A posteriori discontinuous Galerkin error estimator for linear elasticity

Bird, Robert; Coombs, William M.

doi:10.1016/j.amc.2018.08.039

Cited by 9 publications

(8 citation statements)

References 24 publications

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“…The MRDGCFE method combines very well with results from recent papers to tackle the two mentioned problems. In particular, the MRDGCFE method could be used to solve problems involving composite materials combining it with the error estimator presented in [7].…”

Section: Discussionmentioning

confidence: 99%

An adaptive composite discontinuous Galerkin method for elliptic problems on complicated domains with discontinuous coefficients

Giani

2020

Adv Comput Math

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In this paper, we introduce the Multi-Region Discontinuous Galerkin Composite Finite Element Method (MRDGCFEM) with hp-adaptivity for the discretization of second-order elliptic partial differential equations with discontinuous coefficients. This method allows for the approximation of problems posed on computational domains where the jumps in the diffusion coefficient form a micro-structure. Standard numerical methods could be used for such problems but the computational effort may be extremely high. Small enough elements to represent the underlying pattern in the diffusion coefficient have to be used. In contrast, the dimension of the underlying MRDGCFE space is independent of the complexity of the diffusion coefficient pattern. The key idea is that the jumps in the diffusion coefficient are no longer resolved by the mesh where the problem is solved; instead, the finite element basis (or shape) functions are adapted to the diffusion pattern allowing for much coarser meshes. In this paper, we employ hp-adaptivity on a series of test cases highlighting the practical application of the proposed numerical scheme.

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Section: Discussionmentioning

confidence: 99%

An adaptive composite discontinuous Galerkin method for elliptic problems on complicated domains with discontinuous coefficients

Giani

2020

Adv Comput Math

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“…This results in exponential convergence of the error with DOF for problems containing non-regular parts in the solution. 61,62 The DG norm for linear elasticity is…”

Section: Error Estimatementioning

confidence: 99%

“…This is consistent with observations on the performance of hp-adaptivity schemes for elliptic problems containing point singularities. 61,62 When a triangular element is refined in h it is refined homogeneously, as in Figure 1. The parent element is split into four new child triangle elements, all of which are similar to their parent and inherit the material properties and polynomial order of their parent.…”

Section: Hp-adaptivitymentioning

confidence: 99%

Adaptive configurational force‐based propagation for brittle and fatigue crack analysis

Bird

Coombs

2022

Numerical Meth Engineering

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This article presents a hpr-adaptive crack propagation method for highly accurate 2D crack propagation paths which requires no a priori knowledge of the tip solution. The propagation method is designed to be simple to implement, only hr-adaptivity is required, with the propagation step size independent of the initial mesh allowing users to obtain high fidelity crack path predictions for domains containing multiple cracks propagating at different rates. The proposed method also includes a crack path derefinement scheme, where elements away from the crack tip are derefined whilst elements close to the crack tip are small so capture the fidelity of the crack path. The result is that the propagation of cracks over an increasingly larger distances has negligible increased computational effort and effect on the propagation path prediction. The linear elastic problem is solved using the hp discontinuous Galerkin symmetric interior penalty finite element method, which is post-processed to obtain the configurational force at each tip to a user defined accuracy. Several numerical examples are used to demonstrate the accuracy, efficiency, and capability of the method. Due to the method's high accuracy crack path solutions of benchmark problems that are prolifically used in the literature are challenged.

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“…The symmetric version of the method, SIPDG, dates back to a paper by Douglas 31 and has been thoroughly studied. 5,32 The main drawback of this approach is the necessity to evaluate the penalty (or stability) parameters. In previous studies, significant emphasis has been placed on the stability of various versions of the IPDG method.…”

Section: Introductionmentioning

confidence: 99%

Penalty‐free discontinuous Galerkin method

Jaśkowiec,

Sukumar

2024

Numerical Meth Engineering

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In this article, we present a new high‐order discontinuous Galerkin (DG) method, in which neither a penalty parameter nor a stabilization parameter is needed. We refer to this method as penalty‐free DG. In this method, the trial and test functions belong to the broken Sobolev space, in which the functions are in general discontinuous on the mesh skeleton and do not meet the Dirichlet boundary conditions. However, a subset can be distinguished in this space, where the functions are continuous and satisfy the Dirichlet boundary conditions, and this subset is called admissible. The trial solution is chosen to lie in an augmented admissible subset, in which a small violation of the continuity condition is permitted. This subset is constructed by applying special augmented constraints to the linear combination of finite element basis functions. In this approach, all the advantages of the DG method are retained without the necessity of using stability parameters or numerical fluxes. Several benchmark problems in two dimensions (Poisson equation, linear elasticity, hyperelasticity, and biharmonic equation) on polygonal (triangles, quadrilateral, and weakly convex polygons) meshes as well as a three‐dimensional Poisson problem on hexahedral meshes are considered. Numerical results are presented that affirm the sound accuracy and optimal convergence of the method in the norm and the energy seminorm.

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A posteriori discontinuous Galerkin error estimator for linear elasticity

Cited by 9 publications

References 24 publications

An adaptive composite discontinuous Galerkin method for elliptic problems on complicated domains with discontinuous coefficients

An adaptive composite discontinuous Galerkin method for elliptic problems on complicated domains with discontinuous coefficients

Adaptive configurational force‐based propagation for brittle and fatigue crack analysis

Penalty‐free discontinuous Galerkin method

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