基于凝聚有限元的虚单元法提高弹性动力学模拟的临界时间步长
Virtual elements on agglomerated finite elements to increase the critical time step in elastodynamic simulations
摘要：
在本文中，我们使用一阶虚拟单元法（VEM）调查的多面体的形状质量的影响，在显式三维弹性动力学有限元（FE）模拟的临界时间步长的估计。低质量的有限元是常见的网格化现实复杂的组成部分，虽然四面体网格技术通常是强大的，网格算法不能保证高质量的网格任意几何形状或非水密的计算机辅助设计模型。为了在这种网格上进行可靠的模拟，我们考虑了具有四面体和棱柱体单元的有限元网格，这些单元具有形状不好的单元$-$四面体，其二面角接近$0^\circ$和$180^\circ$，以及具有三角形面的细长棱柱，三角形面具有短边$-$，并将这些“坏”单元与相邻单元聚集在一起，以形成更大的多面体虚拟单元。在每个单元上，利用单元特征值不等式估计临界时间步长。对于一套说明性的有限元网格与$ε$是一个网格坐标参数，导致网格质量差，我们表明，采用VEM的凝聚多面体产生的关键时间步长是不敏感的$ε\rightarrow 0$。显式动力学模拟表明，在网格上的解决方案的时间显着减少凝聚虚拟元素vis- $à$ -vis四面体网格上的锥形梁。
英文摘要：
In this paper, we use the first-order virtual element method (VEM) to investigate the effect of shape quality of polyhedra in the estimation of the critical time step for explicit three-dimensional elastodynamic finite element (FE) simulations. Low-quality finite elements are common when meshing realistic complex components, and while tetrahedral meshing technology is generally robust, meshing algorithms cannot guarantee high-quality meshes for arbitrary geometries or for non-water-tight computer-aided design models. For reliable simulations on such meshes, we consider FE meshes with tetrahedral and prismatic elements that have badly-shaped elements$-$tetrahedra with dihedral angles close to $0^\circ$ and $180^\circ$, and slender prisms with triangular faces that have short edges$-$and agglomerate such `bad' elements with neighboring elements to form a larger polyhedral virtual element. On each element, the element-eigenvalue inequality is used to estimate the critical time step. For a suite of illustrative finite element meshes with $ε$ being a mesh-coordinate parameter that leads to poor mesh quality, we show that adopting VEM on the agglomerated polyhedra yield critical time steps that are insensitive as $ε\rightarrow 0$. The significant reduction in solution time on meshes with agglomerated virtual elements vis-$à$-vis tetrahedral meshes is demonstrated through explicit dynamics simulations on a tapered beam.
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