Constructions of some minimal finite element systems

Within the framework of finite element systems, we show how spaces of differential forms may be constructed, in such a way that they are equipped with commuting interpolators and contain prescribed functions, and are minimal under these constraints. We show how various known mixed finite element spaces fulfill such a design principle, including trimmed polynomial differential forms, serendipity elements and TNT elements. We also comment on virtual element methods and provide a dimension formula for minimal compatible finite element systems containing polynomials of a given degree on hypercubes.Comment: Various minor changes, based on suggestions of paper referee

Some basic formulations of the virtual element method (VEM) for finite deformations

Abstract We present a general virtual element method (VEM) framework for finite elasticity, which emphasizes two issues: element-level volume change (volume average of the determinant of the deformation gradient) and stabilization. To address the former issue, we provide exact evaluation of the average volume change in both 2D and 3D on properly constructed local displacement spaces. For the later issue, we provide a new stabilization scheme that is based on the trace of the material tangent modulus tensor, which captures highly heterogeneous and localized deformations. Two VEM formulations are presented: a two-field mixed and an equivalent displacement-based, which is free of volumetric locking. Convergence and accuracy of the VEM formulations are verified by means of numerical examples, and engineering applications are demonstrated

3D mixed virtual element formulation for dynamic elasto-plastic analysis

The virtual element method (VEM) for dynamic analyses of nonlinear elasto-plastic problems undergoing large deformations is outlined within this work. VEM has been applied to various problems in engineering, considering elasto-plasticity, multiphysics, damage, elastodynamics, contact- and fracture mechanics. This work focuses on the extension of VEM formulations towards dynamic elasto-plastic applications. Hereby low-order ansatz functions are employed in three dimensions with elements having arbitrary convex or concave polygonal shapes. The formulations presented in this study are based on minimization of potential function for both the static as well as the dynamic behavior. Additionally, to overcome the volumetric locking phenomena due to elastic and plastic incompressibility conditions, a mixed formulation based on a Hu-Washizu functional is adopted. For the implicit time integration scheme, Newmark method is used. To show the model performance, various numerical examples in 3D are presented

p- and hp- virtual elements for the Stokes problem

We analyse the p- and hp-versions of the virtual element method (VEM) for the the Stokes problem on a polygonal domain. The key tool in the analysis is the existence of a bijection between Poisson-like and Stokes-like VE spaces for the velocities. This allows us to re-interpret the standard VEM for Stokes as a VEM, where the test and trial discrete velocities are sought in Poisson-like VE spaces. The upside of this fact is that we inherit from [7] an explicit analysis of best interpolation results in VE spaces, as well as stabilization estimates that are explicit in terms of the degree of accuracy of the method. We prove exponential convergence of the hp-VEM for Stokes problems with regular right-hand sides. We corroborate the theoretical estimates with numerical tests for both the p- and hp-versions of the method