Super-Localized Orthogonal Decomposition Method for Heterogeneous Linear Elasticity

TL;DR

The paper introduces the Super-Localized Orthogonal Decomposition (SLOD) method for efficient numerical homogenization of multiscale linear elasticity problems, improving sparsity and reducing oversampling without losing accuracy.

Contribution

It develops a novel superlocalization technique for the SLOD, enhancing computational efficiency and scalability for complex multiscale elasticity problems.

Findings

SLOD achieves higher sparsity than traditional methods.

The method reduces oversampling significantly.

Numerical experiments confirm high accuracy and efficiency.

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.