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Mass lumping and outlier removal strategies for complex geometries in isogeometric analysis

Yannis Voet, Espen Sande, Annalisa Buffa

TL;DR

The paper addresses the CFL bottleneck in explicit isogeometric dynamics caused by outlier eigenfrequencies. It develops mass lumping techniques that extend from simple geometries to nontrivial single-patch, multipatch, and trimmed geometries, and pairs them with algebraic deflation to remove persistent outliers via low-rank perturbations. The approach preserves or improves the CFL condition while enabling efficient solves through block, hierarchical, and multipatch lumping, with an explicit strategy to scale the lumped mass using Woodbury-type inverses. Numerical experiments across single-patch, multipatch, and trimmed geometries show substantial reductions in iteration counts and computational time for long-time simulations, with modest impacts on accuracy, making the method attractive for industrially relevant complex geometries.

Abstract

Mass lumping techniques are commonly employed in explicit time integration schemes for problems in structural dynamics and both avoid solving costly linear systems with the consistent mass matrix and increase the critical time step. In isogeometric analysis, the critical time step is constrained by so-called "outlier" frequencies, representing the inaccurate high frequency part of the spectrum. Removing or dampening these high frequencies is paramount for fast explicit solution techniques. In this work, we propose mass lumping and outlier removal techniques for nontrivial geometries, including multipatch and trimmed geometries. Our lumping strategies provably do not deteriorate (and often improve) the CFL condition of the original problem and are combined with deflation techniques to remove persistent outlier frequencies. Numerical experiments reveal the advantages of the method, especially for simulations covering large time spans where they may halve the number of iterations with little or no effect on the numerical solution.

Mass lumping and outlier removal strategies for complex geometries in isogeometric analysis

TL;DR

The paper addresses the CFL bottleneck in explicit isogeometric dynamics caused by outlier eigenfrequencies. It develops mass lumping techniques that extend from simple geometries to nontrivial single-patch, multipatch, and trimmed geometries, and pairs them with algebraic deflation to remove persistent outliers via low-rank perturbations. The approach preserves or improves the CFL condition while enabling efficient solves through block, hierarchical, and multipatch lumping, with an explicit strategy to scale the lumped mass using Woodbury-type inverses. Numerical experiments across single-patch, multipatch, and trimmed geometries show substantial reductions in iteration counts and computational time for long-time simulations, with modest impacts on accuracy, making the method attractive for industrially relevant complex geometries.

Abstract

Mass lumping techniques are commonly employed in explicit time integration schemes for problems in structural dynamics and both avoid solving costly linear systems with the consistent mass matrix and increase the critical time step. In isogeometric analysis, the critical time step is constrained by so-called "outlier" frequencies, representing the inaccurate high frequency part of the spectrum. Removing or dampening these high frequencies is paramount for fast explicit solution techniques. In this work, we propose mass lumping and outlier removal techniques for nontrivial geometries, including multipatch and trimmed geometries. Our lumping strategies provably do not deteriorate (and often improve) the CFL condition of the original problem and are combined with deflation techniques to remove persistent outlier frequencies. Numerical experiments reveal the advantages of the method, especially for simulations covering large time spans where they may halve the number of iterations with little or no effect on the numerical solution.
Paper Structure (17 sections, 14 theorems, 66 equations, 20 figures, 2 tables, 1 algorithm)

This paper contains 17 sections, 14 theorems, 66 equations, 20 figures, 2 tables, 1 algorithm.

Table of Contents

  1. Introduction and background
  2. Model problem and its discretization
  3. Mass lumping
  4. Row-sum mass lumping and its generalization
  5. Block mass lumping
  6. Hierarchical mass lumping
  7. Multipatch mass lumping
  8. Solving linear systems with the lumped mass matrix
  9. Outlier removal
  10. Deflation techniques
  11. Eigenvalue and eigenvector computations
  12. Numerical experiments
  13. Single-patch geometries
  14. Multipatch geometries
  15. Trimmed geometries
  16. ...and 2 more sections

Key Result

Lemma 3.6

Let $\mathcal{B} \in \mathbb{R}^{n_1n_2\dots n_d \times n_1n_2\dots n_d}$ be $d$-level banded with bandwidths $\mathbf{b}=(b_1,b_2,\dots,b_d)$ and block sizes $\mathbf{r}=(r_1,r_2,\dots,r_d)$. Then $\mathcal{B}$ has bandwidth

Figures (20)

  • Figure 3.1: Block tridiagonal matrix $\mathcal{P}_2$ constructed from a block septadiagonal matrix $\mathcal{B}$
  • Figure 3.2: Sparsity patterns
  • Figure 4.1: Truncation of the largest eigenvalues
  • Figure 5.1: Stretched square
  • Figure 5.2: Quarter of a plate with a hole
  • ...and 15 more figures

Theorems & Definitions (51)

  • Definition 3.1: Lumping operator
  • Remark 3.2
  • Definition 3.3: Loewner partial order
  • Definition 3.4: $1$-level banded matrix
  • Definition 3.5: $d$-level banded matrix
  • Lemma 3.6
  • proof
  • Definition 3.7
  • Definition 3.8
  • Lemma 3.9
  • ...and 41 more