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

1. 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

2. 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

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

4. 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

5. 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

6. 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

7. 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

8. 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

9. 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

10. 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

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

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

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

14. 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

15. 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

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

17. 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

18. 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

19. 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

20. 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

21. 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

22. 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

23. 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

24. 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

25. 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

26. 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

27. 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

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

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

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

31. 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

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

33. 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

34. 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

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

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

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

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

39. 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

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

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

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

43. 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

44. 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

45. 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

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

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

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

49. 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

50. 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

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

52. 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

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

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

55. 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

56. 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

57. 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

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

59. 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

60. Skeleton Modeling Radial Basis Functions Radial Basis Function: with33

61. Skeleton Modeling
RBF: How to Obtain Width? 34

62. Skeleton Modeling
RBF: How to Obtain Width? 35

63. 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

64. Skeleton Modeling
RBF: Effects of Design Variables 37

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

66. 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

67. 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

68. 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