MultiPhysics Inverse Homogenization for the Design of Innovative Cellular Materials: Application to ThermoElastic Problems
Abstract
:1. Introduction
2. Methods
2.1. Inverse Homogenization
2.2. Discretization on Anisotropic Adapted Meshes
2.3. MultiPhysics Optimization Algorithm
Algorithm 1 MultiPmicroSIMPATY 

3. Results
3.1. Design Case 1
3.2. Design Case 2
3.3. Design Case 3
4. Discussion of Results
4.1. Comparison with OffTheShelf Designs
4.2. Comparison with Standard Inverse Homogenization
5. Conclusions and Perspectives
 (i)
 The MultiPmicroSIMPATY algorithm provides original design solutions, complying also with conflicting requirements;
 (ii)
 The good performance of microSIMPATY has been confirmed also in a thermoelastic context. Standard issues typical of topology optimization, such as the presence of intermediate densities, of jagged boundaries, and of too complex structures, is mitigated by the employment of a mesh customized to the design process (see Figure 7 and Table 3);
 (iii)
 The new cellular materials have been successfully compared with consolidated solutions in terms of mechanical and thermal properties (see Table 2);
 (iv)
 Filtering can be considerably limited thanks to the use of mesh adaptation. This turns into an improvement in terms of accuracy of the optimization process (see Figure 8);
 (v)
 The employment of an anisotropic mesh adaptation provides advantages with a view to a manufacturing phase. Indeed, the unit cells designed by MultiPmicroSIMPATY exhibit very smooth geometries which demand for a very limited postprocessing;
 (vi)
 The procedure here settled turns out to be fully general with respect to the selected multiphysics context.
Author Contributions
Funding
Acknowledgments
Conflicts of Interest
References
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${\mathit{E}}_{1111}^{\mathit{H}}$  ${\mathit{E}}_{1212}^{\mathit{H}}$  $\frac{{\mathit{E}}_{2222}^{\mathit{H}}}{{\mathit{E}}_{1111}^{\mathit{H}}}$  ${\mathit{k}}_{11}^{\mathit{H}}$  $\frac{{\mathit{k}}_{22}^{\mathit{H}}}{{\mathit{k}}_{11}^{\mathit{H}}}$  $\mathcal{M}$  

Design Case 1  
${c}^{u}$  0.080  0.080  2.000  1.000  0.580  
c  0.038  0.056  1.299  0.199  0.566  
${c}^{l}$  0.050  0.055  1.000  0.010  0.000  0.292 
Design Case 2  
${c}^{u}$  0.350  0.150  2.000  1.000  2.000  
c  0.250  0.086  0.299  0.317  0.597  
${c}^{l}$  0.230  0.080  0.300  0.300  0.000  0.412 
Design Case 3  
${c}^{u}$  0.150  0.100  1.100  0.400  1.100  
c  0.151  0.083  1.074  0.260  1.002  
${c}^{l}$  0.100  0.080  1.000  0.250  1.000  0.415 
${\mathit{E}}_{\mathit{x}}^{\mathit{H}}$  ${\mathit{E}}_{\mathit{y}}^{\mathit{H}}$  ${\mathit{G}}^{\mathit{H}}$  ${\mathit{k}}_{11}^{\mathit{H}}$  ${\mathit{k}}_{22}^{\mathit{H}}$  

Design Case 1  
D1  0.012  0.015  0.056  0.200  0.113  
A  0.009  0.009  0.075  0.163  0.163  
B  0.095  0.042  0.059  0.198  0.131  
Design Case 2  
D2  0.126  0.039  0.082  0.317  0.126  
C  0.341  0.116  0.002  0.432  0.125  
Design Case 3  
D3  0.070  0.070  0.082  0.260  0.261  
L  0.188  0.188  0.072  0.255  0.255 
D1  D2  D3  

MultiPmicroSIMP  0.330  0.443  0.486 
MultiPmicroSIMPATY  0.292  0.412  0.415 
Mass reduction [%]  11.5%  7.0%  14.6% 
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Gavazzoni, M.; Ferro, N.; Perotto, S.; Foletti, S. MultiPhysics Inverse Homogenization for the Design of Innovative Cellular Materials: Application to ThermoElastic Problems. Math. Comput. Appl. 2022, 27, 15. https://doi.org/10.3390/mca27010015
Gavazzoni M, Ferro N, Perotto S, Foletti S. MultiPhysics Inverse Homogenization for the Design of Innovative Cellular Materials: Application to ThermoElastic Problems. Mathematical and Computational Applications. 2022; 27(1):15. https://doi.org/10.3390/mca27010015
Chicago/Turabian StyleGavazzoni, Matteo, Nicola Ferro, Simona Perotto, and Stefano Foletti. 2022. "MultiPhysics Inverse Homogenization for the Design of Innovative Cellular Materials: Application to ThermoElastic Problems" Mathematical and Computational Applications 27, no. 1: 15. https://doi.org/10.3390/mca27010015