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

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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:

C^H_{2D}=
\begin{bmatrix}
C^H_{11} & C^H_{12} & C^H_{13}\\
C^H_{12} & C^H_{22} & C^H_{23} \\
C^H_{13} & C^H_{23} & C^H_{33} 
\end{bmatrix}

• We apply test strain fields \varepsilon^{0(i)} to the unit cell.
• Due to the symmetry it is sufficient to apply three test strain fields.
• The test fields consider normal strain state in the x_1 and x_2 directions as well as a state of pure shear.

\varepsilon^{0(11)}=
\begin{bmatrix}
1 \\
0 \\
0 
\end{bmatrix} \ , \ 
\varepsilon^{0(22)}=
\begin{bmatrix}
0 \\
1 \\
0 
\end{bmatrix} \ , \ 
\varepsilon^{0(12)}=
\begin{bmatrix}
0 \\
0 \\
1 
\end{bmatrix} 

The element contributions to the strain energy q^e_{ij} can be calculated from the nodal displacement vectors:

q_{ij}^e= \frac{1}{|\Omega|}(d_0^{e(i)}-d^{e(i)})^TK^e(d_0^{e(j)}-d^{e(j)})

• d_0^{e(i)} and d^{e(i)} are the applied and resulting nodal displacements for the element e corresponding to the test field (i)
• K^e element stiffness matrix
• Then the element contribution is normalized by the size of the unit cell
• The components of the effective constitutive matrix are found as the sum of all element contributions to the strain energy withing the unit cell

C^H_{ij}=\sum_{e \in \Omega}q^e_{ij}

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Mechanical Properties and Symmetry

For square symmetry: