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Topology Optimization for Localizing Design Problems: An
Explorative ReviewChris Reichard
Supervisors: Fred van Keulen Matthijs Langelaar
Shinji Nishiwaki
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Outline
• Introduction Topology Optimization Heat Conduction Problem
• Research Project Research Problem and Objective Skeleton Modeling Sub-Structuring
• Findings / Results
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Sub-Structuring
Skeleton Modeling
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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
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Automotive Control Arm
Source: Example by Abaqus software
Heat Conduction
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Heat Conduction Optimization
• Optimization Problem: Heat conduction Uniform heat applied
• Objective: Minimize temperature
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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
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Heat Conduction Optimization
• 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
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Design Variables
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Heat Conduction Optimization
• 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
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Design Variables
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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
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Research Problem
• Main Issue: Need many small elements to define structure
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• Localization: Small, local details Structure in fraction of design area
Sparse design
Design Variables
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Research Problem
• Main Issue: Need many small elements to define structure Increase resolution, dramatic increase time
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• Localization: Small, local details Structure in fraction of design area
Sparse design
Design Variables
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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
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Finite Element Analysis (FEA) is the main issue!
Efficiency Issue
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Finite Element Analysis (FEA) is the main issue!• Time increases due to increase in elements
Efficiency Issue
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Characteristics of Local Problem
• Develops into bar like structure
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30% of Total Volume 10% of Total Volume 1% of Total Volume
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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
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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
Source: A Fully Automatic Rigging Algorithm for 3D Character Animation Masanori Sugimoto, University of Tokyo
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Skeleton Modeling
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Indirect Representation of Skeleton
• Structure Boundary known from surface level
• Need to extract skeleton from surface
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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
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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
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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
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• 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
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• 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
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• Connectivity of Skeleton Points How are the skeleton points connected? Need extra Information
Skeleton Modeling
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Challenges with Direct Representation
Source: printactivities.com
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• Connectivity of Skeleton Points How are the skeleton points connected? Need extra Information
Skeleton Modeling
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Challenges with Direct Representation
Source: printactivities.com
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• Connectivity of Skeleton Points How are the skeleton points connected? Need extra Information
• Differentiability: Needed to update design Structure is non-continuous
Skeleton Modeling
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Challenges with Direct Representation
Source: printactivities.com
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Summary Findings / Results
• Benefits: Simplified representation
which exploits sparse structure
Reduced number of elements
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Source: A Fully Automatic Rigging Algorithm for 3D Character Animation Masanori Sugimoto, University of Tokyo
Skeleton Modeling
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Summary Findings / Results
• 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
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Source: A Fully Automatic Rigging Algorithm for 3D Character Animation Masanori Sugimoto, University of Tokyo
Skeleton Modeling
Combine
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Characteristics of Local Problem
• Develops into bar like structure
• Elements with changing material
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30% of Total Volume 10% of Total Volume 1% of Total Volume
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Sub-Structuring
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Definition• Current methods: Structured groupings Using multiple processors
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Sub-Structuring
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• Current methods: Structured groupings Using multiple processors• Idea: Separate elements into groups Groups: Changing vs. static elements
Definition
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Sub-Structuring
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• Achieved By: Invert static matrix separate from changing
KT q1T K q
Expensive in terms of time
Definition
:K n n
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Sub-Structuring
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• Achieved By: Invert static matrix separate from changing Benefit: Reduction of number of variables
needed to be inverted every iteration
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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
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Sub-Structuring
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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
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Sub-Structuring
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• 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
Estimated Improvement
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
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Sub-Structuring
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• 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
fixed s c c c cfixed fixed fixed
n n nCost iter n n n n n
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
Estimated Improvement
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Sub-Structuring
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Buffer Zone• Issues: Groups of elements change each iteration
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Sub-Structuring
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StructureAreas of Design Change
Buffer Zone• Issues: Groups of elements change each iteration
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Sub-Structuring
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Radial Buffer
Buffer Zone• Issues: Groups of elements change each iteration• Solution: Buffer zone to reduce updates
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Sub-Structuring
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Radial Buffer Sensitivity Buffer
Buffer Zone• Issues: Groups of elements change each iteration• Solution: Buffer zone to reduce updates
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Sub-Structuring
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• Issues: Groups of elements change each iteration• Solution: Buffer zone to reduce updates
Radial Buffer Sensitivity Buffer Combined Buffer
Buffer Zone
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Sub-Structuring
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Example Implementation
High conductive structure
Low conductive region
Static Domain
Buffered changing domain
Elements with changing material
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Summary Findings / Results
• Benefits: Reduced size of matrix to
invert every iteration Time savings
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Sub-Structuring
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Summary Findings / Results
• Benefits: Reduced size of matrix to
invert every iteration Time savings Buffer method is low cost
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Sub-Structuring
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Summary Findings / Results
• Benefits: Reduced size of matrix to
invert every iteration Time savings Buffer method is low cost
• 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
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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!
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Topology Optimization for Localizing Design Problems: An
Explorative ReviewChris Reichard
Supervisors: Fred van Keulen Matthijs Langelaar
Shinji Nishiwaki
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Introduction
• Thesis Performed at: TU Delft, Netherlands Kyoto University, Japan
Experiences
• Guidance By: Fred van Keulen Matthijs Langelaar Shinji Nishiwaki
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What is Topology Optimization?
Objective: Minimize displacement for given
load
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What is Topology Optimization?
Objective: Minimize displacement for given
load
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Build approximate model: Through many small elements Material is varied in elements Displacement solved in each
element
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What is Topology Optimization?
Objective: Minimize displacement for given
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
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Test CaseHeat Conduction
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Structure
No Structure
Max. Temp.
Min. Temp.
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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
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Finite Element Analysis (FEA) is the main issue!
Efficiency Issue
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Summary Findings / Results
• Benefits: Reduced size of matrix to
invert every iteration Time savings Buffer method is low cost
• 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
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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
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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
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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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Skeleton ModelingRBF: How to Obtain Width?
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Skeleton ModelingRBF: How to Obtain Width?
w/ n being the number of full spaces in between level set grid
points
B LSLS
B A
dw n d
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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
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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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