPE turned on). It also causes SOLID46 elements to be displayed with layers. • /PSYMB, LAYR,n (Utility Menu> PlotCrls> Symbols) followed by EPLOT displays layer number n for all selected layered elements. This can be used to display and verify each individual layer across the entire model. • /PSYMB, ESYS,1 followed by EPLOT displays the element coordinate system triad for those elements whose default coordinate system has been changed. • LAYLIST (Utility Menu> List> Elements> Layered Elements) lists the layer stacking sequence from real constants and any two material properties for SHELL99, SHELL91, SOLID46, and SOLID191 elements. You can specify a range of layer numbers for the listing. LIST LAYERS 1 TO 4 IN REAL SET 1 FOR ELEMENT TYPE 1 TOTAL LAYERS = 4 LSYM = 1 LP1 = 0 LP2 = 0 EFS = .000E+00 NO. ANGLE THICKNESS MAT

Figure 2.14.  Example of an element display.

Composite Materials   •  37

--- ----- ---------- --1 45.0 0.250 1 2 -45.0 0.250 2 3 -45.0 0.250 2 4 45.0 0.250 1 -----------------------SUM OF THK 1.00 • LAYPLOT (Utility Menu> Plot> Layered Elements) displays the layer stacking sequence from real constants in the form of a sheared deck of cards (see Figure 2.15). The layers are crosshatched and color coded for clarity. The hatch lines indicate the layer angle (real constant THETA) and the color indicates layer material number (MAT). You can specify a range of layer numbers for the display. • SECPLOT (Main Menu> Preprocessor> Sections> Shell> Plot Section) displays the section stacking sequence from sections in the form of a sheared deck of cards (see Figure 2.15). The sections are crosshatched and color coded for clarity. The hatch lines indicate the layer angle (THETA) and the color indicates layer material number (MAT) defined by the SECDATA command. You can specify a range of layer numbers for the display. 4. By default, only data for the bottom of the first (bottom) layer, top of the last (top) layer, and the layer with the maximum failure criterion value are written to the results file. If you are interested in data for all layers, set KEYOPT(8) = 1. Be aware, though, that this may result in a large results file. Layer#

Theta

Material# 1

1

2

2

2

45

3 –45

4

1

–45 45

Figure 2.15.  Sample LAYPLOT display for [45/−45/−45/45] sequence.

38  •  Using ANSYS for Finite Element Analysis

5. Use the ESEL S,LAYER command to select elements that have a certain layer number. If an element has a zero thickness for the requested layer, the element is not selected. For energy output, the results are applicable only to the entire element; you cannot get output results for individual layers. 6. Use the LAYER command (Main Menu> General Postproc> Options for Outp) in POST1 (or LAYERP26 (Main Menu> TimeHist Postpro> Define Variables) in POST26) to specify the layer number for which results are to be processed. The SHELL command (Main Menu> General Postproc> Options for Outp or Main Menu> TimeHist Prostpro> Define Variables) specifies a TOP, MID, or BOT location within the layer. The default in POST1 is to store results for the bottom of the bottom layer, and the top of the top layer, and the layer with the maximum failure criterion value. In POST26, the default is layer 1. If KEYOPT(8) = 1 (i.e., data stored for all layers), the LAYER and LAYERP26 commands store the TOP and BOT results for the specified layer number. MID values are then calculated by average TOP and BOT values. If KEYOPT (8) = 2 is set for SHELL181 during SOLUTION, then LAYER and LAYERP26 commands store the TOP, BOTTOM, and MID results for the specified layer number. In this case, MID values are directly retrieved from the results file. For transverse shear stresses with KEYOPT(8) = 0, therefore, POST1 can only show a linear variation, whereas the element solution printout or KEYOPT(8) = 2 can show a parabolic variation. 7. By default, POST1 displays all results in the global Cartesian coordinate system. Use the RSYS command (Main Menu> General Postproc> Options for Outp) to transform the results to a different coordinate system. In particular, RSYS,SOLU allows you to display results in the layer coordinate system if LAYER is issued with a nonzero layer number.

2.3 Tutorial 3: Simply Supported Laminated Plate Under Pressure In this tutorial, a simply supported square cross-ply laminated plate is subjected to a uniform pressure po. The stacking sequence of the plies is symmetric about the middle plane. Determine the center deflection δ (Z-direction) of the plate due to the pressure load and the von Mises stress distribution.

Composite Materials   •  39 Y uz = 0;

Z ux = 0; h X uz = 0; uy = 0

a uz = 0; uy = 0 uz = 0;

Po 0° 90° 90° 0°

t t

X

ux = 0 a

Geometric properties a = 10 m h = 0.1 m t = 0.025 m

Material properties Ex = 25 × 106 N/m2 Ey = 1 × 106 N/m2 υxy = 0.25 (Major Poisson’s ratio) Gxy = G = 0.5 × 106 N/m2 Gyz = 0.2 × 106 N/m2

Loading

po = 1.0 N/m2

2.3.1  Approach and Assumptions A quarter of the plate is modeled due to symmetry in geometry, material orientation, loading, and boundary conditions. One model using Linear Layered Structural Shell Elements is analyzed. 3-D 8-node Layered Structural Solid Elements SOLID46 elements can also be used. Note that PRXY is used to directly input the major Poisson’s ratio. EZ (explicitly input) is assumed to be equal to EY. 2.3.2  Summary of Steps 1. 2. 3. 4. 5. 6. 7. 8.

Set preferences. Define element types and options. Define real constants. Define material properties. Build geometry. Generate mesh. Verification of data. Apply displacement constraints.

40  •  Using ANSYS for Finite Element Analysis Y

UX = 0

UZ = 0;

Symmetric boundary condition (ux=0)

X

a

UZ = 0; Symmetric boundary condition (uy=0)

UZ = 0; UY = 0

UZ = 0;

UY = 0

UX = 0 a

9. Apply pressure load. 10. Obtain solution. 11. Review results. 12. Exit the ANSYS program. 2.3.3  Step-By-Step Analysis Utility Menu >File>Change title Enter Simply Supported Laminated Plate (SHELL99) then OK.

1. Set preferences In preparation for defining materials, you will set preferences so that only materials and elements that pertain to a structural analysis are available for you to choose. To set preferences:

Composite Materials   •  41

Main Menu> Preferences

Turn on structural filtering. OK to apply filtering and close the dialog box.

42  •  Using ANSYS for Finite Element Analysis

Before going to the next step, enter preprocessor by selecting Preprocessor from the ANSYS Main Menu.

2. Define element types and options In any analysis, you need to select from a library of element types and define the appropriate ones for your analysis. For this analysis, you will first use element type, SHELL99. SHELL99 may be used for layered applications of a structural shell model. It allows up to

Composite Materials   •  43

250 layers. If more than 250 layers are required, a user-input constitutive matrix is available. Main Menu> Preprocessor Element Type> Add/Edit/Delete

Add an element type.

44  •  Using ANSYS for Finite Element Analysis

Structural shell family of elements. Choose the 8-noded layered shell element (SHELL99). OK to apply the element type and close the dialog box.

Close the element type dialog box.

3. Define real constants For this analysis, since the assumption is plane stress with thickness, you will enter the thickness as a real constant for PLANE82.

Composite Materials   •  45

To find out more information about PLANE82, you will use the ANSYS Help System in this step by clicking on a Help button from within a dialog box. Main Menu> Preprocessor> Real Constants> Add/Edit/Delete

Add a real constant set. OK for SHELL99.

OK to real constant set number 1.

46  •  Using ANSYS for Finite Element Analysis

Enter 4 for number of layers and 1 for symmetric.

Enter material number, angle, and thickness for each layer. Because of symmetry (0/90/90/0), enter data only for the first two layers. Layer 1: 1, 0, 0.025 Layer 2: 1, 90, 0.025 OK to define the real constant and close the dialog box. Close the real constant dialog box.

Composite Materials   •  47

4. Define material properties Main Menu> Preprocessor> Material Props> Material Models

Double-click on Structural, Linear, Elastic, Orthotropic. Enter 25.0E6 for EX. Enter 1.0E6 for EY and EZ. Enter 0.25 for PRXY and PRXZ. Enter 0.01 for PRYZ. Enter 0.5E6 for GXY and GXZ. Enter 0.2E6 for GYZ. OK to define material property set and close the dialog box. Material> Exit 5. Build geometry There are several ways to create the model geometry within ANSYS, some more convenient than others. Decide where the origin will be located and then define the rectangle primitive relative

48  •  Using ANSYS for Finite Element Analysis

to that origin. The location of the origin is arbitrary. Here, use the center of the plate. ANSYS does not need to know where the origin is. Simply begin by defining a rectangle relative to that location. In ANSYS, this origin is called the global origin. Main Menu> Preprocessor> Modeling> Create> Areas> Rectangle> By Dimensions

Enter the following: X1=0, X2=5, Y1=0, Y2=5 OK to close the dialog box.

Composite Materials   •  49

6. Generate mesh One nice feature of the ANSYS program is that you can automatically mesh the model without specifying any mesh size controls. This is using what is called a default mesh. If you’re not sure how to determine the mesh density, let ANSYS try it first. Instead you will specify a global element size to control overall mesh density. Main Menu> Preprocessor> Meshing> Mesh Tool

50  •  Using ANSYS for Finite Element Analysis

Set Global Size control.

Type in 0.5. OK

Composite Materials   •  51

Choose Area Meshing. Click on Mesh. Pick All for the area to be meshed (in picking menu). Close any warning messages that appear. Close the Mesh Tool.

52  •  Using ANSYS for Finite Element Analysis

7. Verification of data Since a large amount of input data is required for composites, you should verify the data before proceeding with the solution. Several commands are available for this purpose. Some of them are demonstrated as follows: EPLOT (Utility Menu> Plot> Elements) displays all selected elements. Using the /ESHAPE,1 command (Utility Menu> ­PlotCtrls> Style> Size and Shape) before EPLOT causes shell elements to be displayed as solids with the layer thicknesses obtained from real constants or section definition. Utility Menu> PlotCtrls>Style>Size and Shape

Composite Materials   •  53

Turn on display of element key then Ok

Utility Menu> Plot Elements Select isometric view

54  •  Using ANSYS for Finite Element Analysis

Zoom in to clearly see the four layers along the thickness direction. Pick the zoom tablet. Select the region Enlarged view of the corner showing layers.

LAYPLOT (Utility Menu> Plot> Layered Elements) This displays the layer stacking sequence from real constants in the form of a sheared deck of cards. The layers are crosshatched

Composite Materials   •  55

and color coded for clarity. The hatch lines indicate the layer angle (real constant THETA) and the color indicates layer material number (MAT). You can specify a range of layer numbers for the display. Utility Menu> Plot> Layered Elements Select the element to be displayed (only one element can be selected at one time).

Enter the range of layers to be displayed for the selected element. OK.

56  •  Using ANSYS for Finite Element Analysis

Figure shows the configuration 0/90/90/0 for the layered element.

2.3.4  Apply Loads The beginning of the solution phase. A new, static analysis is the default, so you will not need to specify analysis type for this problem. Also, there are no analysis options for this problem. Utility Menu > Plotctrls > Numbering Select LINE numbers

Composite Materials   •  57

Utility Menu > Plot > Lines

8. Apply displacement constraints You can apply symmetric displacement constraints directly to two lines. Main Menu> Solution> Define Loads> Apply> Structural> Displacement> Symmetry B.C.>On Lines

Pick the two lines of symmetry (Line numbers 1 and 4). OK (in picking menu).

58  •  Using ANSYS for Finite Element Analysis

Now, apply displacement on the two simply supported plate edges. Main Menu> Solution> Define Loads> Apply> Structural> Displacement>On Lines

Pick the line at the top (Line numbers 3). OK (in picking menu).

Composite Materials   •  59

Click on UX. Enter 0 for zero displacement. APPLY to apply constraints and return to dialog box.

Pick the line at the top (Line numbers 3). OK (in picking menu). Click on UZ. Enter 0 for zero displacement. APPLY to apply constraints and return to dialog box.

60  •  Using ANSYS for Finite Element Analysis

Pick the line at the right (Line numbers 2). OK (in picking menu). Click on UY. Enter 0 for zero displacement. APPLY to apply constraints and return to dialog box.

Pick the line at the right (Line numbers 2). OK (in picking menu). Click on UZ. Enter 0 for zero displacement. OK to apply constraints and return to dialog box.

Composite Materials   •  61

Utility Menu> Plot Lines

9. Apply pressure load Now apply the uniform pressure load on the top surface. The ANSYS convention for pressure loading is that a positive load value represents pressure into the surface (compressive). Main Menu> Solution> Define Loads> Apply> Structural> Pressure> On Areas

62  •  Using ANSYS for Finite Element Analysis

Pick the area. OK.

Enter 100 for VALUE.

Composite Materials   •  63

Pressure distribution is shown by face-out lines. It can be changed to arrows demonstrated as follows.

Utility Menu > Plotctrls > Symbols Select Arrows for Pressure. OK.

64  •  Using ANSYS for Finite Element Analysis

10. Obtain solution Main Menu> Solution> Solve> Current LS

Review the information in the status window, then choose File> Close (Windows), OK to begin the solution. Choose Yes to any Verify messages that appear. Then Close the information window when solution is done. ANSYS stores the results of this one load step problem in the database and in the results file, Jobname.RST (or Jobname.RTH for thermal, Jobname.RMG for magnetic, and Jobname.RFL for fluid analyses). The database can actually contain only one set of results at any given time, so in a multiple load step or multiple substep analysis, ANSYS stores only the final solution in the database. ANSYS stores all solutions in the results file.

Composite Materials   •  65

11. Review results The beginning of the postprocessing phase. Main Menu> General Postproc> Plot Results> Contour Plot> Nodal Solu Choose DOF Solution item to be contoured. SCROLL DOWN AND CHOOSE Z- COMPONENT OF DISPLACEMENT (UZ).

Main Menu> General Postproc> Plot Results> Contour Plot> Nodal Solu Choose Stress item to be contoured. Scroll down and choose von Mises (SEQV).

66  •  Using ANSYS for Finite Element Analysis

To see the stress distribution along the thickness, First select bottom view, then Rotate the view once about positive x-axis.

Zoom the region of interest.

Composite Materials   •  67

List reaction solution: Main Menu> General Postproc> List Results> Reaction Solu OK to list all items and close the dialog box. Scroll down and find the total vertical force, FZ. Total FZ = Pressure × Area = 100 × (5 × 5) = 2,500 N File> Close (Windows). PRINT FZ REACTION SOLUTIONS PER NODE NODE FZ 2 -43.064 22 85.605 23 -171.83 . . 58 -34.133 59 -69.352 60 -34.873 61 -70.354 TOTAL VALUES VALUE -2500.0

68  •  Using ANSYS for Finite Element Analysis

12. Exit the ANSYS program When exiting the ANSYS program, you can save the geometry and loads portions of the database (default), save geometry, loads, and solution data (one set of results only), save geometry, loads, solution data, and postprocessing data (i.e., save everything), or save nothing. You can save nothing here, but you should be sure to use one of the other save options if you want to keep the ANSYS data files. 1. Toolbar: Quit. 2. Choose Quit—No Save! 3. OK. Congratulations! You have completed this tutorial.

Chapter 3

Probabilistic Design Analysis 3.1  Probabilistic Design Probabilistic design is an analysis technique for assessing the effect of uncertain input parameters and assumptions on your model. A probabilistic analysis allows you to determine the extent to which uncertainties in the model affect the results of a finite element analysis. An uncertainty (or random quantity) is a parameter whose value is impossible to determine at a given point in time (if it is time-dependent) or at a given location (if it is location-dependent). An example is ambient temperature; you cannot know precisely what the temperature will be one week from now in a given city. In a probabilistic analysis, statistical distribution functions (such as the Gaussian or normal distribution, the uniform distribution, etc.) describe uncertain parameters. Computer models are expressed and described with specific numerical and deterministic values; material properties are entered using certain values, the geometry of the component is assigned a certain length or width, and so on. An analysis based on a given set of specific numbers and values is called a deterministic analysis. Naturally, the results of a deterministic analysis are only as good as the assumptions and input values used for the analysis. The validity of those results depends on how correct the values were for the component under real-life conditions. In reality, every aspect of an analysis model is subjected to scatter (in other words, is uncertain in some way). Material property values are different if one specimen is compared to the next. This kind of scatter is inherent for materials and varies among different material types and material properties. For example, the scatter of the Young’s modulus for many materials can often be described as a Gaussian distribution with

70  •  Using ANSYS for Finite Element Analysis

standard deviation of ±3–5 percent. Likewise, the geometric properties of components can only be reproduced within certain manufacturing tolerances. The same variation holds true for the loads that are applied to a finite element model. However, in this case the uncertainty is often due to a lack of engineering knowledge. For example, at elevated temperatures the heat transfer coefficients are very important in a thermal analysis, yet it is almost impossible to measure the heat transfer coefficients. This means that almost all input parameters used in a finite element analysis are inexact, each associated with some degree of uncertainty. It is neither physically possible nor financially feasible to eliminate the scatter of input parameters completely. The reduction of scatter is typically associated with higher costs either through better and more precise manufacturing methods and processes or increased efforts in quality control; hence, accepting the existence of scatter and dealing with it rather than trying to eliminate it makes products more affordable and production of those products more cost-effective. To deal with uncertainties and scatter, you can use the ANSYS Probabilistic Design System (PDS) to answer the following questions: • If the input variables of a finite element model are subjected to scatter, how large is the scatter of the output parameters? How robust are the output parameters? Here, output parameters can be any parameter that ANSYS can calculate. Examples are the temperature, stress, strain, or deflection at a node, the maximum temperature, stress, strain, or deflection of the model, and so on. • If the output is subjected to scatter due to the variation of the input variables, then what is the probability that a design criterion given for the output parameters is no longer met? How large is the probability that an unexpected and unwanted event takes place (what is the failure probability)? • Which input variables contribute the most to the scatter of an output parameter and to the failure probability? What are the sensitivities of the output parameter with respect to the input variables? Probabilistic design can be used to determine the effect of one or more variables on the outcome of the analysis. In addition to the probabilistic design techniques available, the ANSYS program offers a set of strategic tools that can be used to enhance the efficiency of the probabilistic design process. For example, you can graph the effects of one input variable versus an output parameter, and you can easily add more samples and additional analysis loops to refine your analysis.

Probabilistic Design Analysis   •  71

3.1.1  T  raditional (Deterministic) VERSUS Probabilistic Design Analysis Methods In traditional deterministic analyses, uncertainties are either ignored or accounted for by applying conservative assumptions. Uncertainties are typically ignored if the analyst knows for certain that the input parameter has no effect on the behavior of the component under investigation. In this case, only the mean values or some nominal values are used in the analysis. However, in some situations the influence of uncertainties exists but is still neglected; for example, the Young’s modulus mentioned earlier or the thermal expansion coefficient, for which the scatter is usually ignored. Let’s assume that you are performing a thermal analysis and you want to evaluate the thermal stresses (thermal stresses are directly proportional to the Young’s modulus as well as to the thermal expansion coefficient of the material). The equation is: σtherm = E α ΔT If the Young’s modulus alone has a Gaussian distribution with a 5 percent standard deviation, then there is almost a 16 percent chance that the stresses are more than 5 percent higher than what you would think they are in a deterministic case. This figure increases if you also take into account that, typically, the thermal expansion coefficient also follows a Gaussian distribution.

Random input variables taken into account Young’s ­modulus (Gaussian ­distribution with 5% standard deviation) Young’s modulus and thermal expansion coefficient (each with Gaussian distribution with 5% standard deviation)

Probability that the thermal stresses are more than 5% higher than expected

Probability that the thermal stresses are more than 10% higher than expected

~16%

~2.3%

~22%

~8%

When a conservative assumption is used, this actually tells you that uncertainty or randomness is involved. Conservative assumptions are usually expressed in terms of safety factors. Sometimes regulatory b­odies

72  •  Using ANSYS for Finite Element Analysis

demand safety factors in certain procedural codes. If you are not faced with such restrictions or demands, then using conservative assumptions and safety factors can lead to inefficient and costly overdesign. You can avoid overdesign by using probabilistic methods while still ensuring the safety of the component. Probabilistic methods even enable you to quantify the safety of the component by providing a probability that the component will survive operating conditions. Quantifying a goal is the necessary first step toward achieving it. Probabilistic methods can tell you how to achieve your goal. 3.1.2  Reliability and Quality Issues Use probabilistic design when issues of reliability and quality are paramount. Reliability is usually always a concern because product or component failures have significant financial consequences (costs of repair, replacement, warranty, or penalties); worse, a failure can result in injury or loss of life. Although perfection is neither physically possible nor financially feasible, probabilistic design helps you to design safe and reliable products while avoiding costly overdesign and conserve manufacturing resources (machining accuracy, efforts in quality control, and so on). Quality is the perception by a customer that the product performs as expected or better. In a quality product, the customer rarely receives unexpected and unpleasant events where the product or one of its components fails to perform as expected. By nature, those rare “failure” events are driven by uncertainties in the design. Here, probabilistic design methods help you to assess how often “failure” events may happen. In turn, you can improve the design for those cases where the “failure” event rate is above your customers’ tolerance limit. 3.1.3  Probabilistic Design Terminology PDS term

Random input variables (RVs)

Correlation

Description Quantities that influence the result of an analysis. In probabilistic design, RVs are often called ­“drivers” because they drive the result of an analysis. You must specify the type of statistical distribution the RVs follow and the parameter values of their distribution functions. Two (or more) RVs that are statistically dependent on each other.

Probabilistic Design Analysis   •  73

The results of a finite element analysis. The RPs are typically a function of the RVs; that is, changing the values of the RVs should change the value of the RPs. Probabilistic The RVs and RPs are collectively known as design variables ­probabilistic design variables. A unique set of parameter values that represents Sample a particular model configuration. A sample is ­characterized by random input variable values. The collection of all samples that are required or Simulation that you request for a given probabilistic analysis. Random output parameters (RPs)

PDS term

Probabilistic model

Mean value

Description The simulation contains the information used to determine how the component would behave under real-life conditions (with all the existing uncertainties and scatter); therefore, all samples represent the simulation of the behavior. The combination of definitions and specifications for the deterministic model (in the form of the analysis file). The model has these components: •  RVs. •  Correlations. •  RPs. • The selected settings for probabilistic method and its parameters. If you change any part of the probabilistic model, then you will generate different results for the probabilistic analysis (that is, different results values and/or a different number of results). For example, modifying the analysis file may affect the results file. If you add or take away an RV or change its distribution function, you solve a different probabilistic problem (which again leads to different results). If you add an RP, you will still solve the same probabilistic problem, but more results are generated. A measure of location often used to describe the general location of the bulk of the scattering data of a random output parameter or of a statistical distribution function. (Continued  )

74  •  Using ANSYS for Finite Element Analysis

(Continued  ) PDS term

Median value

Standard deviation

Description Mathematically, the mean value is the arithmetic average of the data. The mean value also represents the center of gravity of the data points. Another name for the mean value is the expected value. The statistical point where 50% of the data is below the median value and the 50% is above. For symmetrical distribution functions (Gaussian, uniform, etc.) the median value and the mean value are identical, while for nonsymmetrical distributions they are different. A measure of variability (dispersion or spread) about the arithmetic mean value, often used to describe the width of the scatter of a random output parameter or of a statistical distribution function. The larger the standard deviation, the wider the scatter and the more likely it is that there are data values further apart from the mean value.

3.1.4  S  teps for Probabilistic Design Analysis using ANSYS The usual process for probabilistic design consists of the following general steps: 1. Create an analysis file for use during looping. The file should represent a complete analysis sequence and must do the following: • Build the model parametrically (PREP7). • Obtain the solution(s) (SOLUTION). • Retrieve and assign to parameters the quantities that will be used as RVs and RPs (POST1/POST26). 2. Establish parameters in the ANSYS database, which correspond to those used in the analysis file. This step is typical, but not required (Begin or PDS); however, if you skip this step, then the parameter names are not available for selection in interactive mode. 3. Enter PDS and specify the analysis file (PDS). 4. Declare random input variables (PDS). 5. Visualize random input variables (PDS). Optional. 6. Specify any correlations between the RVs (PDS). 7. Specify random output parameters (PDS).

Probabilistic Design Analysis   •  75

8. Choose the probabilistic design tool or method (PDS). 9. Execute the loops required for the probabilistic design analysis (PDS). 10. Fit the response surfaces (if you did not use a Monte Carlo Simulation method) (PDS). 11. Review the results of the probabilistic analysis (PDS). Since analyzing complex problems can be time-consuming, ANSYS offers you the option of running a probabilistic analysis on a single processor or distributing the analyses across multiple processors. By using the ANSYS PDS parallel run capabilities, you can run many analysis loops simultaneously and reduce the overall run time for a probabilistic analysis. The following Figure 3.1 shows the flow of information during a probabilistic design analysis.

Figure 3.1.  The flow of information during a probabilistic design analysis.

3.2  Probability Distributions Probability distributions are a fundamental concept in statistics. They are used both on a theoretical level and a practical level. Some practical uses of probability distributions are: • To calculate confidence intervals for parameters and to calculate critical regions for hypothesis tests.

76  •  Using ANSYS for Finite Element Analysis

• For univariate data, it is often useful to determine a reasonable distributional model for the data. • Statistical intervals and hypothesis tests are often based on specific distributional assumptions. Before computing an interval or test based on a distributional assumption, we need to verify that the assumption is justified for the given data set. In this case, the distribution does not need to be the best-fitting distribution for the data, but an adequate enough model so that the statistical technique yields valid conclusions. • Simulation studies with random numbers generated from using a specific probability distribution are often needed. The mathematical definition of a continuous probability function, f(x), is a function that satisfies the following properties. 1. The probability that x is between two points a and b is: b

p [ a ≤ x ≤ b ] = ∫ f ( x ) dx a

2. It is non-negative for all real x. 3. The integral of the probability function is one, that is: ∞

∫−∞ f ( x) dx = 1 What does this actually mean? Since continuous probability functions are defined for an infinite number of points over a continuous interval, the probability at a single point is always zero. Probabilities are measured over intervals, not single points. That is, the area under the curve between two distinct points defines the probability for that interval. This means that the height of the probability function can in fact be greater than one. The property that the integral must equal one is equivalent to the property for discrete distributions that the sum of all the probabilities must equal one. Probability distributions are typically defined in terms of the probability density function (pdf). However, there are a number of probability functions used in applications. For a continuous function, the pdf is the probability that the variate has the value x. Since for continuous distributions the probability at a single point is zero, this is often expressed in terms of an integral between two points.

Probabilistic Design Analysis   •  77 b

∫a f ( x) dx = pr [a ≤ X ≤ b] For a discrete distribution, the pdf is the probability that the variate takes the value x. f ( x ) = Pr [ X = x ] Figure 3.2 shows the normal pdf. The cumulative distribution function (cdf) is the probability that the variable takes a value less than or equal to x. That is: F ( x ) = Pr [ X ≤ x ] = ∞ For a continuous distribution, this can be expressed mathematically as: F ( x) = ∫

∞ −∞

f ( m) d m

For a discrete distribution, the cdf can be expressed as: x

F ( x ) = ∑ f (i ) i=0

Figure 3.3 shows the normal cdf. The horizontal axis is the allowable domain for the given probability function. Since the vertical axis is a probability, it must fall between zero and one. It increases from zero to one as we go from left to right on the horizontal axis. 0.4 0.35

Probability

0.3 0.25 0.2 0.15 0.1 0.05 0 -5

-4

-3

-2

-1

0

1

2

X

Figure 3.2.  The normal probability density function (pdf).

3

4

5

78  •  Using ANSYS for Finite Element Analysis 1 0.5 0

Probability

1

0.5 0 1 0.5 0 -4

-3

-2

-1

0 X

1

2

3

4

Figure 3.3.  The normal cumulative distribution function (cdf).

3.2.1  Gallery of Common Continuous Distributions There are a large number of distributions used in statistical applications. Detailed information on a few of the most common distributions is available as follows. 0.4 0.35 0.3

Probability

0.25 0.2

0.15 0.1 0.05 0 -5

-4

-3

-2

-1

0

1

2

3

4

5

7

8

9

10

X

Normal distribution 2 1.8 1.6

Probability

1.4 1.2 1

0.8

0.6 0.4 0.2 0 0

1

2

3

4

5

6

X

Uniform distribution

Probabilistic Design Analysis   •  79 0.5 0.45 0.4

Probability

0.35 0.3

0.25 0.2

0.15 0.1 0.05 0

0

1

2

3

4

5

6

7

8

9

10

1.8

2

9

10

X

Exponential distribution

1.4 1.2

Probability

1 0.8

0.6 0.4 0.2 0 0

0.4

0.6

0.8

1

X

1.2

1.4

1.6

Weibull distribution

0.7 0.6

Probability

0.5 0.4

0.3

0.2 0.1 0

0

1

2

3

4

5 x

6

Lognormal distribution

7

8

80  •  Using ANSYS for Finite Element Analysis 0.7 0.6

Probability

0.5 0.4 0.3 0.2 0.1 0

0

1

2

3

4 x

5

6

7

8

Fatigue life distribution

0.4

Probability

0.2 0

0.6 0.4 0.2 0 700

800

900

1000

1100

1200

1300

X

Gamma distribution

0.2 0.18 0.16

Probability

0.14 0.12 0.1

0.08

0.06 0.04 0.02 0 –15

–10

–5

0 x

5

Double exponential distribution

10

15

Probabilistic Design Analysis   •  81 0.4 0.35 0.3

Probability

0.25 0.2

0.15 0.1 0.05 0 -6

-4

-2

0 X

2

6 x

8

4

6

10

12

Power normal distribution 0.7 0.6

Probability

0.5 0.4 0.3 0.2 0.1 0 0

2

4

Power lognormal distribution

3.2.2  Normal Distribution 3.2.2.1  Probability Density Function The general formula for the probability density function of the normal distribution is: 2

f ( x) =

( )

e −( x − m) / 2s 2

s 2p

where m is the location parameter and is the scale parameter. The case where m = 0 and s = 1 is called the standard normal distribution. The equation for the standard normal distribution is:

82  •  Using ANSYS for Finite Element Analysis

e− x / 2 2

f ( x) =

2p

Figure 3.2 shows the standard normal pdf. 3.2.2.2  Cumulative Distribution Function The formula for the cdf of the normal distribution does not exist in a s­ imple closed formula. It is computed numerically. Figure 3.3 shows the normal cdf. 3.2.2.3  Common Statistics Mean

The location parameter m

Median

The location parameter m

Mode Range

The location parameter m Infinity in both directions

Standard Deviation

The scale parameter s

Coefficient of Variation

s/m

Skewness Kurtosis

0 3

3.2.2.4  Parameter Estimation The location and scale parameters of the normal distribution can be estimated with the sample mean and sample standard deviation, respectively. 3.2.2.5  Comments The Gaussian or normal distribution is a very fundamental and commonly used distribution for statistical matters. It is typically used to describe the scatter of the measurement data of many physical phenomena. Strictly speaking, every random variable follows a normal distribution if it is generated by a linear combination of a very large number of other random effects, regardless of which distribution these random effects originally follow. The Gaussian distribution is also valid if the random variable is a linear combination of two or more other effects if those effects also follow a Gaussian distribution. For both theoretical and practical reasons, the normal distribution is probably the most important distribution in statistics. For example:

Probabilistic Design Analysis   •  83

• Many classical statistical tests are based on the assumption that the data follow a normal distribution. This assumption should be tested before applying these tests. • In modeling applications, such as linear and nonlinear regression, the error term is often assumed to follow a normal distribution with fixed location and scale. • The normal distribution is used to find significance levels in many hypothesis tests and confidence intervals. 3.2.3  Uniform Distribution 3.2.3.1  Probability Density Function The general formula for the probability density function of the uniform distribution is: f ( x) =

1 � for � A≤ x≤ B B− A

where A is the location parameter and B - A is the scale parameter. The case where A = 0 and B = 1 is called the standard uniform distribution. The equation for the standard normal distribution is: f ( x ) = 1 for 0 ≤ x ≤ 1 Since the general form of probability functions can be expressed in terms of the standard distribution, all subsequent formulas in this section are given for the standard form of the function. Figure 3.4 shows the uniform pdf. 2 1.8 1.6

Probability

1.4 1.2 1

0.8

0.6 0.4 0.2 0

0

1

2

3

4

5 X

6

Figure 3.4.  The uniform probability density function.

7

8

9

10

84  •  Using ANSYS for Finite Element Analysis

3.2.3.2  Cumulative Distribution Function The formula for the cdf of the uniform distribution is: F ( x ) = x for 0 ≤ x ≤ 1 Figure 3.5 shows the uniform cdf.

1 0.9 0.8

Probability

0.7 0.6

0.5 0.4 0.3 0.2 0.1 0 0

1

2

3

4

5 X

6

7

8

Figure 3.5.  The uniform cumulative distribution function.

3.2.3.3  Common Statistics Mean Median

( A + B) / 2 ( A + B) / 2

Mode

B−A

Range

( B − A )2 12

Standard Deviation

B− A

3 ( B + A)

Coefficient of Variation

s/m

Skewness Kurtosis

0 9/5

9

10

Probabilistic Design Analysis   •  85

3.2.3.4  Parameter Estimation The method of moments estimators for A and B are: A = x − 3S B = x + 3S The maximum likelihood estimators for A and B are: A = midrang (Y1 , Y2 , …, Yn ) − 0.5 rang (Y1 , Y2 , …, Yn ) B = midrang (Y1 , Y2 , …, Yn ) + 0.5 rang (Y1 , Y2 , …, Yn )

3.2.3.5  Comments The uniform distribution is a very fundamental distribution for cases where no other information apart from a lower and an upper limit exists. It is very useful to describe geometric tolerances. It can also be used in cases where there is no evidence that any value of the random variable is more likely than any other within a certain interval. In this sense, it can be used for cases where “lack of engineering knowledge” plays a role. The uniform distribution defines equal probability over a given range for a continuous distribution. For this reason, it is important as a reference distribution. One of the most important applications of the uniform distribution is in the generation of random numbers. That is, almost all random number generators generate random numbers on the (0, 1) interval. For other distributions, some transformation is applied to the uniform random numbers.

3.2.4  Lognormal Distribution A variable X is lognormally distributed if Y = LN(X) is normally distributed with “LN” denoting the natural logarithm. The general formula for the pdf of the lognormal distribution is:

f ( x) =

e

(

) / 2s 2 ( )

− ln (( x − q ) / m )

( x − q) s

2

2p

� x ≥ q; m, s > 0

86  •  Using ANSYS for Finite Element Analysis

where s is the shape parameter, q is the location parameter, and m is the scale parameter. The case where q = 0 and m = 1 is called the standard lognormal distribution. The equation for the standard normal distribution is: 2

f ( x) =

( ) � x ≥ 0, s > 0

− ln( x) e ( ) / 2s 2

x s 2p

Since the general form of probability functions can be expressed in terms of the standard distribution, all subsequent formulas in this section are given for the standard form of the function. Figure 3.6 shows the lognormal pdf for four values of s. There are several common parameterizations of the lognormal distribution. The form given here is from Evans, Hastings, and Peacock.

lognormal PDF (σ = 0.5)

0.5

0 0

2

4

6 8 x lognormal PDF (σ = 2)

0.5 0

2

4

x

6

8

0.4 0.2 0

2

0

2

3

1

0

0.6

0

10

Probability

Probability

1.5

lognormal PDF (σ = 1)

0.8 Probability

Probability

1

10

4 6 8 x lognormal PDF (σ = 5)

10

2 1 0

4

x

6

8

10

Figure 3.6.  The lognormal probability density function for four values of s.

3.2.4.1  Cumulative Distribution Function The formula for the cdf of the lognormal distribution is:  ln ( x )  F ( x) = Φ  � x ≥ 0, s > 0  s  where F is the cdf of the normal distribution. Figure 3.7 shows the lognormal cdf with the same values of s as the pdf plots above.

Probabilistic Design Analysis   •  87 lognormal PDF (σ = 0.5)

Probability

0.5

0

0

2

4

6 8 x lognormal PDF (σ = 2)

0 0

Probability

0

2

4

x

6

8

2

0.8

0.5

0

0.5

10

Probability

1

lognormal PDF (σ = 1)

1 Probability

1

4 6 8 x lognormal PDF (σ = 5)

10

0.6 0.4 0.2 0 0

10

2

4

6

x

8

10

Figure 3.7.  The lognormal cumulative distribution function for four values of s.

3.2.4.2  Common Statistics Mean

e0.5s

Median

Scale parameter m (= 1 if scale parameter not specified).

Mode

2

1 es

2

Range Standard Deviation

Zero to positive infinity

Coefficient of Variation

( e + 2) e − 1 (e ) + 2 (e ) + 3 (e ) − 3

(

)

es es − 1 2

2

s2

Skewness

s2

4

s2

Kurtosis

s2

3

s2

2

es − 1 2

The maximum likelihood estimates for the scale parameter, m, and the shape parameter s are: ∧



m = exp ( m ) and

∑ i =1( ln ( X i ) − m ) N



s =



N

2

88  •  Using ANSYS for Finite Element Analysis

Where: N�



m=

∑ i =1 lnX i N

If the location parameter is known, it can be subtracted from the original data points before computing the maximum likelihood estimates of the shape and scale parameters. 3.2.4.3  Comments The lognormal distribution is a basic and commonly used distribution. It is typically used to describe the scatter of the measurement data of physical phenomena, where the logarithm of the data would follow a normal distribution. The lognormal distribution is very suitable for phenomena that arise from the multiplication of a large number of error effects. It is also correct to use the lognormal distribution for a random variable that is the result of multiplying two or more random effects (if the effects that get multiplied are also lognormally distributed). If is often used for lifetime distributions; for example, the scatter of the strain amplitude of a cyclic loading that a material can endure until low-cycle-fatigue occurs is very often described by a lognormal distribution. The lognormal distribution is used extensively in reliability applications to model failure times. The lognormal and Weibull distributions are probably the most commonly used distributions in reliability applications. 3.2.5  Weibull Distribution 3.2.5.1  Probability Density Function The formula for the pdf of the general Weibull distribution is: f ( x) =

g  x − m   a a 

(g −1)

(

)

exp − (( x − m) / a ) x ≥ m; g, a > 0 g

Probabilistic Design Analysis   •  89

where g is the shape parameter, m is the location parameter, and a is the scale parameter. The case where m = 0 and a = 1 is called the standard lognormal distribution. The case where m = 0 and a = 1 is called the 2-parameter Weibull distribution. The equation for the standard Weibull distribution reduces to: (g −1)

f ( x) = g ( x)

( )

exp − xg x ≥ 0; g > 0

Weibull PDF (g = 0.5) Probability

0.5

0

0

1

0.8

2

3 x Weibull PDF (g = 2)

4

Probability

Probability

0.2 1

2

x

3

0.4 0.2 1

1

0.4

0

0.6

0 0

5

0.6

0

Weibull PDF (g = 1)

0.8

Probability

1

4

5

2 3 x Weibull PDF (g = 5)

4

5

4

5

0.5

0 0

1

2

x

3

Figure 3.8.  The Weibull probability density function for four values of g.

Since the general form of probability functions can be expressed in terms of the standard distribution, all subsequent formulas in this section are given for the standard form of the function. Figure 3.8 shows the Weibull pdf for four values of g. 3.2.5.2  Cumulative Distribution Function The formula for the cdf of the Weibull distribution is: F ( x) = 1 − e

( ) x ≥ 0; g > 0

− xg

Figure 3.9 shows the Weibull cdf with the same values of as the pdf plots above.

90  •  Using ANSYS for Finite Element Analysis Weibull CDF (g = 0.5)

0.6

0.5

0.4 0.2 0

0

1

2

3 x Weibull CDF (g = 2)

4

1

1

2 3 x Weibull CDF (g = 5)

4

5

4

5

0.5

0.5

0 0

0 0

5

Probability

1 Probability

Weibull CDF (g = 1)

1 Probability

Probability

0.8

1

2

x

3

4

5

0 0

1

2

x

3

Figure 3.9.  The Weibull cumulative distribution function for four values of g.

3.2.5.3  Common Statistics  g + 1 , Γ  g  Mean



Γ ( a ) = ∫t a −1e − t dt a

where Γ is the gamma function Median

ln ( 2)

Mode

 1 1 − g 

1

g 1

g

�g > 1

0 g ≤1 Range Standard Deviation

Coefficient of Variation

Zero to positive infinity.  g + 2    g + 1  − Γ Γ  g    g    g + 2 Γ  g    g + 1   Γ  g  

2

−1

2

Probabilistic Design Analysis   •  91

3.2.5.4  Comments The Weibull distribution is used extensively in reliability applications to model failure times. In engineering, the Weibull distribution is most often used for strength or strength-related lifetime parameters, and it is the standard distribution for material strength and lifetime parameters for very brittle materials (for these very brittle materials the “weakest-link-theory” is applicable).

3.3 Choosing a Distribution for a Random Variable The type and source of the data you have determines which distribution functions can be used or are best suited to your needs. 3.3.1  Measured Data If you have measured data then you first have to know how reliable that data is. Data scatter is not just an inherent physical effect, but also includes inaccuracy in the measurement itself. You must consider that the person taking the measurement might have applied a “tuning” to the data. For example, if the data measured represents a load, the person measuring the load may have rounded the measurement values; this means that the data you receive are not truly the measured values. Depending on the amount of this “tuning,” this could provide a deterministic bias in the data that you need to address separately. If possible, you should discuss any bias that might have been built into the data with the person who provided that data to you. If you are confident about the quality of the data, then how to proceed depends on how much data you have. In a single production field, the amount of data is typically sparse. If you have only few data then it is reasonable to use it only to evaluate a rough figure for the mean value and the standard deviation. In these cases, you could model the random input variable as a Gaussian distribution if the physical effect you model has no lower and upper limit, or use the data and estimate the minimum and maximum limit for a uniform distribution. In a mass production field, you probably have a lot of data, in which case you could use a commercial statistical package that will allow you to actually fit a statistical distribution function that best describes the scatter of the data.

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3.3.2  M  ean Values, Standard Deviation, Exceedence Values The mean value and the standard deviation are most commonly used to describe the scatter of data. Frequently, information about a physical quantity is given in the form that its value is; for example, “100±5.5.” Often, but not always, this form means that the value “100” is the mean value and “5.5” is the standard deviation. To specify data in this form implies a Gaussian distribution, but you must verify this (a mean value and standard deviation can be provided for any collection of data regardless of the true distribution type). If you have more information (e.g., you know that the data must be lognormal distributed), then the PDS allows you to use the mean value and standard deviation for a definition of a lognormal distribution. Sometimes the scatter of data is also specified by a mean value and an exceedence confidence limit. The yield strength of a material is sometimes given in this way; for example, a 99 percent exceedence limit based on a 95 percent confidence level is provided. This means that derived from the measured data we can be sure by 95 percent that in 99 percent of all cases the property values will exceed the specified limit and only in 1 percent of all cases they will drop below the specified limit. The supplier of this information is using mean value, the standard deviation, and the number of samples of the measured data to derive this kind of information. If the scatter of the data is provided in this way, the best way to pursue this further is to ask for more details from the data supplier. Since the given exceedence limit is based on the measured data and its statistical assessment, the supplier might be able to provide you with the details that were used. If the data supplier does not give you any further information, then you could consider assuming that the number of measured samples was large. If the given exceedence limit is denoted with and the given mean value is denoted with then the standard deviation can be derived from the equation:

s=

X1− a / 2 − X m C

where the values for the coefficient C are: Exceedence Probability 99.5% 99.0% 97.5% 95.0% 90.0%

C 2.5758 2.3263 1.9600 1.6449 1.2816

Probabilistic Design Analysis   •  93

3.3.3  No Data In situations where no information is available, there is never just one right answer. Below are hints about which physical quantities are usually described in terms of which distribution functions. This might help you with the particular physical quantity you have in mind. Also below is a list of which distribution functions are usually used for which kind of phenomena. Keep in mind that you might need to choose from multiple options. 3.3.3.1  Geometric Tolerances • If you are designing a prototype, you could assume that the actual dimensions of the manufactured parts would be somewhere within the manufacturing tolerances. In this case it is reasonable to use a uniform distribution, where the tolerance bounds provide the lower and upper limits of the distribution function. • Sometimes the manufacturing process generates a skewed distribution; for example, one half of the tolerance band is more likely to be hit than the other half. This is often the case if missing half of the tolerance band means that rework is necessary, while falling outside the tolerance band on the other side would lead to the part being scrapped. In this case a Beta distribution is more appropriate. • Often a Gaussian distribution is used. The fact that the normal distribution has no bounds (it spans minus infinity to infinity) is theoretically a severe violation of the fact that geometrical extensions are described by finite positive numbers only. However, in practice this is irrelevant if the standard deviation is very small compared to the value of the geometric extension, as is typically true for geometric tolerances. 3.3.3.2  Material Data • Very often the scatter of material data is described by a Gaussian distribution. • In some cases the material strength of a part is governed by the “weakest-link-theory.” The “weakest-link-theory” assumes that the entire part would fail whenever its weakest spot would fail. For material properties where the “weakest-link” assumptions are valid, then the Weibull distribution might be applicable. • For some cases, it is acceptable to use the scatter information from a similar material type. Let’s assume that you know that a material type very similar to the one you are using has a certain material property with a Gaussian distribution and a standard deviation of

94  •  Using ANSYS for Finite Element Analysis

±5 percent around the measured mean value; then let’s assume that for the material type you are using, you only know its mean value. In this case, you could consider using a Gaussian distribution with a standard deviation of ±5 percent around the given mean value. • For temperature-dependent materials it is prudent to describe the randomness by separating the temperature dependency from the scatter effect. In this case, you need the mean values of your material property as a function of temperature in the same way that you need this information to perform a deterministic analysis. If M(T) denotes an arbitrary temperature-dependent material property then the following approaches are commonly used: Multiplication equation: M (T )rand = Crand M (T ) Additive equation: M (T )rand = M (T ) + ∆M rand Linear equation: M (T )rand = Crand M (T ) + ∆M rand • Here, M(T ) denotes the mean value of the material property as a function of temperature. In the “multiplication equation” the mean value function is scaled with a coefficient Crand and this coefficient is a random variable describing the scatter of the material property. In the “additive equation” a random variable ∆Mrand is added on top of the mean value function M(T ). The “linear equation” combines both approaches and here both Crand and ∆Mrand are random variables. However, you should take into account that in general for the “linear equation” approach Crand and ∆Mrand are, correlated. • Deciding which of these approaches is most suitable to describe the scatter of the temperature-dependent material property requires that you have some raw data about this material property. Only by reviewing the raw data and plotting it versus temperature you can tell which approach is the better one. 3.3.3.3  Load Data • For loads, you usually only have a nominal or average value. You could ask the person who provided the nominal value the following

Probabilistic Design Analysis   •  95

questions: If we have 1,000 components that are operated under real-life conditions, what would the lowest load value be that only one of these 1,000 components is subjected to and all others have a higher load? What would the most likely load value be, that is, the value that most of these 1,000 components have (or are very close to)? What would the highest load value be that only one of the 1,000 components is subjected to and all others have a lower load? To be safe you should ask these questions not only of the person who provided the nominal value, but also to one or more experts who are familiar with how your products are operated under reallife conditions. From all the answers you get, you can then consolidate what the minimum, the most likely, and the maximum value probably is. As verification you can compare this picture with the nominal value that you would use for a deterministic analysis. If the nominal value does not have a conservative bias to it then it should be close to the most likely value. If the nominal value includes a conservative assumption (is biased), then its value is probably close to the maximum value. Finally, you can use a triangular distribution using the minimum, most likely, and maximum values obtained. • If the load parameter is generated by a computer program then the more accurate procedure is to consider a probabilistic analysis using this computer program as the solver mechanism. Use a probabilistic design technique on that computer program to assess what the scatter of the output parameters are, and apply that data as input to a subsequent analysis. In other words, first run a probabilistic analysis to generate an output range, and then use that output range as input for a subsequent probabilistic analysis.

3.3.4  Choosing Random Output Parameters Output parameters are usually parameters such as length, thickness, diameter, or model coordinates. The ANSYS PDS does not restrict you with regard to the number of RPs, provided that the total number of probabilistic design variables (i.e., RVs and RPs together) does not exceed 5,000. ANSYS recommends that you include all output parameters that you can think of and that might be useful to you. The additional computing time required to handle more RPs is marginal when compared to the time required to solve the problem. It is better to define RPs that you might not consider important before you start the analysis. If you forgot to include a random output parameter that later turns out to be important, you must redo the entire analysis.

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3.4  Probabilistic Design Techniques The Monte Carlo Simulation method is the most common and traditional method for a probabilistic analysis. This method lets you simulate how virtual components behave the way they are built. One simulation loop represents one manufactured component that is subjected to a particular set of loads and boundary conditions. For Monte Carlo simulations, you can employ either the Direct Sampling method or the Latin Hypercube Sampling method. When you manufacture a component, you can measure its geometry and all of its material properties (although typically, the latter is not done because this can destroy the component). In the same sense, if you started operating the component then you could measure the loads it is subjected to. Again, to actually measure the loads is very often impractical. But the bottom line is that once you have a component in your hand and start using it, then all the input parameters have very specific values that you could actually measure. With the next component you manufacture you can do the same; if you compared the parameters of that part with the previous part, you would find that they vary slightly. This comparison of one component to the next illustrates the scatter of the input parameters. The Monte Carlo Simulation techniques mimic this process. With this method you “virtually” manufacture and operate components or parts one after the other. The advantages of the Monte Carlo Simulation method are: • The method is always applicable regardless of the physical effect modeled in a finite element analysis. It not based on assumptions related to the RPs that if satisfied would speed things up and if violated would invalidate the results of the probabilistic analysis. Assuming the deterministic model is correct and a very large number of simulation loops are performed, then Monte Carlo techniques always provide correct probabilistic results. Of course, it is not feasible to run an infinite number of simulation loops; therefore, the only assumption here is that the limited number of simulation loops is statistically representative and sufficient for the probabilistic results that are evaluated. This assumption can be verified using the confidence limits, which the PDS also provides. • Because of the reason mentioned previously, Monte Carlo simulations are the only probabilistic methods suitable for benchmarking and validation purposes.

Probabilistic Design Analysis   •  97

• The individual simulation loops are inherently independent; the individual simulation loops do not depend on the results of any other simulation loops. This makes Monte Carlo Simulation techniques ideal candidates for parallel processing. The Direct Sampling Monte Carlo technique has one drawback: it is not very efficient in terms of required number of simulation loops. 3.4.1  Direct Sampling Direct Monte Carlo Sampling is the most common and traditional form of a Monte Carlo analysis. It is popular because it mimics natural processes that everybody can observe or imagine and is therefore easy to understand. For this method, you simulate how your components behave based on the way they are built. One simulation loop represents one component that is subjected to a particular set of loads and boundary conditions. The Direct Monte Carlo Sampling technique is not the most efficient technique, but it is still widely used and accepted, especially for benchmarking and validating probabilistic results. However, benchmarking and validating requires many simulation loops, which is not always feasible. This sampling method is also inefficient due to the fact that the sampling process has no “memory.” For example, if we have two RVs X1 and X2 both having a uniform distribution ranging from 0.0 to 1.0, and we generate 15 samples, we could get a cluster of two (or even more) sampling points that occur close to each other if we graphed the two variables (as shown in Figure 3.10). While in the space of all RVs, it can happen that one sample has input values close to another sample, this does not provide new information and 1 X2

0

0

X1

1

Figure 3.10.  The graph of X1 and X2 illustrating bad sample distribution.

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insight into the behavior of a component in a computer simulation if the same (or almost the same) samples are repeated. 3.4.2  Latin Hypercube Sampling The Latin Hypercube Sampling (LHS) technique is a more advanced and efficient form for Monte Carlo Simulation methods. The only difference between LHS and the Direct Monte Carlo Sampling technique is that LHS has a sample “memory,” meaning it avoids repeating samples that have been evaluated before (it avoids clustering samples). It also forces the tails of a distribution to participate in the sampling process. Generally, the LHS technique requires 20 percent to 40 percent fewer simulations loops than the Direct Monte Carlo Simulation technique to deliver the same results with the same accuracy. However, that number is largely problem dependent. Figure 3.11 shows the graph of X1 and X2 illustrating Good Sample Distribution. 1 X2

0

0

X1

1

Figure 3.11.  The graph of X1 and X2 illustrating good sample distribution.

3.5 Postprocessing Probabilistic Analysis Results There are two groups of postprocessing functions in the PDS: statistical and trend. A statistical analysis is an evaluation function performed on a single probabilistic design variable; for example, a histogram plot of a random output parameter. A trend analysis typically involves two or more probabilistic design variables; for example, a scatter plot of one probabilistic design variable versus another.

Probabilistic Design Analysis   •  99

3.5.1  Statistical Postprocessing Statistical postprocessing allows you several options for reviewing your results. 3.5.1.1  Sample History The most fundamental form of postprocessing is directly reviewing the simulation loop results as a function for the number of simulation loops. Here, you can review the simulation values, the mean, minimum, or maximum values, or the standard deviations. It is most helpful to review the mean values and standard deviation history for Monte Carlo Simulation results if you want to decide if the number of simulation loops was sufficient. If the number of simulation loops was sufficient, the mean values and standard deviations for all RPs should have converged. Convergence is achieved if the curve shown in the respective plots approaches a plateau. If the curve shown in the diagram still has a significant and visible trend with increasing number of simulation loops then you should perform more simulation loops. In addition, the confidence bounds plotted for the requested history curves can be interpreted as the accuracy of the requested curve. With more simulation loops, the width of the confidence bounds is reduced. 3.5.1.2  Histogram A histogram plot is most commonly used to visualize the scatter of a probabilistic design variable. A histogram is derived by dividing the range between the minimum value and the maximum value into intervals of equal size. Then the PDS determines how many samples fall within each interval, that is, how many “hits” landed in the intervals. Most likely, you will use histograms to visualize the scatter of your RPs. The ANSYS PDS also allows you to plot histograms of your RVs so you can double check that the sampling process generated the samples according to the distribution function you specified. For RVs, the PDS not only plots the histogram bars, but also a curve for values derived from the distribution function you specified. Visualizing histograms of the RVs is another way to make sure that enough simulation loops have been performed. If the number of simulation loops is sufficient, the histogram bars will: • Be close to the curve that is derived from the distribution f­ unction. • Be “smooth” (without large “steps”). • Not have major gaps.

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3.5.1.3  Cumulative Distribution Function The cdf is a primary review tool if you want to assess the reliability or the failure probability of your component or product. Reliability is defined as the probability that no failure occurs. Hence, in a mathematical sense reliability and failure probability are two sides of the same coin and numerically they complement each other (are additive to 1.0). The cdf value at any given point expresses the probability that the respective parameter value will remain below that point. Figure 3.12 shows the cdf of the random property X: The value of the cdf at the location x0 is the probability that the values of X stay below x0. Whether this probability represents the failure probability or the reliability of your component depends on how you define failure; for example, if you design a component such that a certain deflection should not exceed a certain admissible limit then a failure event occurs if the critical deflection exceeds this limit. Thus for this example, the cdf is interpreted as the reliability curve of the component. On the other hand, if you design a component such that the eigenfrequencies are beyond a certain admissible limit then a failure event occurs if an eigenfrequency drops below this limit. Thus for this example, the cdf is interpreted as the failure probability curve of the component. The cdf also lets you visualize what the reliability or failure probability would be if you chose to change the admissible limits of your design. Often you are interested in visualizing low probabilities and you want to assess the more extreme ends of the distribution curve. In this case, plotting the cdf in one of the following ways is more appropriate: • As a Gauss plot (also called a “normal plot”). If the probabilistic design variable follows a Gaussian distribution then the cdf is ­displayed as a straight line in this type of plot. 100% 90%

F(x)

80% 70% 60% 50% 40% 30% 20% 10% 0%

x1

x2

xi

X

Figure 3.12.  The cumulative distribution function of the random property X.

Probabilistic Design Analysis   •  101

• As a lognormal plot. If the probabilistic design variable follows a lognormal distribution then the cdf is displayed as a straight line in this type of plot. • As a Weibull plot. If the probabilistic design variable follows a Weibull distribution then the cdf is displayed as a straight line in this type of plot.

3.5.1.4  Print Probabilities The PDS offers a function where you can determine the cdf at any point along the axis of the probabilistic design variable, including an interpolation function so you can evaluate the probabilities between sampling points. This feature is most helpful if you want to evaluate the failure probability or reliability of your component for a very specific and given limit value. 3.5.1.5  Print Inverse Probabilities The PDS offers a function where you can probe the cdf by specifying a certain probability level; the PDS tells you at which value of the probabilistic design variable this probability will occur. This is helpful if you want to evaluate what limit you should specify to not exceed a certain failure probability, or to specifically achieve certain reliability for your component.

3.5.2  Trend Postprocessing Trend postprocessing allows you several options for reviewing your results.

3.5.2.1  Sensitivities Probabilistic sensitivities are important in allowing you to improve your design toward a more reliable and better quality product, or to save money while maintaining the reliability or quality of your product. You can request a sensitivity plot for any random output parameter in your model. There is a difference between probabilistic sensitivities and deterministic sensitivities. Deterministic sensitivities are mostly only local gradient information. For example, to evaluate deterministic sensitivities you can vary each input parameters by ±10 percent (one at a time) while keeping all other input parameters constant, then see how the output parameters react to these variations. As illustrated in the following figure, an output ­parameter

102  •  Using ANSYS for Finite Element Analysis

would be considered very sensitive with respect to a certain input parameter if you observe a large change of the output parameter value. Y

Steep gradient=higher sensitivity Y1 ΔX ΔY1

Y2

ΔY2 Lower gradient=lower sensitivity X

These purely deterministic considerations have various disadvantages that are taken into consideration for probabilistic sensitivities, namely: • A deterministic variation of an input parameter that is used to determine the gradient usually does not take the physical range of variability into account. An input parameter varied by ±10 percent is not meaningful for the analysis if ±10 percent is too much or too little compared with the actual range of physical variability and randomness. In a probabilistic approach the physical range of variability is inherently considered because of the distribution functions for input parameters. Probabilistic sensitivities measure how much the range of scatter of an output parameter is influenced by the scatter of the RVs. Hence, two effects have an influence on probabilistic sensitivities: the slope of the gradient, plus the width of the scatter range of the RVs. This is illustrated in the following figures. If a random input variable has a certain given range of scatter, then the scatter of the corresponding random output parameter is larger, and the larger the slope of the output parameter curve is (first illustration). But remember that an output parameter with a moderate slope can have a significant scatter if the RVs have a wider range of scatter (second illustration). Range of scatter X

Y

Y

Y1 Range of scatter Y1

Y2

Range of scatter Y2 X

Range of scatter X

Y2 Scatter range Y2 X

Probabilistic Design Analysis   •  103

• Gradient information is local information only. It does not take into account that the output parameter may react more or less with respect to variation of input parameters at other locations in the input parameter space. However, the probabilistic approach not only takes the slope at a particular location into account, but also all the values the random output parameter can have within the space of the RVs. • Deterministic sensitivities are typically evaluated using a finite-differencing scheme (varying one parameter at a time while keeping all others fixed). This neglects the effect of interactions between input parameters. An interaction between input parameters exists if the variation of a certain parameter has a greater or lesser effect if at the same time one or more other input parameters change their values as well. In some cases interactions play an important or even dominant role. This is the case if an input parameter is not significant on its own but only in connection with at least one other input parameter. Generally, interactions play an important role in 10 percent to 15 percent of typical engineering analysis cases (this figure is problem dependent). If interactions are important, then a deterministic sensitivity analysis can give you completely incorrect results. However, in a probabilistic approach, the results are always based on Monte Carlo simulations, either directly performed using your analysis model or using response surface equations. Inherently, Monte Carlo simulations always vary all RVs at the same time; thus if interactions exist then they will always be correctly reflected in the probabilistic sensitivities. To display sensitivities, the PDS first groups the RVs into two groups: those having a significant influence on a particular random output parameter and those that are rather insignificant, based on a statistical significance test. This tests the hypothesis that the sensitivity of a particular random input variable is identical to zero and then calculates the probability that this hypothesis is true. If the probability exceeds a certain significance level (determining that the hypothesis is likely to be true), then the sensitivity of that random input variable is negligible. The PDS will plot only the sensitivities of the RVs that are found to be significant. However, insignificant sensitivities are printed in the output window. You can also review the significance probabilities used by the hypothesis test to decide which group a particular random input variable belonged to the PDS allows you to visualize sensitivities either as a bar chart, a pie chart, or both. Sensitivities are ranked so the random input variable having the highest sensitivity appears first.

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In a bar chart the most important random input variable (with the highest sensitivity) appears in the leftmost position and the others follow to the right in the order of their importance. A bar chart describes the sensitivities in an absolute fashion (taking the signs into account); a positive sensitivity indicates that increasing the value of the random input variable increases the value of the random output parameter for which the sensitivities are plotted. Likewise, a negative sensitivity indicates that increasing the random input variable value reduces the random output parameter value. In a pie chart, sensitivities are relative to each other. In a pie chart the most important random input variable (with the highest sensitivity) will appear first after the 12 o’clock position, and the others follow in clockwise direction in the order of their importance. Using a sensitivity plot, you can answer the important questions. How can I make the component more reliable or improve its quality? If the results for the reliability or failure probability of the component do not reach the expected levels, or if the scatter of an output parameter is too wide and therefore not robust enough for a quality product, then you should make changes to the important input variables first. Modifying an input variable that is insignificant would be waste of time. Of course you are not in control of all random input parameters. A ­typical example where you have very limited means of control are material properties. For example, if it turns out that the environmental temperature (outdoor) is the most important input parameter then there is probably nothing you can do. However, even if you find out that the reliability or quality of your product is driven by parameters that you cannot control, this has importance—it is likely that you have a fundamental flaw in your product design! You should watch for influential parameters like these. If the input variable you want to tackle is a geometry-related parameter or a geometric tolerance, then improving the reliability and quality of your product means that it might be necessary to change to a more accurate manufacturing process or use a more accurate manufacturing machine. If it is a material property, then there is might be nothing you can do about it. However, if you only had a few measurements for a material property and consequently used only a rough guess about its scatter and the material property turns out to be an important driver of product reliability and quality, then it makes sense to collect more raw data. In this way, the results of a probabilistic analysis can help you spend your money where it makes the most sense—in areas that affect the reliability and quality of your products the most.

Probabilistic Design Analysis   •  105

3.5.2.2  Scatter Plots While the sensitivities point indicate which probabilistic design parameters you need to modify to have an impact on the reliability or failure probability, scatter plots give you a better understanding of how and how far you should modify the input variables. Improving the reliability and quality of a product typically means that the scatter of the relevant RPs must be reduced. The PDS allows you to request a scatter plot of any probabilistic design variable versus any other one, so you can visualize the relationship between two design variables (input variables or output parameters). This allows you to verify that the sample points really show the pattern of correlation that you specified (if you did so). Typically, RPs are correlated because they are generated by the same set of RVs. To support the process of improving the reliability or quality of your product, a scatter plot showing a random output parameter as a function of the most important random input variable can be very helpful. When you display a scatter plot, the PDS plots the sampling points and a trendline. For this trendline, the PDS uses a polynomial function and lets you choose the order of the polynomial function. If you plot a random output parameter as a function of a random input variable, then this trendline expresses how much of the scatter on the random output parameter (Y-axis) is controlled by the random input variable (X-axis). The deviations of the sample points from the trendline are caused and controlled by all the other RVs. If you want to reduce the scatter of the random output parameter to improve reliability and quality, you have two options: • Reduce the width of the scatter of the most important random input variable(s) (that you have control over). • Shift the range of the scatter of the most important random input variable(s) (that you have control over). The effect of reducing and shifting the scatter of a random input variable is illustrated in the following figures. “Input range before” denotes the scatter range of the random input variable before the reduction or shifting, and “input range after” illustrates how the scatter range of the random input variable has been modified. In both cases, the trendline tells how much the scatter of the output parameter is affected and in which way the range of scatter of the random input variable is modified.

Input range before Random input variable

Output range before

Input range after

Output range after

Random output parameter

Input range after Output range after

Output range before

Random output parameter

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Input range before Random input variable

It depends on your particular problem if either reducing or shifting the range of scatter of a random input variable is preferable. In general, reducing the range of scatter of a random input variable leads to higher costs. A reduction of the scatter range requires a more accurate process in manufacturing or operating the product—the more accurate, the more expensive it is. This might lead you to conclude that shifting the scatter range is a better idea, because it preserves the width of the scatter (which means you can still use the manufacturing or operation process that you have). The following points are some considerations if you want to do that: • Shifting the scatter range of a random input variable can only lead to a reduction of the scatter of a random output parameter if the trendline shows a clear nonlinearity. If the trendline indicates a linear trend (if it is a straight line), then shifting the range of the input variables anywhere along this straight line doesn’t make any difference. For this, reducing the scatter range of the random input variable remains your only option. • It is obvious from the second illustration that shifting the range of scatter of the random input variable involves an extrapolation beyond the range where you have data. Extrapolation is always difficult and even dangerous if done without care. The more sampling points the trendline is based on, the better you can extrapolate. Generally, you should not go more than 30–40 percent outside of the range of your data. But the advantage of focusing on the important RVs is that a slight and careful modification can make a difference.

3.5.2.3  Correlation Matrix Probabilistic sensitivities are based on a statistical correlation analysis between the individual probabilistic design variables. The PDS lets you

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review the correlation data that has been used to derive sensitivities and decide if individual sensitivity values are significant or not. This information is collected in the correlation matrix of the RPSs versus the RVs. The PDS also lets you review the correlations that have been sampled between RVs, which are stored in the RVs correlation matrix. The correlations between RPs are important if you want to use the probabilistic results of your probabilistic analysis as input for another probabilistic analysis.

3.6 Tutorial 4: Probabilistic Design Analysis of Circular Plate Bending In this tutorial, a circular plate of thickness t with a center hole is rigidly attached along the inner edge and unsupported along the outer edge. The plate is subjected to bending by a moment Ma applied uniformly along the outer edge. The input parameters are subject to uncertainty. Measurements show that the plate dimensions can vary significantly. Specimen tests show that the material properties can also vary. The applied force is also subject to uncertainty. You will determine the variation of the output parameters given the uncertainty of the plate dimensions, material properties, and applied force. The output parameters that you will study are the maximum deflection of the plate and the maximum equivalent stress at the clamped edge. Y

Z Ma

Ma a b

X b

t

a

Geometric properties Inner radius (b) Outer radius (a) Thickness (t)

Value 100.0±0.1 mm 300.0±0.1 mm 1.0±0.1 mm

Distribution Uniform Uniform Uniform

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Material properties

Nominal value

Young’s modulus (E)

207,000 N/mm2

Poisson’s ratio (ν)

0.3

Loading

Nominal value

Applied Moment (Ma)

100 N-mm/mm

Distribution Gaussian distribution with a mean value equal to the nominal value and a standard deviation of 5% of its mean value. Gaussian distribution with a mean value equal to the nominal value and a standard deviation of 5% of its mean value. Distribution Lognormal distribution with a mean value equal to the nominal value and a standard deviation of 10% of its mean value.

3.6.1  Approach and Assumptions Since the problem is axisymmetric only a small sector of elements is needed. A small angle θ = 30° is used for approximating the circular boundary. The calculated load is equally divided and applied to the outer nodes. 3.6.2  Summary of Steps 1. Enter the PDS and execute the file for the deterministic model (CIRCPLATE.txt). 2. Specify CIRCPLATE.txt as the analysis file for the probabilistic analysis. 3. Define the RVs for your probabilistic analysis. For this problem, you define the inner radius, outer radius and thickness of the plate, Young’s modulus and poisson’s ratio, and applied moment as RVs with various distribution functions. 4. Define the random output variables. You define the maximum deflection and maximum equivalent stress at the fixed edges as the output parameters.

Probabilistic Design Analysis   •  109

5. Obtain solution. Define Monte Carlo as the probabilistic analysis method and execute the Monte Carlo probabilistic simulations. 6. Perform statistical postprocessing to visualize and evaluate the Monte Carlo results. 7. Perform trend postprocessing. 8. Generate HTML report and exit. 3.6.3  Step-by-Step Analysis 1. Enter PDS and specify analysis file. You begin by entering the PDS and executing the file CIRCPLAE. txt for the deterministic model. The CIRCPLAE.txt file contains a complete analysis sequence for a simple plate with a single force load. It uses parameters to define all inputs and outputs. Main Menu > Prob Design

Utility Menu >File>Read Input from Choose the file CIRCPLATE.txt from your working directory. OK. Close.

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File: CIRCPLATE.txt /TITLE, Bending of Circular Plate (PDS by Al-Tabey) /PREP7

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RADIN=100 RADOUT=300 THICK=1 MOMENT=157.08 ! Total moment on a 30 deg segment YOUNG=207000 PRATIO=0.3 ET,1,SHELL63 R,1,THICK MP,EX,1,YOUNG MP,NUXY,1,PRATIO CSYS,1 ! DEFINE CYLINDRICAL C.S. CYL4,0,0,RADIN,0,RADOUT,30 LESIZE,1, , ,30, , , , ,1 LESIZE,3, , ,30, , , , ,1 LESIZE,2, , ,20,0.5, , , ,1 LESIZE,4, , ,20,0.5, , , ,1 MSHKEY,0 AMESH,1 NROTAT,ALL DL,3,1,ALL, DL,2,1,UY, DL,2,1,ROTX, DL,2,1,ROTZ, DL,4,1,UY, DL,4,1,ROTX, DL,4,1,ROTZ, LSEL,S,,,1 NSLL,R,1 F,ALL,MY,-(MOMENT)/31 ALLSEL,ALL FINISH /SOLUTION SOLVE FINISH /POST1 RSYS,1 PLNSOL,U,Z, LSEL,S,,,1 NSLL,S,1 NSORT,U,Z *GET, DEFMAX, SORT, 0, MAX LSEL,S,,,3 NSLL,S,1 NSORT,S,EQV *GET, STRMAX, SORT, 0, MAX ALLSEL, ALL 2. Specify analysis file. Now define CIRCPLATE as the analysis file for the probabilistic analysis. PDS uses CIRCPLATE to create a file for performing analysis loops. PDS uses the CIRCPLATE input and output parameters as RVs and RPs. Main Menu > Prob Design>-Analysis File-Assign Choose the file CIRCPLATE from your working directory. Use the Browse button if you choose.

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Main Menu > Prob Design>- Analysis File-Assign

Choose the file CIRCPLATE from your working directory. Use the Browse button if you choose.

3. Define input variables. You will now define the input parameters and their distribution functions. Main Menu > Prob Design>Prob Definitns>Random Input

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Define Inner Radius as an input variable: Add a random input variable.

Choose RADIN as the parameter. Choose UNIF as the distribution type. OK.

Enter RADIN-0.1 and RADIN+0.1 for the lower and upper boundary, respectively. OK.

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The following figure will appear.

Define Outer Radius as an input variable. Add a random input variable. Choose RADOUT as the parameter. Choose UNIF as the distribution type. OK.

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Enter RADOUT-0.1 and RADOUT+0.1 for the lower and upper boundary, respectively. OK.

The following figure will appear.

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Similarly, define Thickness as an input variable. Add a random input variable. Define thickness as an input variable. Choose THICK as the parameter. Choose UNIF as the distribution type. OK. Enter THICK-0.1 and THICK+0.1 for the lower and upper boundary, respectively. OK.

Define Young’s Modulus as an input variable. Add a random input variable. Choose YOUNG as the parameter. Choose GAUS as the distribution type. OK.

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Enter YOUNG for the mean value and 0.05*YOUNG for the standard deviation. OK.

The following figure will appear.

Similarly, define Poisson’s Ratio as an input variable. Add a random input variable. Choose PRATIO as the parameter. Choose GAUS as the distribution type. OK. Enter PRATIO for the mean value and 0.05*PRATIO for the standard deviation. OK.

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Define Applied Moment as an input variable. Add a random input variable. Choose MOMENT as the parameter. Choose LOG1 as the distribution type. OK.

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Enter MOMENT for the mean value and 0.1*MOMENT for the standard deviation. OK.

Close.

4. Define output parameters. You will now define the maximum deflection of the plate and the maximum equivalent stress at the clamped edges as output parameters. Main Menu> Prob Design> Prob Definitns> Random Output

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Add a random output variable. Choose DEFMAX as the parameter. OK.

Add a random output variable. Choose STRMAX as the parameter. OK.

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Close.

5. Execute Monte Carlo simulations to obtain solution. You will now specify the Monte Carlo simulation method and various options. You will specify the Latin Hypercube sampling tech-

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nique. For the same accuracy, it generally requires fewer simulation loops than the Direct Monte Carlo sampling technique. You will set the number of simulations to 40 and the number of repetitions to 1 to give 40 analysis loops. Forty loops will be a sufficient number for demonstration purposes. Main Menu> Prob Design>-Prob MethodMonte Carlo Sims

Choose the Latin Hypercube sampling method. OK.

Type in 40 for the Number of Simulations and 1 for the Number of Repetitions to give a total of 40 simulations. Choose Random Sampling for the Interval Sampling Option. Choose Execute ALL Sims for the Autostop Option. Choose Use Continue CONT for the Random Seed Option. OK.

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Main Menu> Prob Design>Run>-Exec SerialRun Serial

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Enter Result1 for the Solution Set Label. It is the name for the results set. OK.

Review the information in the dialog box and then choose OK to initiate the solution.

The solution is complete when the statement “LOOP 40 OUT OF 40—CYCLE 1 OUT OF 1 IS FINISHED” appears in the ANSYS Output Window.

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6. Perform Statistical Postprocessing. (a) Sample history You will now review simulation results and mean value plots for maximum deflection. If the number of simulations is sufficient, the mean value plots for random output variables converge (the curve flattens out). Main Menu> Prob Design> Prob Results-Statistics>Sampl History

Choose DEFMAX for the Prob Design Variable. Choose Samples for the Plot Type. OK.

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The following figure will appear.

Main Menu> Prob Design> Prob Results-Statistics> Sampl History

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Choose DEFMAX for the Prob Design Variable. Choose Mean Values for the Plot Type. OK.

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The following figure will appear.

The curve is relatively flat, indicating that the number of simulations is sufficient. (b) Histogram You will also review a histogram plot for the maximum deflection in order to visualize the scatter of this random output parameter. To double check that the number of simulation loops is sufficient, you will plot a histogram of the Young’s modulus random input variable. The ANSYS PDS not only plots the histogram bars, but also plots a curve for values derived from the distribution function that you specified. If the number of simulations is sufficient, the histogram for Young’s modulus will have bars close to the curve derived for the distribution function, which are smooth and without any major gaps.

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Main Menu> Prob Design> Prob Results-Statistics> Histogram

Choose DEFMAX for the Prob Design Variable. OK.

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The following figure will appear.

Main Menu> Prob Design> Prob Results-Statistics> Histogram

Choose YOUNG for the Prob Design Variable. OK.

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The following figure will appear.

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The histogram bars resemble the pdf, indicating that the number of simulations is sufficient. However, further increase in number of simulation to 50 will improve the results. (c) Cumulative DF and probabilities You will also determine the probability that the maximum deflection is below 0.525 mm. Main Menu> Prob Design> Prob Results-Statistics> CumulativeDF

Choose DEFMAX for the Prob Design Variable. OK.

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The following figure will appear.

Note: The curve shows that there is a about a 93 percent probability that the deflection remains below 1.375. Main Menu> Prob Design> Prob Results-Statistics> Probabilities

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Choose DEFMAX for the Prob Design Variable. Choose Less than for the Relation Label. Enter 2.0 for the Limit Value. OK.

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The following figure will appear.

Note: that the probability is 97.8 percent that the maximum deflection is below the value of 2.0 mm. After reviewing the information, choose File>Close. (d) Inverse problem You will also determine the maximum deflection that will give a 90 percent probability that the deflection is below that value. Main Menu> Prob Design> Prob Results-Statistics> Inverse Prob

Choose DEFMAX for the Prob Design Variable. Enter 0.90 for the Target Probability. OK.

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The following figure will appear.

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Note: that there is a 90 percent probability that the maximum deflection is below 1.438 mm. After reviewing the results, choose File>Close. 7. Perform trend postprocessing. You will now request sensitivity plots for DEFMAX and STRMAX to determine which RVs are most significant. Main Menu> Prob Design> Prob Results Trends> Sensitivities

Choose DEFMAX for the Response Param. OK.

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The following figure will appear.

Main Menu > Prob Design> Prob Results Trends> Sensitivities Choose STRMAX for the Response Param. OK.

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Note that the legend indicates that the RVs THICK and MOMENT are important for the random output parameter DEFMAX and STRMAX. You will then request scatter plots of DEFMAX versus the most significant random input variable THICK. Main Menu> Prob Design> Prob Results-Trends> Scatter Plot

Choose THICK for the first parameter. Choose DEFMAX for the second parameter. OK.

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The following figure will appear.

Finally, you will determine the correlation coefficients between the RPs and the RVs. Main Menu> Prob Design> Prob Results-Trends> Correl Matrix

Choose Input Output for the Type of Matrix. OK.

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OK.

The following figure will appear.

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Note: the correlation coefficients. After reviewing the results, choose File>Close. 8. Generate HTML report and exit. You will now generate an HTML report for your probabilistic analysis. It includes deterministic model information, probabilistic model information, and probabilistic analysis results. Main Menu> Prob Design> Prob Results-Report> Report Options

OK (use the default options).

Main Menu> Prob Design> Prob Results-Report> Generate Report

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Enter report file name, your first name, and your last name. OK.

You will find the report and all related files in your current directory in a subdirectory with the report file name. Toolbar: Quit Choose Quit—No Save! OK Congratulations! You have completed this tutorial.

Chapter 4

APDL Programming APDL stands for ANSYS Parametric Design Language, a scripting ­language that you can use to automate common tasks or even build your model in terms of parameters (variables). While all ANSYS commands can be used as part of the scripting language, the APDL commands discussed here are the true scripting commands and encompass a wide range of other features such as repeating a command, macros, if-thenelse branching, do-loops, and scalar, vector, and matrix operations. While APDL is the foundation for sophisticated features such as design optimization, probabilistic design analysis, and adaptive meshing, it also offers many conveniences that you can use in your day-to-day analyses. APDL is also a macro language to create macros. You can record a frequently used sequence of ANSYS commands in a macro file (these are sometimes called command files). Creating a macro enables you to, in effect, create your own custom ANSYS command. In addition to executing a series of ANSYS commands, a macro can call graphical user interface (GUI) functions or pass values into arguments.

4.1  Create the Analysis File The analysis file is a key component and crucial to ANSYS optimization and probabilistic design analysis. The program uses the analysis file to form the loop file, which is used to perform analysis loops. Any type of ANSYS analysis (structural, thermal, magnetic, etc.; linear or nonlinear) can be incorporated in the analysis file. There are two ways to create an analysis file: • Input commands line by line with a system editor. • Create the analysis interactively through ANSYS and use the ANSYS command log as the basis for the analysis file.

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Both methods have advantages and disadvantages. Creating the file with a system editor is the same as creating a batch input file for the analysis. This method allows you full control of parametric definitions through exact command inputs. It also eliminates the need to clean out unnecessary commands later. However, if you are not moderately familiar with ANSYS commands, this method may be inconvenient. You may find it easier to perform the initial analysis interactively, and then use the resulting command log as the basis for the analysis file. In this case, final editing of the log file may be required in order to make it suitable for optimization looping. 4.1.1  Analysis File using Log Files The ANSYS program records every command it executes, whether typed in directly or executed by a function in the GUI, in two places: the session log file and the internal database command log. • The session log file is a text file that is saved in your working ­directory. • The database command log is saved in the ANSYS database. You can copy this log to a file at any time by choosing Utility Menu> File> Write DB Log File. Both files are command logs that can be used as input to the ANSYS program. 4.1.1.1  Session Log File Every ANSYS session produces a session log named Jobname.LOG. The default jobname is FILE or file, depending on the operating system. The program opens the log file when you first enter the program, and closes it when you exit the program. The session log file provides a complete record of your ANSYS session (in terms of commands) and is quite valuable as a means of recovering from a system crash or catastrophic user mistake. By reading in a renamed copy of your log file (or by submitting it as a batch file), you can re-execute every command in your log file, recreating your database exactly as it existed previously. Your log file is also useful as a debugging tool that can help to reveal any mistakes you might have made in an ANSYS session. Should you

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require help from your ASD in debugging an ANSYS session, he or she will almost certainly ask to see a copy of your log file. Each new ANSYS session appends commands to the existing file Jobname.LOG. (That is, the log file is not overwritten during a new ANSYS session, but added to.) A “time stamp,” consisting of the current date and time, is included so that you can identify the start of each session. Use / FILNAME,1 to start a new log file for the session. You can list your entire log file during an interactive run by picking Utility Menu> List> Files> Log File. Since this file is in ASCII format, you can view and edit it readily using an external text editor. 4.1.1.2  Database Command Log ANSYS captures commands generated (or typed in) during an ANSYS session not only in the log file but also in memory. This in-memory version of the command history is called the internal database log. When you save the database, the program saves this command log in the database file (Jobname.DB) along with the other database information. Use either Utility Menu> File> Write DB Log File or the LGWRITE command to write the database command log to a named ASCII file. You can then edit this file, make desired changes, and use the file as command input to the program. This capability is especially useful if you want to use the command history that was created during an interactive session, but have somehow lost or corrupted the session log file (Jobname.LOG) that was associated with your database. If you create your database in multiple sessions by saving and resuming the database file, the ANSYS program keeps the database log continuous by appending each new command that is processed. Therefore, the internal database log is not fragmented; it will represent the complete database. 4.1.2  Using a Command Log File as Input The procedure for re-executing the commands contained in a Jobname. LOG file or in the database log consists of three main steps: 1. Establish the Command Log File The method to do this depends on whether you use the session log file or the database log.

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4.1.2.1  Session Log File To establish a command log file from the session log file (Jobname. LOG), perform these steps: 1. Rename or copy the session log file to a different name. 2. List the log file by choosing Utility Menu> List> Files> Log File. 3. Choose File> Save as from the Log File window. 4.1.2.2  Database Command Log To establish a command log file from the database log, pick Utility Menu> File> Write DB Log File. You can specify a file name or use the default name, Jobname.LGW. You also have the option (with the Kedit field) to write all commands (default), essential commands only (Kedit = REMOVE), or essential commands with nonessential commands commented out (Kedit = COMMENT). 1. Edit the Command Log File Sometimes, you will need to edit your command log file before using it as program input. As you edit your log file, you may want to add comments or indentation to improve its readability. You can add comments to your log file by using the comment character (!). 2. Read in the Edited Log File In an interactive session, pick Utility Menu> File> Read Input from to read in the edited command log file.

4.2 Tutorial 5: Stress Analysis of Bicycle Wrench In this tutorial will be found the von Mises stresses for the bicycle wrench made of steel shown under the given distributed and boundary condition. Material Properties: Modulus of elasticity: E = 200 GPa (Steel) Poisson’s ratio: ν = 0.32 Geometry: Lengths and radii as shown Thickness: 3mm Loading: Distributed load: 88 N/cm Constraints: ux, uy, uz = 0 at left hexagon

APDL Programming   •  149 R = 1.25 cm

7 mm side

88 N / cm

1.5 cm

1 cm Fixed all around

3 cm

1 cm

9 mm side

1. Establish the Command Log File First, solve the problem interactively and then save the database log file. • Start ANSYS File -> Save As -> Tutorial 1 -> OK • Use the structural solid element PLANE82 for FEM modeling: Preprocessor -> Element Type -> Add/Edit/Delete -> Add -> Structural Mass-Solid -> Select 8node 82 -> OK -> Options -> Element Behavior: Select Plane Stress w/thk -> OK -> Close • Enter Real Constants for the element type chosen: Preprocessor -> Real Constants -> Add/Edit/Delete -> Add -> OK -> Enter Thickness THK = 0.3 -> OK -> Close • Enter material property data for specified steel: Preprocessor -> Material Props -> Material Model -> Structural -> Linear -> Elastic -> Isotropic -> Enter Young’s modulus EX = 200e9 and Poisson’s ratio PRXY = 0.32 -> OK -> Close • Create geometry for two similar rectangles 1.5 cm by 3 cm at locations (2.25,0.5) and (7.25, 0.5): Preprocessor -> Modeling -> Create -> Areas-Rectangle -> By 2 Corners -> In dialogue box enter WP X = 2.25, WP Y = 0.5, Width = 3, Height = 1.5 -> Apply -> Enter values for next rectangle: WP X = 7.25, WP Y = 0.5, Width = 3, Height = 1.5 -> OK • Create geometry for three circles, all of 1.25 cm radius: Preprocessor -> Modeling -> Create -> Areas -> Circle -> Solid Circle -> In dialogue box enter WP X = 1.25, WP Y = 1.25, Radius = 1.25 -> Apply -> Enter values for next circle -> WP X = 6.25, WP Y = 1.25, Radius = 1.25 -> Apply ->

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• •



Enter values for next circle -> WP X = 11.25, WP Y = 1.25, Radius = 1.25 -> OK Perform the Boolean operation add to union the areas together: Preprocessor -> Modeling -> Operate -> Booleans -> Add -> Areas -> Pick All Create geometry for three hexagons, two of 7 mm side (at the ends) and one of 9 mm side (in the center): Preprocessor -> Modeling -> Create -> Areas -> Polygon -> Hexagon -> In dialogue box enter WP X = 1.25, WP Y = 1.25, Radius = 0.7 -> Theta = 120 -> Apply -> Enter values for next hexagon -> WP X = 6.25, WP Y = 1.25, Radius = 0.9 -> Theta = 120 -> Apply -> Enter values for next hexagon -> WP X = 11.25, WP Y = 1.25, Radius = 0.7 -> Theta = 120 -> OK Perform the Boolean operation subtract to get the hexagonal holes in the wrench the body: Preprocessor -> Modeling -> Operate -> Booleans -> Subtract -> Areas -> Click on the solid portion of wrench -> Apply -> One by one pick the three hexagonal areas -> OK Now create a mesh in the final wrench shape, first refining the mesh size: Preprocessor -> Meshing -> Size Controls -> ManualSize -> Global > Size -> Enter Size = 0.1 -> OK Preprocessor -> Meshing -> Mesh -> Areas -> Free -> Click on wrench -> OK Apply the boundary conditions and the load: Preprocessor -> Loads -> Analysis Type -> New Analysis -> Static -> OK Preprocessor -> Loads -> Define Loads -> Apply -> Structural -> Displacement -> On Key Points -> Click on the six corner points of the left hexagon -> OK -> Select All DOF -> OK Preprocessor -> Loads -> Define Loads -> Apply -> Structural -> Pressure -> On Lines -> Pick the line indicated in problem statement (top line of right arm) -> OK -> Enter VALUE = 88 -> OK Perform the solution: Solution -> Solve -> Current LS -> OK Start post-processing: Check the deformed shape: General Post Proc -> Plot Results -> Deformed Shape -> Def + undef edge -> OK To establish a command log file from the database log, pick Utility Menu> File> Write DB Log File. You can specify a file name or use the default name, Jobname.LGW.

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2. Edit the Command Log File Using any text editor, you can edit either one of the following files: Session log file: Tutorial 1.log Database command log file: Tutorial 1.lgw Note: One may prefer to edit database command l of file having essential commands with nonessential commands commented out using character. The nonessential comments have been highlighted as follows. /BATCH ! /COM,ANSYS RELEASE 11.0 UP20070125 11:04:23 08/30/2008 /input,menust,tmp,’’,,,,,,,,,,,,,,,,1 ! /GRA,POWER ! /GST,ON ! /PLO,INFO,3 ! /GRO,CURL,ON ! /CPLANE,1 ! /REPLOT,RESIZE WPSTYLE,,,,,,,,0 ! SAVE, Tutorial-1,db,C:\A6_ANS~1\ FINOPT~1\OPTIMI~1\ /PREP7 !* ET,1,PLANE82 !* KEYOPT,1,3,3 KEYOPT,1,5,0 KEYOPT,1,6,0 !* ! SAVE, Tutorial 1,db, !* R,1,0.3, !* ! SAVE, Tutorial 1,db, !* MPTEMP,,,,,,,, MPTEMP,1,0 MPDATA,EX,1,,200e9 MPDATA,PRXY,1,,0.32 ! SAVE, Tutorial 1,db, BLC4,2.25,0.5,3,1.5 BLC4,7.25,.5,3,1.5

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! SAVE, Tutorial 1,db, CYL4,1.25,1.25,1.25 CYL4,6.25,1.25,1.25 CYL4,11.25,1.25,1.25 ! SAVE, Tutorial 1,db, FLST,2,5,5,ORDE,2 FITEM,2,1 FITEM,2,-5 AADD,P51X ! SAVE, Tutorial 1,db, RPR4,6,1.25,1.25,0.7,120 RPR4,6,6.25,1.25,0.9,120 RPR4,6,11.25,1.25,0.7,120 ! SAVE, Tutorial 1,db, ! /REPLOT,RESIZE ! alist, all FLST,3,3,5,ORDE,2 FITEM,3,1 FITEM,3,-3 ASBA, 6,P51X ! SAVE, Tutorial 1,db, ESIZE,0.1,0, MSHKEY,0 CM,_Y,AREA ASEL, , , , 4 CM,_Y1,AREA CHKMSH,’AREA’ CMSEL,S,_Y !* AMESH,_Y1 !* CMDELE,_Y CMDELE,_Y1 CMDELE,_Y2 !* ! SAVE, Tutorial 1,db, !* ANTYPE,0 ! /ZOOM,1,S CRN,-0.465951,-0.036231,-0.465951,0.144092 FLST,2,6,3,ORDE,6 FITEM,2,9

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FITEM,2,13 FITEM,2,15 FITEM,2,19 FITEM,2,21 FITEM,2,-22 !* /GO DK,P51X, , , ,0,ALL, , , , , , ! SAVE, Tutorial 1,db, ! /AUTO,1 ! /REP,FAST FLST,2,1,4,ORDE,1 FITEM,2,7 /GO !* SFL,P51X,PRES,88, ! SAVE, Tutorial 1,db, FINISH /SOL ! /STATUS,SOLU SOLVE ! SAVE, Tutorial 1,db, FINISH /POST1 ! PLDISP,2 ! SAVE, Tutorial 1,db, ! LGWRITE,’Tutorial 1’,’lgw’,’C:\A6_ANS~1\FINOPT~1\ OPTIMI~1\’,COMMENT After removing nonessential comments and adding comments: /PREP7 !* Selection of Element Type ET,1,PLANE82 KEYOPT,1,3,3 ! Option plane with thickness KEYOPT,1,5,0 KEYOPT,1,6,0 !* Enter thickness value as a real constant R,1,0.3, !* Enter material properties MPTEMP,,,,,,,, MPTEMP,1,0

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MPDATA,EX,1,,200e9 MPDATA,PRXY,1,,0.32 ! Create geometric model BLC4,2.25,0.5,3,1.5 BLC4,7.25,.5,3,1.5 CYL4,1.25,1.25,1.25 CYL4,6.25,1.25,1.25 CYL4,11.25,1.25,1.25 ! Add rectangles with three circular areas FLST,2,5,5,ORDE,2 FITEM,2,1 FITEM,2,-5 AADD,P51X ! Create three hexagonal areas RPR4,6,1.25,1.25,0.7,120 RPR4,6,6.25,1.25,0.9,120 RPR4,6,11.25,1.25,0.7,120 ! Subtract hexagonals to create final shape FLST,3,3,5,ORDE,2 FITEM,3,1 FITEM,3,-3 ASBA, 6,P51X ! Meshing of the geometry ESIZE,0.1,0, MSHKEY,0 CM,_Y,AREA ASEL, , , , 4 CM,_Y1,AREA CHKMSH,’AREA’ CMSEL,S,_Y AMESH,_Y1 CMDELE,_Y CMDELE,_Y1 CMDELE,_Y2 !* ANTYPE,0 ! Set analysis type for Static Analysis ! ! Apply displacement boundary conditions FLST,2,6,3,ORDE,6 FITEM,2,9 FITEM,2,13

APDL Programming   •  155

FITEM,2,15 FITEM,2,19 FITEM,2,21 FITEM,2,-22 /GO DK,P51X, , , ,0,ALL, , , , , , ! Apply pressure FLST,2,1,4,ORDE,1 FITEM,2,7 /GO SFL,P51X,PRES,88, FINISH ! Save and Exit the Preprocessor (PREP7) ! /SOL ! Enter Solution phase SOLVE FINISH 3. Read in the Edited Log File In an interactive session, pick Utility Menu> File> Read Input from to read in the edited command log file.

4.3 Tutorial 6: Heat Loss from a Cylindrical Cooling Fin In this tutorial, a very long rod 25 mm in diameter has one end maintained at 100°C. The surface of the rod is exposed to ambient air at 25°C with a convection heat transfer coefficient of 10 W/m2.K. Determine the heat loss from the rod constructed of pure copper. Tb = 100°

T∞ = 25°C h = 10 W/m2.K

Air

D = 2.5 cm

L = 1.32 m

156  •  Using ANSYS for Finite Element Analysis

Geometric properties Diameter Length

Value 25 mm 1320 mm

Material properties

Nominal value

Thermal Conductivity ­Coefficient

398 W/m.k

Convection Heat Transfer Coefficient

10 W/m2.k

Loading Temperature Thermal Load Convection Thermal Load

Nominal Value

Distribution Uniform Uniform

Distribution At Average ­Temperature T = (Tb + T∞)/2 = 62.5C°~355k

Location

100°C

At the base of the fin.

10 W/m2.k

At the cylindrical edge and the flat face at the tip of the fin.

1. Establish the Command Log File First, solve the problem interactively and then save the database log file. • Set preference for thermal analysis: Preference > Thermal > OK • Use the thermal solid element Quad4node55 for FEM modeling: Preprocessor -> Element Type -> Add/Edit/Delete -> Add-> Thermal Mass-Solid -> Select 4node 55 -> OK -> Options -> Element Behavior: Select Axisymmetric -> OK -> Close • Enter material property data for specified steel: Preprocessor -> Material Props -> Material Model -> Thermal > Conductivity > Isotropic > Enter conductivity value KXX = 398 > OK -> Close • Create geometry for one rectangles 12.5E-3 m by 1.32 m at locations (0,0): Preprocessor -> Modeling -> Create -> Areas-Rectangle -> By 2 Corners -> In dialogue box enter WP X = 0, WP Y = 0, Width = 12.5E-3, Height = 1.3 -> OK • Now, to create a mesh, first refining the mesh size by specifying the number of elements along the length (line numbers 2 and 4) = 600 and along radius (line numbers 1 and 3) = 5:

APDL Programming   •  157

Preprocessor -> Meshing -> Size Controls -> ManualSize -> Lines > Picked Lines > type 2 then press Enter > type 4 then press Enter-> OK > Enter # of divisions = 600 > Apply > type 1 then press Enter > type 3 then press Enter-> OK > Enter # of divisions = 5 -> OK • Now create the mesh: Preprocessor -> Meshing -> Mesh -> Areas -> Free -> Click on the area of select Pick All -> OK • Select type of analysis: Solution > Analysis Type -> New Analysis -> Steady-State -> OK • Apply the temperature boundary condition: Solution > Define Loads > Apply -> Thermal > Temperature > On Line > Enter line No. = 1 > Select TEMP and enter value = 100 > OK. • Apply the convective boundary conditions: Solution > Define Loads > Apply -> Thermal > Convection > On Lines > Enter line Numbers = 2, 3 and 4 (after typing each number press ENTER key) > OK > Enter Film Coeff = 10 and Bulk Temperature = 25 > OK. • Perform the solution: Solution -> Solve -> Current LS -> OK • To establish a command log file from the database log, pick Utility Menu> File> Write DB Log File. You can specify a file name or use the default name, Jobname.LGW. 2. Edit the Command Log File Using any text editor, edit the Database command log file: Tutorial 2.lgw /BATCH ! /COM,ANSYS RELEASE 11.0 UP20070125 13:36:28 08/30/2008 /input,start110,ans,’C:\Program Files\ ANSYS Inc\v110\ANSYS\apdl\’,,,,,,,,,,,,,,,,1 !* /NOPR /PMETH,OFF,0 KEYW,PR_SET,1 KEYW,PR_STRUC,0 KEYW,PR_THERM,1 KEYW,PR_FLUID,0 KEYW,PR_ELMAG,0

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KEYW,MAGNOD,0 KEYW,MAGEDG,0 KEYW,MAGHFE,0 KEYW,MAGELC,0 KEYW,PR_MULTI,0 KEYW,PR_CFD,0 /GO !* ! /COM, ! /COM,Preferences for GUI filtering have been set to display: ! /COM, Thermal !* /PREP7 !* ET,1,PLANE55 !* KEYOPT,1,1,0 KEYOPT,1,3,1 KEYOPT,1,4,0 KEYOPT,1,8,0 KEYOPT,1,9,0 !* ! SAVE, Tutorial 2,db, !* MPTEMP,,,,,,,, MPTEMP,1,0 MPDATA,KXX,1,,398 ! SAVE, Tutorial 2,db, BLC4,0,0,12.5e-3,1.32 ! SAVE, Tutorial 2,db, ! /PNUM,KP,0 ! /PNUM,LINE,1 ! /PNUM,AREA,0 ! /PNUM,VOLU,0 ! /PNUM,NODE,0 ! /PNUM,TABN,0 ! /PNUM,SVAL,0 ! /NUMBER,0 !* ! /PNUM,ELEM,0

APDL Programming   •  159

! /REPLOT !* ! /DIST,1,0.729,1 ! /REP,FAST ! /DIST,1,0.729,1 ! /REP,FAST ! /DIST,1,0.729,1 ! /REP,FAST ! /DIST,1,0.729,1 ! /REP,FAST ! /DIST,1,0.729,1 ! /REP,FAST ! /DIST,1,0.729,1 ! /REP,FAST ! /DIST,1,0.729,1 ! /REP,FAST ! /DIST,1,0.729,1 ! /REP,FAST ! /DIST,1,0.729,1 ! /REP,FAST ! /DIST,1,0.729,1 ! /REP,FAST ! /DIST,1,0.729,1 ! /REP,FAST FLST,5,2,4,ORDE,2 FITEM,5,2 FITEM,5,4 CM,_Y,LINE LSEL, , , ,P51X CM,_Y1,LINE CMSEL,,_Y !* LESIZE,_Y1, , ,600, , , , ,1 !* FLST,5,2,4,ORDE,2 FITEM,5,1 FITEM,5,3 CM,_Y,LINE LSEL, , , ,P51X CM,_Y1,LINE CMSEL,,_Y

160  •  Using ANSYS for Finite Element Analysis

!* LESIZE,_Y1, , ,5, , , , ,1 !* MSHKEY,0 CM,_Y,AREA ASEL, , , , 1 CM,_Y1,AREA CHKMSH,’AREA’ CMSEL,S,_Y !* AMESH,_Y1 !* CMDELE,_Y CMDELE,_Y1 CMDELE,_Y2 !* FINISH /SOL !* ANTYPE,0 ! /AUTO,1 ! /REP,FAST ! /ZOOM,1,S CRN,0.362883,-0.778990,0.337117,-0.538560 FLST,2,1,4,ORDE,1 FITEM,2,1 !* /GO DL,P51X, ,TEMP,100,0 FLST,2,3,4,ORDE,2 FITEM,2,2 FITEM,2,-4 /GO !* SFL,P51X,CONV,10, ,25, ! /STATUS,SOLU SOLVE FINISH /POST1 !* ! /EFACET,1 ! PLNSOL, TEMP,, 0

APDL Programming   •  161

 LGWRITE,’Tutorial 2’,’lgw’,’C:\A6_ANS~1\ ! FINOPT~1\OPTIMI~1\’,COMMENT After editing /PMETH,OFF,0 ! Setting preference KEYW,PR_SET,1 KEYW,PR_STRUC,0 KEYW,PR_THERM,1 KEYW,PR_FLUID,0 KEYW,PR_ELMAG,0 KEYW,MAGNOD,0 KEYW,MAGEDG,0 KEYW,MAGHFE,0 KEYW,MAGELC,0 KEYW,PR_MULTI,0 KEYW,PR_CFD,0 /PREP7 ! Enter into preprocessor (PREP7) !* ET,1,PLANE55 ! Set element type with axisymmetric option !* KEYOPT,1,1,0 KEYOPT,1,3,1 KEYOPT,1,4,0 KEYOPT,1,8,0 KEYOPT,1,9,0 MPTEMP,,,,,,,, ! Enter material properties MPTEMP,1,0 MPDATA,KXX,1,,398 BLC4,0,0,12.5e-3,1.32 ! Geometric modeling FLST,5,2,4,ORDE,2 ! Divide lines 2 and 4 into 600 divisions FITEM,5,2 FITEM,5,4 CM,_Y,LINE LSEL, , , ,P51X CM,_Y1,LINE CMSEL,,_Y !* LESIZE,_Y1, , ,600, , , , ,1 !* FLST,5,2,4,ORDE,2 ! Divide lines 1 and 3 into 5 divisions

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FITEM,5,1 FITEM,5,3 CM,_Y,LINE LSEL, , , ,P51X CM,_Y1,LINE CMSEL,,_Y !* LESIZE,_Y1, , ,5, , , , ,1 !* MSHKEY,0 ! Mesh the area CM,_Y,AREA ASEL, , , , 1 CM,_Y1,AREA CHKMSH,’AREA’ CMSEL,S,_Y !* AMESH,_Y1 !* CMDELE,_Y CMDELE,_Y1 CMDELE,_Y2 !* FINISH ! Save and exit preprocessor /SOL ! Enter Solution module !* ANTYPE,0 ! Set analysis type as steady-state FLST,2,1,4,ORDE,1 ! Specify temperature boundary conditions on line 1 FITEM,2,1 DL,P51X, ,TEMP,100,0 FLST,2,3,4,ORDE,2 ! Specify convective b/ conditions on lines, 2,3 & 4 FITEM,2,2 FITEM,2,-4 SFL,P51X,CONV,10, ,25, SOLVE ! Solve the problem FINISH 3. Read in the Edited Log File In an interactive session, pick Utility Menu> File> Read Input from to read in the edited command log file. Further editing for design optimization:

APDL Programming   •  163

• Define fin length (L) and fin radius (R) as PARAMETERS to be used as design variable. • Using APDL commands get the fin heat transfer rate (THEAT) and fin tip temperature (Tempend) to be used in the objective function and state variable. • Define objective function as ObjFunct=C-TOTALHEAT /PMETH,OFF,0 ! Setting preference KEYW,PR_SET,1 KEYW,PR_STRUC,0 KEYW,PR_THERM,1 KEYW,PR_FLUID,0 KEYW,PR_ELMAG,0 KEYW,MAGNOD,0 KEYW,MAGEDG,0 KEYW,MAGHFE,0 KEYW,MAGELC,0 KEYW,PR_MULTI,0 KEYW,PR_CFD,0 ! ! *******************************PREPROCESSING ! /PREP7 ! Enter into preprocessor (PREP7) !* ET,1,PLANE55 ! Set element type with axisymmetric option !* KEYOPT,1,1,0 KEYOPT,1,3,1 KEYOPT,1,4,0 KEYOPT,1,8,0 KEYOPT,1,9,0 MPTEMP,,,,,,,, ! Enter material properties MPTEMP,1,0 MPDATA,KXX,1,,398 L=1.32 ! Set fin length as parameter L R=12.5e-3 ! Set fin radius as parameter R BLC4,0,0,R,L ! Geometric modeling in terms of R and L FLST,5,2,4,ORDE,2 ! Divide lines 2 and 4 into 600 divisions

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FITEM,5,2 FITEM,5,4 CM,_Y,LINE LSEL, , , ,P51X CM,_Y1,LINE CMSEL,,_Y !* LESIZE,_Y1, , ,600, , , , ,1 !* FLST,5,2,4,ORDE,2 ! Divide lines 1 and 3 into 5 divisions FITEM,5,1 FITEM,5,3 CM,_Y,LINE LSEL, , , ,P51X CM,_Y1,LINE CMSEL,,_Y !* LESIZE,_Y1, , ,5, , , , ,1 !* MSHKEY,0 ! Mesh the area CM,_Y,AREA ASEL, , , , 1 CM,_Y1,AREA CHKMSH,’AREA’ CMSEL,S,_Y !* AMESH,_Y1 !* CMDELE,_Y CMDELE,_Y1 CMDELE,_Y2 !* FINISH ! Save and exit preprocessor ! ! **************************************SOLUTION ! /SOL ! Enter Solution module !*

APDL Programming   •  165

ANTYPE,0 ! Set analysis type as steady-state FLST,2,1,4,ORDE,1 ! Specify temperature boundary conditions on line 1 FITEM,2,1 DL,P51X, ,TEMP,100,0 FLST,2,3,4,ORDE,2 ! Specify convective b/ conditions on lines, 2,3 & 4 FITEM,2,2 FITEM,2,-4 SFL,P51X,CONV,10, ,25, SOLVE ! Solve the problem FINISH ! ****************************** POST PROCESSING ! /POST1 FSUM *GET,THEAT,FSUM,HEAT *SET,TOTALHEAT,-THEAT ObjFunct=10000-TOTALHEAT ALLSEL *GET,Tempend,NODE,607,TEMP, FINISH

Chapter 5

Design Optimization Optimizing a structure or manufactured product refers to a sequence of analyses aimed at reducing the weight or cost of the structure or product while maintaining the functionality of the structure or product, and that meets all engineering requirements, and that can be manufactured and assembled. This process may include modifications to the structural or product topology, component thicknesses, and material properties. The optimization process is iterative where the specified design variables are altered, in the course of the optimization process, in a manner that modifies the objective function (structural or product weight or cost) in an advantageous manner. “The term optimization, or mathematical programming, refers to the study of problems in which one seeks to minimize or maximize a real function by systematically choosing the values of real or integer variables from within an allowed set.” Using the computer to achieve the most efficient design of a product, finite element analysis (FEA) and other methods are used. The finite element method (FEM) is among the most powerful tool for nonlinear simulations. In the past, design engineers have performed a combination of manual and automated methods to accomplish design optimization. They have applied FEA to parts of a CAD drawing, determined what components needed work and then redrew the object manually. Increasingly, software that is tightly integrated with the CAD program can perform the analysis and automatically redraw the object.

5.1 Optimum Design The optimum design is the best design in some predefined sense. Among many examples, the optimum design for a frame structure may be the one with minimum weight or maximum frequency; in heat transfer, the minimum temperature; or in magnetic motor design, the maximum peak

168  •  Using ANSYS for Finite Element Analysis

torque. In many other situations minimization of a single function may not be the only goal, and attention must also be directed to the satisfaction of predefined constraints placed on the design (e.g., limits on stress, geometry, displacement, heat flow). The independent variables in an optimization analysis are the design variables. The vector of design variables is indicated by: X = [ X1



X2

X 3  X n ] (5.1)

Design variables are subject to n constraints with upper and lower limits, that is, X i ≤ X i ≤ X i (i = 1, 2, 3, ……, n) (5.2)



Where: n = number of design variables. The design variable constraints are often referred to as side constraints and define what is commonly called feasible design space. Now, minimize: f = f ( x ) (5.3)

Subject to:

gi ( x ) ≤ gi (i = 1, 2, 3, ……, m1 ) (5.4)



hi ( x ) ≤ hi ( x ) (i = 1, 2, 3, ……, m2 ) (5.5)



Wi ≤ Wi ( x ) ≤ Wi (i = 1, 2, 3, ……, m3 ) (5.6)

Where: f = objective function gi, hi, wi = state variables containing the design, with under bar and over bars representing lower and upper bounds respectively. m1 + m2 + m3 = number of state variables constraints with various upper and lower limit values. The state variables can also be referred to as dependent variables in that they vary with the vector x of design variables. (Equation 5.3) through (Equation 5.6) represent a constrained minimization problem whose aim is the minimization of the objective function f under the constraints imposed by (Equations 5.2, 5.4, 5.5, and 5.6). Design configurations that satisfy all constraints are referred to as feasible designs. Design configurations with one or more violations are termed

Design Optimization   •  169

infeasible. In defining feasible design space, a tolerance is added to each state variable limit. So if X* is a given design set defined as:

X * =  X *1

X *2

X *3  X *n  (5.7)

The design is deemed feasible only if:

( )



g *i = gi x* ≤ gi + ai (i = 1, 2, 3, ……, m1 ) (5.8)



hi − bi ≤ h*i = hi x* (i = 1, 2, 3, ……, m2 ) (5.9)



Wi − gi ≤ W *i = Wi x* ≤ Wi − gi (i = 1, 2, 3, ……, m3 ) (5.10)

( )

( )

Where: ai, bi and gi = tolerances And

X i ≤ X *i ≤ X i (i = 1, 2, 3, ……, n) (5.11)

(Equation 5.8) to (Equation 5.11) are the defining statements of a feasible design set. As design sets are generated by methods or tools (discussed as follows) and if an objective function is defined, the best design set is computed and its number is stored. The best set is determined under one of the following conditions. 1. If one or more feasible sets exist the best design set is the feasible one with the lowest objective function value. In other words, it is the set that most closely agrees with the mathematical goals expressed by (Equation 5.3) to (Equation 5.6). 2. If all design sets are infeasible, the best design set is the one closest to being feasible, irrespective of its objective function value.

5.1.1  Optimum Design Fundamentals Some of the fundamental concepts needed to understand optimization process are explained as follows: Problem formulation: Problem formulation is normally the most difficult part of the process. It is the selection of design variables, constraints, objectives, and models of the disciplines. A further consideration is the strength and breadth of the interdisciplinary coupling in the problem.

170  •  Using ANSYS for Finite Element Analysis

Design variables: A design variable is a specification that is controllable from the point of view of the designer. For instance, the thickness of a structural member can be considered a design variable. Another might be the material the member is made out of. Design variables can be continuous (such as a wing span), discrete (such as the number of ribs in a wing), or boolean (such as whether to build a monoplane or a biplane). Design problems with continuous variables are normally solved more easily. Design variables are often bounded, that is, they often have maximum and minimum values. Depending on the solution method, these bounds can be treated as constraints or separately. Constraints: A constraint is a condition that must be satisfied in order for the design to be feasible. An example of a constraint in aircraft design is that the lift generated by a wing must be equal to the weight of the aircraft. In addition to physical laws, constraints can reflect resource limitations, user requirements, or bounds on the validity of the analysis models. Constraints can be used explicitly by the solution algorithm or can be incorporated into the objective using Lagrange multipliers. Objectives: An objective is a numerical value that is to be maximized or minimized. For example, a designer may wish to maximize profit or minimize weight. Many solution methods work only with single objectives. When using these methods, the designer normally weighs the various objectives and sums them to form a single objective. Other methods allow multi-objective optimization. Models: The designer must also choose models to relate the constraints and the objectives to the design variables. These models are dependent on the discipline involved. They may be empirical models, such as a regression analysis of aircraft prices, theoretical models, such as from computational fluid dynamics, or reduced-order models of either of these. In choosing the models the designer must trade off fidelity with analysis time. The multidisciplinary nature of most design problems complicates model choice and implementation. Often several iterations are necessary between the disciplines in order to find the values of the objectives and constraints. As an example, the aerodynamic loads on a wing affect the structural deformation of the wing. The structural deformation in turn changes the shape of the wing and the aerodynamic loads. Therefore, in analyzing a wing, the aerodynamic and structural analyses must be run a number of times in turn until the loads and deformation converge. Standard form: Once the design variables, constraints, objectives, and the relationships between them have been chosen, the problem can be expressed in the following form: find x that minimizes J(x) subject to g(x) ≤ 0, h(x) = 0 and � xlb ≤ x ≤ xub where J is an objective, x is a vector

Design Optimization   •  171

of design variables, g is a vector of inequality constraints, h is a vector of equality constraints, xlb and xub are vectors of lower and upper bounds on the design variables. Maximization problems can be converted to minimization problems by multiplying the objective by constraints can be reversed in a similar manner. Equality constraints can be replaced by two inequality constraints. Optimization Tree: The optimization is divided into three types, namely, the continuous, discrete, and multi-objective optimization. The various ways for optimization along with the procedures to be followed are listed in Figure 5.1. Problem solution: The problem is normally solved using appropriate techniques from the field of optimization. These include gradient-based algorithms, population-based algorithms, or others. Very simple problems can sometimes be expressed linearly; in that case, the techniques of linear programming are applicable. Gradient-based methods

Population-based

Other methods

• Newton’s method • Random search • Genetic algorithms • Steepest descent • Grid search • Memetic • Conjugate • Simulated annealing ­algorithms ­gradient ­sequential harmony search direct • Particle swarm ­quadratic search ­optimization ­programming

Figure 5.1.  Optimization tree listing the optimization methods.

172  •  Using ANSYS for Finite Element Analysis

5.1.2  Applications and Examples 5.1.2.1  Applications 1. Design of structures and manufactured products to reduce weight while maintaining required structural, functional, and manufacturing characteristics. 2. Optimization of component thickness and shape based on user-­ defined objective and constraints. 3. Structural and product design constrained by size, weight, frequency, and so on. 4. Feasible structural and product designs subject to size and shape constraints. 5. Design of structures and manufactured products for minimum cost, weight, or size. 5.1.2.2  Examples A few examples of optimization are listed as follows: 1. Optimization of structural crashworthiness behavior, for example, automotive industry, aviation and aeronautical industry, transportation safety, and so on. 2. Optimization of sheet metal forming processes, for example, optimization of tool geometry with respect to springback compensation. 3. Identification and optimization of material parameters in nonlinear material models.

5.2 Design Optimization Using ANSYS The optimization module is an integral part of the ANSYS program that can be employed to determine the optimum design. The ANSYS program can determine an optimum design, a design that meets all specified requirements yet demands a minimum in terms of expenses such as weight, s­ urface area, volume, stress, cost, and other factors. An optimum design is one that is as effective as possible. Virtually any aspect of design can be optimized:

Design Optimization   •  173

dimensions (such as thickness), shape (such as fillet radii), placement of supports, cost of fabrication, natural frequency, material property, and so on. Among many examples, the optimum design for a frame structure may be the one with minimum weight or maximum frequency; in heat transfer, the minimum temperature; or in magnetic motor design, the maximum peak torque. Any ANSYS item that can be expressed in terms of parameters is a candidate for design optimization. In many other situations minimization of a single function may not be the only goal, and attention must also be directed to the satisfaction of predefined constraints placed on the design (e.g., limits on stress, geometry, displacement, heat flow). 5.2.1  D  esign Optimization Terminology and Information Flow While working toward an optimum design, the ANSYS optimization routines employ three types of variables that characterize the design process: design variables, state variables, and the objective function. These variables are represented by scalar parameters in ANSYS Parametric Design Language (APDL). The use of APDL is an essential step in the optimization process. The independent variables in an optimization analysis are the design variables. To understand the terminology involved in design optimization, consider the following problem: Find the minimum-weight design of a beam of rectangular cross-section subject to the following constraints: • Total stress σ should not exceed σmax [s < smax]. • Beam deflection Δ should not exceed Δmax [∆ < ∆ max]. • Beam height h should not exceed hmax [h < hmax]. Design Variables (DVs): Independent quantities varied to achieve the optimum design. Upper and lower limits are specified to serve as “constraints” on the DVs. These limits define the range of variation for the DV. In the above beam example, width b and height h are obvious candidates for DVs. Both b and h cannot be zero or negative, so their lower limit would be b,h > 0.0. Also, h has an upper limit of hmax. Up to 60 DVs may be defined in an ANSYS design optimization problem. State Variables (SVs): Quantities that constrain the design. Also known as “dependent variables,” they are typically response quantities that are functions of the DVs. A state variable may have a maximum and minimum limit, or it may be “single sided,” having only one limit. Our beam example has two SVs: σ (the total stress) and Δ (the beam deflection). You can define up to 100 SVs in an ANSYS design optimization problem.

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Objective Function: The dependent variable that you are attempting to minimize. It should be a function of the DVs (i.e., changing the values of the DVs should change the value of the objective function). In the beam example, the total weight of the beam could be the objective function (to be minimized). You may define only one objective function in an ANSYS design optimization problem. Optimization Variables: Collectively, the design variables, state variables, and the objective function. In an ANSYS optimization, these variables are represented by user-named variables called parameters. You must identify which parameters in the model are DVs, which are SVs, and which is the objective function. Design Set or Design: A unique set of parameter values representing a given model configuration. Typically, a design set is characterized by the optimization variable values; however, all model parameters (including those not identified as optimization variables) are included in the set. Feasible Design: A design that satisfies all specified constraints (those on the SVs as well as on the DVs). If any one of the constraints is not satisfied, the design is considered infeasible. The best design is the one that satisfies all constraints and produces the minimum objective function value. (If all design sets are infeasible, the best design set is the one closest to being feasible, irrespective of its objective function value.) Analysis File: An ANSYS input file containing a complete analysis sequence (preprocessing, solution, and postprocessing). The file must contain a parametrically defined model, using parameters to represent all inputs and outputs to be used as DVs, SVs, and the objective function. Loop File: An optimization file (named Jobname.LOOP), created automatically via the analysis file. The design optimizer uses the loop file to perform analysis loops. Loop: A single pass through the analysis file. Output for the last loop performed is saved in file Jobname.OPO. An optimization iteration (or simply iteration) is one or more analysis loops that result in a new design set. Typically, an iteration equates to one loop; however, for the first order method, one iteration represents more than one loop. Optimization Database: This contains the current optimization environment, which includes optimization variable definitions, parameters, all optimization, specifications, and accumulated design sets. This database can be saved (to Jobname.OPT) or resumed at any time in the optimizer. The following Figure 5.2 illustrates the flow of information during an optimization analysis. The analysis file must exist as a separate entity. The optimization database is not part of the ANSYS model database.

Design Optimization   •  175

RESUME

SAVE

File.DB ANSYS database file

ANSYS Model database

Analysis file (parametrically OPEXE defined model) OPSAVE

OPRESU

Optimization database

OPEXE OPEXE

File.LOOP Loop file

File.OPO Last loop output

File.OPT optimization data file

Figure 5.2.  Optimization data flow.

5.2.2  Optimization Methods The ANSYS optimization procedure offers several methods and tools that in various ways attempt to address the mathematical problem stated earlier. ANSYS optimization methods perform actual minimization of the objective function of Equation 5.3. It will be shown that they transform the constrained problem into an unconstrained one that is eventually minimized. Design tools, on the other hand, do not directly perform minimization. Use of the tools offer alternate means for understanding design space and the behavior of the dependent variables. Methods and tools are discussed in the following sections. The ANSYS program uses two optimization methods to accommodate a wide range of optimization problems: • The subproblem approximation method is an advanced zero-order method that can be efficiently applied to most engineering problems.

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• The first-order method is based on design sensitivities and is more suitable for problems that require high accuracy. For both the subproblem approximation and first-order methods, the program performs a series of analysis-evaluation-modification cycles. That is, an analysis of the initial design is performed, the results are evaluated against specified design criteria, and the design is modified as necessary. The process is repeated until all specified criteria are met. 5.2.2.1  Subproblem Approximation Method The subproblem approximation method can be described as an advanced zero-order method in that it requires only the values of the dependent variables, and not their derivatives. There are two concepts that play a key role in the subproblem approximation method: the use of approximations for the objective function and state variables, and the conversion of the constrained optimization problem to an unconstrained problem. Approximations: For this method, the program establishes the relationship between the objective function and the DVs by curve fitting. This is done by calculating the objective function for several sets of DV values (i.e., for several designs) and performing a least squares fit between the data points. The resulting curve (or surface) is called an approximation. Each optimization loop generates a new data point, and the objective function approximation is updated. It is this approximation that is minimized instead of the actual objective function. State variables are handled in the same manner. An approximation is generated for each state variable and updated at the end of each loop. Conversion to an Unconstrained Problem: State variables and limits on design variables are used to constrain the design and make the optimization problem a constrained one. The ANSYS program converts this problem to an unconstrained optimization problem because minimization techniques for the latter are more efficient. The conversion is done by adding penalties to the objective function approximation to account for the imposed constraints. Convergence Checking: At the end of each loop, a check for convergence (or termination) is made. The problem is said to be converged if the current, previous, or best design is feasible and any of the following conditions are satisfied: • The change in objective function from the best feasible design to the current design is less than the objective function tolerance. • The change in objective function between the last two designs is less than the objective function tolerance.

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• The changes in all design variables from the current design to the best feasible design are less than their respective tolerances. • The changes in all design variables between the last two designs are less than their respective tolerances. Sometimes the solution may terminate before convergence is reached. This happens if one of the following conditions is true: • The number of loops specified has been performed. • The number of consecutive infeasible designs has reached the specified limit. The default number is 7. 5.2.2.2  First-Order Method Like the subproblem approximation method, the first-order method converts the problem to an unconstrained one by adding penalty functions to the objective function. However, unlike the subproblem approximation method, the actual finite element representation is minimized and not an approximation. The first-order method uses gradients of the dependent variables with respect to the design variables. For each iteration, gradient calculations (which may employ a steepest descent or conjugate direction method) are performed in order to determine a search direction, and a line search strategy is adopted to minimize the unconstrained problem. Thus, each iteration is composed of a number of sub-iterations that include search direction and gradient computations. That is why one optimization iteration for the first-order method performs several analysis loops. Convergence Checking: First-order iterations continue until either convergence is achieved or termination occurs. The problem is said to be converged if, when comparing the current iteration design set to the previous and best sets, one of the following conditions is satisfied: • The change in objective function from the best design to the current design is less than the objective function tolerance. • The change in objective function from the previous design to the current design is less than the objective function tolerance. 5.2.3  Optimization Design Tools In addition to the two optimization techniques, the ANSYS program offers a set of strategic tools that can be used to enhance the efficiency of the design process. For example, a number of random design iterations can be

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performed. The initial data points from the random design calculations can serve as starting points to feed the optimization methods. 5.2.3.1  Single Loop Analysis Tool This is a simple and very direct tool for understanding design space. It is not necessary but it may be useful to compute values of state variables or the objective function. The design variables are all explicitly defined by the user. A single loop is equivalent to one complete FEA. At the beginning of each iteration, the user defines design variable values, X=X*=­design variables defined by the user, and executes a single loop or iteration. If either state variables or the objective function are defined, corresponding state variable and objective function values will result. 5.2.3.2  Random Tool Multiple loops are performed, with random design variable values at each loop. This design tool will fill the design variable vector with randomly generated values for each iteration. Each random design iteration is equivalent to one complete analysis loop. A maximum number of loops and a desired number of feasible loops can be specified. This tool is useful for studying the overall design space, and for establishing feasible design sets for subsequent optimization analysis. Random iterations continue until either one of the following conditions is satisfied: n � r� = N r n f =� N f Where: nr = number of random iterations performed per each execution nf = total number of feasible design sets (including feasible sets from previous executions) Nr = maximum number of iterations Nf = desired number of feasible design sets. 5.2.3.3  Sweep Tool The sweep tool is used to scan global design space that is centered on a user-defined, reference design set. Upon execution, a sweep is made in

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the direction of each design variable while holding all other design variables fixed at their reference values. The state variables and the objective function are computed and stored for subsequent display at each sweep evaluation point. A sweep execution will produce ns design sets calculated from: n � s� = nN s Where: n = number of design variables Ns = number of evaluations to be made in the direction of each design variable.

5.2.3.4  Factorial Tool This is a statistical tool that is used to generate design sets at all extreme combinations of design variable values. This technique is related to the technology known as design of experiment that uses a two-level, full and fractional factorial analysis. The primary aim is to compute main and interaction effects for the objective function and the state variables.

5.2.3.5  Gradient Tool The gradient tool computes the gradient of the state variables and the objective function with respect to the design variables. A reference design set is defined as the point of evaluation for the gradient. Using this tool, you can investigate local design sensitivities.

5.2.4  General Process for Design Optimization One can approach an ANSYS optimization in two ways: as a batch run or interactively via the graphical user interface (GUI). If you are familiar with ANSYS commands, you can perform the entire optimization by creating an ANSYS command input file and submitting it as a batch job. This may be a more efficient method for complex analyses (e.g., nonlinear) that require extensive run time. Alternatively, the interactive features of optimization offer greater flexibility and immediate feedback for review of loop results. When performing optimization through the GUI, it is important to first

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establish the analysis file for your model. Then all operations within the optimizer can be performed interactively, allowing the freedom to probe your design space before the actual optimization is done. The insights you gain from your initial investigations can help to narrow your design space and achieve greater efficiency during the optimization process. (The interactive features can also be used to process batch optimization results.) The process involved in design optimization consists of the following general steps. The steps may vary slightly, depending on whether you are performing optimization interactively (through the GUI), in batch mode, or across multiple machines. 1. Create an analysis file to be used during looping. This file should represent a complete analysis sequence and must do the following: • Build the model parametrically (PREP7). • Obtain the solution(s) (SOLUTION). • Retrieve and assign to parameters the response quantities that will be used as state variables and objective functions (POST1/ POST26). 2. Enter OPT and specify the analysis file (OPT). 3. Declare optimization variables (OPT). 4. Choose optimization tool or method (OPT). 5. Specify optimization looping controls (OPT). 6. Initiate optimization analysis (OPT). 7. Review the resulting design sets data (OPT) and postprocess results (POST1/POST26). 1. Create the analysis file The analysis file is a key component and crucial to ANSYS optimization. The program uses the analysis file to form the loop file, which is used to perform analysis loops. Any type of ANSYS analysis (structural, thermal, magnetic, etc.; linear or nonlinear) can be incorporated in the analysis file. In this file, the model must be defined in terms of parameters (which are usually the DVs), and results data must be retrieved in terms of parameters (for SVs and the objective function). Only numerical scalar parameters are used by the design optimizer. There are two ways to create an analysis file: • Input commands line by line with a system editor. • Create the analysis interactively through ANSYS and use the ANSYS command log as the basis for the analysis file.

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No matter how you intend to create the analysis file, the basic information that it must contain is the same. The steps it must include are explained next. 5.2.4.1  Build the Model Parametrically PREP7 is used to build the model in terms of the DV parameters. For our beam example, the DV parameters are B (width) and H (height), so the element real constants are expressed in terms of B and H: /PREP7 ! Initialize DV parameters: B=2.0 ! Initialize width H=3.0 ! Initialize height ! ET,1,BEAM3 ! 2-D beam element AREA=B*H ! Beam cross-sectional area IZZ=(B*(H**3))/12 ! Moment of inertia about Z R,1,AREA,IZZ,H ! Real constants in terms of DV parameters ! ! Rest of the model: MP,EX,1,30E6 ! Young’s modulus N,1 ! Nodes N,11,120 FILL E,1,2 ! Elements EGEN,10,1,-1 FINISH ! Leave PREP7 5.2.4.2  Obtain the Solution The SOLUTION processor is used to define the analysis type and analysis options, apply loads, specify load step options, and initiate the finite element solution. The SOLUTION input for the beam example could look like this: /SOLU ANTYPE,STATIC ! Static analysis (default) D,1,UX,0,,11,10,UY ! UX=UY=0 at the two ends of the beam SFBEAM,ALL,1,PRES,100 ! Transverse pressure  (load per unit length) = 100 SOLVE FINISH ! Leave SOLUTION

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5.2.4.3  Retrieve Results Parametrically This is where we retrieve results data and assign them to parameters. These parameters usually represent SVs and the objective function. The *GET command (Utility Menu> Parameters> Get Scalar Data), which assigns ANSYS calculated values to parameters, is used to retrieve the data. POST1 is typically used for this step, especially if the data are to be stored, summed, or otherwise manipulated. In our beam example, the weight of the beam is the objective function (to be minimized). Since weight is directly proportional to volume, and assuming uniform density, minimizing the total volume of the beam is the same as minimizing its weight. Therefore, we can use volume as the objective function. The SVs for this example are the total stress and deflection. The parameters for these data may be defined as follows: /POST1 SET,... NSORT,U,Y ! Sorts nodes based on UY deflection *GET,DMAX,SORT,,MAX ! Parameter DMAX = maximum  deflection ! ! Derived data for line elements are accessed through ETABLE: ETABLE,VOLU,VOLU ! VOLU = volume of each element ETABLE,SMAX_I,NMISC,1 ! SMAX_I = max. stress at end I of each element ETABLE,SMAX_J,NMISC,3 ! SMAX_J = max. stress at end J of each element ! SSUM ! Sums the data in each column of the element table *GET,VOLUME,SSUM,,ITEM,VOLU ! Parameter VOLUME  = total volume ESORT,ETAB,SMAX_I,,1 ! Sorts elements based on  absolute value of SMAX_I *GET,SMAXI,SORT,,MAX ! Parameter SMAXI = max.  value of SMAX_I ESORT,ETAB,SMAX_J,,1 ! Sorts elements based on  absolute value of SMAX_J *GET,SMAXJ,SORT,,MAX ! Parameter SMAXJ = max.  value of SMAX_J SMAX=SMAXI>SMAXJ ! Parameter SMAX = greater of  SMAXI and SMAXJ

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FINISH ... 2. Enter OPT and specify the analysis file The remaining steps are performed within the OPT processor. When you first enter the optimizer, any parameters that exist in the ANSYS database are automatically established as design set number 1. To enter the optimizer, use one of these methods: Command(s): /OPT GUI: Main Menu> Design Opt In interactive mode, you must specify the analysis file name. To specify the analysis file name, use one of these methods: Command(s): OPANL GUI: Main Menu> Design Opt> Analysis File> Assign 3. Declare optimization variables The next step is to declare optimization variables, that is, specify which parameters are DVs, which ones are SVs, and which one is the objective function. As mentioned earlier, up to 60 DVs and up to 100 SVs are allowed, but only one objective function is allowed. To declare optimization variables, use one of these methods: Command(s): OPVAR GUI: Main Menu> Design Opt> Design Variables Main Menu> Design Opt> State Variables Main Menu> Design Opt> Objective Minimum and maximum constraints can be specified for SVs and DVs. Constraints are needed for the objective function. Each variable has a tolerance value associated with it, which you may input or let default to a program-calculated value. 4. Choose optimization tool or method In the ANSYS program, several different optimization tools and methods are available. Single loop is the default. To specify a tool or method to be used for subsequent optimization looping, use one of these methods: Command(s): OPTYPE GUI: Main Menu> Design Opt> Method/Tool 5. Specify optimization looping controls Each method and tool has certain looping controls associated with it, such as maximum number of iterations, and so on. All of the commands that you use to set these controls are accessed by the menu path Main Menu> Design Opt> Method/Tool. There are also a number of general controls that affect how data is saved during optimization. They are as follows:

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• To specify the file where optimization data is to be saved (defaults to Jobname.OPT): Command(s): OPDATA GUI: Main Menu> Design Opt> Controls • To determine whether information from the best design set is saved (by default, the database and results files are saved only for the last design set): Command(s): OPKEEP GUI: Main Menu> Design Opt> Controls 6. Initiate optimization analysis After all appropriate controls have been specified, you can initiate looping: Command(s): OPEXE GUI: Main Menu> Design Opt> Run Upon execution of OPEXE, an optimization loop file (Jobname.LOOP) is written from the analysis file. This loop file, which is transparent to the user, is used by the optimizer to perform analysis loops. Looping will continue until convergence, termination (not converged, but maximum loop limit or maximum sequential infeasible solutions encountered), or completion (e.g., requested number of loops for random design generation) has been reached. The values of all optimization variables and other parameters at the end of each iteration are stored on the optimization data file (Jobname.OPT). Up to 130 such sets are stored. When the 130th set is encountered, the data associated with the “worst” design are discarded. 7. Review design sets data After optimization looping is complete, you can review the resulting design sets in a variety of ways using the commands described in this section. These commands can be applied to the results from any optimization method or tool. To list the values of parameters for specified set numbers: Command(s): OPLIST GUI: Main Menu> Design Opt> Design Sets> List To graph specified parameters versus set number so you can track how a variable changed from iteration to iteration: Command(s): PLVAROPT GUI: Main Menu> Design Opt> Graphs/Tables In addition to reviewing the optimization data, you may wish to postprocess the analysis results using POST1 or POST26. By default, results are saved for the last design set in file Jobname.RST (or .RTH, etc., depending on the type of analysis). The results and the database

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for the best design set will also be available if OPKEEP,ON was issued before looping. The “best results” will be in file Jobname.BRST (.BRTH, etc.), and the “best database” will be in Jobname.BDB. 5.2.5  Guidelines for Performing Optimization Analysis 5.2.5.1  Choosing Design Variables DVs are usually geometric parameters such as length, thickness, diameter, or model coordinates. They are restricted to positive values. Some points to remember about DVs are: • Use as few DVs as possible. Obviously, more DVs demand more iterations and, therefore, more computer time. One way to reduce the number of design variables is to eliminate some DVs by expressing them in terms of others, commonly referred to as design variable linking. • Specify a reasonable range of values for the design variables (MIN and MAX). Too wide a range may result in poor representation of design space, whereas too narrow a range may exclude “good” designs. Remember that only positive values are allowed, and that an upper limit must be specified. • Choose DVs such that they permit practical optimum designs.

5.2.5.2  Choosing State Variables SVs are usually response quantities that constrain the design. Examples of SVs are stresses, temperatures, heat flow rates, frequencies, deflections, absorbed energy, elapsed time, and so on. A state variable need not be an ANSYS-calculated quantity; virtually any parameter can be defined as a state variable. Some points to keep in mind while choosing state variables are: • When defining SVs (OPVAR command), a blank input in the MIN field is interpreted as “no lower limit.” Similarly, a blank in the MAX field is interpreted as “no upper limit.” A zero input in either of these fields is interpreted as a zero limit. • Choose enough SVs to sufficiently constrain the design. In a stress analysis, for example, choosing the maximum stress as the only SV may not be a good idea because the location of the maximum stress may change

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from loop to loop. Also avoid the other extreme, which would be to choose the stress in every element as a state variable. The preferred method is to define the stresses at a few key locations as state variables. • For the subproblem approximation method, if possible, choose SVs that have a linear or quadratic relationship with the DVs. • If a state variable has both an upper and lower limit, specify a reasonable range of limit values. Avoid very small ranges, because feasible designs may not exist. A stress range of 500 to 1,000 psi, for example, is better than 900 to 1,000 psi. • If an equality constraint is to be specified (such as frequency = 386.4 Hz), define two state variables for the same quantity and bracket the desired value, illustrated as follows: *GET,FREQ,ACTIVE,,SET,FREQ ! Parameter FREQ = calculated frequency FREQ1=FREQ FREQ2=FREQ /OPT OPVAR,FREQ1,SV,,387 ! Upper limit on FREQ1 = 387 OPVAR,FREQ2,SV,386 ! Lower limit on FREQ2 = 386 • Avoid choosing SVs near singularities (such as concentrated loads) by using selecting before defining the parameters. 5.2.5.3  Choosing the Objective Function The objective function is the quantity that you are trying to minimize or maximize. Some points to remember about choosing the objective function are: • The ANSYS program always tries to minimize the objective function. If you need to maximize a quantity x, restate the problem and minimize the quantity x1 = C − x or x1 = 1/x, where C is a number much larger than the expected value of x. C − x is generally a better way to define the objective function than 1/x because the latter, being an inverse relationship, cannot be as accurately represented by the approximations used in the subproblem approximation method. • The objective function should remain positive throughout the optimization, because negative values may cause numerical problems. To prevent negative values from occurring, simply add a sufficiently large positive number to the objective function (larger than the highest expected objective function value).

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5.2.5.4  Restarting an Optimization Analysis To restart an optimization analysis, simply resume the optimization database file (Jobname.OPT): Command(s): OPRESU GUI: Main Menu> Design Opt> Opt Database> Resume Once the data is read in, you can respecify optimization type, controls, and so on, and initiate looping. (The analysis file corresponding to the resumed database must be available in order to perform optimization.) To initiate looping: Command(s): OPEXE GUI: Main Menu> Design Opt> Run You can use (Main Menu> Design Opt> Opt Database> Resume) in an interactive session to resume optimization data. If there is data in the optimization database at the time you want to resume, you should first clear the optimization database. To clear the optimization database: Command(s): OPCLR GUI: Main Menu> Design Opt> Opt Database> Clear & Reset Since the ANSYS database is not affected by the OPCLR command, it may also be necessary to clear the ANSYS database if the resumed optimization problem is totally independent of the previous one. To clear the ANSYS database: Command(s): /CLEAR GUI: Utility Menu> File> Clear & Start New 5.2.6  Sample Optimization Analysis In the following example, you will perform an optimization analysis of a hexagonal steel plate using the GUI for the analysis. Problem Description: You will build a parametric model of a hexagonal steel plate, using thickness t1 and fillet radius fil as the parameters. All other dimensions are fixed. This example uses a 2-D model and takes advantage of symmetry. Problem Specifications: The loading for this example is tensile pressure (traction) of 100 MPa at the three flat faces. The following material properties are used for this analysis: ­Thickness = 10 mm Young’s modulus (E) = 2.07e5 MPa Poisson’s ratio (υ) = 0.3 1. Test analysis file To test the analysis file, you clear the database and then read input from the hexplate.lgw file.

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1. Choose menu path Utility Menu> File> Clear & Start New. Click on OK. 2. When the Verify dialog box appears, click Yes. 3. Change the jobname. Choose menu path Utility Menu> File> Change Jobname. The Change Jobname dialog box appears. 4. Change the jobname to hexplate and click on OK. 5. Choose menu path Utility Menu> File> Read Input from. In the Files list, click on hexplate.lgw. Then click on OK. You see a replay of the entire analysis. Click on Close when the “Solution is done!” message appears. ! ******************************* ! First Pass: Create analysis file. ! ******************************** *create,hexplate ! ! GEOMETRY (in mm) !----------------*afun,deg !Degree units for trig. functions inrad=200*cos(30)-20 ! Inner radius t1=30 ! Thickness fil=10 ! Fillet radius /prep7 ! Create the three bounding annuli cyl4,-200,,inrad,-30,inrad+t1,30 cyl4,200*cos(60),200*sin(60),inrad,90,inrad+t1,-150 cyl4,200*cos(60),200*sin(60),inrad,90,inrad+t1,150 aplot aadd,all adele,all ! Delete area, keep lines lplot ! Fillets on inner slot lsel,,radius,,inrad+t1 ! Select inner arcs l1 = lsnext(0) ! Get their line numbers l2 = lsnext(l1) l3 = lsnext(l2) lfillet,l1,l2,fil ! Fillets lfillet,l2,l3,fil lfillet,l3,l1,fil

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lsel,all lplot ! Keep only symmetric portion wprot,,90 lsbw,all wprot,,,60 lsbw,all csys,1 lsel,u,loc,y,0,60 ldele,all,,,1 lsel,all ksll ksel,inve kdele,all ! Delete unnecessary keypoints ksel,all lplot ! Create missing lines and combine right edge lines csys,0 ksel,,loc,y,0 lstr,kpnext(0),kpnext(kpnext(0)) ! Bottom symmetry edge ksel,all csys,1 ksel,,loc,y,60 lstr,kpnext(0),kpnext(kpnext(0)) ! 60-deg. symm. edge ksel,all csys,0 lsel,,loc,x,100 lcomb,all ! Add lines at the right edge lsel,all ! Create the area al,all aplot ! MESHING ! ------et,1,82,,,3 ! Plane stress with thickness r,1,10 ! Thickness mp,ex,1,2.07e5 ! Young’s modulus of steel, MPa

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mp,nuxy,1,0.3 ! Poisson’s ratio smrt,3 amesh,all eplot finish ! LOADING ! ------/solu csys,1 lsel,u,loc,y,1,59 dl,all,,symm ! Symmetry b.c. csys,0 lsel,,loc,x,100 sfl,all,pres,-50 ! Pressure load (MPa) lsel,all lplot ! SOLUTION ! -------eqslv,pcg solve ! POSTPROCESSING ! -------------/post1 plnsol,s,eqv ! Equivalent stress contours /dscale,,off ! Displacement scaling off /expand,6,polar,half,,60 !Symmetry expansion /replot /expand !Retrieve max equivalent stress & volume nsort,s,eqv *get,smax,sort,,max ! smax = max. equivalent stress etable,evol,volu ssum *get,vtot,ssum,,item,evol ! vtot = total volume finish 2. Enter the optimizer and identify analysis file In the next several steps of this problem, you optimize the design. The overdesigned steel plate under tension loading of 100 MPa needs to

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be optimized for minimum weight subject to a maximum von Mises stress limit of 150 MPa. You are allowed to vary the thickness t1 and fillet radius fil. First, enter the optimizer and identify the analysis file. 1. Choose menu path Main Menu> Design Opt> Analysis File> Assign. The Assign Analysis File dialog box appears. 2. In the Files list, click once on hexplate.lgw and then click on OK. 3. Identify the optimization variables 1. Choose menu path Main Menu> Design Opt> Design Variables. The Design Variables dialog box appears. 2. Click on Add. The Define a Design Variable dialog box appears. 3. In the list of parameter names, click on T1. Type 20.5 in the MIN field and 40 in the MAX field. Click on Apply. 4. In the list of parameter names, click on FIL. Type 5 in the MIN field and 15 in the MAX field. Click on OK. 5. Click on Close to close the Design Variables dialog box. 6. Choose menu path Main Menu> Design Opt> State Variables. The State Variables dialog box appears. 7. Click on Add. The Define a State Variable dialog box appears. 8. In the list of parameter names, click on SMAX. Type 150 in the MAX field. Click on OK. 9. Click on Close to close the State Variables dialog box. 10. Choose menu path Main Menu> Design Opt> Objective. The Define Objective Function dialog box appears. 11. In the list of parameter names, click on VTOT. Set the TOLER field to 1.0. Click on OK. 4. Run the optimization This step involves specifying run time controls and the optimization method, saving the optimization database, and executing the run. 1. Choose menu path Main Menu> Design Opt> Controls. The Specify Run Time Controls dialog box appears. 2. Change the OPKEEP setting from “Do not save” to “Save.” Click on OK. 3. Specify an optimization method. Choose menu path Main Menu> Design Opt> Method/Tool. The Specify Optimization Method dialog box appears. 4. Choose Subproblem. Click on OK. Click OK again. 5. Save the optimization database. Choose menu path Main Menu> Design Opt> Opt Database> Save. In the Selection field, type hexplate.opt0. Click on OK.

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6. Execute the run. Choose menu path Main Menu> Design Opt> Run. Review the settings and click on OK. (If you receive any warning messages during the run, close them.) 7. Notes will appear to let you know which design set ANSYS is currently running. When the run converges, review the Execution Summary. Click on OK. 5. Review the results In this step, you start by listing design sets, then graph the objective function and state variables versus set number. 1. Choose menu path Main Menu> Design Opt> Design Sets> List. The List Design Sets dialog box appears. 2. Verify that the ALL Sets option is selected. Click on OK. 3. Review the information that appears in the window. Click on Close. 4. Choose menu path Main Menu> Design Opt> Graphs/ Tables. The Graph/List Tables of Design Set Parameters dialog box appears. 5. For X-variable parameter, select Set number. For Y-variable parameters, select VTOT. Click on OK. Review the graph. 6. Choose menu path Main Menu> Design Opt> Graphs/ Tables. The Graph/List Tables of Design Set Parameters dialog box appears. 7. For X-variable parameter, select Set number. For Y-variable parameters, select SMAX. Unselect VTOT by clicking on it. Click on OK. Review the graph. 6. Restore the best design In this step, you restore the best design. First, however, save the optimization database to a file. 1. Choose menu path Main Menu> Design Opt> Opt Database> Save. The Save Optimization Data dialog box appears. 2. In the Selection field, type hexplate.opt1. Then click on OK. 3. Choose menu path Main Menu> Finish. 4. Issue the following commands in the ANSYS Input window. After you type each command in the window, press ENTER resume, hexplate, bdb/post1 file,hexplate, brst lplot. 5. Choose menu path Main Menu> General Postproc> Read Results> First Set. 6. Choose menu path Main Menu> General Postproc> Plot Results> Contour Plot> Nodal Solu. The Contour Nodal Solution Data dialog box appears. 7. Choose Stress from the list on the left. Choose von Mises SEQV from the list on the right. Click on OK. Review the plot.

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8. Choose menu path Utility Menu> PlotCtrls> Style> Displacement Scaling. For DMULT, select 0.0 (off). Click on OK. 9. Choose menu path Utility Menu> PlotCtrls> Style> Symmetry Expansion> User-Specified Expansion. The Expansion by Values dialog box appears. 10. Fill in the 1st Expansion of Symmetry section. For NREPEAT, type 6. For TYPE, choose Polar. For PATTERN, choose Alternate Symm. Type 0, 60, and 0 in the DX, DY, and DZ fields, respectively. Click on OK. 7. Exit ANSYS Click on Quit in the ANSYS Toolbar. Select an option to save, then click on OK.

5.3 Tutorial 7: Design Optimization Tutorial In this tutorial will be addressed the optimization of heat transfer rate from a cylindrical pin fin. A very long rod 25 mm in diameter has one end maintained at 100°C. The surface of the rod is exposed to ambient air at 25°C with a convection heat transfer coefficient of 10 W/m2.K. Determine the heat loss from the rod constructed of pure copper.

Geometric properties Diameter Length

Value   25 mm 1320 mm

Distribution Uniform Uniform

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Nominal Material properties value Distribution Thermal Conductivity At Average ­Temperature 398 W/m.K ­Coefficient T = (Tb + T∞)/2 = 62.5°C ~ 355k Convection Heat 10 W/m2.K ­Transfer Coefficient

Loading Temperature Thermal Load Convection Thermal Load

Nominal value 100°C 10 W/m2.K

Location At the base of the fin. At the cylindrical edge and the flat face at the tip of the fin.

5.3.1  Approach and Assumptions • Since the problem is axisymmetric only a small sector of elements is needed. A rectangular strip is used for approximating the cylindrical fin. • The analysis is carried out for steady-state conditions. • Material properties are considered to be constant. • One-dimensional conduction along the rod. • Negligible radiation exchange with surroundings. • Uniform heat transfer coefficient. 5.3.2  Solutions/Results Results Units Total Heat Loss

FEA Watts (W) 29.09103

Book (Incropera) Watts (W) 29.4

5.3.3  Input File Using Log File You can perform the example optimization analysis of a cylindrical fin using the ANSYS commands shown as follows. Items prefaced with an exclamation point (!) are comments. The analysis happens in essentially two passes. In the first pass, you create the analysis file. In the second pass, you create the optimization input. If you prefer, you can perform the second pass of the example analysis using the GUI method rather than ANSYS commands.

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/BATCH ! *************************************** /PREP7 ! ET,1,PLANE55 !* KEYOPT,1,1,0 KEYOPT,1,3,1 KEYOPT,1,4,0 KEYOPT,1,8,0 KEYOPT,1,9,0 ! ! ************ Material Properties (Conduction  and Convection Coefficients): Pure Copper !* At Degree : 355 kelvin MPTEMP,,,,,,,, MPTEMP,1,0 MPDATA,KXX,1,,398 MPTEMP,,,,,,,, MPTEMP,1,0 MPDATA,HF,1,,10 ! ! ******************************************** Geometry *SET,L,1.32 ! LENGTH IN M *SET,R,12.5E-3 ! RADIUS IN M BLC4,0,0,R,L ! ! ******************************************** Meshing MSHKEY,0 CM,_Y,AREA FLST,5,2,4,ORDE,2 FITEM,5,2 FITEM,5,4 CM,_Y,LINE LSEL, , , ,P51X CM,_Y1,LINE CMSEL,,_Y !* LESIZE,_Y1, , ,600, , , , ,1

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!* FLST,5,2,4,ORDE,2 FITEM,5,1 FITEM,5,3 CM,_Y,LINE LSEL, , , ,P51X CM,_Y1,LINE CMSEL,,_Y !* LESIZE,_Y1, , ,5, , , , ,1 !* CM,_Y,AREA ASEL, , , , 1 CM,_Y1,AREA CHKMSH,’AREA’ CMSEL,S,_Y !* MSHKEY,1 AMESH,_Y1 MSHKEY,0 !* CMDELE,_Y CMDELE,_Y1 CMDELE,_Y2 FINISH ! ******************************************** SOLUTION /SOLU ! ANTYPE,0 ! Steady-State Thermal Analysis ! FLST,2,1,4,ORDE,1 FITEM,2,1 DL,P51X, ,TEMP,100,0 ! Prescribed temperature at the left end ! FLST,2,2,4,ORDE,2 FITEM,2,2 FITEM,2,-3 SFL,P51X,CONV,10, ,25, ! Convective boundary condition !

Design Optimization   •  197

SOLVE FINISH ! ! ******************************************** ***** POST PROCESSING /POST1 FSUM *GET,THEAT,FSUM,HEAT *SET,TOTALHEAT,-THEAT ObjFunct=10000-TOTALHEAT ALLSEL *GET,Tempend,NODE,607,TEMP, FINISH 5.3.4  Optimization Problem Using a BATCH file for the analysis describes this example optimization analysis as consisting of two passes. In the first you create an analysis file, and in the second you create the optimization input. It is better to avoid graphical picking operations when defining a parametric model. Thus, the GUI method is not recommended for performing the first pass of the example analysis and will not be presented here. However, it is acceptable to perform the optimization pass of the cylindrical fin example using the GUI method instead of the ANSYS commands shown earlier. The GUI procedure for performing the optimization pass follows. 5.3.5  Summary of Steps 1. Enter the Design Optimization (Design Opt) and execute the file for the deterministic model (Fin.txt). 2. Specify Fin.txt as the analysis file for the design optimization ­analysis. 3. Define the design variables, state variables, and objective function for your design optimization analysis. For this problem, you define the length and radius of the cylindrical fin as design variables, temperature at the tip of the fin as state variable, and total heat dissipated as the objective function. 4. Run the optimization. Select a suitable method for optimization. 5. Obtain solution. 6. Perform postprocessing to visualize and evaluate the design sets of optimization results.

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7. Save the optimization database. 8. Generate HTML report and exit. 5.3.6  Step-by-Step Analysis 1. Test analysis file To test the analysis file, you clear the database and then read input from the Fin.lgw file. Choose menu path Utility Menu> File> Read Input from. In the Files list, click on Fin.lgw. Then click on OK. You see a replay of the entire analysis. Click on Close when the “Solution is done!” message appears. Main Menu > Design Opt

Utility Menu >File>Read Input from. Choose the file Fin.txt from your working directory. OK.

Design Optimization   •  199

2. Enter the optimizer and identify analysis file In the next several steps of this problem, you optimize the design. The previously designed cylindrical fin under thermal loading of 100°C at the base of the fin needs to be optimized for maximum heat loss through the surface subjected to a maximum temperature limit at the tip of the fin to 40°C. You are allowed to vary the length L and the radius R. First, enter the optimizer and identify the analysis file. Choose menu path Main Menu> Design Opt> Analysis File> Assign. The Assign Analysis File dialog box appears. In the Files list, click once on Fin.lgw and then click on OK. Main Menu> Design Opt> Analysis File> Assign

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Choose the file Fin from your working directory. Use the Browse button if you choose.

3. Identify the optimization variables You will now define the Design Variables (DV) and their limits. Main Menu> Design Opt> Design Variables

Define Length and Radius as design variable. Add a Design variable.

Choose L as the parameter. Type 0.3 m in the MIN field. Type 1.5 m in the MAX field Apply

Design Optimization   •  201

Choose R as the parameter. Type 0.01 m in the MIN field. Type 0.05 m in the MAX field. OK.

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Close.

Define State Variables and the constraining limits. Main Menu> Design Opt> State Variables.

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Add a State variable.

Click on TEMPEND. Type 40°C in the MAX field. OK.

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Define Objective Function and its tolerance. Main Menu> Design Opt> Objective.

Click on TOTALHEAT. OK.

4. Run the optimization This step involves specifying run time controls and the optimization method, saving the optimization database, and executing the run. Main Menu> Design Opt> Controls.

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Change OPKEEP setting from “Do not save” to “Save.” OK.

Main Menu> Design Opt> Method/Tool.

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Choose Sub-Problem OK

OK.

Main Menu> Design Opt> Opt Database> Save

Design Optimization   •  207

To save the optimization database, type Fin.opt0 OK.

Main Menu> Design Opt> Run

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OK.

Solution Converges Click OK.

5. Review the results In this step, you start by listing design sets, then graph the objective function and state variables versus set number. Main Menu> Design Opt> Design Sets> List

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Select ALL Sets. OK.

Review results and Close.

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Main Menu> Design Opt> Graphs/Tables

For X-variable select Set number For Y-variable select ObjFunct OK.

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Review the Graph for Objective function

Main Menu> Design Opt> Graphs/Tables

For X-variable select Set number. For Y-variable select TOTALHEAT. Unselect OBJFUNCT. OK.

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Review the Graph.

6. Restore the best design Main Menu> Design Opt> Opt Database> Save.

Design Optimization   •  213

Type Fin.opt1. OK.

Main Menu> Finish.

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Type in Ansys Input Window. Resume,Fin,bdb/post1 File,Fin,brth lplot.

Utility Menu> PlotCtrls> Style> Symmetry Expansion> User-Specified Expansion.

Main Menu> General Postproc> Plot Results >Contour Plot> Nodal Solu. Review results.

Design Optimization   •  215

7. Exit ANSYS. Toolbar: Quit Choose Quit—Select an option to Save! OK. Congratulations! You have completed this tutorial.

Bibliography Cook, R.D. 1995. Finite Element Modeling for Stress Analysis, 1st ed. H ­ oboken, NJ: John Wiley & Sons. Incropera, F.P. 1985. Fundamentals of Heat and Mass Transfer, Example 3.7, 2nd ed., 104. Singapore: John Wiley & Sons. Reddy, J.N. 1993. An Introduction to the Finite Element Method, New York: McGrawHill. Logan, D.L. 2001. A First Course in the Finite Element Method, 3rd ed. Mason, OH: Thomas Learning Publishing. Reddy, J.N. 1972. “Exact Solutions of Moderately Thick Laminated Shells.” ­Journal Engineering Mechanics 110, no. 5, pp. 794–805. Timoshenko, S. 1956. Strength of Material, Part II, Elementary Theory and ­Problems, 3rd ed., 111. New York, NY: D. Van Nostrand Co., Inc.

About the Authors Wael A. Altabey is an assistant professor in the department of mechanical engineering, faculty of engineering, Alexandria University, ­Alexandria, Egypt and has been a postdoctoral researcher at the International Institute for Urban Systems Engineering, Southeast University, Nanjing, Jiangsu, China. Dr Altabey’s research has been in the area of composite structures and utilization of artificial neural networks for damage detection. Email: [email protected] Mohammad Noori is a professor of mechanical engineering at ­California Polytechnic State University in San Luis Obispo, California, USA, and a fellow of the American Society of Mechanical Engineers. Dr Noori has over 34 years of experience as a scholar and educator. He has also been a distinguished visiting professor at the International Institute for Urban Systems Engineering, Southeast University, Nanjing, China. Dr Noori’s research has been in the area of mechanical and random vibrations and structural health monitoring. Email: [email protected] Libin Wang is a professor and the dean of the school of civil ­engineering at Nanjing Forestry University, in Nanjing, China. He has been an e­ ducator and scholar, for over 20 years, and has taught the subject of finite element analysis both at the undergraduate and graduate level. Dr Wang’s research is in the field of engineering mechanics, hazard mitigation, and bridge engineering, as well as timber structures. Email: [email protected]

Index A Analysis file, ADPL programming advantages and disadvantages, 145–146 using command log files, 147–148 using log files, 146–147 ANSYS Parametric Design Language (APDL) programming creating analysis file, 145–148 heat loss from cylindrical cooling fin, 155–165 stress analysis of bicycle wrench, 148–155 B BEAM188, 29 BEAM189, 29 C cdf. See Cumulative distribution function Ceramic matrix composites (CMCs), 20 Circular plate bending, 107–143 CMCs. See Ceramic matrix composites Compression, 22 Constitutive matrices, 31 Constraints, 170 Continuous distributions, 78–81

Continuous probability function, 76 Convergence checking, 177 Correlation, 72 Correlation matrix, 106–107 Cumulative distribution function (cdf), 77–78 D Design optimization, ANSYS program general process for, 179–185 guidelines for performing, 185–187 optimization design tools, 177–179 optimization methods, 175–177 sample optimization analysis, 187–193 terminology and information flow, 173–174 Design optimization tutorial approach and assumptions, 194 description, 193–194 input file using log file, 194–197 optimization problem, 197 solutions/results, 194 step-by-step analysis, 198–215 summary of steps, 197–198 Design set, 174 Design variables (DVs), 170, 173, 185

222  •  Index

Direct Monte Carlo sampling, 97–98 DVs. See Design variables Dynamic analysis description of, 1 harmonic analysis of structure, 1–11 modal analysis of structure, 11–17 static analysis vs., 1 Dynamic loads, 1 E Exceedence values, 92 F Factorial tool, 179 Feasible design, 174 Finite strain shell (SHELL181), 27 First-order method, 177 Flexural loads, 23 G Gradient tool, 179 H Harmonic analysis of structure assigning loads and solving, 3–6 command log file, 3 description of, 1–2 postprocessing-viewing results, 6–11 preprocessing-defining problem, 2 Heat loss from cylindrical cooling fin, 155–165 I Individual layer properties, 29–31 L Latin hypercube sampling, 98 Layered structural solid element (SOLID191), 28–29

Linear layered structural shell element (SHELL99), 27 Lognormal distribution, 85–88 Loop, 174 Loop file, 174 M Mean value, 73 Median value, 74 Metal matrix composites (MMCs), 19 MMCs. See Metal matrix composites Modal analysis of structure assigning loads and solving, 11–13 description of, 11 postprocessing-viewing results, 13–15 preprocessing-defining problem, 11 using reduced method, 16–17 Modeling composites using ANSYS failure criteria, 33–35 layered configuration, 29–33 modeling and postprocessing guidelines, 35–38 proper element type selection, 26–29 Monte Carlo techniques advantages, 96–97 direct sampling, 97–98 Latin hypercube sampling, 98 N Node offset, 32–33 Nonlinear layered structural shell element (SHELL91), 27 Normal distribution, 81–83 O Objective function, 174, 186 Optimization database, 174

Index   •   223

Optimization data flow, 175 Optimization design tools factorial tool, 179 gradient tool, 179 random tool, 178 single loop analysis tool, 178 sweep tool, 178–179 Optimization methods description of, 175–176 first-order method, 177 subproblem approximation method, 176–177 Optimization tree, 171 Optimization variables, 174 Optimum design applications, 172 description, 167–169 examples, 172 fundamental concepts, 169–171 P pdf. See Probability density function PMCs. See Polymer matrix composites Polymer matrix composites (PMCs) definition of, 19 description of, 20–22 loading of, 22–23 with other structural materials, 23–26 properties of, 20 Postprocessing probabilistic analysis statistical postprocessing, 99–101 trend postprocessing, 101–107 PREP7, 181 Probabilistic design analysis circular plate bending, 107–143 definition of, 69–70 reliability and quality issues, 72 steps using ANSYS, 74–75

terminology, 72–74 traditional (deterministic) vs., 71–72 variables, 73 Probabilistic design techniques advantages, 96–97 direct Monte Carlo sampling, 97–98 Latin hypercube sampling, 98 Probabilistic model, 73 Probabilistic sensitivities, 101–104 Probability density function (pdf), 76–77 Probability distributions gallery of common continuous distributions, 78–81 lognormal distribution, 85–88 normal distribution, 81–83 practical uses of, 75–76 uniform distribution, 83–85 Weibull distribution, 88–91 Problem formulation, 169 Problem solution, 171 Q Quality, probabilistic design, 72 R Random input variables, 72 Random output variables, 73 Random tool, 178 Random variable distribution exceedence values, 92 mean values, 92 measured data, 91 no data, 93–95 output parameters, 95 standard deviation, 92 Reduced method, modal analysis, 16–17 Reliability, probabilistic design, 72 S Sample, 73

224  •  Index

Sample optimization analysis, 187–193 Scatter plots, 105–106 Shear loads, 22–23 SHELL63, 28 SHELL91 (nonlinear layered structural shell element), 27 SHELL99 (linear layered structural shell element), 27 SHELL181 (finite strain shell), 27 Simply supported laminated plate under pressure applying loads, 56–68 approach and assumptions, 39 step-by-step analysis, 40–56 summary of steps, 39–40 Simulation, 73 Single loop analysis tool, 178 SOLID46 (3-D layered structural shell element), 27–28 SOLID65, 29 SOLID95, 28 SOLID191 (layered structural solid element), 28–29 Standard deviation, 74 Standard lognormal distribution, 86, 89 Standard normal distribution, 81 Standard uniform distribution, 83 State variables (SVs), 174, 185–186

Static analysis vs. dynamic analysis, 1 Statistical postprocessing technique cumulative distribution function, 100–101 histogram, 99 print inverse probabilities, 101 print probabilities, 101 sample history, 99 Stress analysis of bicycle wrench, 148–155 Subproblem approximation method, 176–177 SVs. See State variables Sweep tool, 178–179 T Tension, 22 3-D layered structural shell element (SOLID46), 27–28 Trend postprocessing technique correlation matrix, 106–107 scatter plots, 105–106 sensitivities, 101–104 U Uniform distribution, 83–85 W Weibull distribution, 88–91

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THE CONTENT

Dynamic, Probabilistic Design and Heat Transfer Analysis, Volume II Wael A. Altabey • Mohammad Noori • Libin Wang Over the past two decades, the use of finite element method as a design tool has grown rapidly. Easy to use commercial software, such as ANSYS, have become common tools in the hands of students as well as practicing engineers. The objective of this book is to demonstrate the use of one of the most commonly used Finite Element Analysis software, ANSYS, for linear static, dynamic, and thermal analysis through a series of tutorials and examples. Some of the topics covered in these tutorials include development of beam, frames, and Grid Equations; 2-D elasticity problems; dynamic analysis; composites, and heat transfer problems. These simple, yet, fundamental tutorials are expected to assist the users with the better understanding of finite element modeling, how to control modeling errors, and the use of the FEM in designing complex load bearing components and structures. These tutorials would supplement a course in basic finite element or can be used by practicing engineers who may not have the advanced training in finite element analysis. Wael A. Altabey is an assistant professor in the department of mechanical engineering, faculty of engineering, Alexandria University, Alexandria, Egypt and has been a postdoctoral researcher at the

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For further information, a free trial, or to order, contact:  [email protected]

Mohammad Noori is a professor of mechanical engineering at California Polytechnic State University in San Luis Obispo, California, USA, and a fellow of the American Society of Mechanical Engineers. Dr Noori has over 34 years of experience as a scholar and educator. He has also been a distinguished visiting professor at the International Institute for Urban Systems Engineering, Southeast University, Nanjing, China. Libin Wang is a professor and the dean of the school of civil engineering at Nanjing Forestry University, in Nanjing, China. He has been an educator and scholar, for over 20 years, and has taught the subject of finite element analysis both at the undergraduate and graduate level.

ISBN: 978-1-94708-322-6

Using ANSYS for Finite Element Analysis, Volume II

• Manufacturing Engineering • Mechanical & Chemical Engineering • Materials Science & Engineering • Civil & Environmental Engineering • Advanced Energy Technologies

Using ANSYS for Finite Element Analysis

ALTABEY • NOORI • WANG

EBOOKS FOR THE ENGINEERING LIBRARY

SUSTAINABLE STRUCTURAL SYSTEMS COLLECTION Mohammad Noori, Editor

Using ANSYS for Finite Element Analysis Dynamic, Probabilistic Design and Heat Transfer Analysis Volume II

Wael A. Altabey Mohammad Noori Libin Wang