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A low-rank solver for conforming multipatch Isogeometric Analysis

Monica Montardini, Giancarlo Sangalli, Mattia Tani

TL;DR

This work tackles the linear elasticity problem in Isogeometric Analysis on complex multipatch geometries by introducing a low-rank solver based on Tucker tensor representations. The domain is decomposed into subdomains formed by unions of neighboring patches, enabling local tensor-product solves and a block-diagonal overlapping Schwarz preconditioner used within a truncated preconditioned conjugate gradient method. The authors provide a theoretical bound on the relative error introduced by the low-rank data approximation and demonstrate memory savings of up to two orders of magnitude while achieving iteration counts that are nearly independent of mesh size and spline degree. Numerical experiments on several challenging geometries validate robustness and substantial memory reductions, and the framework offers a path toward efficient real-world IgA simulations and potential extensions to nonlinear problems.

Abstract

In this paper, we propose an innovative isogeometric low-rank solver for the linear elasticity model problem, specifically designed to allow multipatch domains. Our approach splits the domain into subdomains, each formed by the union of neighboring patches. Within each subdomain, we employ Tucker low-rank matrices and vectors to approximate the system matrices and right-hand side vectors, respectively. This enables the construction of local approximate fast solvers. These local solvers are then combined into an overlapping Schwarz preconditioner, which is utilized in a truncated preconditioned conjugate gradient method. Numerical experiments demonstrate the significant memory storage benefits and a uniformly bounded number of iterations with respect to both mesh size and spline degree.

A low-rank solver for conforming multipatch Isogeometric Analysis

TL;DR

This work tackles the linear elasticity problem in Isogeometric Analysis on complex multipatch geometries by introducing a low-rank solver based on Tucker tensor representations. The domain is decomposed into subdomains formed by unions of neighboring patches, enabling local tensor-product solves and a block-diagonal overlapping Schwarz preconditioner used within a truncated preconditioned conjugate gradient method. The authors provide a theoretical bound on the relative error introduced by the low-rank data approximation and demonstrate memory savings of up to two orders of magnitude while achieving iteration counts that are nearly independent of mesh size and spline degree. Numerical experiments on several challenging geometries validate robustness and substantial memory reductions, and the framework offers a path toward efficient real-world IgA simulations and potential extensions to nonlinear problems.

Abstract

In this paper, we propose an innovative isogeometric low-rank solver for the linear elasticity model problem, specifically designed to allow multipatch domains. Our approach splits the domain into subdomains, each formed by the union of neighboring patches. Within each subdomain, we employ Tucker low-rank matrices and vectors to approximate the system matrices and right-hand side vectors, respectively. This enables the construction of local approximate fast solvers. These local solvers are then combined into an overlapping Schwarz preconditioner, which is utilized in a truncated preconditioned conjugate gradient method. Numerical experiments demonstrate the significant memory storage benefits and a uniformly bounded number of iterations with respect to both mesh size and spline degree.
Paper Structure (19 sections, 3 theorems, 95 equations, 12 figures, 7 tables, 3 algorithms)

This paper contains 19 sections, 3 theorems, 95 equations, 12 figures, 7 tables, 3 algorithms.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. B-Splines
  4. Isogeometric spaces on a multipatch domain
  5. Model problem
  6. The Tucker format
  7. Low-rank multipatch method
  8. The linear system
  9. The preconditioner
  10. The TPCG method with block-wise truncation
  11. Subdomains as unions of patches
  12. Analysis of the low-rank approximation of the linear system
  13. Numerical experiments
  14. L-shaped domain
  15. 3D-cross domain
  16. ...and 4 more sections

Key Result

Lemma 1

Let $\underline{v}_h,\underline{w}_h \in \left[ V_h \right]^3$, and let $\boldsymbol{v},\boldsymbol{w}$ be block vectors representing these functions with respect to $\left[ V_h^{(1)} \right]^3,\ldots,\left[ V_h^{(\mathcal{N}_{sub})} \right]^3$ and their respective bases. Then where $\widehat{\underline{v}}_{ptc}^{(m)} = \underline{v}_h \circ \mathcal{F}_{m}$, $\widehat{\underline{w}}_{ptc}^{(m)

Figures (12)

  • Figure 1: Examples of boundary conditions that are not allowed for the L-shaped domain. A remedy is proposed in Remark \ref{['rmk:boundary_cond']}
  • Figure 2: L-shaped domain.
  • Figure 3: Results for the L-shaped domain.
  • Figure 4: 3D-cross domain.
  • Figure 5: Results for the 3D-cross domain.
  • ...and 7 more figures

Theorems & Definitions (9)

  • Remark 1
  • Remark 2
  • Remark 3
  • Lemma 1
  • proof
  • Theorem 1
  • proof
  • Theorem 2
  • proof