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<p>This paper introduces a mathematical framework for explicit structural dynamics, employing approximate dual functionals and row-sum mass lumping. We demonstrate that the Petrov–Galerkin method developed in our previous work—utilizing row-sum mass lumping—can be interpreted as a Galerkin method with a customized higher-order accurate mass matrix. Unlike prior work, our method correctly incorporates Dirichlet boundary conditions, while preserving optimal spatial accuracy. The mathematical analysis is substantiated by spectral analysis, two-dimensional linear benchmarks illustrating robustness under mesh distortion and a three-dimensional geometrically non-linear forced vibration analysis. The obtained 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.</p>

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

  • 期刊论文

  • 出版模式：

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  • 发表日期：

    2025年1月25日

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  • R. R. Hiemstra,
  • T. H. Nguyen,
  • S. Eisenträger,
  • W. Dornisch,
  • D. Schillinger

摘要

This paper introduces a mathematical framework for explicit structural dynamics, employing approximate dual functionals and row-sum mass lumping. We demonstrate that the Petrov–Galerkin method developed in our previous work—utilizing row-sum mass lumping—can be interpreted as a Galerkin method with a customized higher-order accurate mass matrix. Unlike prior work, our method correctly incorporates Dirichlet boundary conditions, while preserving optimal spatial accuracy. The mathematical analysis is substantiated by spectral analysis, two-dimensional linear benchmarks illustrating robustness under mesh distortion and a three-dimensional geometrically non-linear forced vibration analysis. The obtained 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.