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- Permanent Link:
- https://ufdc.ufl.edu/UFE0046753/00001
## Material Information- Title:
- Methods to Avoid Numerical Instabilities in Topology Optimization Using Nodal Design Variables
- Creator:
- Zhang, Yiming
- Place of Publication:
- [Gainesville, Fla.]
Florida - Publisher:
- University of Florida
- Publication Date:
- 2014
- Language:
- english
- Physical Description:
- 1 online resource (84 p.)
## Thesis/Dissertation Information- Degree:
- Master's ( M.S.)
- Degree Grantor:
- University of Florida
- Degree Disciplines:
- Mechanical Engineering
Mechanical and Aerospace Engineering - Committee Chair:
- KUMAR,ASHOK V
- Committee Co-Chair:
- KIM,NAM HO
- Graduation Date:
- 5/3/2014
## Subjects- Subjects / Keywords:
- Checkerboards ( jstor )
Data smoothing ( jstor ) Design analysis ( jstor ) Design optimization ( jstor ) Mathematical variables ( jstor ) Mechanism design ( jstor ) Objective functions ( jstor ) Projective geometry ( jstor ) Shape optimization ( jstor ) Topology ( jstor ) Mechanical and Aerospace Engineering -- Dissertations, Academic -- UF filter -- implicit -- instabilities -- optimization -- projection -- topology - Genre:
- bibliography ( marcgt )
theses ( marcgt ) government publication (state, provincial, terriorial, dependent) ( marcgt ) born-digital ( sobekcm ) Electronic Thesis or Dissertation Mechanical Engineering thesis, M.S.
## Notes- Abstract:
- Topology denotes the material layout of mechanical component. Topology optimization determines where to remove material from a structure to optimize it. Topology optimization emerged 20 years ago and has developed rapidly since then. Now topology optimization is a popular research topic in both academia and industry for conceptual design. The research on topology optimization focuses on developing efficient algorithms and innovative application. Density-based topology optimization algorithm, which is the most popular algorithm for topology optimization, has been systematically studied in this thesis. In this thesis, density-based topology optimization using nodal design variables is developed to improve the numerical performance of topology optimization algorithm. Previous work on this algorithm has been applied to several kind of design problems and proved to be more effective than those using element-wise design variables. The experimental platform is a structured mesh based FEA software. Topology optimization can be implemented more efficiently using a structured mesh with regular shaped elements than using traditional FEA mesh. Topology optimization using nodal density variables is studied and tested with several numerical techniques such as: RAMP Method, Two-level Mesh Method, Filter Method, Projection Method, Gradient Smoothing Method. These methods have been compared for reducing numerical instabilities: mesh independence, non-convexity, checkerboard, low manufacturability. Filter Method and Projection Method, which were originally developed using constant density per element, have been implemented to compliance minimization problem, compliant mechanism design, stress constrained minimizing weight problem. ( en )
- General Note:
- In the series University of Florida Digital Collections.
- General Note:
- Includes vita.
- Bibliography:
- Includes bibliographical references.
- Source of Description:
- Description based on online resource; title from PDF title page.
- Source of Description:
- This bibliographic record is available under the Creative Commons CC0 public domain dedication. The University of Florida Libraries, as creator of this bibliographic record, has waived all rights to it worldwide under copyright law, including all related and neighboring rights, to the extent allowed by law.
- Thesis:
- Thesis (M.S.)--University of Florida, 2014.
- Local:
- Adviser: KUMAR,ASHOK V.
- Local:
- Co-adviser: KIM,NAM HO.
- Electronic Access:
- RESTRICTED TO UF STUDENTS, STAFF, FACULTY, AND ON-CAMPUS USE UNTIL 2015-05-31
- Statement of Responsibility:
- by Yiming Zhang.
## Record Information- Source Institution:
- UFRGP
- Rights Management:
- Applicable rights reserved.
- Embargo Date:
- 5/31/2015
- Classification:
- LD1780 2014 ( lcc )
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REPORT xmlns http:www.fcla.edudlsmddaitss xmlns:xsi http:www.w3.org2001XMLSchema-instance xsi:schemaLocation http:www.fcla.edudlsmddaitssdaitssReport.xsd INGEST IEID E81VBSHXK_3W8SVR INGEST_TIME 2014-10-03T21:54:56Z PACKAGE UFE0046753_00001 AGREEMENT_INFO ACCOUNT UF PROJECT UFDC FILES PAGE 1 METHODS TO AVOID NUMERICAL INSTABILITIES IN TOPOLOGY OPTIMIZATION USING NODAL DESIGN VARIABLE S By YIMING ZHANG A THESIS PRESENTED TO THE GRADUATE SCHOOL OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF SCIENCE UNIVERSITY OF FLORIDA 201 4 PAGE 2 2 2014 Yiming Zhang PAGE 3 3 T o my family PAGE 4 4 ACKNOWLEDGMENTS I would like to thank Dr. Ashok V. Kumar sincerely for his patience He met me at least once a week to teach me necessary skills and give feedback during the research. His diligence and knowledgeab ility inspire me to make progress all the time. I learnt how to perform systematic study of a new field from this thesis. His guidance and inspiration I would like to thank Dr. Nam H. Kim as the member of my supervisory committee. It is my honor to have him in my committee and be guided for my thesis. I am grateful for his willingness to review this thesis and provide valuable suggestions. I thank University of Florida, and Department of Mechanical and Aerospace Engineering. I have gained professional skills and values which are beneficial lifetime. Finally, I thank my family, especially my mother, for their support and trust all the time. PAGE 5 5 TABLE OF CONTENTS page ACKNOWLEDGMENTS ................................ ................................ ................................ .. 4 LIST OF TABLES ................................ ................................ ................................ ............ 7 LIST OF FIGURES ................................ ................................ ................................ .......... 8 LIST OF ABBREV IATIONS ................................ ................................ ........................... 10 ABSTRACT ................................ ................................ ................................ ................... 11 CHAPTER 1 INTRODUCTION ................................ ................................ ................................ .... 13 1.1 O verview ................................ ................................ ................................ ........... 13 1.2 Goals and Objectives ................................ ................................ ........................ 16 1.2.1 Goal ................................ ................................ ................................ ......... 16 1.2.2 Objectives ................................ ................................ ................................ 16 1.3 Outline ................................ ................................ ................................ .............. 17 2 TOPOLOGY OPTIMIZATION ................................ ................................ ................. 18 2.1 General Introduction of Topology Optimization ................................ ................. 18 2.2 Density based Methods ................................ ................................ .................... 19 2.2.1 Mathematical Problem Statement ................................ ........................... 19 2.2.2 Various Problem Forms ................................ ................................ ........... 22 2.2.3 Numerical Instabilities ................................ ................................ .............. 22 2.3 Applications of Topology Optimization ................................ .............................. 25 2.3.1 Compliance oriented Problem ................................ ................................ 25 2.3.2 Material with Nonlinear Properties ................................ ........................... 26 2.3.3 Stress constrained Topology Optimization ................................ .............. 26 2.3.4 Design under Structure Dependent Loading ................................ ........... 26 2.3.5 Compliant Mechanism Design ................................ ................................ 26 2.3.6 Reliability based Topology Optimization ................................ .................. 26 2.3.7 Topology Optimization of Multidisciplinary Problems ............................... 27 2.4 Previous Work of Topology Optimization Using Nodal Design Variables .......... 27 2.4.1 Implicit Boundary Finite Element Method ................................ ................ 28 2.4.2 Compliant Mechanism Design ................................ ................................ 29 2.4.3 Stress Constrained Topology Optimization ................................ ............. 30 2.4. 4 Gradient Smoothing Method ................................ .............................. 31 3 IMPLICIT BOUNDARY TOPOLOGY OPTIMIZATION ................................ ............ 33 3.1 Penalty Scheme ................................ ................................ ................................ 33 PAGE 6 6 3.2 Two le vel Mesh Method ................................ ................................ .................... 35 3.3 Filter Method ................................ ................................ ................................ ..... 37 3.3.1 Compliance Minimization Problem with Filter ................................ .......... 39 3.3.2 Compliant Mechanism Design with Filter ................................ ................. 41 3.3.3 Stress Constrained Topology Optimization with Filter ............................. 42 3.4 Projection Method ................................ ................................ ............................. 44 3.4.1 Exponent Projection ................................ ................................ ................ 45 3.4.2 Tan Projection ................................ ................................ ......................... 46 4 RESULTS AND DISCUSSIONS ................................ ................................ ............. 48 4.1 Penalty Schemes of Topology Optimization ................................ ..................... 48 4.2 Mesh Independence ................................ ................................ ......................... 51 4.2.1 Two level Mesh Method ................................ ................................ .......... 51 4.2.2 Gradient Smoothing Method ................................ .............................. 53 4.2.3 Filter Method ................................ ................................ ........................... 54 4.3 Leng th Scale Control ................................ ................................ ........................ 56 4.3.1 Filter Method ................................ ................................ ........................... 57 4.3.2 Two level Mesh Method ................................ ................................ .......... 59 4.4 Smoothing Results ................................ ................................ ............................ 61 4.4.1 B spline Element with Two level Mesh Method ................................ ....... 61 4.4.2 Gradient Smoothing Method ................................ .............................. 63 4.5 Accurate Boundary ................................ ................................ ........................... 64 4.6 Ap plication of Implicit Boundary Topology Optimization ................................ ... 68 4.6.1 Compliance oriented Topology Optimization ................................ ........... 69 4.6.2 Topology Optimization of Compliant Mechanism ................................ .... 71 4.6.3 Stress constra ined Minimizing Weight Topology Optimization ................ 73 5 CONCLUSIONS ................................ ................................ ................................ ..... 76 5.1 Summary ................................ ................................ ................................ .......... 76 5.2 Scope of Future Work ................................ ................................ ....................... 77 LIST OF REFERENCES ................................ ................................ ............................... 79 BIOGRAPHICAL SKETCH ................................ ................................ ............................ 84 PAGE 7 7 LIST OF TABLES Table page 4 1 Influence of Two level Mesh on Computation Time ................................ ............ 63 PAGE 8 8 LIST OF FIGURE S Figure page 2 1 Three categories of structural optimization ................................ ........................ 18 2 2 Comparison of SIMP, RAMP, SINH intermediate density penalization models ................................ ................................ ................................ ............... 21 2 3 Flow chart of density based topology optimization problem ............................... 21 2 4 Gray region of MBB beam topology optimization. ................................ ............... 23 2 5 Checkerboard problem of MBB beam. ................................ ............................... 23 2 6 Dependence of topology optimization results on FEA element size .................. 24 2 7 SIMP Method analysis with various penalty factor value ................................ .... 24 3 1 Refine process of Two level Mesh Method. ................................ ........................ 36 3 2 Sketch of filter ................................ ................................ ................................ ..... 38 3 3 Subdivision of boundary element. ................................ ................................ ....... 45 3 4 The influence of on projected design variable from exponent projection function ................................ ................................ ................................ ............... 46 3 5 The influence of on projected design variable from tan projection function ................................ ................................ ................................ ............... 47 4 1 MBB Beam model ................................ ................................ .............................. 49 4 2 SIMP Method analysis with various penalty factor value ................................ .... 50 4 3 RAMP Method analysis with various penalty factor value ................................ .. 50 4 4 L shaped model ................................ ................................ ................................ 52 4 5 Two level Mesh Method while fixing plotting mesh. ................................ ............ 53 4 6 Compliance minimization with Gradient Smoothing Method using various element number. ................................ ................................ ................................ 54 4 7 Cantilever Beam model ................................ ................................ ..................... 55 4 8 Filter Method analysis with various element number ................................ .......... 56 4 9 Topology optimization of cantilever structure with various filter size. .................. 57 PAGE 9 9 4 10 Topology optimizati on using circle filter and square filter with various filter size ................................ ................................ ................................ ..................... 58 4 11 Two level Mesh Method using Quad 4 element with various element size ......... 60 4 12 Two level Mesh Method using B spline 9 element with various element size. ... 61 4 13 Topology optimization using SIMP Method with various element types. ............ 62 4 14 Topology optimization using SIMP Method and Gradient Smoothing Method with various weight factor ................................ ................................ ... 64 4 15 Stress constrained topology optimization using Exponent Projection ................. 65 4 16 Stress constrained topology optimization using Tan Projection ......................... 66 4 17 Stress co nstrained topology optimization using Tan Projection and .. 67 4 18 Stress constrained topology optimization ................................ ........................... 68 4 19 Topology optimization of knee structure ................................ ............................. 69 4 20 Topology optimization of bridge structure ................................ ........................... 70 4 21 Topology optimization of cantilever beam ................................ .......................... 70 4 22 Topology optimization of MBB beam ................................ ................................ .. 71 4 23 Topology optimization of gripper ................................ ................................ ......... 72 4 24 Topology optimization of inverter ................................ ................................ ........ 73 4 25 Topology optimization under shear load in rectangular domain ......................... 74 4 26 Topology optimization under shear load in L shaped domain ............................ 75 PAGE 10 10 LIST OF ABBREVIATIONS FE A Finite Element Analysis IBFEM Implicit Boundary Finite Element Method IBTO Implicit Boundary Topology Optimization MB Method Moving Barrier Method MBB Beam Messerschmitt Blkow Blohm beam MMA Method Method of Moving Asymptotes NURBS Non Uniform Rational B Splines RAMP Rational approximation of material properties SIMP Solid Isotropic Microstructure with Penalization PAGE 11 11 Abstract of Thesis Presented to the Graduate School of the University of Florida in Partial Fulfillment of the Requirements for the Degree of Ma ster of Science METHODS TO AVOID NUMERICAL INSTABILITIES IN TOPOLOGY OPTIMIZATION USING NODAL DESIGN VARIABLES By Yiming Zhang May 2014 Chair: Ashok V. Kumar Major: Mechanical Engineering Topology denotes the material layout of mechanical component Topology optimization determines where to remove material from a structure to optimize it. Topology optimization emerged 20 years ago and has developed rapidly since then. Now topology optimization is a popular research topic in both academi a and indust r y for conceptual design. The research on topology optimization focuses on developing efficient algorithms and innovative application Density based topology optimization algorithm, which is the most popular algorithm for topology optimization, has been sys tematically studied in this thesis. In this thesis, d ensity based topology optimization using nodal design variables is developed to improve the numerical performance of topology optimization algorithm. Previous work on this algorithm has been applied to several kind of design problems and proved to be more effective than those using element wise design variables. The experiment al platform is a structure d mesh based FEA software T opology optimization can be implemented more efficient ly using a structured mesh with regular shaped elements than using traditional FEA mesh PAGE 12 12 T opology optimization using nodal density variables is studied and tested with several numerical techniques such as: RAMP Method, Two level Mesh Method, Filter Method, Projection Method, Gradient Smoothing Method. These methods have been compared for reducing numerical instabilities: mesh independence, non convexity, checkerboard, low manufacturability. Filter Method and Projection Method which were originally develop ed using constant density per element, h ave been implemented to comp liance minimization problem, compliant mechanism design, stress constrained minimizing weight problem. PAGE 13 13 CHAPTER 1 INTRODUC TION 1.1 O verview Structur al optimization techniques aim at i mproving the performance of mechanical structures under c ertain constraints (e.g. cost) which include shape optimization, size optimization and topology optimization. Shape optimization means the selection of appropriate shape of holes and structures (e.g. triangle, circle). Size optimization represents the optimization of specified shapes (e.g. radius of circle). Topology denotes the location and shape of sub structure s and holes. Topology optimization is a combination of shape and size optimization which is an art of determining where to dig holes from material. Topology optimization algorithm based on finite element analysis is emerged around 1990s and has become a po pular topic since then. Plenty of inspiring research has been published in this field The applications of topology optimization in several fields have proved to be very effective and success ful including heat transfer, design under dynamic constraint, des ign under stress constraint, wave propagation, compliant mechanism design, bio structure design, design under static load. Topology optimization techniques are now popular in both academic and industrial implementations for conceptual design. There are two trends in the research on topology optimization. One is the development of more efficient and s t able algorithm. The other is the innovative application to structure design. Topology optimization algorithm computes the shape or material distribution with in in a specified domain namely feasible domain a. Topology optimization algorithm determines where to introduce holes or remove material to improve the desired performance of the structure. PAGE 14 14 The mathematic form of topology optimization is essentially the optimization of an objective function under certain constraints (e.g. side constraints of design variables). Objective functions are calculated based on finite element analysis. The arguments of objective function are design variables which represent optim um structures. H ere are four popular methods to realize topology optimization. Firs t one is density based method. This method assigns a density value (positive constant) to each element. Ideally the density should be 0 or 1. 0 means hole, 1 means solid. Ph ysical properties function. Density is the design variable. Second one is hard kill method. This method works by gradually removing (or adding) a finite amount of material from the design domain. The choice of the material to be removed or added is based on heuristic criteria. Here density is also the design variable but not related to physical properties. A criterion function is calculated for each element and removal of the ele ment is applied to those with low criteria values. Third one is boundary variation method. Structural boundaries are implicitly represented by a scalar function. The movement of boundaries are expressed by the scalar function and motivated by the objective function. Fo u rth one is bio inspired cellular division based method. This method is inspired by the cellular division processes of living organisms. Topology of the structures are implicitly governed by a developmental program that when executed completes a sequence of tasks that develop the topology in stages. This method is driven by genetic algorithm. PAGE 15 15 Among all these methods, density based met hod is the mos t widely used one Besides its popularity, there are certain shortages constraining the spread of topology optimization such as checkerboard problem (including islandin g), mesh dependent results problem, non global results and low manufacturability. Checkerboard problem describes the sub structure having no connectivity with surroundings. Mesh dependent problem means the optimum results depend on the element size of FEA. Non global results denote the problem that while changing the parameter of topology optimization algorithm, optimum results changes correspondingly. Low manufacturability represents the problems such as non smooth boundary and extremely large or small sub structures. This thesis studies the numerical instabilities of density based topology optimization systematically and compares the effect of several method to reduce them. Comparing with traditional topology optimization algorithms, topology optimization algorithm is implemented using structural grid finite element method. Finite element method is developed in order to simulate and analyze engineering structures. FEM discretizes the domain and boundaries of a problem by elements and connect these elements by nodes. There are two major shortages of traditional FEM. The first one is expensive computation and difficult to create high quality mesh for analysis. daptive analysis which requires remeshing the domain. Mesh independent method is developed to eliminate above two problems. Mesh independent uti lizes structural grid for analysis, the mesh can be generated automatically and the boundaries can be represented precisely PAGE 16 16 Implicit boundary topology optim ization is first named and introduced in this thesis. Besides implementing using structural grid, IBTO use nodal design variables rather than element wise design variables, the boundary is implicitly expressed and could be calculated more accurately. Sever al numerical method has been implemented and experimented systematically to test the performance of IBTO for compliance oriented problem, compliant mechanism design and stress constrained topology optimization. 1.2 Goals and Objectives 1.2.1 Goal The main goal of this thesis is to systematically study numerical instabilities of density based topology optimization and experiments using numerical techniques to reduce them. Related algorithms are implemented using Implicit Boundary Topology Optimizati on. The performance of IBTO has been experimented in this thesis. 1.2.2 Objectives The main objectives of the thesis are listed: 1. Research on convexity of convexity of compliance Implement and experiment with SIMP Method, RAMP Method and Gradient Smoothi ng Method to obtain the performance of above methods 2. Research on mesh dependence problem. Implement Two level Mesh Method and Filter Method. Experiment with Two level Mesh Method, Filter Method and Gradient Smoothing Method to study mesh dependence 3. Exp eriment with Filter Method to obtain the performance of minimum length scale control. 4. Research on the smoothness of optimum boundaries. Experiment with B spline elements and Gradient Smoothing Method / 5. Implement projection method and experiment to obtain the performance for generating accurate boundaries. PAGE 17 17 6. Apply the above numerical methods to compliance minimization problem, compliant mechanism design, and stress constrained topology optimization. 1.3 Outline The remaining chapters are organized as follow s: In Chapter 2, fundamental concepts and past research on topology optimization are introduced. Previous work on topology optimization using nodal variables as well as mesh independent analysis using i mplicit boundary finite element method (IBFEM) are explained. In Chapter 3, the theory of Two level Mesh Method, interpolation technique of d ensity based method, Filter Method and Projection Method are introduced. Algorithm for implementation and corresponding derivation are given. In Chapter 4, sever al numerical instabilities and corresponding methods to reduce them have been studied In Chapter 5 the summary of the work and conclusions are provided. The future work prospect is also given in this chapter. PAGE 18 18 CHAPTER 2 TOPOLOGY OPTIMIZATION 2. 1 General Introduction of T opology O ptimization In general, mechanical structural optimization problems can be stated as sizing, shape or topology optimization Sizing problems usually involves find ing optimal thickness dimensions or cross sectional area. The goal of shape problems is to find optimum shape of a specified domain. Topology optimization algorithm is the process for finding the optimal layout of material and holes inside specified domain [ 1 ] of the structure includes information on the topology, shape and sizing of the structure [ 2 ] Topology optimization problem is a method for determining where to introduce holes in a structure. Figure 2 1 is cited from [ 2 ] to illustrate sizing, shape and topology optimization problems. Figure 2 1 Three categories of structural optimization A) Sizing optimization of a truss structure B) Shape optimization C) Topology optimization. The initial problems are shown at the left hand side and the optimal solutions are shown at the right. The topology optimization method for continuum structures has developed rapidly since the 19 90s and it has been implemented for solving in dustrial problems. A ) B) C) PAGE 19 19 Topology optimization has great significance for early conceptual design It has been the most active research area in structural and multidisciplinary optimization in the past two decades [ 1 ] The practical scope of topology optimization has increased beyond a few linear structural responses to include combinations of structures, hear transfer, acoustics, fluid flow, aeroelasticity, mate rials design and other multiphysics disciplines [ 1 ] There are four popular methods for achieving topology optimization: density based methods [ 2 5 ] hard kill methods [ 1 6 ] boundary variation methods [ 7 12 ] bio inspired cellular division based method [ 13 ] Brief introduction is given of four methods in following paragraphs. 2. 2 Density based M ethods 2.2.1 Mathematical P roblem S tatement Density based methods are the most widely used algorithm for topology optimization [ 2 5 ] These methods discretize a specified domain into finite elements and usually treat each element as having a constant but variable density An objective function is defined based on the response of the structure under certain loads. T he objective function is minimized t by varying the element density to obtain the optim al structure. It is essentially Assigning integer 0/1 to each elements would raise programming challenges. Continuous density values ranging from 0~1 is adopted to represent the solidness at each element. Penalty methods are utilized to force solutions to reach 0/1 distribution where 0 means void, and 1 means solid. SIMP and RAMP methods are the most well known density based algorithm. The problem statement based on linear static finite ele ment analysis is given [ 1 ] : PAGE 20 20 (2 1) Here f is the objective function, is the vector of density design variables, is the displacement vector, K is the stiffness matrix, F is the force vector, and are constants. For the example of compliance minimization problem f=c= c is compliance, V is structu re material volume, is design domain volume, is allowable volume fraction. Density based methods adopt an interpolation function and penalization technique to express the physical quantities of the structure as a function of continuous design variables. Each element is assigned a design variable, namely density, where 0 means hole, 1 means solid. Density variables are penalized by a power law and multiplied with physical quantities such as material stiffness, cost, or conductivity [ 14 ] where is Y oung is Y oung material, is the penalization function. Figure 2 2 is cited from [ 1 ] to illustrate three penalization functions. p is usually a fixed integer during optimization process Equation (2 1) is the standard form for structural optimization problem [ 15 ] which could be solved using mult iple optimality criteria method s such as CONLIN, MBMethod, MMA, SNOPT. The volume of the structure, V is the sum of design variables, and the design variables are densit ies of each elements. Figure 2 3 is the flow chart of general topology optimization process which is cited fro m [ 2 ] A 99 line MATLAB codes for topology optimization a based on such a density approach could be accessed online [ 16 ] More details of SIMP method could refer to [ 1 17 18 ] PAGE 21 21 Figure 2 2 Comparison of SIMP, RAMP, SINH intermediate density penalization models A) SIMP B) RAMP C) SINH Figure 2 3 Flow chart of density based topology optimization problem Initialize (setting parameters) Finite Element Analysis Sensitivity Analysis Regulation (e.g. Filter Method) Optimization Algorithm (e.g. MB Method) Update design variables c onver ged Yes No Plot results PAGE 22 22 2.2.2 Various Problem F orms The simplicity and relative efficiency of density based method has led to its widespread use in both academia and industr y Major aspects and literatures are listed here for general understanding and further study. S everal objectives and constraints could be formulated into equation (2 1) including compliance, stresses, frequency, d isplacements. Compliance is the most commonly used objective function for topology optimiz ation. Compliance is defined as where f is the vector of node loads, u is the vector of node displacement. Compliance is a measure of the capacity of a struc ture to deform under applied loads. Besides linear static mechanical structure design, topology optimization has been applied to eigenvalue problems, s tructures with non linear response, structure s under stress constraint, structures under dependent load ing, compliant mechanism design, and multiphysics problems [ 1 ] 2.2.3 Numerical Instabilities Although density based topology optimization is a popular and relatively mature algorithm, several inherent numerical instabilities still exist and debase the quality of optimal results including gray region, checkerboard, mesh dependence, ragged boundary, local minima and expensive calculation All these concepts have been discussed in this section. Ideally density of each elements should be 0/1, but in practice is a small number ( e.g. 10e 3 ) which is r equired to avoid singularity of finite element analysis and instability of optimizer. Also intermediate value of between 0~1 could generate physically meaningless material, namely gray region, which lead to inaccurate boundary of optimum structure Fig ure 2 4 illustrates gray regions of optimal PAGE 23 23 results. Gray regions are usually emphasized by filter method and smoothing technique [ 19 ] Several techniques have been developed to eliminate gray region such as projection methods, morphology filters and density slope control [ 20 24 ] Figure 2 4 Gray region of MBB beam topology optimization. Mesh dependence problem is the condition that while refining FEA element size, optimal topology results would change correspondingly. Figure 2 6 is cited from [ 2 ] to demonstrate results changing while refining element size. Design engineers are only interested in the most accurate results Checkerboard denotes the structure topology similar to a checkerboard which is physically meaningless and cannot be manufactured. Figure 2 5 is cited from [ 2 ] to illustrates the checkerboard topology of MBB beam Several solutions have been proposed such as global gradient, slope control, local gradient control, regularized penalty methods, patch control, integral methods [ 1 ] Filter methods are the most efficient and straightforward methods which include sensitivity filter and density filter [ 25 27 ] Figure 2 5 Checkerboard problem of MBB beam. PAGE 24 24 A B C Figure 2 6 Dependence of topology optimization results on FEA element size. A) Element number 10 00 B) Element number 4 00 0 C) Element number 9000. Topology optimization searches for the global minimum value of objective fu nctions. But SIMP method can only guarantee local minimum [ 28 ] While changing the penalization parameter p, different solutions would appear. Figure 2 7 demonstrates c hanging results for different p using SIMP methods. Again, design engineers are only interested in the most accurate results. Continuation method and RAMP method are developed to deal with this issue [ 2 5 ] A B Figure 2 7. SIMP Method analysis with various penalty factor value. A) p= 3 B) p =7 PAGE 25 25 Computation cost is a major issue of simulation. Especially for topology optimization algorithms, iteratively analysis would exaggerate the drawback of certain algorithm such as bio inspired cellular division based method and continuation method. I mprov ing the efficiency of topology optimization algorithms is very necessary. For density type filters are developed to improve calculation time [ 26 ] Considering that topology optimization includes so many fascinating research topics, great research progress, vast applicati introduction of topology optimization in this thesis cannot cover all the concepts in detail. Only basic information of topology optimization is introduced and referred literatures are listed carefully for further review. 2.3 Applications of Topology O ptimization Topology optimization algorithms have been proved to be effective and success ful in several fields. Several popular applications have been introduced in this section. 2.3.1 Compliance oriented P roblem Compliance minimization seems to be the most popular objective function for topology optimization problems [1, 2]. Compliance is a global measurement of structures ability to deform under certain loads. Mathematical form of compliance is ( 2 2 ) W here is the nodal force vector and is nodal displacement vector PAGE 26 26 2.3.2 Material with Nonlinear P roperties Topology optimization has been applied in numerical examples for elastic materials. Topology optimization using material with nonlinear elastic property is based on nonlinear finite element analysis. Unexpected numerical instabilities are faced due to the nature of nonlinear deformation. Research on this specific problem concentrates on i mproving the numerical instabilities while performing nonlinear FEA [38, 39, 40] 2.3.3 Stress constrained Topology O ptimization Stress is an essential consideration while designing mechanical structures. Three methods have been developed for this problem: local method, global method, and block aggregation techniques. Stress constrained could be realized in both objective functions or as additional constraints [ 41, 42 ] Fatigue and damage criteria are emerging recently. 2.3.4 Design under S tructure D epende nt L oading Structure dependent loading denotes loads changing during the optimization proce ss such as pressure loading, self weight loading and multi component structure [43 46] 2.3.5 Compliant Mechanism Design Compliant Mechanism denotes an independent component that can behave like a mechanism by deforming in the desired fashion, such as gripper. Popular objective functions of this problem focus on desired movement of certain points [1, 2] 2.3.6 Reliability b ased Topology O ptimization This kind of problem combines probabilistic theory with topology optimization. Statistical and probabilistic methods consider the variation of design and improve the quality of massive product structures [47, 48] PAGE 27 27 2.3.7 Topology O ptimization of M ultidisciplinary P roblems Besides compliance oriented problems, topology optimization works well for certain other objective functions such as heat transfer constrained, fluid flow constrained, dynamic response constrained and biomedical design [2, 49, 50] 2.4 Previous W ork of T opology O ptimization U sing N odal D esign V ariables Traditional density based topology optimization algorithms are based on element wise variables. Implicit boundary topology optimization is based on nodal design variables which could generate high quality results. Density inside one element could be interpolated usin g shape functions which are independent from those for FEA. The physical significance could be interpreted as: s uppose FEA elements are constituted of these points constitute of the whole structure. A n efficient linear optimizer Moving Barrier Method [ 29 ] is utilized to solve Implicit Boundary Topology Optimization. By treating nodal density as design variables and define structure boundary as a contour at =0.5 optimum results with smoothing boundary would be obtained even using sparser mesh compared with traditional topology algorithm Topology optimization using nod al design variables has been improved using gradient smoothing method and applied to compliance oriented problem, compliant mechanism design, stress constrained topology optimization problem. Considering this thesis extends above algorithms and tries to reduce numerical instabilities while applying topology optimization algorithm, above methods are introduced briefly here. PAGE 28 28 2.4.1 Implicit Boundary Finite Element Method IBFEM is a finite element method us ing structured mesh to discretize analysis domain. P revious work ha s demonstrated successful applications to structural analysis, electromagnetism and heat transfer analysis [ 19 30 34 ] IBFEM is a mesh independen t method, which uses structur ed mesh to di scretize the domain. The boundary of the domain is implicitly described by scalar equations. The solution structure of IBFEM could be written in the following forms. ( 2 3 ) W here is a specified step function of the bounda ry is the essential boundary condition is a grid variable which is defined by piecewise interpolation or approximation. E quation ( 2 3 ) could satisfy on the boundary which is defined by T he function value must be positive and non zero inside the domain. An approximate step function of the boundary can be defined as : ( 2 4 ) W here, is the normal distance between point s to the boundary lines and is the width defined for step function After substitut ing the step function to the integra l form o f governing equation, remaining process is identical to traditional FEM. PAGE 29 29 2.4.2 Compliant Mechanism Design Compliant structures, designed to function as mechanisms, are often referred to as flexures or compliant mechanisms. Unlike traditional mechanisms that are made of rigid l inkages, compliant mechanisms are structures that produce motion by elastic deformation. Topology optimization has been shown to be an effective approach for designing compliant mechanisms. Mathematic problem statement is equation (2 5 ), objective function is defined as least square error at certain points. min s.t. (2 5 ) s.t. s.t. Here and are the actual and prescribed output displacements corresponding to the input displacements or forces and is the number of data points at which desir ed output is specified. Therefore, the square of difference between the actual displacement for a given shape and the desired displacement is minimized. The mass of the structure is computed by integrating the density function over the feasible region and it is constrained to be less than a specified value SIMP approach (2 6 ) The gradient of the objective function is given as: PAGE 30 30 (2 7 ) The gradient of the nodal displacements can be computed by the standard design sensitivity analysis methods. Defining an adjoint variable as : (2 8 ) where is the global stiffness matrix and the right hand side vector components are defined as for nodes at which displacement output is specified and at all other nodes. Using this adjoint variable, the gradient of the objective function can be computed as given in equation (2 9 ) (2 9 ) 2.4.3 Stress C onstrained Topology O ptimization The optimization problem can be restated using a global stress measure constructed using the above stated bound on compliance. We use this global measure to construct an objective function that penalizes viol ation of stress constraints as follows: min s.t. (2 10 ) s.t. is chosen to be: PAGE 31 31 (2 11 ) Considering the bound on Von Mises stress: (2 12 ) W here O bjective function could be modified as (2 1 3 ) 2.4.4 Gradient Smoothing Method Gradient Smoothing Method [ 19 ] was developed to smoothen structur al boundaries. Numerical examples also illustrate mesh independence while using this method wit h appropriate parameter. This method add a smoothing term to the objective functions : (2 14 ) Where f is the original objective function w is a constant decided by heuristic search Appropriate w could generate mesh independence results. Denote the smoothing term as P, t he sensitivity of the smoothing term with respect to the nodal density can be computed as given in Equation (2 15 ) : (2 15 ) PAGE 32 32 w is deci ded by a weight factor w hich denotes the value of smoothing term is set equal to a percentage of the whole objective function: (2 16 ) PAGE 33 33 CHAPTER 3 IMPLICIT BOUNDARY TOPOLOGY OPTIMIZATION Fundamental concepts and previous work on Implicit Boundary Topology Optimization (IBTO) has been introduced in Chapter 2. Four characteristic s of IBTO are: implemented using structural grid, adopt nodal design variables rather than element wise design variables, boundary of optimum results is defined implicitly at objective and constraint functions are accurately calculated inside the bo undary defined by The development of IBTO focuses on how to solve topology optimization problems efficiently and accurately. In this thesis RAMP Method, Filter Method, and Projection Method have been combined with IBTO to achieve results with clea r boundary and better manufacturability. Two level Methods have been derived and experimented. Derivation and implementation of above methods are given in this section. 3 1 Penalty Scheme For isotropic materials, design variables are related using interpolation functions : (3 1) ( 3 2) ( 3 3 ) ( 3 4 ) Where modulus used for topology design. Ideally should be 0/1. In practice is chos en PAGE 34 34 to be some continuous functions for penalty as discussed in Chapter 2. SIMP Method s and RAMP me thods are experimented with IBTO. denotes SIMP scheme, denotes RAMP scheme. nodes # denotes nodes number per FEA elements. is shape function such as Lagrange shape functions. is design variable at element node i. Gaussian integration is a dopted for FEA. Equation (3 1) is used at Gaussian points to approximate stiffness matrix of elements. The stiffness matrix of an element is computed as: (3 5) Where [B] is the strain displacement matrix, [ ] is the constitutive matrix which contains elastic constants. Moving Barrier Method is a s ensitivity guided optimizer. The gradient of stiffness matrix with respect to the nodal design variables is : (3 6) (3 7) (3 8) PAGE 35 35 Above matrices are calculated inside each elements and assemble d to get global matrices. FEA elements nodes and design nodes are the same, but the shape functions are independent. 3.2 Two level Mesh Method Adopting B spline elements for IBTO would generate more accurate and smooth results compare d with traditional ele ments (e.g. quad 4 element ). But the computation is expensive considering the h igh order of B spline elements. The procedure of FEA analysis is essentially matrix manipulation Matrix manipulation takes up major part of calculation time. Two level Mesh is developed for improving analysis efficiency an d providing mesh independence. Mesh A is created for FEA analysis and mesh P is created for interpolating the density function Element size of mesh A and mesh P are and respectively. The basic procedur e is 1. Set element types for topology optimization ( e.g. B spline elements ) element size refine parameter and discretize analysis region using quad 4 element with element size is positive integer. Each element of mesh A includes elements from mesh P. 2. IBTO design variables are only defined in mesh P Objective function (e.g. complia nce), sensitivity analysis stiffness matrix and displacements are computed from mesh A Mesh P is used for plotting optimum results. 3. Optimizer would solve for design variables and assign nodal design variables to mesh P. PAGE 36 36 4. Identify global coordinates of Gaussian points in all elements A and corresponding local coordinates in elements P. Interpolate density value at these Gaussian points from Mesh P and update stiffness matrix 5. Repeat FEA and optimizer process until meeting the condition of convergence. A B C Figure 3 1 Refine process of Two level Mesh Method Blue denotes the boundary of plotting mesh. Orange denotes the boundary of analysis mesh. A) refine factor r=1 B) refine factor r=2 C) refine factor r=3. While fixing element size of mesh P and increasing r, FEA and optimum results would be more accurate. Two level Mesh is implemented using connectivity table. Each element has a unique connectivity table storing local node number and corresponding global node number. Connectivity table is used for assembling matrix. While a pply ing Two level Mesh for compliance oriented problems, the gradient of stiffness matrix must be computed with respect to nodal design variables of the upper level mesh P. PAGE 37 37 ( 3 9 ) ( 3 10 ) calculated at node j (3 11) Where denotes variables defined in Mesh P and denotes variables defined in Mesh A. is stiffness matrix of one element. Global stiffness matrix is assembled by m is node number per element of Mesh A. n is node number per element of Mesh P. N is shape function. 3.3 Filter Method Filter was originally developed to reduce checkerboard and mesh dependence situation. As a popular technique to improve numerical insta bilities of topology optimization, filter method was implemented and tested using nodal design variables. Efficiency of filter is improved using IBFEM. Derivation and implementation of filter are given in this section. The intuition idea of filter is to a verage the nodal design variables ( not density ) in a specified domain and to adopt updated nodal design variables f or computing objective functions, gradient and plotting the optimum. ( 3 12 ) W here is a distance based weight function and here. PAGE 38 38 T he sketch of filter is illustrated in Figure 3 2 Filter domain radius is set before FEA and diameter of domain radius is approximately equal to the minimum length scale of optimum result. Figure 3 2 Sketch of filter, blue X marks central node of filter, orange points denote nodes covering in the filter of central node. Filter domain is a circle or sphere region centered at node The shape of filter has not been investigated before. Square filter and circle filter are implemented and compared in this thesis. In traditional FEA, i dentification of the nodes inside a filter requires nodes distance which is computationally expensive This situation improves in IBFEM. Considering the element type in IBFEM is square, search for filter nodes could be processed from element to element like this: 1. Specify global coordinates of target node element size a and filter radius r. PAGE 39 39 2. E lement s adjacent to target node and affiliated nodes are called first layer. Elements and affiliated nodes adjacent to nodes from 1 st layer are called 3 rd layer, etc. Searching for filter nodes are processed layer by layer using connectivity table. The number of layers required for radius r is r/a. For square filter, distance of nodes is not required at all. For circle filter, only distance from neighboring nodes to target nodes is required. The more nodes there are in the mesh the more efficien t this implementation is Compliance minimization pro blem, compliant mechanism design, and stress constraint topology optimization using IBFEM have been introduced in Chapter 2. Formulations for implementing Filter Method in above three problems are derived in following paragraph s. 3.3.1 Compliance M inimization P roblem with Filter Mathematic statement of c ompliance oriented problem is given: min: ( 3 13 ) s.t. ( 3 14 ) ( 3 15 ) (3 16) PAGE 40 40 Where denotes structure boundary, body force is neglected. denotes feasible domain. denotes filtered design variables. [u] and [ f ] are independent of design variables Discrete form of equation (3 15) is: (3 17) is obtained from FEA. For isotropic materials, [K] is symmetric. Discrete form of e quation (3 14) is: ( 3 18 ) Where e denotes element. Assemble to get n denotes nodes per elements. Equation (3 18) is a linear combination of T ake derivation of equation (3 17): (3 19) Substitute equation (3 19) for t he gradient of L over : (3 20) PAGE 41 41 Considering equation (3 5) ]= [K] is assembled from ]. is further derived: (3 21) W here nni denotes nodes number of filter centered at node i and : (3 22) (3 23) (3 24) 3.3.2 Compliant Mechanism Design with Filter Mathematic statement of compliant mechanism design is given: min: (3 25) T he constraints are the same as those given in equation (3 14), (3 15) (3 16) The fundamental concepts have been introduced in Chapter 2. Substitute equation (3 19) for the gradient of over is given : PAGE 42 42 (3 26) Where denotes the number of specified data points. Data points are specified as global coordinates. denotes desired displacement at data points. denotes shape function of the element which includes data point m. Construct the adjoint variable for all nodes in the elements for which output displacements are specified: (3 27) Let then : (3 28) Gradient is now updated as: (3 29) Where is give n in equation (3 21). 3.3.3 Stress C onstraine d Topology O ptimization with Filter Mathematic statement of stress constrained minimizing weight problem is given: min: (3 30) PAGE 43 43 s.t. s.t. Where and T he stress function is chosen to be: (3 31) (3 32) T he gradient of weight term has been given in equation (3 18). The gradient of stress term is given: (3 33) Where, (3 34) (3 35) Substitute equation (3 34) and (3 35) into (3 33): (3 36) (3 37) (3 38) Where is given in equation (3 21) is calculated using adjoint method: PAGE 44 44 (3 39) (3 40) is given in (3 21). 3.4 Projection M ethod Projection Method was firs t developed to eliminate gray region caused by filter. Averaging scheme of Filter Method would blur the boundary and generate intermediate density value which is called gray region problem. The fundamental me chanism of Projection Method is to pass the filtered design variable through an appropriate project ion function The projection function transform s the intermediate (or filtered) density to which are close to 1 Ideally, projection function should be step function such as Heaviside step function: (3 41) But, discontinuity of Heaviside step function would result in failure of optimizer. Certain regions would become complete 0 or 1 and would have zero sensitivity Th e i nterpolated values of the density at Gaussian points rather than nodal design variables are projected (pass through projection function). In order to obtain accurate boundary, boundary elements are split into two parts along optimum boundary and both parts have independent Gaussian points. Figure 3 3 (A) is the geometry of non boundary eleme nt, projection is implemented to the Gaussian points of the element. Figure 3 3 (b) illustrates a typical boundary element which contains the element was subdivided into region 1 and region 0 while performing numerical integratio n. Density in region 1 is larger than 0.5 and density in region 0 is smaller than 0.5. PAGE 45 45 Projection is implemented at the Gaussian points of region 1 and region 0 rather than the original boundary element. The exponent projection function invented by Guest i s introduced here. The exponent projection function invented by Guest is here. Tan p rojection function is developed in this thesis. A B Figure 3 3 Subdivision of boundary element A) Geometry of non boundary element, B) Geometr y of boundary element. 3.4.1 Exponent Projection Exponent projection funct ion is: (3 42) Where, is constant The influence of is illustrated in Figure (3 2) Derivative of over is: (3 43) Where denotes the number of nodes inside the filter which is centered at node I, is given in equation (3 23) is given in equation (3 24) and: PAGE 46 46 (3 44) Figure 3 4 The influence of on projected design variable from exponent projection function (a) (b) (c) (d) 3.4.2 Tan Projection The i deal projection function is: (3 45) Tan projection function tends towards this ideal projection function for large value of and is defined as: (3 46) PAGE 47 47 Figure 3 5 The influence of on projected design variable from tan projection function (a) (b) (c) The influence of the constant, on this projection function is illustrated in Figure 3 5 Derivative of over is the same as equation (3 43), where is given in equation (3 23) is given in equation (3 24) and: ( 3 47) Compliance minimization problem, compliant mechanism design, and stress constrained topology optimization were modified to use the projected density is modified as: (3 48) PAGE 48 48 CHAPTER 4 RESULTS AND DISCUSSIONS Implicit Boundary Topology Optimization could generate high quality result using sparser mesh compared with traditional topology optimization [ 19 ] In this section several numerical behavior of topology optimization algorithm is studied. Section 4 .1 looks at the c onvexity of objective function 4 2 m esh i ndependence, 4 3 l ength Scale Control, 4 4 s moothing r esults and finally in section 4 5 we presen t result obtained using projection and accurate boundary integration Methods aimed at improving various numerical instabilities have been implemented and tested in IBFEM software During the implementation of these algorithms, theoretical value of gradien t for objective has been verified using finite difference calculation. 4 .1 Penalty Schemes of Topology Optimization SIMP m ethod and RAMP m ethod are the two most popular penalty schemes used for density based topology optimization model. Continuation method is commonly used for obtaining global results [ 5 28 35 ] The essence of topology optimization problem is the optimization of a scalar function. Convexity is influenced by objective function and constraint. While performing topology o ptimization using multiple element types (e.g. Quad 9, Quad 16, B spline 9, B spline 16) optimum results seems to be different T his is caused by numerical approximation as well as non convexity Influence of penalty factor is tested using quad 4 element and different element size while fixing element size and element type When penalty factor is large enough, topology optimization problem using RAMP m ethod could have unique optimum results. But for SIMP m ethod, when using l arge penalty factors the results can still vary SIMP m ethod and RAMP m ethod usin g nodal design variables are first discussed in this section. PAGE 49 49 RAMP m ethod has been shown to generate unique topology design for several examples [ 5 ] When penalty factor p is large enough, topology optimization problem becomes convex. This characteristic has been tested here using a bridge structur e and minimum compli ance topology optimization algorithm. For the feasible region in Figure 4 1, length is 300 m, height is 100 m, Y modulus is 1 .0 load is applied at q uarter spot s of the bottom edge. SIMP penalty is 4, mass constraint is 0.7 and e lement type is 2D Quad 4. Element number is 12 0* 40 (element size 2 .5). Penalty factor p from SIMP m ethod changes from 2 to 100 gradually in Figure 4 2 SIMP m ethod could generate distinctive optimum results while increasing penalty factor. In general, continuation method is adopted for generating a global results. Continuation method increases p gradually from 2 to 5 and find s corresponding results with lowest compliance. This method involves additional computational costs Figure 4 1. MBB Beam model PAGE 50 50 A B C D Figure 4 2. SIMP Method analysis with various penalty factor value. A) p=2, B) p= 4 C) p= 7 D) p= 12 Penalty factor p from RAMP Method changes from 0 to 1 e8 gradually in Figure 4 3. RAMP Method could generate unique convergent results theoretically while p is large enough. But when p is extremely large (p=1e8), no results are obtained. This may be because for la r ger p, the optimizer oscillates and diverge. But when p is reasonabl y large (from 100 to 1000), islanding and checkerboard problems are improved. R elatively stable results are generated. A B C D E Figure 4 3. RAMP Method analysis with various penalty factor value. A) p= 2 B) p= 5 C) p= 10 D) p= 100 E) p=1 000 PAGE 51 51 SIMP m ethod is the most popular topology optimization model both in academi a area and industr y [ 1 ] RAMP m ethod seems to be more stable than SIMP m ethod using nodal design variables. Possible risk of RAMP m ethod i s impractical to find appropriate p by trial in a large range ( from 0 to 1e8 ) co nsidering different structures require different value of p. For RAMP Method, if p is small, results vary with different p, if p is too large, no results would be generated. Also while p=3, 4, 5 in SIMP Method, the topology results are similar to those generated by RAMP Methods [ 1 ] Topology results improve dramatically while combining SIMP Metho d with additional techniques. 4 2 Mesh Independence Upon refining element size, thinner structures can be represented by the mesh and FEA result s become more accurate. T opology design using SIMP Method changes correspondingly while re fining element size for FEA [ 1 37 ] This situation is called mesh dependence. Designers are only interested in the best result from multiple outcomes with reasonable element size. In order to achieve mesh independen t results, Two level Mesh Method, Gradient Smoothin g Method and Filter Met hod have been tested and discussed in this section. 4.2.1 Two level Mesh Method In this approach, two different mesh are used, one for FEA analysis which we nalysis mesh and the other is for interpolating the density for shape repr esentation which we have referred to as the lot mesh used for plotting results. T hese t wo meshes could have different element type and element size. In order to illustrate mesh independence only 2D Quad 4 element s are used for both me sh Compliance minimization for an L shaped feasible region is PAGE 52 52 analyzed here. For L shaped region in Figure 4 4 length is 30 m, height is 12 m elastic modulus is 200 G 3 point force on right edge is 2 00 N, top edge is fixed. V olume constraint is 0. 7 SIMP penalty is 4, and M B Method constant is 3 Fix element number of plot mesh as 25 25 and refining analysis mesh gradually in Figure 4 5. Figure 4 4 L shaped model PAGE 53 53 A B C D Figure 4 5 Two level Mesh Method while fixing plotting mesh A) Analysis element number 40 40, B) Analysis element number 80 80, C) Analysis element number 120 120 and D) Analysis element number 200 200. With a fixed plotting mesh refining the analysis mesh, does not change the optimal topology and the quality of the analysis results improve because the FEA is more accurate with a refined mesh 4.2.2 Gradient Smoothing Method Compliance minimization with Gradient Smoothing m ethod could remove small structures qualitatively [ 19 ] L shaped region in Figure 4 4 has been analyzed to test this characteristic. Mesh element number changes from 20*20 gradually in Figure 4 6 Smoothing weight factor is determined to be 0. 25 by trial to obtain clear and mesh independent PAGE 54 54 results. For large results become blur, for small or mesh independe nt A B C D Figure 4 6 Compliance minimization with Gradient Smoothing Method using various element number. A) 20 20, B) 40 4 0, C) 80 80 and D) 12 0 12 0. 4.2.3 Filter Method Filter m ethod is one of the most popular techniques for removing checker board, realizing mesh independence and length scale control for traditional topology optimization [ 1 21 25 27 37 ] Filters have been combined with Implicit Boundary Topology Optimization and tested. One significant weakness of the Filter m ethod is long calculation time for identifying elements within the filter radius of each node. This situation could be dramatically improved using structural mesh. Nodes within the filter can been found PAGE 55 55 using the connectivity table. The more elements for a nalysis, the more calculation time Implicit Boundary Topology Optimization could save compa red with traditional topology optimization. Compliance minimization of a c antilever ed b eam has been analyzed in order to illustrate the mesh independence of results obtained using Filter m ethod and Implicit Topology Optimization in Figure 4 8 The c antilever Beam is shown in Figure 4 7 whose length is 40 m, height is 25 m, Y .0 0.25, left edge is fixed, and a 1 N point load i s applied at the middle of right edge. SIMP penalty is 3, MB Method constant is 3. Element type is 2D Quad 4. Filter radius is fixed to be 2 m. The mesh is refined as shown in Figure 4 8 to show that the topology is the same regardless of the number of ele ments It is seen that the filter guarantee s mesh independence a nd the optimal results become smoother while refining element size. Figure 4 7 Cantilever Beam model PAGE 56 56 A B C D Figure 4 8 Filter Method analysis with various element number. A) 30 18, B) 60 37, C) 90 56 and D) 12 0 75 4 3 Length Scale Control Len gth scale is the minimum size of the structure which influence s manufacturability directly. A too large structure is difficult to handle and a to o small structure is difficult to manufacture. Fil ter method has been illustrated to control length scale of op timum structures. Cantilever beam example from Figure 4 7 is analyzed here, whose length is 40 m, height is 25 m, Y .0 MPa, fixed, 1 N point load is applied at the middle of right edge. SIMP penalty is 3, MB Method constant is 3.Element type is 2D Quad 4. Element number is f ixed to be 1 6 0* 100 (element size 0. 25 ) PAGE 57 57 4.3.1 Filter Method Filter size has been increase d gradually from 0 in Figure 4 9 The minimum length scale is approximately the diameter of filter. It is noticed that while filter radius is small (comparing with element size), filter method works well. While filter size is large, results become more and more blurry Suppose the filter size is infinite large, all the design variables would have same value. A B C D E Figure 4 9 Topology optimization of cantilever structure with various filter size. A) 0, B) 0.5, C) 0.75, D) 1.25, E) 3. PAGE 58 58 Most literature adopt circle as filter shape. Square filter is implemented and tested here. Filter radius square filter is half size of a side. In Figure 4 1 0 square filter and circle filter is compared. The optimum results are slightly different while changing filter shape. Intuitively circle filter could treat all the directions evenly comparing with square filter. A B C D E F Figu re 4 1 0 Topology optimization using circle filter and square filter with various filter size. A) circle 0.5, B) square 0.5, C) circle, 0.75, D) square, 0.75, E) circle 1.25, F) square 1.25 PAGE 59 59 4.3.2 Two level Mesh Method Two level Mesh Me thod is originally developed to control the length scale of optimum structures. Structure inside each element is plotted using shape functions which are polynomials. F or different shape functions, there exists a fixed function range, h ence the plotted stru cture inside the element has a minimum length scale ( determined by the element size). In Figure 4 1 1 the e ffect of Quad 4 element and element size on length scale control is illustrated. In Figure 4 1 2 the effect of B spline 9 element and element size on length scale control is illustrated. Cantilever beam is analyzed here. Analysis element size is fixed to be 0.17, namely, element number is 40*25 for providing same FEA results. Minimum lengths scale decreased while refining element size. But for differe nt element type, the minimum length scale is different and the minimum length scale could not be controlled quantitatively comparing with filter method. PAGE 60 60 A B C D E Figure 4 1 1 Two level Mesh Method using Quad 4 element with various element size A) 2, B) 1 C) 0.5, D) 0.3, E) 0.17 PAGE 61 61 A B C D Figure 4 1 2 Two level Mesh Method using B spline 9 element with various element size. A) 2, B) 1, C) 0.5, D) 0.3 4 4 Smoothing Results In this section, non smooth or wavy boundary problem is discussed. Non smooth boundaries are not desirable and are difficult to manufacture. Interpolation scheme significa ntly influence s the smoothness of boundary. Several methods for generating smooth boundary have been illustrated in this section. 4.4.1 B spline Element with Two level Mesh Method Several element types are analyzed here with similar node numbers to demonstrate smoothness of boundary Cantilever beam design ( Figure 4 7 ) is analyzed here with Quad 4, Quad 9, B spline 9, B spline 16 element using SIMP Method alone Results are demonstrated in Figure 4 1 3 PAGE 62 62 A B C D Figur e 4 1 3 Topology optimization using SIMP Method with various element types. Element number is 60*37. A) Quad 4 B) Quad 9 C) B spline 9 and D) B spline 16 Using high order elements for topology optimization (e.g. B spline element) could generate results with smooth boundaries and prevent most checkerboard situations. But computation time increases dramatically while using high order elements. In T able 4 1, c antilever beam is chosen as numerical example, all element number s are fixed to be 40*25, computation time for each iteration before and after using Two level M esh are compared. Refine factor n means shrink element size to 1/n. Two level Mesh Method could improve computation time efficiently Computation time varies from computer by computer and the weight of FEA analysis in the whole computation process computation time adapted here is the average value of random selected 5 iterations and one digit accura cy could be guaranteed PAGE 63 63 Table 4 1. Influence of Two level Mesh on Computation T ime ( unit: millisecond ) Element type One level Mesh Two level Mesh (refine 1) Two level Mesh (refine 2) Two level Mesh (refine 3) Quad 4 1180 1200 3 385 78 00 Quad 9 8800 5200 6400 16085 B spline 9 7800 3480 7092 1 6387 B spline 16 38700 4300 9 487 25028 IBTO c ould generate relatively smooth results comparing with traditional topology optimization algorithm (element wise boundary) even using sparse mesh. High order interpolation functions alone would generate high quality results. Using Two level Mesh method could improve the computation time especially for high order elements such as B spline elements. 4.4.2 Gradient Smoothing Method The effect of Gradient Smoothing Method could not be defined quanti ta tively In general while weight factor is smaller, less gray region and rough boundary are obtained In Figure 4 1 4 increases gradually. Sparse mesh (40*25) is used here to amplify the non smoothness of boundary (while using dense mesh, element size is too small comparing with structure size and waviness of the boundary seems less significant). The results seem smooth and clear while PAGE 64 64 A B C D E F Figure 4 1 4 Topology optimization using SIMP Method and Gradient Smoothing Method with various weight factor A) Element boundary, B) 0, C) 0.1, D) 0.2, E) 0.3, F) 0.4 4.5 Accurate Boundary Gray region problem is mainly discussed in this section. Gray region could blur the boundary. Ideally, optimum result should be black/white, intermediate density of gray region would influence the accuracy of optimum results. In IBTO, boundary is implicitl y defined at Gray region mainly appear after Filter Method and Gradient Smoothing Method. PAGE 65 65 Projection Method is implemented after Filter Method. Considering Projection Method is implemented at Gaussian points, mass constraint is no longer a linear here for compliance oriented problems and compliant mechanism design. In this section, Exponent Projection and Tan Projection were tested for stress constrained problem. I n Figure 4 1 5 Exponent Projection had been tested for L shaped structure from Figure 4 4 Element number was 75*75 (element size 0.004), Quad 4 element, filter radius was 0.012. A B C Figure 4 1 5 Stress constrained topology optimization using Exponent Projection A) B) C) Exponent Projection seems not working for IBTO. This may be because that for IBTO, boundary is defined implicitly at rather than PAGE 66 66 A B C D E F G H Figure 4 1 6 Stress constrained topology optimization using Tan Projection A) B) C) D) E) F) G) H) PAGE 67 67 In Figure 4 1 6 Tan Projection was tested with the same stress constrained problem. Gray region was eliminated, namely, boundary became more accurate. But the optimum results became not unique. For Tan Projection, while is small, the penalty is too large, the optimizer would be unstable. And according to the experiments, are appropriate values. In Figure 4 1 7 same stress constrained problem was analyzed, is fixed t o be 1000 and while refining element size, the boundary became smoother and mesh independent. A B Figure 4 1 7 Stress constrained topology optimization using Tan Projection and A) Element number 75*75, B ) Element number 120*120 One important issue had been noticed, the boundary narrowed down and became rugged after projection. In Figure 4 1 8 Effect of Filter Method and Projection Method had been compared. PAGE 68 68 A B C Figure 4 1 8 Stress constrained topology optimization. A) Without projection, B) Using Tan Projection, plotting filtered density C) Using Tan Projection, ,plotting projected density The computation process of Figure 4 1 8 (B) and (C) are the same, and the only difference of these two pictures is plotting scheme. For T a n Projection, the boundary should be the same while plotting and But it's not here as we observed. So the plotting schemes of IBFEM software could be modified to plot the right boundary. 4 6 Application of Implicit Boundary Topology Optimization In section 4.5 all numerical examples are compliance oriented. IBTO could generate acceptable results with sparse mesh and the results generated from IBTO are similar with those from traditional topology optimization algorithm. Gradient PAGE 69 69 Smoothing Method and Filter Method turns out to be effective using IBTO. Several t opology designs has been performed in this section. 4.6.1 Compliance oriented Topology Optimization Example 1 Feasible region of a knee structure is illustrated in Figure 4 19 length is 30 m, height is 6 m, Y .0 0.25, top edge is fixed, 200 N downward point load is applied at the middle of right edge. SIMP penalty is 4 mass constraint is 0.7. MB Method constant is 3. Element type is 2D Quad 4. Element number is 80 *80 (element size 0. 375 ). Filter radius is 0.7. A B Figure 4 19 Topology optimization of knee structure A) Feasible region, B) optimum topology Example 2 Feasible region of a bri d ge structure is illustrated in Figure 4 2 0 length is 300 m, 100 N downward point load is applied at q uarter spot s of the bottom edge. SIMP penalty PAGE 70 70 is 4, mass constraint is 0.7. MB Method constant is 3.Element type is 2D Quad 4. Element number is 120 40 (element size 0.375). Filter radius is 5. A B Figure 4 2 0 Topology optimization of bridge structure A) Feasible region, B) optimum t opology Example 3 Cantilever b eam region is illustrated in Figure 4 2 1 length is 40 m, height is 25 m, Y is applied at the middle of right edge. SIMP penalty is 3, MB Met hod constant is 3. Mass constraint is 0.5. Element type is 2D Quad 4. Filter radius is fixed to be 2 m. A B Figure 4 2 1 Topology optimization of cantilever beam A) Feasible region, B) optimum topology Example 4 Feasible region for MBB b eam is illustrated in Figure 4 2 2 length is 120 m, PAGE 71 71 right corner is only free in x direction. 1 N point load is applied at the middle of uppe r edge. SIMP penalty is 4, MB Method constant is 3. Mass constraint is 0.5. Element type is 2D Quad 4. Element number is 120*20 (element size is 1). Filter radius is fixed to be 2 m. A B Figure 4 2 2 Topology optimization of MBB beam A) Feasible region, B) optimum topology 4.6.2 Topology Optimization of Compliant Mechanism Example 5 Push gripper is a benchmark problem of compliant mechanism design Figure 4 2 3 shows the feasible region of compliant mechanism In this problem the goal is to make the points A and B move towards each other to grip a work piece when a pushing force is applied on the left edge. The compliant mechanism is supported at the two corners on the left edge of the feasible domain and the actuating force ap plied at the middle of the edge. M agnitude of the force is 150000 N. Two opposite loads of magnitude 5000 N are applied at the poin ts A and B PAGE 72 72 A B Figure 4 2 3 Topology optimization of gripper A) Feasible region, B) optimum topology he material equal s to and the P 0.3. SIMP penalty is 4. Mass constraint is 0.7. Element number is 60*60 (element size is 0.002). Filter radius is 0.004. Example 6 The feasible domain of inverter problem is shown in Figure 4 2 4 Here the desired motion is that the displacement at the output point should be opposite to the displacement at the input point T he mechanism is supported at the two left co rners of the feasible domain. Input force has a magnitude of 150000 N and is applied at the middle of the left edge. Th e size of the design domain is A displacement of magnitude (in the negative x direction) is specified as the desi red output displacement and a displacement of magnitude is specified at the input port where the input forces are applied. The material of the domain is assumed to be steel with modulus of elasticity equal to 30 e6 psi and the Poi Mass A B PAGE 73 73 constraint is 0.8. SIMP penalty is 4. Element number is 80*80 (element size is 0.125). Filter radius is 0.125. A B Figure 4 2 4 Topology optimization of inverter A) Feasible region, B) optimum topology 4.6.3 Stress constrained Minimizing Weight Topology Optimization Example 7 A rectangular domain with dimension 0.2m*0.6m is defined as feasible region. Left edge is fixed and a shear force of magnitude 1000 KN is applied on right edge. The modulus of ela Element number is 30*90 (element size is 0.67). Filter radius is 0.7. Feasible domain and topology result are shown in Figure 4 2 5 PAGE 74 74 A B Figure 4 2 5 Topolog y optimization under shear load in rectangular domain. A) Feasible region, B) optimum topology Example 8 Knee structure is analyzed here. Top edge is fixed and a shear load of 1000 KN Allowable stress is 200 MPa. Element number is 75*75 (element size is 0.004). Filter radius is 0 .008. Feasible domain and optimum topology are demonstrated in Figure 4 2 6 PAGE 75 75 A B Figure 4 2 6 Topology optimization under shear load in L shaped domain. A) Feasible region, B) optimum topology PAGE 76 76 CHAPTER 5 CONCLUSIONS 5 .1 Summary I n this thesis, density based topology optimization algorithm was studied systematically. Fundamental concepts of topology optimization and details of density based topology optimization algorithm were introduced first followed by a discussion of previous work related to topology optimizati on using nodal design variables S everal methods, including RAMP Method, Filter Method and Projection Method, which are effective using element wise design variables has been modified and implemented using nodal design va riables A Two level Mesh Method has been studied and shown to be effective way to attain mesh independence and also to significantly reduce the computational cost All the derivations and implementation procedure were illustrated. These numerical methods were experimented for compliance minimization problem first. RAMP Method was implemented and compared with SIMP Method for convexity. It is noted that the quality of results improve while just increasing penalty factor of RAMP Method. SIMP Method could obt ain local optima and continuation method maybe required for obtaining global optima. Two level Mesh Method was initially developed for controlling length scale of optimum structures. It was found to be not very reliable in controlling length scale conside quantitatively Two level Mesh Method wa s later tested for improving the efficiency of calculation while performing FEA analysis using quad 4 element and plotting using various element s Two level Mesh Method could improve computation time while obtaining high quality results. Also Two level Mesh Method could realize mesh independence. PAGE 77 77 Filter Method, as one of the most popular numerical method for density based topology optimization alg orithm, was modified for nodal design variables. Considering the experimental platform is structural grid based FEA software, the identification of the nodes inside a filter is more efficient. Filter Method was later tested for improving mesh dependence, s olving checkerboard problem and improve manufacturability. Projection Method aimed at reducing gray region and obtaining accurate bou ndary. Projection was performed at Gaussian points which could generate more accurate boundary theoretically. This process requires additional segmentation of boundary elements. Tan projection function was introduced and compared with exponent projection function. Topology optimization using tan projection could reduce gray region and generate accurate boundary. After the implementation, Gradient Smoothing Method was also compared with above methods. Filter Method and Projection Method were extended to compliant mechanism design and stress constrained weight minimization problems. Implicit Boundary T opology Optimization is introduced in this thesis to describe an approach that has three characteristics: Design variables are nodal density values boundary is implicitly defined at structural properties are accurately computed using this boundary 5 .2 Scope of Future Work Linear optimizer M B Method was adopted in this thesis considering its efficient for structural optimization problems. But linear optimizer limited the performance of Projection Method. Non linear optimizer such as MMA may be implemented in the future. PAGE 78 78 Projection aims at reducing gray region for Filter Method which could also be implemented with Gradient Smoothing Method. The basic frame of IBTO has been establish ed and IBTO has been proved to be effective for several popular topology optimization algorithms. 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PAGE 84 84 BIOGRAPHICAL SKETCH Yiming Zhang was born and brought up in a small city called Yongcheng in Henan Province, China. He got admitted to Henan Experimental High School in Zhengzhou and left his hometown for further education since then. He entered Shanghai Jiao Tong University, China in 2008 and graduate d with a Bachelor of Science in m echanical engineering and a utomation in June 2012. His research interest was product development and manufacturing, development of automation equipment during undergraduate study. He enrolled in the University of Florida in August 2012 and received a Master of Scie nce degree in mech anical e ngineering in May 2014. During the study at University of Florida, his area of specialization include finite element method, numerical method, optimization algorithms, surrogate modeling, mechanical structural design and analysis, and object orie nt ation programming technique |