时域谱元法的质量特性模型及其构建方法

The mass property model and its implementation in the time-domain spectral element method

研究了构建时域谱单元质量特性模型的数学机制,针对时域切比雪夫谱单元和勒让德谱单元建立了一种直接导出谱单元一致质量矩阵和集中质量矩阵的统一数学方法,对比分析两种谱单元质量特性模型的特征,并从物理角度探讨了谱单元质量特性模型的合理性.研究表明,数值积分点与谱单元节点选取是否一致是决定时域谱单元形成一致质量模型或集中质量模型的根本原因,当采用高斯-勒让德积分计算谱单元质量模型时将导出一致质量矩阵,而采用高斯-洛巴托积分则导出集中质量矩阵.而集中质量模型更具有物理合理性,两种谱单元质量特性模型优劣相当,均可取得很好的动力问题分析结果.

The mathematical mechanism of constructing mass property model for the timedomain spectral elements is studied in this paper. A unified mathematical method for directly deriving consistent and lumped mass matrix is established for the time-domain Chebyshev and Legendre spectral elements. The characteristics of two mass property models of the spectral elements are analyzed through comparison. Meanwhile,the rationality of mass property model of spectral element is discussed from physical perspective. This study reveals that the formation of consistent or lumped mass matrix in time-domain spectral elements depends on whether the quadrature points are coincident with the element nodes or not. To be specific,the GaussLegendre quadrature results in consistent mass matrix for spectral elements,and the GaussLobatto quadrature leads to lumped mass matrix. The lumped mass matrix is more reasonable in physics. The two mass property models of spectral elements have comparable performance and they can both achieve good results for dynamic problems.