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

TL;DR

The paper tackles the memory and computational challenges of solving 2D linear elasticity on complex domains by marrying the Finite Element Method with low-rank Quantized Tensor Train (QTT) representations in a domain-partitioned framework. It introduces a comprehensive redesign of FE operations—mesh discretization, node/DOF ordering (including Z-order), stiffness/mass assembly, and algebraic solves—so that subdomain solutions can be concatenated into a global QTT system solved by the AMEn algorithm. Key contributions include a detailed domain-partitioning strategy with replicated interface nodes, canonical and Z-order DOF arrangements, explicit subdomain stiffness construction, global assembly via block Kronecker products, and Dirichlet boundary treatment within the QTT context. Numerical experiments on cantilever, SEN, and L-shaped domains demonstrate memory savings, exponential convergence with respect to discretization levels, and controlled QTT rank growth, confirming that QTT-FEM can deliver high-fidelity elasticity solutions on geometries beyond simple squares with substantially reduced memory footprints. The work highlights the practical viability of TT/QTT-based solvers for large-scale, complex-geometry elasticity problems and lays groundwork for further enhancements and preconditioning strategies within a tensor-structured FE pipeline.

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.

Figures (8)

• Figure 1: Mesh partition for the Jacobian calculation
• Figure 2: Geometry, boundary conditions, and representative meshes for the test cases of Section \ref{['sec5:numerical']}. $(a)$ Cantilever beam test case assuming $d=2$ levels and $q=20$ subdomains. $(b)$ Single edge notch tensile test case with $d=2$ levels and $q=2$ subdomains. $(c)$ L-shaped domain test case with $d=3$ levels and $q=3$ subdomains. Red vertical and horizontal lines separate adjacent subdomains; each subdomain is partitioned by a $3\times3$-square grid, and denoted by $\Omega^{(m)}$.
• Figure 3: Cantilever beam test case: bilogarithmic plot of QTT and FEniCS energy seminorm $(a)$ and $\mathbb{L}^{2}$-norm $(b)$ errors for different AMEn approximation accuracy.
• Figure 4: Energy seminorm error assuming different AMEn approximation accuracies for the single edge notch tensile test $(a)$ and the L-shaped panel $(b)$.
• Figure 5: Cantilever beam test case: $(a)$ memory storage versus the number of degrees of freedom for different approximation accuracy $\varepsilon$ and $(b)$ effective rank $r_e$ versus the number of degrees of freedom. Single edge notch tensile test: $(c)$ memory storage versus degrees of freedom for different values of $\varepsilon$, and $(d)$ effective rank $r_e$ versus the number of degrees of freedom. L-shaped panel: $(e)$ memory storage versus the number of degrees of freedom for different approximation accuracy $\varepsilon$, and $(f)$ effective rank $r_e$ versus the number of degrees of freedom.

Theorems & Definitions (2)

• Remark 4.1
• Remark 4.2