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

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

Automotive Control Arm

Source: Example by Abaqus software

Heat Conduction

Heat Conduction Optimization

• Optimization Problem: Heat conduction Uniform heat applied

• Objective: Minimize temperature

• Optimization Problem: Heat conduction Uniform heat applied

• Objective: Minimize temperature

• Achieved by: Placement of two materials kH: moves heat efficiently kL: moves heat inefficiently

• 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

Design Variables

• 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

Design Variables

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

Sparse design

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

• 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

• 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

Finite Element Analysis (FEA) is the main issue!

Efficiency Issue

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

Efficiency Issue

Characteristics of Local Problem

• Develops into bar like structure

Skeleton Modeling

• Idea: Skeleton Model Computer graphics, medical imaging,

scientific visualization Model structure through skeleton

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

Definition

Skeleton Modeling

Indirect Representation of Skeleton

• Structure Boundary known from surface level

• Need to extract skeleton from surface

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

Skeleton Modeling

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

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

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

Ambiguous on how to connect points

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

Skeleton Modeling

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

Skeleton Modeling

Summary Findings / Results

• Benefits: Simplified representation

which exploits sparse structure

Reduced number of elements

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

Skeleton Modeling

Combine

Characteristics of Local Problem

• Develops into bar like structure

• Elements with changing material

Sub-Structuring

Definition• Current methods: Structured groupings Using multiple processors

Sub-Structuring

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

Definition

Sub-Structuring

• Achieved By: Invert static matrix separate from changing

KT q1T K q

Expensive in terms of time

Definition

:K n n

Sub-Structuring

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

needed to be inverted every iteration

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

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

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

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

• 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

2 32 2, , 2 2

3 3s s s

iter iter iter

Sub-Structuring

• 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

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

Buffer Zone• Issues: Groups of elements change each iteration

Sub-Structuring

StructureAreas of Design Change

Buffer Zone• Issues: Groups of elements change each iteration

Sub-Structuring

Radial Buffer

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

Sub-Structuring

Radial Buffer Sensitivity Buffer

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

Sub-Structuring

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

Radial Buffer Sensitivity Buffer Combined Buffer

Buffer Zone

Sub-Structuring

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

Sub-Structuring

invert every iteration Time savings Buffer method is low cost

Sub-Structuring

• Challenges: Developing matrix structure

Sub-Structuring

CC C CS

S SS S

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

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!

Topology Optimization for Localizing Design Problems: An

Explorative ReviewChris Reichard

Supervisors: Fred van Keulen Matthijs Langelaar

Shinji Nishiwaki

Introduction

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

Experiences

• Guidance By: Fred van Keulen Matthijs Langelaar Shinji Nishiwaki

Objective: Minimize displacement for given

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

element

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

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

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

Efficiency Issue

Finite Element Analysis (FEA) is the main issue!

Efficiency Issue

• Challenges: Developing matrix structure

Sub-Structuring

CC C CS

S SS S

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.

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

Principal Curvatures

Source: Eric Gaba. Wikipedia. Principal Curvatures

Skeleton Modeling

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

Principal Curvatures

, ,u u u v

u v v v

S S S SE FI

S S S SF G

, ,uu uv

S n S nL MII

S n S nM N

A EG F

B FM GK EN

C LN M

B B AC

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

S n S nL MII

S n S nM N

1I IIW F F

A EG F

B FM GK EN

C LN M

B B AC

Skeleton Modeling Radial Basis Functions

Radial Basis Function:

i is s e

, , 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

Skeleton ModelingRBF: How to Obtain Width?

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

points

B LSLS

dw n d

Skeleton ModelingRBF: Effects of Design Variables

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

s ss s sc c

c cc c cs s

T k F k T

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

1sT k F k T

ss s sc c

1K ss scK

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:

T Tss sc ss ss cs

Tcs cc cs cc cc

K K L L LK

K K L L L

1ij ik jk

c cc cs ss sc c cc cs csk

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

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 K y F

1c c c

Tc c c

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

Sub-Structuring

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

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

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