Finite strain primal interface formulation with consistently evolving stabilization

Abstract

A stabilized discontinuous Galerkin method is developed for general hyperelastic materials at finite strains. Starting from a mixed method incorporating Lagrange multipliers along the interface, the displacement formulation is systematically derived through a variational multiscale approach whereby the numerical fine scales are modeled via edge bubble functions. Analytical expressions that are free from user-defined parameters arise for the weighted numerical flux and stability tensor. In particular, the specific form taken by these derived quantities naturally accounts for evolving geometric nonlinearity as well as discontinuous material properties. The method is applicable both to problems containing nonconforming meshes or different element types at specific interfaces and to problems consisting of fully discontinuous numerical approximations. Representative numerical tests involving large strains and rotations are performed to confirm the robustness of the method. FINITE STRAIN INTERFACE FORMULATION WITH EVOLVING STABILIZATION 279The key factors impacting the robustness and efficiency of the DG method are the design of the so-called numerical flux and the penalty or stabilization parameter. Regarding the penalty parameter, studies for embedded interface problems [5] as well as discontinuous discrete approximations [2,13] have shown that selecting a value outside of an optimal range leads to issues with accuracy and stability. Choosing a value that is too low leads to loss of coercivity and results in an ill-posed discrete problem. On the other hand, choosing a value that is too high leads to ill conditioning in the stiffness matrix as well as to overly strict enforcement of the displacement jump condition. When the jump condition is strictly enforced, the computed response approaches that of a continuous Galerkin method, and the rationale for employing DG is lost. Dimensional analyses indicate that the stability parameter is a function of the element geometry, polynomial order, material properties, and the local interface topology; however, its magnitude can be elusive. The classical approach in the context of linear problems is to estimate the parameter through eigenvalue analyses [14,15]. More recently, in the context of embedded interface problems [4], the values of the parameter for linear simplex elements were determined by conducting a mathematical analysis of the coercivity condition. In particular, definitions for the numerical flux and penalty parameter that involved a weighting of the element size and material properties across the interface emerged. In contrast, the standard definition for the numerical flux in DG methods [12] assumes an equal weighting of the flux field from each side of the interface, which is postulated based on the assumption of mesh uniformity. Other interface methods have previously employed either area weighting [16] or stiffness weighting [5] alone. Additional techniques for defining the penalty parameter for linear problems include developments using bubble functions [17,1...

On the algorithmic and implementational aspects of a Discontinuous Galerkin method at finite strains

“…Recently, a stabilized Discontinuous Galerkin (DG) method was developed by Truster et al [1] for modeling large strain solid mechanics problems. The method, denoted herein as the Variational Multiscale Discontinuous Galerkin (VMDG) method, is consistently derived from an underlying Lagrange multiplier interface formulation and possesses a form analogous to the symmetric interior penalty Galerkin method [2].…”

“…However, amongst all the preceding methods, the design of the stability parameter is crucial to obtaining stable computed response, particularly in the nonlinear context. The idea of adapting [10] or evolving [1] the stability parameter with solution nonlinearity has received little attention in the literature. In summary, the preceding developments for solid mechanics DG methods would be greatly enhanced by qualitatively investigating the effects of method attributes such as symmetry and stability parameter definition upon the method performance such as accuracy and number of required Newton-Raphson iterations.…”

“…The objective of this paper is to propose a family of algorithmic modifications to the VMDG method [1] and systematically analyze their mathematical and algorithmic properties. These modifications consist of selectively neglecting terms in the interface nonlinear and incremental weak forms, resulting in both symmetric and nonsymmetric formulations reminiscent of other existing methods [3,4,8].…”

“…While studies on adjoint consistency have been performed for nonlinear fluid mechanics [7], the suboptimal L 2 convergence has not been demonstrated or investigated for DG methods for nonlinear solid mechanics. Fewer nonlinear DG methods employ symmetric tangent matrices; examples include [1,8]. Another noteworthy symmetric DG formulation was developed by Lew and co-workers [9], for which extensive stability and accuracy analyses were conducted in [10].…”

A discontinuous Galerkin method for non-linear electro-thermo-mechanical problems: application to shape memory composite materials

“…Truster et al [63,64] have derived a Variational Multiscale Discontinuous Galerkin (VMDG) method to account for geometric and material non-linearities in which computable expressions emerge during the course of the derivation for the stability tensor and numerical flux weighting tensors. Recently, DG has been used to solve coupled problems.…”