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

Abstract

a b s t r a c tIn this work, algorithmic modifications are proposed and analyzed for a recently developed stabilized finite strain Discontinuous Galerkin (DG) method. The distinguishing feature of the original method, referred to as VMDG, is a consistently derived expression for the numerical flux and stability tensor that account for evolving material and geometric nonlinearity in the vicinity of the interface. Herein, the proposed modifications involve simplifications to the residual force vector and tangent stiffness matrix of the VMDG method that lead to formulations similar to other existing DG methods but retain the enhanced definition for the stability parameters. The primary objective is to reduce the costs associated with implementing the method as well as executing simulations while retaining accuracy and flexibility, thereby making the formulation amenable to boarder material classes such as inelasticity. Each simplification has associated implications on the mathematical and algorithmic properties of the method, such as L 2 convergence rate, solution accuracy, continuity enforcement, and stability of the nonlinear equation solver. These implications are carefully quantified and assessed through a comprehensive numerical performance study. The range of two and three dimensional problems under consideration involves both isotropic and anisotropic materials. Both triangular and quadrilateral element types are employed along with h and p refinement. The ability of the proposed methods to produce stable and accurate results for such a broad class of problems is highlighted.

A numerical method for finite-strain mechanochemistry with localised chemical reactions treated using a Nitsche approach

“…since it is more convenient to work with the derivative of the transposed first Piola-Kirchhoff stress tensor when the transformation between the configurations is performed. Weak form (33) is similar to the weak form of [38,39], however, with different interface stabilisation terms. In [38,39], the stabilised discontinuous Galerkin (DG) method was proposed for large deformations, while in [3], this approach was extended to include models of damage and debonding.…”

Discontinuous Galerkin Method for Frictional Interface Dynamics

“…Details concerning the integration of the interface terms and the definitions of the bubble functions used within the stability tensors in Eq. (18) are found in Masud et al (2012), Truster et al (2015b), and Truster and Masud (2014). For the frictional problem, the Newton-Raphson algorithm with consistent linearization is employed to solve Eq.…”

“…The first simulation on this mesh will employ the definitions of the stability tensor and flux weights defined in Eqs. (18) and (23), and the results will be labeled as VMDG according to the name given in Truster et al (2015b). Second, the results produced by the classical parameters from the DG method (Arnold et al 2002) are also investigated.…”

“…(18) for τ ðαÞ s provided excellent numerical performance in the tests in this paper. Herein, the simple polynomial bubble functions provided inTruster et al (2015b) andTruster and Masud (2014) are chosen to represent the fine-scale bubbles b ðαÞ s within Eq. (18).…”

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.…”