Adaptive stabilization of discontinuous Galerkin methods for nonlinear elasticity: Motivation, formulation, and numerical examples

The goal of this paper is to motivate, introduce and demonstrate a novel approach to stabilizing discontinuous Galerkin (DG) methods in nonlinear elasticity problems. The stabilization term adapts to the solution of the problem by locally changing the size of a penalty term on the appearance of discontinuities, with the goal of better approximating the solution. Consequently, it is called an adaptive stabilization strategy. The need for such a strategy is motivated through two- and three-dimensional examples in nonlinear elasticity. The proposed scheme is simple to implement and compute, and its performance is demonstrated with two- and three-dimensional numerical examples. The accuracy of the proposed method is compared against a conforming method of the same order and a DG method with a traditional form of stabilization. Results for trilinear hexahedral elements indicate that the new stabilization strategy is more robust and more accurate when compared to a traditional form of stabilization. However, conforming trilinear hexahedral elements proved to be more computationally efficient for the examples shown here. A two-dimensional example with linear triangular elements showed comparable performances between the proposed method and a conforming one.