topology optimization using phase field method and ... · topology optimization using phase field...
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Topology Optimization using Phase Field Method and
Polygonal Finite Elements
Arun Lal Gain, Glaucio H. Paulino
Department of Civil and Environmental EngineeringUniversity of Illinois at Urbana-Champaign
11th U.S. National Congress on Computational Mechanics
07/25/2011 2Topology Optimization using Phase Field Method and Polygonal Finite Elements
Motivation Implicit function method such as level-set function, although attractive, require periodic
reinitializations to maintain signed distance characteristics for numerical convergence.
Reinitializations often performed heuristically.
Phase field method doesn’t require any reinitialization.
Allaire G. and Jouve F. (2004) Structural optimization using sensitivity analysis and a level-set method. Journal of Computational Physics, 194:363-393
07/25/2011 3Topology Optimization using Phase Field Method and Polygonal Finite Elements
Motivation Traditionally uniform grids are used for topology optimization which suffer from
numerical anomalies such as checkerboard patterns and one-node connections.
Constrained geometry of structured grids can bias the orientation of the members,
leading to mesh dependent, sub-optimal designs.
Talischi C., Paulino G. H., Pereira A., and Menezes I. F. M. (2010) Polygonal finite elements for topology optimization: A unifying paradigm.International Journal for Numerical Methods in Engineering, 82: 671-698
T6 elements Polygonal elements
07/25/2011 4Topology Optimization using Phase Field Method and Polygonal Finite Elements
Motivation Explore general and curved domains rather than the traditional Cartesian domains
(box-type) that have been extensively used for topology optimization.
07/25/2011 5Topology Optimization using Phase Field Method and Polygonal Finite Elements
Contents Motivation
Introduction
- Polygonal Finite Elements
- Phase Field Method
Centroidal Voronoi Tessellation (CVT) based finite volume method
Implementation Flow Chart
Numerical Examples
Summary & Conclusions
Future Extensions
07/25/2011 6Topology Optimization using Phase Field Method and Polygonal Finite Elements
Isotropic linear elastic system:
Objective:
( )
( )
in on
on
− ⋅ = Ω
= Γ
⋅ = Γ
0 D
N
ε
ε
A f u
A n g
∇
( ) ( ) ( )( )inf L J PλΩ
Ω = Ω + Ω
( )
( )
Ω Γ
Ω
Ω = ⋅ + ⋅
Ω =
∫ ∫
∫
dx ds,
dx,
N
J
P
f u g u Compliance
Volume
Problem description
07/25/2011 7Topology Optimization using Phase Field Method and Polygonal Finite Elements
Polygonal finite elements
( ) ( )( ) ( ) ( )
( )2α
αα
Ρ
= = ∈ℜ∑
, ,i ii i
j i
sN
hx x
x x xx x
( ) ( ) ( )0 1 1δΡ
≤ ≤ = =∑, ,i i j ij iN N Nx x x
( )Ρ
=∑ i iNx x x
Simple approach to discretize complex geometries using polygonal/polyhedral meshes
Finite element space of polygonal elements is constructed using natural neighbor scheme based Laplace interpolants
1 2=where, , , . . . , nP p p p
Sukumar N. and Tabarraei A. (2004) Conforming polygonal finite elements. International Journal of Numerical Methods in Engineering, 61(12):2045-2066
07/25/2011 8Topology Optimization using Phase Field Method and Polygonal Finite Elements
Phase field method1
0
10 1
0
φφ ξ
φ
= ∈Ω< < ∈= ∈Ω
Diffuse interface,
,,
x x
x
( ) ( )1
0
φ φ ξ∈Ω
= ∈ ∈Ω
*
min
,,
,k
k
A xA A x
A x
( ) ( )( ) ( )12
1
11 30 12
φφ κ φ φ φ φ η φ φφ
∂= ∇ + − − − −
∂
''
t
t
Jt J
( ) ( ) ( )( ) ( ) ( )1
1
0 0 1 0 1 0φ
ηφ
= = = ='
, , ' ''
t
t
Jf f f f
Jwhere,
Takezawa A., Nishiwaki S., and Kitamura M. (2010) Shape and topology optimization based on the phase field method and sensitivity analysis.Journal of Computational Physics, 229: 2697-2718
( )2 ftφ κ φ φ∂ ′= ∇ −∂
Allen-Cahn equation: 0φ∂= ∂
∂with on D
n( )φf
φ
( )( )
1
1
φη
φ''
t
t
JJ
07/25/2011 9Topology Optimization using Phase Field Method and Polygonal Finite Elements
Centroidal Voronoi Tessellation (CVT) based finite volume method
( )2 ftφ κ φ φ∂ ′= ∇ −∂
Vasconcellos J. F. V. and Maliska C. R. (2004) A finite-volume method based on voronoi discretization for fluid flow problems. Numerical HeatTransfer, Part B, 45: 319-342
( )φ κ φ φΩ Γ Ω
∂Ω = ⋅ Γ − Ω
∂∫ ∫ ∫, , ,
't t t
d dt d dt f d dtt
∇ n
Allen-Cahn equation:
( )2φ κ φ φΩ Ω Ω
∂Ω = ∇ Ω − Ω
∂∫ ∫ ∫, , ,
't t t
d dt d dt f d dttIntegral form:
07/25/2011 10Topology Optimization using Phase Field Method and Polygonal Finite Elements
CVT based finite volume method
Simplifying each term:
•
•
•
( ) ( )1 1φ φ φ φ φ+ +
Ω Ω
∂Ω = − Ω = − Ω
∂∫ ∫,
n n n np p p
t
d dt dt
3φκ φ κ φ κ
Ρ ΡΓ
∂ ⋅ Γ = ⋅ = ∆ = ∂ ∑ ∑∫ ∫
, , i
nn
iit t p p
d dt s dt s t Pn
∇ ∇n n
φ φφ − ∂= ∂ ,
i
i
n nnp p
ip pn h
( ) ( )( ) ( ) ( )( ) ( ) ( )
1
1
1 0
1 0
φ φ φ φφ φ
φ φ φ φ
+
+Ω
− ≤Ω =Ω ∆ =Ω ∆ − >
∫for
for, , , ,
, , , , ,
' 'n n n ni j i j i j i jn
p p n n n nt i j i j i j i j
r rf d dt t f t
r r
( )φ κ φ φΩ Γ Ω
∂Ω = ⋅ Γ − Ω
∂∫ ∫ ∫, , ,
't t t
d dt d dt f d dtt
∇ nIntegral form:
07/25/2011 11Topology Optimization using Phase Field Method and Polygonal Finite Elements
CVT based finite volume method
( ) ( )( ) ( )
( )( )( )( ) ( )
3
1
3
01 1
10
1
φφ
φ φφ
φ φφ
φ φ
+
Ω +≤
Ω − − ∆= Ω + ∆ +
>Ω + ∆
for
for
,,
, ,
,, ,
,, ,
,
,
np i j n
i jn np i j i j
ni j n n
p i j i j ni jn n
p i j i j
Pr
r t
r t Pr
r t
Semi-implicit updating scheme:
( ) ( )( ) ( )1
1
1 30 12
φφ φ η φ φ
φ= − − −, , , ,
''
tn n n ni j i j i j i j
t
Jr
J
where,
07/25/2011 12Topology Optimization using Phase Field Method and Polygonal Finite Elements
Implementation flow chartStart
Generate mesh (Polymesher)
FE analysis: U = K\F
End
Update , Allen-Cahn equation,
FV Semi-implicit scheme
φ
Initialize phase field function
Sensitivity analysis
07/25/2011 13Topology Optimization using Phase Field Method and Polygonal Finite Elements
Numerical example 1
Objective: Compliance minimization
Domain size: 2x1 with 3200 polygonal elements
FE iterations: 50
For each FE iteration, 20 Allen-Cahn equation updates using CVT based FV method
5 410 10 10 10κ η− −= × = =min, , k
Cantilever beam problem
07/25/2011 14Topology Optimization using Phase Field Method and Polygonal Finite Elements
Numerical example 1Cantilever beam problem
Q4 Elements(80x40)
Polygonal Elements
(3200)
Initial guess Topology after 50 iterations
07/25/2011 15Topology Optimization using Phase Field Method and Polygonal Finite Elements
Numerical example 2Non-Cartesian domains
Objective: Compliance minimization
Domain size: 3000 polygonal elements
FE iterations: 200
For each FE iteration, 20 Allen-Cahn equation updates using CVT based FV method
5 410 10 10 10κ η− −= × = =min, , k
Cantilever beam problem on circular segment design domain
07/25/2011 16Topology Optimization using Phase Field Method and Polygonal Finite Elements
Numerical example 2
Initial guess Topology after 200 iterations
Cantilever beam problem on circular segment design domain
Non-Cartesian domains
07/25/2011 17Topology Optimization using Phase Field Method and Polygonal Finite Elements
Numerical example 3Non-Cartesian domains
Objective: Compliance minimization
Domain size: 3840 polygonal elements
FE iterations: 100
For each FE iteration, 20 Allen-Cahn equation updates using CVT based FV method
5 410 10 10 10κ η− −= × = =min, , k
Bridge problem on semi-circular design domain
07/25/2011 18Topology Optimization using Phase Field Method and Polygonal Finite Elements
Numerical example 3
Initial guess Topology after 100 iterations
Non-Cartesian domains
Bridge problem on semi-circular design domain
07/25/2011 19Topology Optimization using Phase Field Method and Polygonal Finite Elements
Numerical example 4Non-Cartesian domains
Objective: Compliance minimization
Domain size: 3200 polygonal elements
FE iterations: 100
For each FE iteration, 20 Allen-Cahn equation updates using CVT based FV method
5 410 10 10 10κ η− −= × = =min, , k
Doubly curved cantilever beam problem
07/25/2011 20Topology Optimization using Phase Field Method and Polygonal Finite Elements
Numerical example 4Non-Cartesian domains
Doubly curved cantilever beam problem
07/25/2011 21
Numerical example 4
Initial guess Topology after 100 iterations
Non-Cartesian domains
Doubly curved cantilever beam problem
Topology Optimization using Phase Field Method and Polygonal Finite Elements
07/25/2011 22Topology Optimization using Phase Field Method and Polygonal Finite Elements
Numerical example 5Non-Cartesian domains
Objective: Compliant mechanism
Domain size: 6000 polygonal elements
FE iterations: 300
For each FE iteration, 20 Allen-Cahn equation updates using CVT based FV method
5 410 10 10 10κ η− −= × = =min, , k
Inverter problem on circular segment design domain
07/25/2011 23Topology Optimization using Phase Field Method and Polygonal Finite Elements
Numerical example 5Non-Cartesian domains
Inverter problem on circular segment design domain
07/25/2011 24Topology Optimization using Phase Field Method and Polygonal Finite Elements
Numerical example 5
Initial guess Topology after 300 iterations
Non-Cartesian domains
Inverter problem on circular segment design domain
07/25/2011 25Topology Optimization using Phase Field Method and Polygonal Finite Elements
Summary & Conclusions Implicit function-based phase field method using polygonal finite elements offers a
general framework for topology optimization on arbitrary domains.
Meshes based in simplex geometry such as quads/bricks or triangles/tetrahedra
introduce intrinsic bias in standard FEM, but polygonal/polyhedral meshes do not.
Polygonal/polyhedral meshes based on Voronoi tessellation not only remove
numerical anomalies such as one-node connections and checkerboard pattern but
also provide greater flexibility in discretizing non-Cartesian design domains.
Future extensions
07/25/2011 26Topology Optimization using Phase Field Method and Polygonal Finite Elements
Courtesy: Stromberg et. al.
Extension to 3D using polyhedral
meshes
Phase field method using polygonal
meshes paves the way for medical engineering applications including craniofacial segmental
bone replacement
Phase field method with sharpness control of
diffuse interfaces offer an attractive framework for phononic metamaterial
cloaking designCourtesy: Pendry et. al., Science 312
Sutradhar A, Paulino GH, Miller MJ, Nguyen TH (2010) Topology optimization for designing patient-specific large craniofacial segmental bonereplacements. Proceedings of the National Academy of Sciences 107(30): 13222-13227