inverse_homogenization.md

Inverse Homogenization

Based on 1 and 2.

Elasticity (Constitutive) Tensor

Optimization Problem Formulation

\begin{aligned}
& \underset{\phi}{\text{minimize}}
& & f(C^H(\phi,D^{(i)})) \\
& \text{subject to}
& & K (\phi)d^{(i)}=F(\phi)^{(i)} \ \ \ \forall \ i  \\
& & & g(C^H(\phi,D^{(i)})) \geq g_{min} \\
& & & \sum_{e\in \Omega}{\rho_e}v_e \leq V_{max} \\
& & & \phi_{min}\leq \phi_n \leq \phi_{max} \ \  \forall \ \  n \in \Omega \\
& & & d^{(i)} \ \  is \ \  \Omega-periodic \\
\end{aligned}

• f is the objective function
• \phi design variables
• K(\phi) global stiffness matrix
• g constraints like square symmetry or isotropy
• C^H homogenized constitutive matrix
• Applying strain fields, d^{(i)} displacements f^{(i)} force vectors associated with the strain field

---

Numerical Optimization

For a linear elastic homogeneous material:

• the stress tensor is symmetric (\alpha_{ij}=\alpha_{ji})
• constitutive matrix also symmetric C^H_{ij}=C^H_{ji}
• constitutive matrix for the mechanical properties in 2D: