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SUMMARYThis paper presents the formulation and a partial analysis of a class of discontinuous Galerkin methods for quasistatic nonlinear elasticity problems. These methods are endowed with several salient features. The equations that define the numerical scheme are the Euler-Lagrange equations of a onefield variational principle, a trait that provides an elegant and simple derivation of the method. In consonance with general discontinuous Galerkin formulations, it is possible within this framework to choose different numerical fluxes. Numerical evidence suggests the absence of locking at near incompressible conditions in the finite deformations regime when piecewise linear elements are adopted. Finally, a conceivable surprising characteristic is that, as demonstrated with numerical examples, these methods provide a given accuracy level for a comparable, and often lower, computational cost than conforming formulations.Stabilization is occasionally needed for discontinuous Galerkin methods in linear elliptic problems. In this paper we propose a sufficient condition for the stability of each linearized nonlinear elastic problem that naturally includes material and geometric parameters; the latter needed to account for buckling. We then prove that when a similar condition is satisfied by the discrete problem, the method provides stable linearized deformed configurations upon the addition of a standard stabilization term.We conclude by discussing the complexity of the implementation, and propose a computationally efficient approach that avoids looping over both elements and element faces. Several numerical examples are then presented in two and three dimensions that illustrate the performance of a selected discontinuous Galerkin method within the class.

Summary.A brief overview of our recent work on applications of discontinuous Galerking methods in solid mechanics is provided. The discussion is light on technical details, and rather emphasizes key ideas, advantages and disadvantages of the approach, illustrating these with several numerical examples.

This paper proposes and analyzes an adaptive stabilization strategy for enhanced strain (ES) methods applied to quasistatic non-linear elasticity problems. The approach is formulated for any type of enhancements or material models, and it is distinguished by the fact that the stabilization term is solution dependent. The stabilization strategy is first constructed for general linearized elasticity problems, and then extended to the non-linear elastic regime via an incremental variational principle. A heuristic choice of the stabilization parameters is proposed, which in the numerical examples proved to provide stable approximations for a large range of deformations, different problems and material models. We also provide explicit lower bounds for the stabilization parameters that guarantee that the method will be stable. These are not advocated, since they are generally larger than the ones based on heuristics, and hence prone to deteriorate the locking-free behavior of ES methods. Numerical examples with two different non-linear elastic models in thin geometries and incompressible situations show that the method remains stable and locking free over a large range of deformations. Finally, the method is strongly based on earlier developments for discontinuous Galerkin methods, and hence throughout the paper we offer a perspective about the similarities between the two.A. T. EYCK AND A. LEW INTRODUCTIONThis paper proposes and analyzes an adaptive stabilization technique for enhanced strain (ES) methods applied to problems in non-linear elasticity. In this approach the stabilization term is solution dependent, and hence it is termed as adaptive stabilization. The idea and applicability of the method are not specific to any particular material model or type of enhancement.ES methods construct approximations to solutions of elasticity problems by enriching the space that approximates strains within each element. The added strain fields are not required to satisfy any kinematic compatibility condition across element boundaries; the method is hence incompatible. ‡ The extra degrees of freedom provide unquestionable advantages, such as the absence of volumetric and shear locking for incompressible and very thin geometries, respectively, when low-order elements are adopted (see, e.g.[1]). However, the use of incompatible approximations exposes the method to the appearance of numerical instabilities. These instabilities would not be observed in the exact solution, or in approximate ones obtained with most conforming methods.ES methods are not the only incompatible finite element methods. For example, discontinuous Galerkin (DG) methods also suffer from this artifact [2]. This defect not only degrades the benefits of incompatible methods but also their overall versatility and reliability as well. The heart of the problem can be easily described, at least in the linearized elasticity case. In the exact linearized elasticity problem, we seek a minimizer of the potential energy functional among all kinematically compatible str...