Topology Optimization using Phase Field Method and ... · Topology Optimization using Phase Field...

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


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


Motivation Explore general and curved domains rather than the traditional Cartesian domains

(box-type) that have been extensively used for topology optimization.


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


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


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


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


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:


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:


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,


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


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


Numerical example 1Cantilever beam problem

Q4 Elements(80x40)

Polygonal Elements

(3200)

Initial guess Topology after 50 iterations


Numerical example 2Non-Cartesian domains


Domain size: 3000 polygonal elements

FE iterations: 200


5 410 10 10 10κ η− −= × = =min, , k

Cantilever beam problem on circular segment design domain


Numerical example 2


Cantilever beam problem on circular segment design domain

Non-Cartesian domains





FE iterations: 100


5 410 10 10 10κ η− −= × = =min, , k

Bridge problem on semi-circular design domain


Numerical example 3



Bridge problem on semi-circular design domain





FE iterations: 100


5 410 10 10 10κ η− −= × = =min, , k

Doubly curved cantilever beam problem




07/25/2011 21

Numerical example 4




Topology Optimization using Phase Field Method and Polygonal Finite Elements



Objective: Compliant mechanism


FE iterations: 300


5 410 10 10 10κ η− −= × = =min, , k

Inverter problem on circular segment design domain





Numerical example 5





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


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

Thank You !


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