A Low-Rank QTT-based Finite Element Method for Elasticity Problems

TL;DR

This paper introduces a novel low-rank tensor-based finite element method for 2D elasticity problems, significantly reducing memory and computational costs while handling more general domains.

Contribution

It combines Quantized Tensor Train format with domain partitioning, enabling efficient elasticity problem solutions beyond square domains, with detailed implementation strategies.

Findings

Reduces memory usage compared to traditional FEM implementations

Achieves lower tensor ranks in numerical solutions

Demonstrates effectiveness on various test cases

Abstract

We present an efficient and robust numerical algorithm for solving the two-dimensional linear elasticity problem that combines the Quantized Tensor Train format and a domain partitioning strategy. This approach makes it possible to solve the linear elasticity problem on a computational domain that is more general than a square. Our method substantially decreases memory usage and achieves a notable reduction in rank compared to established Finite Element implementations like the FEniCS platform. This performance gain, however, requires a fundamental rethinking of how core finite element operations are implemented, which includes changes to mesh discretization, node and degree of freedom ordering, stiffness matrix and internal nodal force assembly, and the execution of algebraic matrix-vector operations. In this work, we discuss all these aspects in detail and assess the method's performance in the numerical approximation of three representative test cases.