Elastoplasticity with linear tetrahedral elements: A variational multiscale method

Abstract

Summary We present a computational framework for the simulation of J2‐elastic/plastic materials in complex geometries based on simple piecewise linear finite elements on tetrahedral grids. We avoid spurious numerical instabilities by means of a specific stabilization method of the variational multiscale kind. Specifically, we introduce the concept of subgrid‐scale displacements, velocities, and pressures, approximated as functions of the governing equation residuals. The subgrid‐scale displacements/velocities are scaled using an effective (tangent) elastoplastic shear modulus, and we demonstrate the beneficial effects of introducing a subgrid‐scale pressure in the plastic regime. We provide proofs of stability and convergence of the proposed algorithms. These methods are initially presented in the context of static computations and then extended to the case of dynamics, where we demonstrate that, in general, naïve extensions of stabilized methods developed initially for static computations seem not effective. We conclude by proposing a dynamic version of the stabilizing mechanisms, which obviates this problematic issue. In its final form, the proposed approach is simple and efficient, as it requires only minimal additional computational and storage cost with respect to a standard finite element relying on a piecewise linear approximation of the displacement field.

An energy‐stable mixed formulation for isogeometric analysis of incompressible hyperelastodynamics

“…In the meantime, the stabilized finite element method, as a technique initially developed for computational fluid dynamics, has been extended to solid mechanics based on various variational formulations. [12][13][14][15][16][17][18] Using the stabilized formulation allows one to interpolate physical quantities with equal-order interpolations. This feature gives practitioners maximum flexibility in mesh generation and numerical implementation and allows for low-order elements, which are more robust than their higher-order counterparts.…”

“…We invoke the generalized-method 42 for the temporal discretization of the weak form problem (10)- (12). The time interval [0, T] is divided into a set of n ts subintervals of size Δt n ∶= t n+1 −t n delimited by a discrete time vector {t n } n ts n=0 .…”

A blended transient/quasistatic Lagrangian framework for salt tectonics simulations with stabilized tetrahedral finite elements

“…An additional element of difference of our approach with respect to the work in References 37,38 is the use of advanced triangular/tetrahedral finite element discretizations for solid mechanics in the numerical solution of the discrete variational equations. The authors of the present work, over the course of many years, have developed a framework of advanced tetrahedral discretizations 44‐53 for solid mechanics based on the variational multiscale method 54‐56 . The proposed class of triangular/tetrahedral finite elements provides a more flexible and robust computational framework with respect to standard quadrilateral/hexahedral elements used in large deformation problems, and has also the potential for improved integration with mesh adaptation strategies, although this last aspect is deemed beyond the scope of the present work.…”

Novel quadratic Bézier triangular and tetrahedral elements using existing mesh generators: Extension to nearly incompressible implicit and explicit elastodynamics in finite strains

“…Some notable contributions towards finite element schemes for performing the simulations of problems modelled with nearly incompressible and elastoplastic material models using triangular/tetrahedral elements are as follows: fractional‐step–based projection schemes by Zienkiewicz and collaborators; averaged nodal pressure approach by Bonet and Burton; node‐based uniform strain elements by Dohrmann et al; stabilised nodally integrated elements by Puso and Solberg; smoothed FEM by He et al; F‐bar patch for triangular and tetrahedral elements by de Souza Neto et al; 15‐node tetrahedral element with reduced‐order integration by Danielson; mean‐strain 10‐node tetrahedral with energy‐sampling stabilisation by Pakravan and Krysl; discontinuous Galerkin methods by Hansbo and Larson, Noels and Radovitzky, and Nguyen and Peraire; mixed‐stabilised formulations for solid mechanics problems by Franca et al, Maniatty and collaborators, Masud and Xia, Chiumenti and Cervera group, and Scovazzi et al; and schemes based on first‐order conservation laws by Bonet and co‐workers …”