0 challenge the future topology optimization for localizing design problems: an explorative review...

1Challenge the future

Topology Optimization for Localizing Design Problems: An

Explorative ReviewChris Reichard

Supervisors: Fred van Keulen Matthijs Langelaar

Shinji Nishiwaki


Outline

• Introduction Topology Optimization Heat Conduction Problem

• Research Project Research Problem and Objective Skeleton Modeling Sub-Structuring

• Findings / Results

1

Sub-Structuring

Skeleton Modeling


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

2

Automotive Control Arm

Source: Example by Abaqus software

Heat Conduction


Heat Conduction Optimization

• Optimization Problem: Heat conduction Uniform heat applied

• Objective: Minimize temperature

3



• 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



• How is Optimization Performed?1. Discretize problem into small elements Small elements = design variables

2. Provide initial structure

3. Solve temperatures in elements

4. Update design through approximations

4

Design Variables


Research Problem• Localization: Small, local details Structure in fraction of design area

Sparse design

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30% of Total Volume 10% of Total Volume 1% of Total Volume

Increasing sparseness


Research Problem

• Main Issue: Need many small elements to define structure

5

• Localization: Small, local details Structure in fraction of design area

Sparse design

Design Variables


Research Problem

• Main Issue: Need many small elements to define structure Increase resolution, dramatic increase time

5

• Localization: Small, local details Structure in fraction of design area

Sparse design

Design Variables


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

Research Objective

6


Finite Element Analysis (FEA) is the main issue!

Efficiency Issue

7


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

Efficiency Issue

7


Characteristics of Local Problem

• Develops into bar like structure

8



Skeleton Modeling

• Idea: Skeleton Model Computer graphics, medical imaging,

scientific visualization Model structure through skeleton

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Definition

Source: A Fully Automatic Rigging Algorithm for 3D Character Animation Masanori Sugimoto, University of Tokyo


Skeleton Modeling

• Idea: Skeleton Model Computer graphics, medical imaging,

scientific visualization Model structure through skeleton

• How? Global: Background mesh Skeleton Structure: Bar elements

• Obtaining Skeleton Indirect representation Direct representation

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Definition



Skeleton Modeling

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Indirect Representation of Skeleton

• Structure Boundary known from surface level

• Need to extract skeleton from surface


Skeleton Modeling

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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


Skeleton Modeling

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Direct Representation of Skeleton

• Skeleton curve already known and used to develop surface function

• Need to extract width of structure from surface

Surface FunctionSkeleton Curve


Skeleton Modeling

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Direct Representation of Skeleton

• Skeleton curve already known and used to develop surface function

• Need to extract width of structure from surface

Skeleton CurveSurface Function

Structure Boundary


• Connectivity of Skeleton Points How are the skeleton points connected?

Skeleton Modeling

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Challenges with Direct Representation

Ambiguous on how to connect points

Source: printactivities.com


• Connectivity of Skeleton Points How are the skeleton points connected?

Skeleton Modeling

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Ambiguous on how to connect points



• Connectivity of Skeleton Points How are the skeleton points connected? Need extra Information

Skeleton Modeling

12





Skeleton Modeling

12





• Differentiability: Needed to update design Structure is non-continuous

Skeleton Modeling

12




Summary Findings / Results

• Benefits: Simplified representation

which exploits sparse structure

Reduced number of elements

13


Skeleton Modeling



• Benefits: Simplified representation

which exploits sparse structure

Reduced number of elements

• Challenges: Complexity of the methodo Feasibility?o Efficiency Improvement? Combining models to obtain

temperature Updating the structure

13


Skeleton Modeling

Combine


Characteristics of Local Problem

• Develops into bar like structure

• Elements with changing material

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Sub-Structuring

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Definition• Current methods: Structured groupings Using multiple processors


Sub-Structuring

15

• Current methods: Structured groupings Using multiple processors• Idea: Separate elements into groups Groups: Changing vs. static elements

Definition


Sub-Structuring

16

• Achieved By: Invert static matrix separate from changing

KT q1T K q

Expensive in terms of time

Definition

:K n n


Sub-Structuring

16

• Achieved By: Invert static matrix separate from changing Benefit: Reduction of number of variables

needed to be inverted every iteration

1 11

CS SS SC CS SSCC C SCT K qK K K K K q 1 1

SS S SCS S S CT q K tK K

CC C CS

S SS S

C

C S

K K T q

K K T q

KT q1T K q

Expensive in terms of time

Terms calculated every few iterations!

Definition

n n 1n 1n

:K n n

:CC c cK n n

:SS s sK n n

:SC s cK n n

:CS c sK n n


Sub-Structuring

17

Estimated Improvement

• Cost Adaptive Sub-structuring Method: Full Implementation: 3 2 2 31 1

,3 3s c s s c c s cCost n n n n n n n n

c sn n n

3 2

2 32 2, , 2 2

3 3s s s

fixed s c c c cfixed fixed fixed

n n nCost iter n n n n n

iter iter iter


Sub-Structuring

17

• 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

3 2 2 31 1,

3 3s c s s c c s cCost n n n n n n n n

c sn n n


3 2

2 32 2, , 2 2

3 3s s s



iter iter iter


Sub-Structuring

17

• 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

3 2

2 32 2, , 2 2

3 3s s s



iter iter iter

3 2 2 31 1,

3 3s c s s c c s cCost n n n n n n n n

c sn n n

50 Iterations fixed10 Iterations fixed5 Iterations fixed2 Iterations fixed1 Iterations fixedFull Implementation



Sub-Structuring

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Buffer Zone• Issues: Groups of elements change each iteration


Sub-Structuring

18

StructureAreas of Design Change

Buffer Zone• Issues: Groups of elements change each iteration


Sub-Structuring

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Radial Buffer

Buffer Zone• Issues: Groups of elements change each iteration• Solution: Buffer zone to reduce updates


Sub-Structuring

18

Radial Buffer Sensitivity Buffer

Buffer Zone• Issues: Groups of elements change each iteration• Solution: Buffer zone to reduce updates


Sub-Structuring

18

• Issues: Groups of elements change each iteration• Solution: Buffer zone to reduce updates

Radial Buffer Sensitivity Buffer Combined Buffer

Buffer Zone


Sub-Structuring

19

Example Implementation

High conductive structure

Low conductive region

Static Domain

Buffered changing domain

Elements with changing material



• Benefits: Reduced size of matrix to

invert every iteration Time savings

20

Sub-Structuring




invert every iteration Time savings Buffer method is low cost

20

Sub-Structuring





• Challenges: Developing matrix structure

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Sub-Structuring

CC C CS

S SS S

C

C S

K K T q

K K T q


Recommendations

Skeleton Modeling• 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

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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


Topology Optimization for Localizing Design Problems: An

Explorative ReviewChris Reichard

Supervisors: Fred van Keulen Matthijs Langelaar

Shinji Nishiwaki

23


Introduction

• Thesis Performed at: TU Delft, Netherlands Kyoto University, Japan

Experiences

• Guidance By: Fred van Keulen Matthijs Langelaar Shinji Nishiwaki

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Objective: Minimize displacement for given

load

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load

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Build approximate model: Through many small elements Material is varied in elements Displacement solved in each

element




load

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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


Test CaseHeat Conduction

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Structure

No Structure

Max. Temp.

Min. Temp.


Research Plan

• Investigate research problem Examine how structure develops Determine characteristics of localization

• Research known techniques Optimization Modelling

• Develop ideas to exploit problem Investigate ideas Assess feasibility

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

Efficiency Issue

28


Finite Element Analysis (FEA) is the main issue!

Efficiency Issue

7





• Challenges: Developing matrix structure

20

Sub-Structuring

CC C CS

S SS S

C

C S

K K T q

K K T q


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.

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Skeleton Modeling• Skeleton defined as ridge of LSF• Principal curvature to obtain ridge• At each point: and• Need critical point of • Critical pt. = Ridge pt.

max min

max

Principal Curvatures

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Source: Eric Gaba. Wikipedia. Principal Curvatures


Skeleton Modeling

• Principal curvature developed through First and Second Fundamental Form of tangent plane of surface

u v

u v

S Sn

S S

Principal Curvatures

, ,

, ,u u u v

u v v v

S S S SE FI

S S S SF G

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, ,

, ,uu uv

uv vv

S n S nL MII

S n S nM N

2

2

A EG F

B FM GK EN

C LN M

2

1

2

2

4

2

4

2

B B AC

A

B B AC

A

S


Principal Curvature

• First Fundamental Form, I

• Second Fundamental Form, II

• Weingarten Operator (Shape Operator)

• Principal Curvature (Roots of characteristic equation)

How to Obtain it?

2 22I Edu Fdudv Gdv u u u v

u v v v

X X X XE FI

X X X XF G

2 22II Ldu Mdudv Ndv , ,

, ,uu uv

uv vv

S n S nL MII

S n S nM N

1I IIW F F

2

2

A EG F

B FM GK EN

C LN M

2

1

2

2

4

2

4

2

B B AC

A

B B AC

A

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Skeleton Modeling Radial Basis Functions

Radial Basis Function:

with

2

2

,i

iwb

i is s e

x x

x

, , 1,...,i ii

s s i n x x

min maxis s s

: i Ri x x

max Point on Skeleton Curveis s

max Control Width of Structureis s

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Skeleton ModelingRBF: How to Obtain Width?

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35



w/ n being the number of full spaces in between level set grid

points

B LSLS

B A

dw n d

36


Skeleton ModelingRBF: Effects of Design Variables

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SubstructuringDirect Solve

• Solving equations directly is rather inefficient• Results in full matrix for computation of:

cc cs c c

sc ss s s

k k T F

k k T F

1

1

s ss s sc c

c cc c cs s

T k F k T

T k F k T

11 1cc cs ss sc c cs ss sT k k k k F k k F

c

1sT k F k T

ss s sc c

1K ss scK

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SubstructuringModified 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:

0

0

T Tss sc ss ss cs

Tcs cc cs cc cc

K K L L LK

K K L L L

1

1ij ik jk

m

c cc cs ss sc c cc cs csk

K K K K K K K L L with j i

1

1j jk

m

c c cs ss s c c sc sk

F F K K F F F L y

0ss s s

sc c c c

L y F

L K y F

1c c c

Tc c c

y K F

T K y

1Ts ss ss c sc cT L L F K T

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Sub-Structuring

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Method Description Num. of Updates

Percentage of Iter. Fixed (%)

Est. Overall Time Reduction (%)

By Elements By Nodes

Radial BufferPass: 1 25 77 7.81 3.67Pass: 2 14 84 32.19 27.96Pass: 3 10 86 38.87 36.11

Sensitivity Buffer

τ = 0.3 56 52 -36.57 -36.34τ = 0.5 57 53 -36.73 -38.65τ = 0.7 59 54 -38.06 -40.2

Combined Buffer

τ = 0.3, Pass: 1 3 83 43.3 38.77τ = 0.3, Pass: 2 2 70 27.78 21.62τ = 0.5, Pass: 1 8 86 47.78 45.58τ = 0.5, Pass: 2 6 88 47.84 44.61τ = 0.7, Pass: 1 14 84 32.68 30.84τ = 0.7, Pass: 2 10 86 40.54 37.71

Results


Findings / Results

Volume Fraction

Number of Updates

Iterations Fixed

Total Iterations

Estimated Overal Time Reduction (%)

0.2 7 96 116 37.200.1 8 125 145 47.78

0.05 6 145 161 56.290.01 3 128 136 66.81

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