Presenter: Xiao Wang

Supervisor: Prof. Shikui Chen

Computational Modeling Analysis and Design Optimization Research Lab(CMADO)

Department of Mechanical Engineering

Stony Brook University

1. Topology Optimization

2. Level-set Representation and Problem Formulation

3. Numerical Examples

4. Summary

Presentation outline

1. Topology Optimization

2. Level-set Representation and Problem Formulation

3. Numerical Examples

4. Summary

• A technique determining optimum

material distribution inside a given

design domain.

• Allows greater design freedom than

size and shape optimization

• Broad range of applications includes

structural, heat transfer, acoustic, fluid

flow and other multiphysics

disciplines.

size

Topology Optimization

Van Dijk, Nico P., et al. "Level-set methods for structural topology optimization: a review." Structural and Multidisciplinary Optimization 48.3 (2013): 437-472.

Hagishita, T., and M. Ohsaki. "Topology optimization of trusses by growing ground structure method." Structural and Multidisciplinary Optimization 37.4 (2009): 377-393.

Topology optimization: State of The Art

1. Topology Optimization

2. Level-set Representation and Problem Formulation

3. Numerical Examples

4. Summary

( ) 0x ( ) 0x

( ) 0x

( ) 0x

( ) 0x

( ) 0x

( ) 0, (material)

( ) 0, (boundary)

( ) 0, (D\ )(void)

x x

x x

x x

( ) ( ) : ( ( ), )S t x t x t t k

( , )( , ) 0n

x tx t V

t

Hamilton-Jacobi Equation

◊ provide crisp and smooth edges

◊ the movement of structural boundaries,

formation, disappearance, and merge of void

regions, which defines true topological design.

Level Set Representation

Osher and Sethian, 1988

Minimize

* 2

, , , 1

1( )

2

dH

ijkl ijkl ijkl

i j k l

J w C C

( , , ) ( , ), (Y)a x v l v v U

vV Y f

B

T

11 111111 1122

22 222211 2222

12 1212 12

0

0

0 0 2

H H

H H

H

C C

C C

C

1

2

T H

ijklU V C 1

2ijkl ij ijkl klU C d

Problem formulation

Elastic material microstructure unit cell

1 0 1

0 , 1 , 1

0 0 0

ij

1111 1111 1212 1212 1122 1122 1111 2222

2222 2222 2323 2323 2233 2233 2222 3333

3333 3333 1313 1313 1133 1133 1111 3333

2 ,C 2 ,

2 ,C 2 ,C

2 ,C 2 ,

H H H

H H H

H H H

C U U C U U U

C U U U U U

C U U C U U U

Shape sensitivity analysis

The derivative of the objective function with respect to the pseudo-time t :

*

, , , 1

HdijklH

ijkl ijkl ijkl

i j k l

dCdJw C C

dt dt

Week imposition of Dirichlet boundary conditions:

0D

T

ij ijkl klg u C v u u vds

H T

ijkl ij ijkl klC u C u d

' ' '2D

T T

ij ijkl kl ij ijkl kl

T T

ij ijkl kl n ij ijkl kl n

dLu C u d u C v u vds

dt

u C u V ds u C v V ds

Adjoint equation

2 ,

0 ,

D

u inv

onsteepest-decent method

T

n ij ijkl klV u C u

L J g

Lagrange multiplier

Initialdesign

1 0.2 0.2 -0.04 50% -0.2 B

2 0.2 0.2 -0.04 50% -0.2 A

3 0.2 0.2 -0.1 50% -0.5 A

4 0.2 0.2 -0.1 40% -0.5 A

*

1111C *

2222C *

1122C vf

Initial design A Initial design B

Numerical examples

Example 1 Example 2

Example 3 Example 4

Unit cell 3×3 Structure TO process 2.5D Unit cell Elastic tensor

0.195 0.039

0.039 0.195

0.01

Example 1

0.151 0.047 0

0.047 0.144 0

0 0 0.01

0.149 0.072 0

0.072 0.15 0

0 0 0.012

0.104 0.499 0

0.499 0.829 0

0 0 0.005

Example 2

Example 3

Example 4

Summary

Propose a level-set based topology optimization method for the design of

mechanical metamaterials.

Calculate the effective elastic tensor using strain energy functional method.

Imposing the weak form of Dirichlet boundary condition.

Demonstrate the performance of level-set method four examples.