Super-Localized Orthogonal Decomposition Method for Heterogeneous Linear Elasticity

TL;DR

The paper addresses numerical homogenization for linear elasticity with multiscale, highly heterogeneous coefficients without relying on periodicity or scale separation. It extends the Super-Localized Orthogonal Decomposition (SLOD) framework to vector-valued PDEs, constructing rapidly decaying, stable basis functions via a localized augmentation of the LOD basis. A detailed numerical analysis provides energy-error and conditioning bounds, and a deal.II-based implementation demonstrates that SLOD achieves high accuracy with markedly reduced local problem sizes, yielding substantial computational savings in multiscale elasticity simulations. The results indicate that SLOD offers a robust, scalable approach for large-scale, heterogeneous elasticity problems with practical impact in engineering and materials science.

Abstract

We present the Super-Localized Orthogonal Decomposition (SLOD) method for the numerical homogenization of linear elasticity problems with multiscale microstructures modeled by a heterogeneous coefficient field without any periodicity or scale separation assumptions. Compared to the established Localized Orthogonal Decomposition (LOD) and its linear localization approach, SLOD achieves significantly improved sparsity properties through a nonlinear superlocalization technique, leading to computationally efficient solutions with significantly less oversampling - without compromising accuracy. We generalize the method to vector-valued problems and provide a supporting numerical analysis. We also present a scalable implementation of SLOD using the deal.II finite element library, demonstrating its feasibility for high-performance simulations. Numerical experiments illustrate the efficiency and accuracy of SLOD in addressing key computational challenges in multiscale elasticity.

Figures (6)

• Figure 3.1: Illustration of two second-order patches $\omega^{(2)}_T$, one built around an element far-enough from the boundary and another around an element touching the boundary. The fine mesh is presented in gray.
• Figure 6.1: Error of SLOD compared to the classical FEM (blue line). The dashed gray line indicates the theoretical FEM convergence rate, $\mathcal{O}(H^2)$ (left) and $\mathcal{O}(H)$ (right).
• Figure 6.2: Exponential decay of SLOD (solid lines). Dashed lines show the LOD error for the same values of $H$ given in the legend.
• Figure 6.3: Realizations of $\lambda$ and $\mu$ drawn from $\mathcal{U}[1, 100]$ on a grid of characteristic length $\eta = 2^{-6}$.
• Figure 6.4: Error behavior of SLOD (with oversampling $m=2,3,4$) and standard FEM (blue line) in the case of randomly varying Lamé coefficients and constant $\boldsymbol{f}$. The dashed gray lines represent the theoretical FEM convergence orders, $\mathcal{O}(H^2)$ (left) and $\mathcal{O}(H)$ (right).

Theorems & Definitions (14)

• Remark 2.1
• Remark 2.2
• Lemma 3.1
• proof
• Theorem 4.1
• Lemma 4.2
• Remark 4.3
• Lemma 5.1
• Theorem 5.2
• proof