Variationally consistent inertia templates for B-spline- and NURBS-based FEM: Inertia scaling and customization
Variationally consistent inertia templates for B-spline- and NURBS-based FEM: Inertia scaling and customization
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DOI：
10.1016/j.cma.2017.08.035
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发表时间：
2017-11
影响因子：
7.2
通讯作者：
Anne-Kathrin Schaeuble;A. Tkachuk;M. Bischoff
中科院分区：
文献类型：
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作者：
Anne-Kathrin Schaeuble;A. Tkachuk;M. Bischoff
在这篇文章中，变分一致的惯性模板B样条和基于NURBS的有限元提出了一个统一的概念，为两个不同的目的：定制的模板允许建设质量和倒数质量所需的属性，如高阶精度或改善色散行为;惯性缩放允许显式动力学通过增加临界时间步长来显著加速。场parametrized功能在以前的作品作者的小组，但修改的主要变量，即位移，速度和质量特定的线性动量。后者允许在整个域中保持非恒定密度的质量，因此是对Tkachuk和Bischoff（2015）中提出的公式的增强。由于专注于B样条和基于NURBS的有限元，所提出的模板提供了行和集中质量矩阵的替代方案，该矩阵仅具有与多项式阶数无关的二阶精度。早先提出的代数构造的高阶质量的文献可以重建在变分设置这里描述的特殊情况。此外，可以从模板构建更高阶的倒数质量。它们对于显式动力学特别有吸引力，因为与线性问题的集中质量相比，每个时间步长没有额外的费用。对于非线性问题，预计只有很小的开销，但本文只关注线性问题，主要是未变形的网格。将该方法调整到惯性缩放，在本文的示例中获得了最大本征频率降低25%-40%，而精度高于集中或一致质量。
In this contribution, variationally consistent inertia templates for B-spline and NURBS-based finite elements are proposed as a unified concept for two different purposes:Customizationof the template allows construction of masses and reciprocal masses with desired properties like higher-order accuracy or improved dispersion behavior;Inertia scalingallows substantial speed-up for explicit dynamics by increased critical time steps.The derivation of the template is based on a three-field parametrized functional as in previous works of the authors’ group, but with modified primary variables, namely displacement, velocity and mass-specific linear momentum. The latter allows for mass-preservation for non-constant density throughout the domain and is therefore an enhancement to the formulation proposed in Tkachuk and Bischoff (2015).With the focus on B-spline and NURBS-based finite elements, the proposed template provides alternatives to the row-sum-lumped mass matrix, which is only 2nd order accurate independent of the polynomial order. Earlier proposed algebraically constructed higher order masses from the literature can be reconstructed in the variational setting described here as special instances. Furthermore, higher-order reciprocal masses can be constructed from the template. They are especially attractive for explicit dynamics as there is no extra expense per time step compared with lumped mass for linear problems. For non-linear problems only small overhead is expected, but this paper focuses on linear problems only and mainly undistorted meshes. Tuning the method towards inertia scaling, a reduction of the maximum eigenfrequency by 25%–40% is obtained in the examples herein, whereas the accuracy is higher than for lumped or consistent mass.