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Higher order accurate mass lumping for explicit isogeometric methods based on approximate dual basis functions

Rene Hiemstra, Thi-Hoa Nguyen, Sascha Eisentrager, Wolfgang Dornisch, Dominik Schillinger

TL;DR

This paper introduces a mathematical framework for explicit structural dynamics, employing approximate dual functionals and rowsum mass lumping, and reveals that the approach achieves accuracy and robustness comparable to a traditional Galerkin method employing the consistent mass formulation, while retaining the explicit nature of the lumped mass formulation.

Abstract

This paper introduces a mathematical framework for explicit structural dynamics, employing approximate dual functionals and rowsum mass lumping. We demonstrate that the approach may be interpreted as a Petrov-Galerkin method that utilizes rowsum mass lumping or as a Galerkin method with a customized higher-order accurate mass matrix. Unlike prior work, our method correctly incorporates Dirichlet boundary conditions while preserving higher order accuracy. The mathematical analysis is substantiated by spectral analysis and a two-dimensional linear benchmark that involves a non-linear geometric mapping. Our results reveal that our approach achieves accuracy and robustness comparable to a traditional Galerkin method employing the consistent mass formulation, while retaining the explicit nature of the lumped mass formulation.

Higher order accurate mass lumping for explicit isogeometric methods based on approximate dual basis functions

TL;DR

This paper introduces a mathematical framework for explicit structural dynamics, employing approximate dual functionals and rowsum mass lumping, and reveals that the approach achieves accuracy and robustness comparable to a traditional Galerkin method employing the consistent mass formulation, while retaining the explicit nature of the lumped mass formulation.

Abstract

This paper introduces a mathematical framework for explicit structural dynamics, employing approximate dual functionals and rowsum mass lumping. We demonstrate that the approach may be interpreted as a Petrov-Galerkin method that utilizes rowsum mass lumping or as a Galerkin method with a customized higher-order accurate mass matrix. Unlike prior work, our method correctly incorporates Dirichlet boundary conditions while preserving higher order accuracy. The mathematical analysis is substantiated by spectral analysis and a two-dimensional linear benchmark that involves a non-linear geometric mapping. Our results reveal that our approach achieves accuracy and robustness comparable to a traditional Galerkin method employing the consistent mass formulation, while retaining the explicit nature of the lumped mass formulation.
Paper Structure (21 sections, 5 theorems, 38 equations, 5 figures)

This paper contains 21 sections, 5 theorems, 38 equations, 5 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Model problem
  4. Semi-discrete equations of motion
  5. Explicit time stepping methods
  6. Efficient update through mass lumping
  7. Approximate $L^2$ dual bases
  8. Univariate B-splines
  9. $L^2$ dual basis
  10. $L^2$ approximate dual basis
  11. Treatment of Dirichlet boundary conditions
  12. Connection with rowsum mass lumping
  13. Higher order accurate mass lumping
  14. Extension to arbitrary isoparametric mappings
  15. Higher order accuracy under mass lumping
  16. ...and 6 more sections

Key Result

Proposition 3.1

Let $f \in L^2(\Delta)$. The function $u(x) = \sum_{i=1}^{n} \langle f, \tilde{B}_{i} \rangle \, B_{i}(x)$ is the best spline approximation to $f$ in the $L^2$-norm. In other words, it is the minimizer of the convex minimization problem

Figures (5)

  • Figure 1: Normalized frequency spectra associated with consistent mass (black), higher order approximated mass (blue), and rowsum lumped mass (red). The results are obtained for polynomial degrees $p=2-5$ with $N=250$. The outlier frequencies have been removed using the technique presented in Hiemstra_outlier_2021.
  • Figure 2: Normalized frequency errors associated with consistent mass (black), higher order approximated mass (blue), and rowsum lumped mass (red). The results are obtained for polynomial degrees $p=2-5$ with $N=250$. The outlier frequencies have been removed using the technique presented in Hiemstra_outlier_2021.
  • Figure 3: Problem description for explicit dynamics on an annulus.
  • Figure 4: Relative $L^2$ error in the vertical displacement field $u$ as a function of the square root of number of degrees of freedom $N$. Higher order mass lumping maintains the accuracy of the consistent mass, see Figure a. Outlier removal does not negatively affect the accuracy of the methods, see Figure b.
  • Figure 5: Increase of the critical time-step size of the (higher order) lumped mass compared with consistent mass (a) and the increase in timestep due to outlier removal (b). The compound effect of the chosen mass and outlier removal is obtained by multiplying these numbers.

Theorems & Definitions (11)

  • Proposition 3.1: $L^2$-projection
  • proof
  • Definition 3.2: Approximate dual basis
  • Definition 3.3: Discrete inner product on splines
  • Theorem 3.4: Quasi $L^2$-projection
  • proof
  • Theorem 3.5: Quasi $L^2$-projection with a homogeneous boundary condition
  • proof
  • Lemma 3.6
  • proof
  • ...and 1 more