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Mass lumping and stabilization for immersogeometric analysis

Yannis Voet, Espen Sande, Annalisa Buffa

TL;DR

This work analyzes mass lumping in immersogeometric analysis and shows that, although lumping cures the CFL restriction for small trimmed elements, it can activate spurious low-frequency modes that degrade the dynamic solution. It introduces a polynomial-extension stabilization that extends basis functions from well-behaved (good) elements into badly cut (bad) trimmed elements, producing stabilized matrices $\tilde{K}$ and $\tilde{M}$ and enabling lumped solves that exhibit accuracy and CFL behavior comparable to boundary-fitted discretizations. The stabilization successfully removes the deleterious low-energy modes for both smooth IGA and standard $C^0$ finite elements across diverse test geometries, including 1D trims, rotated squares, extruded plates, and perforated plates. The work also discusses the limitations of conventional row-sum lumping for high-order splines and highlights potential directions, such as extended spaces and advanced lumping schemes, to further improve robustness in trimmed geometries.

Abstract

Trimmed (multi-patch) geometries are the state-of-the-art technology in computer-aided design for industrial applications such as automobile crashworthiness. In this context, fast solution techniques extensively rely on explicit time integration schemes in conjunction with mass lumping techniques that substitute the consistent mass with a (usually diagonal) approximation. For smooth isogeometric discretizations, Leidinger [1] first showed that mass lumping removed the dependency of the critical time-step on the size of trimmed elements. This finding has attracted considerable attention but has unfortunately overshadowed another more subtle effect: mass lumping may disastrously impact the accuracy of low frequencies and modes, potentially inducing spurious oscillations in the solution. In this article, we provide compelling evidence for this phenomenon and later propose a stabilization technique based on polynomial extensions that restores a level of accuracy comparable to boundary-fitted discretizations.

Mass lumping and stabilization for immersogeometric analysis

TL;DR

This work analyzes mass lumping in immersogeometric analysis and shows that, although lumping cures the CFL restriction for small trimmed elements, it can activate spurious low-frequency modes that degrade the dynamic solution. It introduces a polynomial-extension stabilization that extends basis functions from well-behaved (good) elements into badly cut (bad) trimmed elements, producing stabilized matrices and and enabling lumped solves that exhibit accuracy and CFL behavior comparable to boundary-fitted discretizations. The stabilization successfully removes the deleterious low-energy modes for both smooth IGA and standard finite elements across diverse test geometries, including 1D trims, rotated squares, extruded plates, and perforated plates. The work also discusses the limitations of conventional row-sum lumping for high-order splines and highlights potential directions, such as extended spaces and advanced lumping schemes, to further improve robustness in trimmed geometries.

Abstract

Trimmed (multi-patch) geometries are the state-of-the-art technology in computer-aided design for industrial applications such as automobile crashworthiness. In this context, fast solution techniques extensively rely on explicit time integration schemes in conjunction with mass lumping techniques that substitute the consistent mass with a (usually diagonal) approximation. For smooth isogeometric discretizations, Leidinger [1] first showed that mass lumping removed the dependency of the critical time-step on the size of trimmed elements. This finding has attracted considerable attention but has unfortunately overshadowed another more subtle effect: mass lumping may disastrously impact the accuracy of low frequencies and modes, potentially inducing spurious oscillations in the solution. In this article, we provide compelling evidence for this phenomenon and later propose a stabilization technique based on polynomial extensions that restores a level of accuracy comparable to boundary-fitted discretizations.

Paper Structure

This paper contains 10 sections, 6 theorems, 72 equations, 32 figures, 1 algorithm.

Table of Contents

  1. Introduction and background
  2. Model problem and discretization
  3. Motivation
  4. Analysis
  5. Stabilization technique
  6. Numerical examples
  7. Conclusion
  8. Error analysis
  9. Elliptic boundary-value problems
  10. Hyperbolic initial boundary-value problems

Key Result

Lemma 4.1

The general solution of the ODE with $f \in C^0([0,T])$, $\lambda \in \mathbb{R}^*_+$ and $u_0,v_0 \in \mathbb{R}$ is given by

Figures (32)

  • Figure 3.1: Trimmed line segment
  • Figure 3.2: Ratio of approximate over exact eigenvalues for consistent and lumped mass approximations
  • Figure 3.3: Close-up on the first $14$ eigenvalues. The eigenvalues of $(K,\mathcal{L}(M))$ are relabeled based on the closest exact eigenvalue.
  • Figure 3.4: Three first and two last eigenfunctions of $(K,M)$ (top row) and $(K,\mathcal{L}(M))$ (bottom row) for $p=3$. The labeling of eigenfunctions matches the labeling of eigenvalues.
  • Figure 3.5: Function $q(x)$ defined in \ref{['eq: function_q(x)']}
  • ...and 27 more figures

Theorems & Definitions (22)

  • Example 3.1: 1D counter-example
  • Remark 3.2
  • Example 3.3: Rotated square
  • Example 3.4: Plate with extrusion
  • Example 3.5: Perforated plate
  • Lemma 4.1
  • proof
  • Corollary 4.2
  • proof
  • Theorem 4.3
  • ...and 12 more