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

Eyck, Alex Ten; Celiker, Fatih; Lew, Adrián J.

doi:10.1016/j.cma.2008.02.020

Cited by 48 publications

(64 citation statements)

References 19 publications

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“…In this section we develop an energetically rigorous and computationally efficient way to introduce a crack in the non-local implicit CDM resolution by combining the extrinsic cohesive law with a discontinuous Galerkin (DG) approach. The DG method is developed to solve the non-local CDM weak formulation, following classical derivations of DG methods for (non-linear) elliptic equations, see [52,53,54,55,56] among many others, while accounting for the existence of cracked surfaces within the continuum as suggested in [22,23,24,25]. In this hybrid non-local DG/ECL method, interface elements are therefore inserted between bulk elements at the beginning of the simulation and continuity before the transition to the CZM is ensured by having recourse to the consistent DG interface terms.…”

Section: Hybrid Non-local Implicit Discontinuous Galerkin/extrinsic Cmentioning

confidence: 99%

Elastic damage to crack transition in a coupled non-local implicit discontinuous Galerkin/extrinsic cohesive law framework

Becker

Noels

2014

Computer Methods in Applied Mechanics and Engineering

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One current challenge related to computational fracture mechanics is the modeling of ductile fracture and in particular the damage to crack transition. On the one hand, continuum damage models, especially in their non-local formulation which avoids the loss of solution uniqueness, can capture the material degradation process up to the localization of the damage, but are unable to represent a discontinuity in the structure. On the other hand cohesive zone methods can represent the process zone at the crack tip governing the crack propagation, but cannot account for the diffuse material damaging process.In this paper we propose to combine, in a small deformations setting, a non-local elastic damage model with a cohesive zone model. This combination is formulated within a discontinuous Galerkin finite element discretization. Indeed this DG weak formulation can easily be developed in a non-local implicit form and naturally embeds interface elements that can be used to integrate the traction separation law of the cohesive zone model. The method remains thus consistent and computationally efficient as compared to other cohesive element approaches.The effects of the damage to crack transition and of the mesh discretization are respectively studied on the compact tension specimen and on the double-notched specimen, demonstrating the efficiency and accuracy of the method.

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Section: Hybrid Non-local Implicit Discontinuous Galerkin/extrinsic Cmentioning

confidence: 99%

Elastic damage to crack transition in a coupled non-local implicit discontinuous Galerkin/extrinsic cohesive law framework

Becker

Noels

2014

Computer Methods in Applied Mechanics and Engineering

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“…As a first step towards a more comprehensive study, elastin was assumed to be an isotropic linear elastic material immersed in incompressible water. With proper preconditioning of the linear system [17], the deformation Fig. 3.…”

Section: Examplesmentioning

confidence: 99%

“…The standard strategy in this case is to add a stabilization term to (5), in the form of a potential energy cost for each discontinuity in the solution, see [16,17]. When the energetic cost of jumps is large enough, a stable scheme is recovered.…”

Section: Application: Nonlinear Elasticitymentioning

confidence: 99%

“…For classical linear elasticity the "right" size of the stabilization term is known [13], but in general, its fully automatic and efficient selection is still an open problem. We have recently made some progress by introducing the idea of adaptive stabilization [16,17], but more comprehensive solutions may still be possible, see Fig. 6.…”

Section: Application: Nonlinear Elasticitymentioning

confidence: 99%

“…This is illustrated in Fig. 4 from [17], for a two-dimensional case. The contrast in the performance of both methods is often larger for three-dimensional computations.…”

Section: Examplesmentioning

confidence: 99%

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Some Applications of Discontinuous Galerkin Methods in Solid Mechanics

2008

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A discontinuous Galerkin formulation of non‐linear Kirchhoff–Love shells

Noels

2008

Numerical Meth Engineering

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SUMMARYDiscontinuous Galerkin (DG) methods provide a means of weakly enforcing the continuity of the unknownfield derivatives and have particular appeal in problems involving high-order derivatives. This feature has previously been successfully exploited (Comput. Methods Appl. Mech. Eng. 2008; 197:2901-2929 to develop a formulation of linear Kirchhoff-Love shells considering only the membrane and bending responses. In this proposed one-field method-the displacements are the only unknowns, while the displacement field is continuous, the continuity in the displacement derivative between two elements is weakly enforced by recourse to a DG formulation. It is the purpose of the present paper to extend this formulation to finite deformations and non-linear elastic behaviors. While the initial linear formulation was relying on the direct linear computation of the effective membrane stress and effective bending couplestress from the displacement field at the mid-surface of the shell, the non-linear formulation considered implies the evaluation of the general stress tensor across the shell thickness, leading to a reformulation of the internal forces of the shell. Nevertheless, since the interface terms resulting from the discontinuous Galerkin method involve only the resultant couple-stress at the edges of the shells, the extension to non-linear deformations is straightforward.

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

Cited by 48 publications

References 19 publications

Elastic damage to crack transition in a coupled non-local implicit discontinuous Galerkin/extrinsic cohesive law framework

Elastic damage to crack transition in a coupled non-local implicit discontinuous Galerkin/extrinsic cohesive law framework

Some Applications of Discontinuous Galerkin Methods in Solid Mechanics

A discontinuous Galerkin formulation of non‐linear Kirchhoff–Love shells

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