{"title":"A Hu-Washizu variational approach to self-stabilized quadrilateral Virtual Elements: 2D linear elastodynamics","authors":"","doi":"10.1007/s00466-023-02438-0","DOIUrl":null,"url":null,"abstract":"<h3>Abstract</h3> <p>A recent mixed formulation of the Virtual Element Method in 2D elastostatics, based on the Hu-Washizu variational principle, is here extended to 2D elastodynamics. The independent modeling of the strain field, allowed by the mixed formulation, is exploited to derive first order quadrilateral Virtual Elements (VEs) not requiring a stabilization (namely, self-stabilized VEs), in contrast to the standard VEs, where an artificial stabilization is always required for first order quads. Lumped mass matrices are derived using a novel approach, based on an integration scheme that makes use of nodal values only, preserving the correct mass in the case of rigid-body modes. In the case of implicit time integration, it is shown how the combination of a self-stabilized stiffness matrix with a self-stabilized lumped mass matrix can produce excellent performances both in the compressible and quasi-incompressible regimes with almost negligible sensitivity to element distortion. Finally, in the case of explicit dynamics, the performances of the different types of derived VEs are analyzed in terms of their critical time-step size. </p>","PeriodicalId":55248,"journal":{"name":"Computational Mechanics","volume":"58 1","pages":""},"PeriodicalIF":3.7000,"publicationDate":"2024-02-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Computational Mechanics","FirstCategoryId":"5","ListUrlMain":"https://doi.org/10.1007/s00466-023-02438-0","RegionNum":2,"RegionCategory":"工程技术","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q1","JCRName":"MATHEMATICS, INTERDISCIPLINARY APPLICATIONS","Score":null,"Total":0}
引用次数: 0
Abstract
A recent mixed formulation of the Virtual Element Method in 2D elastostatics, based on the Hu-Washizu variational principle, is here extended to 2D elastodynamics. The independent modeling of the strain field, allowed by the mixed formulation, is exploited to derive first order quadrilateral Virtual Elements (VEs) not requiring a stabilization (namely, self-stabilized VEs), in contrast to the standard VEs, where an artificial stabilization is always required for first order quads. Lumped mass matrices are derived using a novel approach, based on an integration scheme that makes use of nodal values only, preserving the correct mass in the case of rigid-body modes. In the case of implicit time integration, it is shown how the combination of a self-stabilized stiffness matrix with a self-stabilized lumped mass matrix can produce excellent performances both in the compressible and quasi-incompressible regimes with almost negligible sensitivity to element distortion. Finally, in the case of explicit dynamics, the performances of the different types of derived VEs are analyzed in terms of their critical time-step size.
摘要 基于Hu-Washizu变分原理的二维弹性力学中虚拟元素法的最新混合公式在此被扩展到二维弹性力学中。混合公式允许应变场的独立建模，利用这种独立建模可以推导出不需要稳定化的一阶四边形虚拟元素（VE）（即自稳定虚拟元素），与标准虚拟元素相反，一阶四边形虚拟元素总是需要人工稳定化。我们采用一种新方法导出了集合质量矩阵，该方法基于一种仅使用节点值的积分方案，在刚体模态情况下保留了正确的质量。在隐式时间积分的情况下，演示了自稳定刚度矩阵与自稳定块状质量矩阵的结合如何在可压缩和准不可压缩状态下产生出色的性能，对元素变形的敏感性几乎可以忽略不计。最后，在显式动力学情况下，根据临界时间步长分析了不同类型衍生 VE 的性能。
期刊介绍：
The journal reports original research of scholarly value in computational engineering and sciences. It focuses on areas that involve and enrich the application of mechanics, mathematics and numerical methods. It covers new methods and computationally-challenging technologies.
Areas covered include method development in solid, fluid mechanics and materials simulations with application to biomechanics and mechanics in medicine, multiphysics, fracture mechanics, multiscale mechanics, particle and meshfree methods. Additionally, manuscripts including simulation and method development of synthesis of material systems are encouraged.
Manuscripts reporting results obtained with established methods, unless they involve challenging computations, and manuscripts that report computations using commercial software packages are not encouraged.
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