Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

A mathematical theory for mass lumping and its generalization with applications to isogeometric analysis

Yannis Voet, Espen Sande, Annalisa Buffa

TL;DR

The paper addresses the need for a general, discretization-independent theory of mass lumping in explicit time integration by focusing on row-sum lumping and its spectral consequences. It develops a rigorous generalized-eigenproblem framework, shows that lumping yields controlled underestimation of eigenvalues, and extends the concept to generalized preconditioners and Kronecker-structured matrices, enabling efficient solves via banded or tensor-product forms. It further proposes a nearest Kronecker product approximation as a practical two-level strategy to approximate mass matrices in multidimensional and non-separable settings, supported by perturbation bounds and numerical experiments in isogeometric analysis. The findings demonstrate improved stability (larger critical time steps) and scalable solvers while maintaining accurate low-frequency behavior, and outline directions to address spectrum outliers inherent in high-order isogeometric discretizations, with implications for structural dynamics andIGA-based simulations.

Abstract

Explicit time integration schemes coupled with Galerkin discretizations of time-dependent partial differential equations require solving a linear system with the mass matrix at each time step. For applications in structural dynamics, the solution of the linear system is frequently approximated through so-called mass lumping, which consists in replacing the mass matrix by some diagonal approximation. Mass lumping has been widely used in engineering practice for decades already and has a sound mathematical theory supporting it for finite element methods using the classical Lagrange basis. However, the theory for more general basis functions is still missing. Our paper partly addresses this shortcoming. Some special and practically relevant properties of lumped mass matrices are proved and we discuss how these properties naturally extend to banded and Kronecker product matrices whose structure allows to solve linear systems very efficiently. Our theoretical results are applied to isogeometric discretizations but are not restricted to them.

A mathematical theory for mass lumping and its generalization with applications to isogeometric analysis

TL;DR

The paper addresses the need for a general, discretization-independent theory of mass lumping in explicit time integration by focusing on row-sum lumping and its spectral consequences. It develops a rigorous generalized-eigenproblem framework, shows that lumping yields controlled underestimation of eigenvalues, and extends the concept to generalized preconditioners and Kronecker-structured matrices, enabling efficient solves via banded or tensor-product forms. It further proposes a nearest Kronecker product approximation as a practical two-level strategy to approximate mass matrices in multidimensional and non-separable settings, supported by perturbation bounds and numerical experiments in isogeometric analysis. The findings demonstrate improved stability (larger critical time steps) and scalable solvers while maintaining accurate low-frequency behavior, and outline directions to address spectrum outliers inherent in high-order isogeometric discretizations, with implications for structural dynamics andIGA-based simulations.

Abstract

Explicit time integration schemes coupled with Galerkin discretizations of time-dependent partial differential equations require solving a linear system with the mass matrix at each time step. For applications in structural dynamics, the solution of the linear system is frequently approximated through so-called mass lumping, which consists in replacing the mass matrix by some diagonal approximation. Mass lumping has been widely used in engineering practice for decades already and has a sound mathematical theory supporting it for finite element methods using the classical Lagrange basis. However, the theory for more general basis functions is still missing. Our paper partly addresses this shortcoming. Some special and practically relevant properties of lumped mass matrices are proved and we discuss how these properties naturally extend to banded and Kronecker product matrices whose structure allows to solve linear systems very efficiently. Our theoretical results are applied to isogeometric discretizations but are not restricted to them.
Paper Structure (12 sections, 12 theorems, 52 equations, 25 figures, 2 algorithms)

This paper contains 12 sections, 12 theorems, 52 equations, 25 figures, 2 algorithms.

Table of Contents

  1. Introduction and background
  2. Mass lumping in explicit dynamics
  3. Mathematical properties of mass lumping and its generalization
  4. Generalized eigenvalue problems
  5. Mathematical properties of mass lumping
  6. A generalization of mass lumping
  7. Multidimensional problems
  8. Nearest Kronecker product approximation
  9. Approximation for 2D problems
  10. Approximation for 3D problems
  11. Conclusion
  12. Appendix

Key Result

Lemma 3.1

Let $(A,B)$ be a symmetric pencil with $B$ positive definite. Then, all generalized eigenvalues are real and there exists an invertible matrix $U \in \mathbb{R}^{n \times n}$ such that where $D=\mathop{\mathrm{\operatorname{diag}}}\nolimits(\lambda_1, \dots, \lambda_n)$ is a real diagonal matrix containing the eigenvalues.

Figures (25)

  • Figure 3.1: Eigenvalue ratios $\frac{\lambda_k(K,\tilde{M})}{\lambda_k(K,M)}$ when using the lumped mass matrix and the preconditioner of Loli et al. loli2021easy
  • Figure 3.2: Comparison between $\Lambda(K,M)$ and $\Lambda(K,\tilde{M})$ obtained using the lumped mass matrix and the preconditioner of Loli et al. loli2021easy
  • Figure 3.3: Relative eigenvalue error for the first 20 discrete eigenvalues and $p=6$
  • Figure 3.4: Comparison of \ref{['eq: Bauer_Fike_1']} and \ref{['eq: Crawford_1976']} for the row-sum lumped mass matrix and $p=1$
  • Figure 3.5: Comparison of $\Lambda(K, M)$ and $\Lambda(K, P_i)$ for $i=1,2,3$ and $p=3$
  • ...and 20 more figures

Theorems & Definitions (45)

  • Lemma 3.1
  • Lemma 3.2
  • proof
  • Definition 3.3: Equivalent pencils
  • Theorem 3.4
  • proof
  • Definition 3.5: Loewner partial order
  • Corollary 3.6
  • proof
  • Definition 3.7: Lumping operator
  • ...and 35 more