1 Introduction
2 Topology optimization formulations based on the proposed method
2.1 Homogenization with perturbation analysis
2.2 Imposition of the periodic boundary conditions using the penalty methods
2.3 Sensitivity analysis
2.4 Comparison
The proposed method | ||
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Periodic boundary conditions | Penalty methods | The direct manner (Unrestricted) |
Homogenization | \(\frac{{\varvec{I}}_n^i}{V}\sum _{e}\int _{v} {\varvec{C}}^{e} {\varvec{B}}{\varvec{u}}^e(\varvec{\varepsilon }_{ \mathrm M}^{j}) {\rm{dv}}\) | \(\frac{1}{V}\sum _{e}({\varvec{u}}_e^{A(ij)})^T {\varvec{k}}_e{\varvec{u}}_e^{A(kl)}\) |
Sensitivity | \(\begin{array}{c}\frac{{\varvec{I}}_n^i}{V} p\rho _k^{p-1}(f_{0}-f_{min}) {\varvec{C}}_0 \tilde{{\varvec{B}}}{\varvec{u}}^k(\varvec{\varepsilon }_{\textrm{M}}^{j}) \\ - \frac{{\varvec{I}}_n^i}{V} {\varvec{l}}^T {\varvec{M}}^k(\varvec{\varepsilon }_{\textrm{M}}^{j}) \end{array}\) | \(\frac{1}{V}p\rho _e^{p-1}(f_{0}-f_{min})({\varvec{u}}_e^{A(ij)})^T {\varvec{k}}_0{\varvec{u}}_e^{A(kl)}\) |
3 Extension to the design of piezoelectric materials
3.1 Homogenization
3.2 Periodic boundary conditions
3.3 Sensitivity analysis
4 Implementation
Homogenized stiffness tensors (2D and 3D) | Piezoelectric materials |
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Step 1: Define parameters for topology optimization, i.e., nelx, nely, (nelz), \(\vartheta\), p, \(r_{\mathrm{{min}}}\) | Step 1: Define parameters for topology optimization, i.e., nelx, nely, \(\vartheta\), p, \(r_{\mathrm{{min}}}\) |
Step 2: Preparation (\(\tilde{{\varvec{B}}}\), \({\varvec{K}}_p\), \({\varvec{F}}\), \({\varvec{K}}_b\), \({\varvec{J}}\), \({\varvec{M}}_s\) ) | Step 2: Preparation (\(\tilde{{\varvec{B}}}_u\), \(\tilde{{\varvec{B}}}_\phi\), \({\varvec{K}}_{pu}\), \({\varvec{K}}_{p\phi }\), \({\varvec{F}}_U\), \({\varvec{F}}_\Phi\), \({\varvec{K}}_b\), \({\varvec{s}}\), \({\varvec{M}}_s\)) |
Step 3: Loop of topology optimization | Step 3: Loop of topology optimization |
(1) Assembly of the global stiffness matrices \({\varvec{K}}_b\) and \({\varvec{K}}\) | (1) Assembly of the global stiffness matrices \({\varvec{K}}_b\) and \({\varvec{K}}\) |
(2) FEM solution | (2) FEM solution |
(3) Homogenization and objective function computation | (3) Homogenization and objective function computation |
(4) Sensitivity analysis (\({\varvec{J}}\), \({\varvec{l}}\), \({\varvec{M}}_s\)) | (4) Sensitivity analysis (\({\varvec{s}}\), \({\varvec{t}}\), \({\varvec{M}}_s\)) |
(5) Update the design variables with OC | (5) Update the design variables with MMA |
(6) Density filter | (6) Density filter |
Step 4: Check for convergence | Step 4: Check for convergence |
5 Numerical examples
5.1 2D microstructured materials with negative Poisson’s ratio
5.2 3D microstructured materials with maximum bulk modulus
5.3 2D microstructured materials with maximum hydrostatic coupling coefficient
\(\lambda\) | G | \(e_{31}\) | \(e_{33}\) | \(e_{15}\) | \(\kappa _{11}\) | \(\kappa _{33}\) |
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179 | 54 | \(-2.7\) | 3.65 | 21.3 | 12.5 | 14.4 |