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<p>Despite the extraordinary efficiency advantage of Greville quadrature (GQ) in isogeometric analysis (IGA), the related integration weights depend on specific meshes and have to be re-calculated for each discretization, as makes it non-trivial to theoretically justify the accuracy of IGA with GQ. To address these issues, an equivalent explicit Greville quadrature (EEGQ) is firstly proposed through solving a local linear moment fitting problem, where the integration rules are mesh independent and a pre-evaluation can be readily implemented. The explicit form of EEGQ enables a theoretical analysis of the integration precision for GQ, which indicates a GQ rule resulting from <i>p</i>th degree basis functions also has a <i>p</i>th order integration precision. This insufficient integration precision then limits the algorithmic convergence orders of IGA with GQ to be 3, 2 and 4 regarding the <i>L</i><sub>2</sub>, <i>H</i><sub>1</sub> and frequency error measures, regardless of the basis degree. Subsequently, this convergence obstacle associated with GQ is resolved by introducing an augmented explicit Greville quadrature (AEGQ) toward improving the integration precision. The proposed AEGQ is closely related to EEGQ for interior elements, but the integration rules of AEGQ for the near boundary elements are re-constructed to guarantee the precision of the overall domain integration. It is rationally shown that IGA with AEGQ leads to an optimal convergence for cubic basis functions, and a sub-optimal convergence for quartic and quintic basis functions. These theoretical predictions are consistently demonstrated by numerical results.</p>

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Equivalent and augmented explicit Greville quadrature for isogeometric analysis

  • 期刊论文

  • 发表日期：

    2026年2月2日

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  • Songyang Hou,
  • Xiaolan Xu,
  • Dongdong Wang,
  • Ao Shen,
  • Zhuangjing Sun

摘要

Despite the extraordinary efficiency advantage of Greville quadrature (GQ) in isogeometric analysis (IGA), the related integration weights depend on specific meshes and have to be re-calculated for each discretization, as makes it non-trivial to theoretically justify the accuracy of IGA with GQ. To address these issues, an equivalent explicit Greville quadrature (EEGQ) is firstly proposed through solving a local linear moment fitting problem, where the integration rules are mesh independent and a pre-evaluation can be readily implemented. The explicit form of EEGQ enables a theoretical analysis of the integration precision for GQ, which indicates a GQ rule resulting from pth degree basis functions also has a pth order integration precision. This insufficient integration precision then limits the algorithmic convergence orders of IGA with GQ to be 3, 2 and 4 regarding the L2, H1 and frequency error measures, regardless of the basis degree. Subsequently, this convergence obstacle associated with GQ is resolved by introducing an augmented explicit Greville quadrature (AEGQ) toward improving the integration precision. The proposed AEGQ is closely related to EEGQ for interior elements, but the integration rules of AEGQ for the near boundary elements are re-constructed to guarantee the precision of the overall domain integration. It is rationally shown that IGA with AEGQ leads to an optimal convergence for cubic basis functions, and a sub-optimal convergence for quartic and quintic basis functions. These theoretical predictions are consistently demonstrated by numerical results.