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A theoretical analysis of mass scaling techniques

Yannis Voet, Espen Sande, Annalisa Buffa

Abstract

Mass scaling is widely used in finite element models of structural dynamics for increasing the critical time step of explicit time integration methods. While the field has been flourishing over the years, it still lacks a strong theoretical basis and mostly relies on numerical experiments as the only means of assessment. This contribution thoroughly reviews existing methods and connects them to established linear algebra results to derive rigorous eigenvalue bounds and condition number estimates. Our results cover some of the most successful mass scaling techniques, unraveling for the first time well-known numerical observations.

A theoretical analysis of mass scaling techniques

Abstract

Mass scaling is widely used in finite element models of structural dynamics for increasing the critical time step of explicit time integration methods. While the field has been flourishing over the years, it still lacks a strong theoretical basis and mostly relies on numerical experiments as the only means of assessment. This contribution thoroughly reviews existing methods and connects them to established linear algebra results to derive rigorous eigenvalue bounds and condition number estimates. Our results cover some of the most successful mass scaling techniques, unraveling for the first time well-known numerical observations.

Paper Structure

This paper contains 17 sections, 14 theorems, 74 equations, 12 figures.

Table of Contents

  1. Introduction and background
  2. Mass scaling
  3. Global mass scaling
  4. Preliminaries
  5. Linear fractional transformations
  6. Global deflation
  7. Local mass scaling
  8. Preliminaries
  9. Conventional mass scaling
  10. Local deflation
  11. Method of [Olovsson et al., 2005]
  12. Method of [Hoffmann et al., 2023]
  13. Numerical validation
  14. Ad hoc local SMS
  15. Local deflation
  16. ...and 2 more sections

Key Result

Lemma 2.3

Let $(A,B) \in \mathcal{S}_n \times \mathcal{S}_n^+$. Then, all generalized eigenvalues of $(A,B)$ are real nonnegative 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 nonnegative diagonal matrix containing the eigenvalues.

Figures (12)

  • Figure 2.1: Truncation of the largest eigenvalues
  • Figure 3.1: Values of $Q_e(\mathbf{u}_k)$ for $k=7,\dots,m$ and eigenvalues of $(K_e,M_e)$ and $(K_e,\overline{M}_e)$ for $\beta=1$ and the method of Olovsson et al.
  • Figure 3.2: Values of $Q_e(\mathbf{u}_k)$ for $k=7,\dots,m$ and eigenvalues of $(K_e,M_e)$ and $(K_e,\overline{M}_e)$ for $\beta=1$ and the method of Hoffmann et al.
  • Figure 3.3: Hexahedral finite element mesh
  • Figure 3.4: Increase of the critical time step for the methods of Olovsson et al. and Hoffmann et al.
  • ...and 7 more figures

Theorems & Definitions (32)

  • Definition 2.1: Loewner partial order
  • Definition 2.2
  • Lemma 2.3: stewart1990matrix
  • Lemma 2.4: voet2024robust
  • Definition 2.5: Linear factional transformation
  • Remark 2.6
  • Definition 2.7: Non-degenerate LFT
  • Lemma 2.8
  • proof
  • Definition 2.9: Scaled matrix pair
  • ...and 22 more