A mid-node mass lumping scheme is proposed to formulate the lumped mass matrices of serendipity elements for accurate structural vibration analysis. Since the row-sum technique leads to unacceptable negative lumped mass components for serendipity elements, the diagonal scaling HRZ method is frequently employed to construct lumped mass matrices of serendipity elements. In this work, through introducing a lumped mass matrix template that includes the HRZ lumped mass matrix as a special case, an analytical frequency accuracy measure is rationally derived with particular reference to the classical eight-node serendipity element. The theoretical results clearly reveal that the standard HRZ mass matrix actually does not offer the optimal frequency accuracy in accordance with the given lumped mass matrix template. On the other hand, by employing the nature of non-negative shape functions associated with the mid-nodes of serendipity elements, a mid-node lumped mass matrix (MNLM) formulation is introduced for the mass lumping of serendipity elements without corner nodal mass components, which essentially corresponds to the optimal frequency accuracy in the context of the given lumped mass matrix template. Both theoretical and numerical results demonstrate that MNLM yields better frequency accuracy than the standard HRZ lumped mass matrix formulation for structural vibration analysis.

中文翻译：

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一种利用偶然性有限元进行精确结构振动分析的中节点质量集总方案

提出了一种中间节点质量集总方案来制定偶然性元素的集总质量矩阵，以进行精确的结构振动分析。由于行和技术导致意外元素的不可接受的负集中质量分量，因此经常采用对角缩放 HRZ 方法来构建意外元素的集中质量矩阵。在这项工作中，通过引入包含 HRZ 集总质量矩阵作为特例的集总质量矩阵模板，特别参考经典的八节点偶然性元素，合理地推导了解析频率精度度量。理论结果清楚地表明，标准 HRZ 质量矩阵实际上并没有根据给定的集总质量矩阵模板提供最佳频率精度。另一方面，通过利用与偶发元素中节点相关的非负形状函数的性质，引入了中节点集总质量矩阵 (MNLM) 公式，用于没有角节点质量分量的偶发元素的质量集总，它基本上对应于在给定的集总质量矩阵模板的上下文中的最佳频率精度。理论和数值结果都表明，MNLM 比用于结构振动分析的标准 HRZ 集总质量矩阵公式产生更好的频率精度。这基本上对应于给定集中质量矩阵模板的上下文中的最佳频率精度。理论和数值结果都表明，MNLM 比用于结构振动分析的标准 HRZ 集总质量矩阵公式产生更好的频率精度。这基本上对应于给定集中质量矩阵模板的上下文中的最佳频率精度。理论和数值结果都表明，MNLM 比用于结构振动分析的标准 HRZ 集总质量矩阵公式产生更好的频率精度。