[Paper Review] Optimal design of mixtures of ferromagnetic interactions

Andrea Braides, Leonard Kreutz|arXiv (Cornell University)|Oct 20, 2016

Advanced Mathematical Modeling in Engineering15 references4 citations

TL;DR

This paper establishes a rigorous framework for the optimal design of surface energies in networks with mixtures of ferromagnetic interactions by deriving sharp bounds for $γ$-convergence homogenization. It proves a localization principle enabling reduction to periodic microgeometries, and characterizes the full range of possible homogenized surface tensions as convex, positively homogeneous functions bounded by explicit expressions involving interaction strengths and volume fractions.

ABSTRACT

We provide a general framework for the optimal design of surface energies on networks. We prove sharp bounds for the homogenization of discrete systems describing mixtures of ferromagnetic interactions by constructing optimal microgeometries, and we prove a localization principle which allows to reduce to the periodic setting in the general nonperiodic case.

Motivation & Objective

• To develop a general framework for optimal design of surface energies on networks with mixtures of ferromagnetic interactions.
• To derive sharp bounds for the homogenization of discrete systems with ferromagnetic interactions under fixed volume fraction constraints.
• To establish a localization principle that reduces the analysis of non-periodic systems to the periodic setting.
• To characterize the set of all possible homogenized surface tensions as convex, positively homogeneous functions of degree one.
• To provide a complete description of the G-closure problem for surface tensions arising from mixtures of two interaction strengths.

Proposed method

• Uses a discrete-to-continuum approach with scaled energies $E_\varepsilon(u) = \frac{1}{4}\sum_{\xi\in V}\sum_{i\in\mathbb{Z}^d}\varepsilon^{d-1}c^{\varepsilon}_{i,\xi}(u_i - u_{i+\xi})^2$ to model Ising-type systems on a cubic lattice.
• Applies $\Gamma$-convergence to derive the continuum limit energy $F(u) = \int_{\partial^*\{u=1\}} \varphi(x,\nu_u)\,d\mathcal{H}^{d-1}$, representing surface energy with anisotropic surface tension $\varphi$.
• Introduces a localization principle that allows reduction of general non-periodic systems to periodic microgeometries via a diagonal argument.
• Constructs optimal microgeometries using a hierarchical construction with shrinking cubes $Q_{2^{-k}(1-\delta)}(x_n)$ and periodic approximations.
• Employs a two-step approximation: first approximating the energy in each cube via periodic configurations, then refining the approximation using a diagonal sequence in $\varepsilon$ and $N$.
• Uses the $\Gamma$-limit of discrete energies to characterize the homogenized surface tension $\varphi$ as a limit of surface tensions from periodic configurations with prescribed volume fractions $\theta_{\xi}$.

Experimental results

Research questions

• RQ1 What is the full set of possible homogenized surface tensions that can arise from mixtures of ferromagnetic interactions with two distinct bond strengths?
• RQ2 How can the optimal design of network microstructures be characterized under fixed volume fraction constraints on interaction coefficients?
• RQ3 Can the homogenization of non-periodic systems be reduced to the periodic case via a localization principle?
• RQ4 What are the sharp upper and lower bounds for the surface tension $\varphi(\nu)$ in terms of the interaction strengths $\alpha_\xi$, $\beta_\xi$ and volume fractions $\theta_\xi$?
• RQ5 How does the $\Gamma$-limit of discrete energies with locally varying coefficients converge to a continuum surface energy functional?

Key findings

• The set of all possible homogenized surface tensions $\varphi$ is characterized as the set of all symmetric, convex, positively homogeneous functions of degree one satisfying the double inequality $\sum_{\xi\in V}\alpha_\xi|\langle\nu,\xi\rangle| \leq \varphi(\nu) \leq \sum_{\xi\in V}(\beta_\xi\theta_\xi + (1-\theta_\xi)\alpha_\xi)|\langle\nu,\xi\rangle|$.
• For any $\varphi$ in this set, there exists a sequence of periodic microgeometries with volume fractions $\theta_{\xi}^n \to \theta_\xi$ such that the $\Gamma$-limit of the corresponding discrete energies converges to $\varphi$.
• The localization principle ensures that the $\Gamma$-limit of non-periodic systems can be approximated by periodic ones, enabling reduction to the periodic setting.
• The construction of optimal microgeometries involves a hierarchical refinement using cubes $Q_{2^{-k}(1-\delta)}(x_n)$ and periodic approximations $c^{k,n,N}_{i,\xi}$ with period $T$, such that the volume fractions converge weakly-* to the target $\theta_{\xi}$.
• The $\Gamma$-limit of the discrete energies with coefficients $c^{k,\delta,\varepsilon}_{i,\xi}$ converges to the target energy $E_k^\delta(u)$, with the surface tension $\varphi^\delta_k$ matching the desired form in the bulk and switching to $\sum \beta_\xi|\langle\nu,\xi\rangle|$ near boundaries.
• The full $\Gamma$-limit is recovered via a diagonal argument over $k$, $N$, and $\delta$, ensuring convergence of the energies and local volume fractions to the desired limit.