Presentation IDETC
Transcript of Presentation IDETC
![Page 1: Presentation IDETC](https://reader036.fdocuments.net/reader036/viewer/2022062401/5a66785a7f8b9a316b8b4637/html5/thumbnails/1.jpg)
Presenter: Xiao Wang
Supervisor: Prof. Shikui Chen
Computational Modeling Analysis and Design Optimization Research Lab(CMADO)
Department of Mechanical Engineering
Stony Brook University
![Page 2: Presentation IDETC](https://reader036.fdocuments.net/reader036/viewer/2022062401/5a66785a7f8b9a316b8b4637/html5/thumbnails/2.jpg)
1. Topology Optimization
2. Level-set Representation and Problem Formulation
3. Numerical Examples
4. Summary
Presentation outline
![Page 3: Presentation IDETC](https://reader036.fdocuments.net/reader036/viewer/2022062401/5a66785a7f8b9a316b8b4637/html5/thumbnails/3.jpg)
1. Topology Optimization
2. Level-set Representation and Problem Formulation
3. Numerical Examples
4. Summary
![Page 4: Presentation IDETC](https://reader036.fdocuments.net/reader036/viewer/2022062401/5a66785a7f8b9a316b8b4637/html5/thumbnails/4.jpg)
• 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
![Page 5: Presentation IDETC](https://reader036.fdocuments.net/reader036/viewer/2022062401/5a66785a7f8b9a316b8b4637/html5/thumbnails/5.jpg)
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
![Page 6: Presentation IDETC](https://reader036.fdocuments.net/reader036/viewer/2022062401/5a66785a7f8b9a316b8b4637/html5/thumbnails/6.jpg)
1. Topology Optimization
2. Level-set Representation and Problem Formulation
3. Numerical Examples
4. Summary
![Page 7: Presentation IDETC](https://reader036.fdocuments.net/reader036/viewer/2022062401/5a66785a7f8b9a316b8b4637/html5/thumbnails/7.jpg)
( ) 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
![Page 8: Presentation IDETC](https://reader036.fdocuments.net/reader036/viewer/2022062401/5a66785a7f8b9a316b8b4637/html5/thumbnails/8.jpg)
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
![Page 9: Presentation IDETC](https://reader036.fdocuments.net/reader036/viewer/2022062401/5a66785a7f8b9a316b8b4637/html5/thumbnails/9.jpg)
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
![Page 10: Presentation IDETC](https://reader036.fdocuments.net/reader036/viewer/2022062401/5a66785a7f8b9a316b8b4637/html5/thumbnails/10.jpg)
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
![Page 11: Presentation IDETC](https://reader036.fdocuments.net/reader036/viewer/2022062401/5a66785a7f8b9a316b8b4637/html5/thumbnails/11.jpg)
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
![Page 12: Presentation IDETC](https://reader036.fdocuments.net/reader036/viewer/2022062401/5a66785a7f8b9a316b8b4637/html5/thumbnails/12.jpg)
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
![Page 13: Presentation IDETC](https://reader036.fdocuments.net/reader036/viewer/2022062401/5a66785a7f8b9a316b8b4637/html5/thumbnails/13.jpg)
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.
![Page 14: Presentation IDETC](https://reader036.fdocuments.net/reader036/viewer/2022062401/5a66785a7f8b9a316b8b4637/html5/thumbnails/14.jpg)