Presentation IDETC
- 1. Presenter: Xiao Wang Supervisor: Prof. Shikui Chen Computational Modeling Analysis and Design Optimization Research Lab(CMADO) Department of Mechanical Engineering Stony Brook University
- 2. 1. Topology Optimization 2. Level-set Representation and Problem Formulation 3. Numerical Examples 4. Summary Presentation outline
- 3. 1. Topology Optimization 2. Level-set Representation and Problem Formulation 3. Numerical Examples 4. Summary
- 4. • 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
- 5. 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
- 6. 1. Topology Optimization 2. Level-set Representation and Problem Formulation 3. Numerical Examples 4. Summary
- 7. ( ) 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 t x 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
- 8. Minimize * 2 , , , 1 1 ( ) 2 d H 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 11 1111 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 2 ijkl 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
- 9. Shape sensitivity analysis The derivative of the objective function with respect to the pseudo-time t : * , , , 1 Hd ijklH ijkl ijkl ijkl i j k l dCdJ w C C dt dt Week imposition of Dirichlet boundary conditions: 0 D T ij ijkl klg u C v u u vds H T ijkl ij ijkl klC u C u d ' ' ' 2 D T T ij ijkl kl ij ijkl kl T T ij ijkl kl n ij ijkl kl n dL u 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 in v on steepest-decent method T n ij ijkl klV u C u L J g Lagrange multiplier
- 10. Initial design 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
- 11. 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
- 12. 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
- 13. 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.
Editor's Notes
- #5: Topology optimization is the process of determining the optimal layout of material and connectivity inside a design domain. Compared with size and shape optimization which deal with variables such as thicknesses or cross-sectional areas, topology optimization allows greater design freedom. It’s able to achieve the optimal design without depending on designers’ a priori knowledge. TO has a broad range of applications include structural, heat transfer, acoustic, fluid flow and other multiphysics disciplines.
- #6: The idea of ground structure method is to discretize design domain into a finite spatial distribution of nodes, then connect all nodes using truss elements, retain only the most vital members to optimize with respect to prescribed loadings or performance criteria. The homogenization method is proposed for computing the effective material properties for materials with microstructures, so it closely connected with the studies of materials bounds for composites. The method allows us to predict the topology of the structural but it only results in a nonsmooth estimate of the exact form of its boundary. Traditional boundary variations optimization is also needed. SIMP method operate on a fixed domain of finite elements with basic goal of min\max an objective function by identifying whether each element should consist of solid material or void. The continuous design variables are explicitly interpreted as the material density of each element, penalty methods are then utilized to force solutions to solid/void topologies.
- #8: Level set methods were originally introduced by Osher and Sethian in 1988 as a numerical scheme for tracking fluid fronts propagating. It specifies an implicit iso-surface embedded in a higher dimensional level set function, which divide the computational domain into two sub-areas according to the sign of the function. The optimization procedure can be described by letting the level set function dynamically change with time, thus the dynamic function is expressed as: --------------. Take the derivative with respect to pseudo time t, we obtain the so-called Hamiton-Jacobi equation. Vn is the design velocity field, which is computed based on shape sensitivity analysis.
- #9: Suppose an elastic material consist of microstructures, and the microstructure is periodically assembled by unit cell. The stress and strain relationship can be expressed by generalized Hooke’s law, which is the average stress equals to the effective elastic tensor multiply the average strain. By applying three unit strain independently, we manage to find that the each entry of the elastic tensor is actually the strain energy under specific loading cases. Considering the implementation issue, we transfer the unit strain field into a simple Dirichlet boundary conditions. The strain energy is computed using finite element method, by take the integration of the strain energy density over the whole domain. The metamaterial design problem is then formulated to minimize the difference between the effective elastic tensor and the targets, subject to linear elastic equilibrium equation and volume constraint.
- #10: The most important part is how to find the design velocity field. The objective function and governing equation was combined using Lagrange multiplier method. By using the weak form of the Dirichlet boundary condition, the derivative of the Lagrange equation with respect to the pseudo-time t is given as this equation. The bracket equation is called adjoint equation, by making this equation equals to 0, we may solve the adjoint variables v equals to -2u, replace the adjoint v with -2u, with the steepest-decent method, we managed to find that the design velocity is strain energy density under different loading cases.
- #11: These are some of results of auxetic microstructures, first example start with the initial design B, with the desired Poisson’s ratio equals to -0.2, the volume is constraint at 50%, the second example has the exactly the same problem setting but using a different initial deign A, here is the results. Compared with the second example, next one targeted at poisson’s ratio -0.5, the last example, the volume fraction is changed to 40%.
- #12: From left to right, it shows the unit cell, and its 3x3 structure, optimization process, 2.5D unit cell and calculated elastic tensor. Here we can see the volume ratio and the elastic tensor decreased gradually to reach the target.