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On the prospects of interpolatory spline bases for accurate mass lumping strategies in isogeometric analysis

Yannis Voet, Espen Sande

TL;DR

The paper investigates restoring interpolation for spline bases in isogeometric analysis to enable high-quality mass lumping for explicit structural dynamics. It analyzes three classical lumping techniques and introduces interpolatory spline bases with a moment-fitting quadrature to obtain a diagonal lumped mass, highlighting the challenge of ensuring positive weights and CFL improvements. Numerical experiments in 1D and 2D reveal that Greville points often produce negative weights and poorer eigenfrequency convergence, while Demko points tend to keep weights positive and offer improved accuracy, though a general theory remains open. The work provides an algorithmic framework and Hong Kong-style path toward robust high-order mass lumping in IGA, with future directions including theory, multi-patch extensions, and more reliable quadrature rules.

Abstract

While interpolatory bases such as the Lagrange basis form the cornerstone of classical finite element methods, they have been replaced in the more general finite element setting of isogeometric analysis in favor of other desirable properties. Yet, interpolation is a key property for devising accurate mass lumping strategies that are ubiquitous in explicit dynamic analyses of structures. In this article, we explore the possibility of restoring interpolation for spline bases within isogeometric analysis for the purpose of mass lumping. Although reminiscent of the spectral element method, this technique comes with its lot of surprises and challenges, which are critically assessed.

On the prospects of interpolatory spline bases for accurate mass lumping strategies in isogeometric analysis

TL;DR

The paper investigates restoring interpolation for spline bases in isogeometric analysis to enable high-quality mass lumping for explicit structural dynamics. It analyzes three classical lumping techniques and introduces interpolatory spline bases with a moment-fitting quadrature to obtain a diagonal lumped mass, highlighting the challenge of ensuring positive weights and CFL improvements. Numerical experiments in 1D and 2D reveal that Greville points often produce negative weights and poorer eigenfrequency convergence, while Demko points tend to keep weights positive and offer improved accuracy, though a general theory remains open. The work provides an algorithmic framework and Hong Kong-style path toward robust high-order mass lumping in IGA, with future directions including theory, multi-patch extensions, and more reliable quadrature rules.

Abstract

While interpolatory bases such as the Lagrange basis form the cornerstone of classical finite element methods, they have been replaced in the more general finite element setting of isogeometric analysis in favor of other desirable properties. Yet, interpolation is a key property for devising accurate mass lumping strategies that are ubiquitous in explicit dynamic analyses of structures. In this article, we explore the possibility of restoring interpolation for spline bases within isogeometric analysis for the purpose of mass lumping. Although reminiscent of the spectral element method, this technique comes with its lot of surprises and challenges, which are critically assessed.
Paper Structure (17 sections, 15 theorems, 118 equations, 27 figures, 2 tables, 1 algorithm)

This paper contains 17 sections, 15 theorems, 118 equations, 27 figures, 2 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Problem statement
  3. Review of mass lumping
  4. Row-sum technique
  5. Diagonal scaling
  6. Nodal quadrature
  7. High order mass lumping
  8. Interpolatory spline bases
  9. Negative weights
  10. Truncated interpolatory spline basis
  11. Critical time step
  12. Algorithm
  13. Numerical experiments
  14. Conclusion
  15. Proof of \ref{['lem: spectral_fem']}
  16. ...and 2 more sections

Key Result

Lemma 3.2

For elementwise constant density and affine tensor product spectral elements, $\widehat{M} \succeq M$ and

Figures (27)

  • Figure 4.2: Original and truncated Lagrange cubic spline function for the Demko points on a uniform mesh of $25$ elements and a truncation tolerance of $10^{-4}$
  • Figure 5.1: Different spline bases. Black dots mark knot locations and interpolation points for the B-spline and Lagrange spline bases, respectively.
  • Figure 5.2: Interpolation error $\|f-Pf\|$
  • Figure 5.3: Quadrature error $|I(f)-Q(f)|$
  • Figure 5.4: Relative eigenfrequency error for the Laplace eigenvalue problem on the unit line with homogeneous Dirichlet boundary conditions
  • ...and 22 more figures

Theorems & Definitions (41)

  • Remark 3.1
  • Lemma 3.2
  • Theorem 4.1: Schoenberg-Whitney theorem
  • proof
  • Remark 4.2
  • Definition 4.3: Mass matrices
  • Definition 4.4: Lumped mass matrix
  • Lemma 4.5
  • proof
  • Corollary 4.6
  • ...and 31 more