Supervisors: Fred van Keulen Topology Optimization for Localizing Design Problems: An Explorative Review Chris Reichard Supervisors: Fred van Keulen Matthijs Langelaar Shinji Nishiwaki

Outline Introduction Research Project Findings / Results Skeleton Modeling Outline Introduction Topology Optimization Heat Conduction Problem Research Project Research Problem and Objective Skeleton Modeling Sub-Structuring Findings / Results Sub-Structuring 1

What is Topology Optimization? Topology optimization is a tool to optimize a layout of a structure in a given design space based on: Applied loads Boundary Conditions Performance Criteria Automotive Control Arm Heat Conduction Source: Example by Abaqus software 2

Heat Conduction Optimization Optimization Problem: Heat conduction Uniform heat applied Objective: Minimize temperature 3

Heat Conduction Optimization Optimization Problem: Heat conduction Uniform heat applied Objective: Minimize temperature Achieved by: Placement of two materials kH: moves heat efficiently kL: moves heat inefficiently 3

Heat Conduction Optimization How is Optimization Performed? Discretize problem into small elements Small elements = design variables Provide initial structure Solve temperatures in elements Update design through approximations Design Variables 4

Heat Conduction Optimization How is Optimization Performed? Discretize problem into small elements Small elements = design variables Provide initial structure Solve temperatures in elements Update design through approximations Design Variables 4

Increasing sparseness Research Problem Localization: Small, local details Structure in fraction of design area Sparse design Increasing sparseness 30% of Total Volume 10% of Total Volume 1% of Total Volume 5

Research Problem Localization: Main Issue: Small, local details Structure in fraction of design area Sparse design Main Issue: Need many small elements to define structure Design Variables 5

Research Problem Localization: Main Issue: Small, local details Structure in fraction of design area Sparse design Main Issue: Need many small elements to define structure Increase resolution, dramatic increase time Design Variables 5

Research Objective Improve the implementation of the optimization process for the design of sparse structures based on: Improved efficiency by reducing number of design variables Exploit local features of sparse problem Assess feasibility of developed methods 6

Efficiency Issue Finite Element Analysis (FEA) is the main issue! 7

Efficiency Issue Finite Element Analysis (FEA) is the main issue! Time increases due to increase in elements 7

Characteristics of Local Problem 30% of Total Volume 10% of Total Volume 1% of Total Volume Develops into bar like structure 8

Skeleton Modeling Definition Idea: Skeleton Model Computer graphics, medical imaging, scientific visualization Model structure through skeleton Source: A Fully Automatic Rigging Algorithm for 3D Character Animation Masanori Sugimoto, University of Tokyo 9

Skeleton Modeling Definition Idea: Skeleton Model How? Computer graphics, medical imaging, scientific visualization Model structure through skeleton How? Global: Background mesh Skeleton Structure: Bar elements Obtaining Skeleton Indirect representation Direct representation Source: A Fully Automatic Rigging Algorithm for 3D Character Animation Masanori Sugimoto, University of Tokyo 9

Skeleton Modeling Indirect Representation of Skeleton Structure Boundary known from surface level Need to extract skeleton from surface 10

Skeleton Modeling Indirect Representation of Skeleton Structure Boundary known from surface level Need to extract skeleton from surface Skeleton curve is smooth and continuous but implicit Issue: how to update design 10

Skeleton Modeling Direct Representation of Skeleton Surface Function Skeleton Curve Skeleton curve already known and used to develop surface function Need to extract width of structure from surface 11

Skeleton Modeling Direct Representation of Skeleton Surface Function Skeleton Curve Structure Boundary Skeleton curve already known and used to develop surface function Need to extract width of structure from surface 11

Skeleton Modeling Challenges with Direct Representation Connectivity of Skeleton Points How are the skeleton points connected? Source: printactivities.com Ambiguous on how to connect points 12

Skeleton Modeling Challenges with Direct Representation Connectivity of Skeleton Points How are the skeleton points connected? Source: printactivities.com Ambiguous on how to connect points 12

Source: printactivities.com Skeleton Modeling Challenges with Direct Representation Connectivity of Skeleton Points How are the skeleton points connected? Need extra Information Source: printactivities.com 12

Source: printactivities.com Skeleton Modeling Challenges with Direct Representation Connectivity of Skeleton Points How are the skeleton points connected? Need extra Information Source: printactivities.com 12

Source: printactivities.com Skeleton Modeling Challenges with Direct Representation Connectivity of Skeleton Points How are the skeleton points connected? Need extra Information Differentiability: Needed to update design Structure is non-continuous Source: printactivities.com 12

Summary Findings / Results Skeleton Modeling Benefits: Simplified representation which exploits sparse structure Reduced number of elements Source: A Fully Automatic Rigging Algorithm for 3D Character Animation Masanori Sugimoto, University of Tokyo 13

Summary Findings / Results Skeleton Modeling Benefits: Simplified representation which exploits sparse structure Reduced number of elements Challenges: Complexity of the method Feasibility? Efficiency Improvement? Combining models to obtain temperature Updating the structure Source: A Fully Automatic Rigging Algorithm for 3D Character Animation Masanori Sugimoto, University of Tokyo Combine 13

Characteristics of Local Problem 30% of Total Volume 10% of Total Volume 1% of Total Volume Develops into bar like structure Elements with changing material 14

Sub-Structuring Definition Current methods: Structured groupings Using multiple processors 15

Sub-Structuring Definition Current methods: Idea: Structured groupings Using multiple processors Idea: Separate elements into groups Groups: Changing vs. static elements 15

Expensive in terms of time Sub-Structuring Definition Achieved By: Invert static matrix separate from changing Expensive in terms of time 16

Sub-Structuring Definition Achieved By: Invert static matrix separate from changing Benefit: Reduction of number of variables needed to be inverted every iteration Terms calculated every few iterations! Expensive in terms of time 16

Sub-Structuring Estimated Improvement Cost Adaptive Sub-structuring Method: Full Implementation: 17

Sub-Structuring Estimated Improvement Cost Adaptive Sub-structuring Method: Full Implementation: Assumptions for sub-structuring Matrix structure is in a less optimal form Solution of equations is less efficient 17

Sub-Structuring Estimated Improvement Cost Adaptive Sub-structuring Method: Full Implementation: Assumptions for sub-structuring Matrix structure is in a less optimal form Solution of equations is less efficient Savings determined for FEA only 50 Iterations fixed 10 Iterations fixed 5 Iterations fixed 2 Iterations fixed 1 Iterations fixed Full Implementation 17

Sub-Structuring Buffer Zone Issues: Groups of elements change each iteration 18

Sub-Structuring Buffer Zone Issues: Groups of elements change each iteration Structure Areas of Design Change 18

Sub-Structuring Buffer Zone Issues: Solution: Groups of elements change each iteration Solution: Buffer zone to reduce updates Radial Buffer 18

Sub-Structuring Buffer Zone Issues: Solution: Groups of elements change each iteration Solution: Buffer zone to reduce updates Radial Buffer Sensitivity Buffer 18

Sub-Structuring Buffer Zone Issues: Solution: Groups of elements change each iteration Solution: Buffer zone to reduce updates Radial Buffer Sensitivity Buffer Combined Buffer 18

Sub-Structuring Example Implementation Static Domain Low conductive region Static Domain Buffered changing domain High conductive structure Elements with changing material 19

Summary Findings / Results Sub-Structuring Benefits: Reduced size of matrix to invert every iteration Time savings 20

Summary Findings / Results Sub-Structuring Benefits: Reduced size of matrix to invert every iteration Time savings Buffer method is low cost 20

Summary Findings / Results Sub-Structuring Benefits: Reduced size of matrix to invert every iteration Time savings Buffer method is low cost Challenges: Developing matrix structure 20

Recommendations Skeleton Modeling Sub-Structuring Obtaining skeleton Investigate efficient methods to combine models Ideas to update structure Sub-Structuring Determine efficient methods to formulate Matrices Optimal sizing of buffer zone 21

Conclusion Objective: Improve the implementation of topology optimization for sparse design problems Issues of efficiency need to be addressed Skeleton method shows potential Sub-Structuring up to 65% time savings for 1% of total volume! 22

Supervisors: Fred van Keulen Topology Optimization for Localizing Design Problems: An Explorative Review Chris Reichard Supervisors: Fred van Keulen Matthijs Langelaar Shinji Nishiwaki 23

Introduction Experiences Thesis Performed at: Guidance By: TU Delft, Netherlands Kyoto University, Japan Guidance By: Fred van Keulen Matthijs Langelaar Shinji Nishiwaki 24

What is Topology Optimization? Objective: Minimize displacement for given load 25

What is Topology Optimization? Objective: Minimize displacement for given load Build approximate model: Through many small elements Material is varied in elements Displacement solved in each element 25

What is Topology Optimization? Objective: Minimize displacement for given load Build approximate model: Through many small elements Material is varied in elements Displacement solved in each element Update Design: Design is updated through sensitivities Continues until objective is met 25

Test Case Heat Conduction Max. Temp. Structure No Structure Min. Temp. 26

Research Plan Investigate research problem Research known techniques Examine how structure develops Determine characteristics of localization Research known techniques Optimization Modelling Develop ideas to exploit problem Investigate ideas Assess feasibility 27

Efficiency Issue Finite Element Analysis (FEA) is the main issue! Time increases due to increase in elements 28

Efficiency Issue Finite Element Analysis (FEA) is the main issue! 7

Summary Findings / Results Sub-Structuring Benefits: Reduced size of matrix to invert every iteration Time savings Buffer method is low cost Challenges: Developing matrix structure 20

Level – Set Approach How to obtain skeleton Structure? The issues of obtaining skeleton structure is often seen in areas such as pattern recognition, computer graphics, shape design, etc. 29

Source: Eric Gaba. Wikipedia. Principal Curvatures Skeleton Modeling Principal Curvatures Skeleton defined as ridge of LSF Principal curvature to obtain ridge At each point: and Need critical point of Critical pt. = Ridge pt. Source: Eric Gaba. Wikipedia. Principal Curvatures 30

Skeleton Modeling Principal Curvatures S Principal curvature developed through First and Second Fundamental Form of tangent plane of surface S 31

Principal Curvature How to Obtain it? First Fundamental Form, I Second Fundamental Form, II Weingarten Operator (Shape Operator) Principal Curvature (Roots of characteristic equation) 32

Skeleton Modeling Radial Basis Functions Radial Basis Function: with 33

Skeleton Modeling RBF: How to Obtain Width? 34

Skeleton Modeling RBF: How to Obtain Width? 35

w/ n being the number of full spaces in between level set grid points Skeleton Modeling RBF: How to Obtain Width? w/ n being the number of full spaces in between level set grid points 36

Skeleton Modeling RBF: Effects of Design Variables 37

Substructuring Direct Solve Solving equations directly is rather inefficient Results in full matrix for computation of: 38

Substructuring Modified Cholesky Decomposition Formation of subcomponent matrices as part of Cholesky solution process Decomposition of substructure Formulation of subcomponent equations for changing domain Forward substitution process  Temperature response of changing domain  Recovery of static temperatures: 39

Sub-Structuring Results Method Description Num. of Updates Percentage of Iter. Fixed (%) Est. Overall Time Reduction (%) By Elements By Nodes Radial Buffer Pass: 1 25 77 7.81 3.67 Pass: 2 14 84 32.19 27.96 Pass: 3 10 86 38.87 36.11 Sensitivity Buffer τ = 0.3 56 52 -36.57 -36.34 τ = 0.5 57 53 -36.73 -38.65 τ = 0.7 59 54 -38.06 -40.2 Combined Buffer τ = 0.3, Pass: 1 3 83 43.3 38.77 τ = 0.3, Pass: 2 2 70 27.78 21.62 τ = 0.5, Pass: 1 8 47.78 45.58 τ = 0.5, Pass: 2 6 88 47.84 44.61 τ = 0.7, Pass: 1 32.68 30.84 τ = 0.7, Pass: 2 40.54 37.71 40

Estimated Overal Time Reduction (%) Findings / Results Volume Fraction Number of Updates Iterations Fixed Total Iterations Estimated Overal Time Reduction (%) 0.2 7 96 116 37.20 0.1 8 125 145 47.78 0.05 6 161 56.29 0.01 3 128 136 66.81 41