Computational networks and systems-homogenization of self-adjoint differential operators in variational form on periodic networks and micro-architectured systems

Figure 1.  Homogenization theory: The limit process

Figure 2.  Periodic networks: The network $\mathcal{N}^\Omega_\varepsilon$ is the segment of the infinite network $\mathcal{N}_\varepsilon$ contained in the polyhedral domain $\Omega$, that is obtained by copying and scaling from the reference graph $\mathbb{G}$ in the unit cell $[0, 1)^d$

Figure 3.  Covering with cells: The $\varepsilon$-cells and the corresponding graphs

Figure 4.  Feasible networks: The network $\mathcal{N}^\Omega_1$ (left figure) and the network $\mathcal{N}^\Omega_{\frac{{1}}{{2}}}$ are feasible networks where the corresponding $\varepsilon$-cells cover the domain $\Omega$

Figure 5.  Two-scale transform: The function $x:\Omega \times \mathbb{G} \rightarrow \mathcal{N}^\Omega_\varepsilon$ is surjective, but not injective

Figure 6.  Two-scale transform: Mapping from $\mathcal{N}^\Omega_\varepsilon$ to the product $\Omega \times \mathbb{G}$

Figure 7.  Illustrative example: The reference graph $\mathbb{G}$

Figure 8.  Illustrative example: Diffusion coefficient and source term

Figure 9.  Illustrative example: The periodic network $\mathcal{N}^\Omega_1$

Figure 10.  Illustrative example: Solutions of the microscopic model for different lengths of periodicity $\varepsilon$

Figure 11.  Illustrative example: The approximate homogenized model and the solution for $\varepsilon=0.2$