Galerkin Method For Linear Elasticity Research Articles

A high order unfitted hybridizable discontinuous Galerkin method for linear elasticity

Abstract This work analyses a high-order hybridizable discontinuous Galerkin (HDG) method for the linear elasticity problem in a domain not necessarily polyhedral. The domain is approximated by a polyhedral computational domain where the HDG solution can be computed. The introduction of the rotation as one of the unknowns allows us to use the gradient of the displacements to obtain an explicit representation of the boundary data in the computational domain. The boundary data is transferred from the true boundary to the computational boundary by line integrals, where the integrand depends on the Cauchy stress tensor and the rotation. Under closeness assumptions between the computational and true boundaries, the scheme is shown to be well-posed and optimal error estimates are provided even in the nearly incompressible case. Numerical experiments in two dimensions are presented.

A locking-free stabilized embedded discontinuous Galerkin method for linear elasticity with strong symmetric stress and continuous displacement trace approximation

A locking-free stabilized embedded discontinuous Galerkin method for linear elasticity with strong symmetric stress and continuous displacement trace approximation

Locking-Free Enriched Galerkin Method for Linear Elasticity

Related DatabasesWeb of Science You must be logged in with an active subscription to view this.Article DataHistorySubmitted: 14 January 2021Accepted: 14 September 2021Published online: 05 January 2022Keywordslinear elasticity, enriched Galerkin method, locking-free, operator preconditioningAMS Subject Headings65N30, 65N55, 65F08, 74S05Publication DataISSN (print): 0036-1429ISSN (online): 1095-7170Publisher: Society for Industrial and Applied MathematicsCODEN: sjnaam

A GPU-Accelerated Hybridizable Discontinuous Galerkin Method for Linear Elasticity

A GPU-Accelerated Hybridizable Discontinuous Galerkin Method for Linear Elasticity

A Mixed Discontinuous Galerkin Method for Linear Elasticity with Strongly Imposed Symmetry

In this paper, we study a mixed discontinuous Galerkin (MDG) method to solve linear elasticity problem with arbitrary order discontinuous finite element spaces in $d$-dimension ($d=2,3$). This method uses polynomials of degree $k+1$ for the stress and of degree $k$ for the displacement ($k\geq 0$). The mixed DG scheme is proved to be well-posed under proper norms. Specifically, we prove that, for any $k \geq 0$, the $H({\rm div})$-like error estimate for the stress and $L^2$ error estimate for the displacement are optimal. We further establish the optimal $L^2$ error estimate for the stress provided that the $\mathcal{P}_{k+2}-\mathcal{P}_{k+1}^{-1}$ Stokes pair is stable and $k \geq d$. We also provide numerical results of MDG showing that the orders of convergence are actually sharp.

A Staggered Cell-Centered DG Method for Linear Elasticity on Polygonal Meshes

We develop a new numerical method, namely, a locking-free staggered cell-centered discontinuous Galerkin method for linear elasticity on fairly general meshes. The method is well suited for general meshes possibly including hanging nodes; in particular, it does not deteriorate when the mesh becomes highly distorted. There are three unknowns involved in our formulation: stress, displacement, and rotation. The continuities of the three unknowns are staggered on the interelement boundaries. In addition, the symmetry of the stress tensor is imposed weakly by the introduction of Lagrange multipliers. Optimal a priori error estimates covering low regularities in $L^2$ errors of stress, displacement, and rotation are given; in addition, the locking-free error estimates are also investigated. Numerical experiments confirm the theoretical findings and verify the flexibility to rough grids and the locking-free property of the proposed method.

A lowest-order weak Galerkin method for linear elasticity

The lowest-order weak Galerkin (WG) method is considered for linear elasticity based on the displacement formulation. The new method approximates the displacement using piecewise constant vector functions both inside and on the boundary of each mesh element and its bilinear form does not require a stabilization term for the existence and uniqueness of the solution. A-priori error estimates of optimal order in the discrete H 1 - and L 2 -norms for the displacement are proved when the solution is smooth. The error estimates are independent of the Lamé constant λ , thus the performance of the new method does not deteriorate as the elastic material becomes incompressible. Further, a simple post-processing technique to obtain a numerical approximation of the stress is presented. A careful error analysis reveals that the L 2 -norm error in the stress is also optimal and independent of λ . Several numerical experiments confirm the locking-free property and optimal convergence rates of the new method.

A superconvergent hybridisable discontinuous Galerkin method for linear elasticity

SummaryThe first superconvergent hybridisable discontinuous Galerkin method for linear elastic problems capable of using the same degree of approximation for both the primal and mixed variables is presented. The key feature of the method is the strong imposition of the symmetry of the stress tensor by means of the well known and extensively used Voigt notation, circumventing the use of complex mathematical concepts to enforce the symmetry of the stress tensor either weakly or strongly. A novel procedure to construct element by element a superconvergent postprocessed displacement is proposed. Contrary to other hybridisable discontinuous Galerkin formulations, the methodology proposed here is able to produce a superconvergent displacement field for low‐order approximations. The resulting method is robust and locking‐free in the nearly incompressible limit. An extensive set of numerical examples is utilised to provide evidence of the optimality of the method and its superconvergent properties in two and three dimensions and for different element types.

An hp-version error analysis of the discontinuous Galerkin method for linear elasticity

An hp-version error analysis is developed for the general DG method in mixed formulation for solving the linear elastic problem. First of all, we give the hp-version error estimates of two L2 projection operators. Then incorporated with the techniques in [11], we obtain the hp-version error estimates in energy norm and L2 norm. Some numerical experiments are provided for demonstrating the theoretical results.

Analysis of a Staggered Discontinuous Galerkin Method for Linear Elasticity

We develop a staggered discontinuous Galerkin method for linear elasticity problems and prove its a priori error estimates. In our variational formulation the symmetry of the stress tensor is imposed weakly via Lagrange multipliers but the resulting numerical stress tensor is strongly symmetric. Optimal a priori error estimates are obtained and the error estimates are robust in nearly incompressible materials. Numerical experiments illustrating our theoretical analysis are included.

Analysis of an HDG method for linear elasticity

SummaryWe present the first a priori error analysis for the first hybridizable discontinuous Galerkin method for linear elasticity proposed in Internat. J. Numer. Methods Engrg. 80 (2009), no. 8, 1058–1092. We consider meshes made of polyhedral, shape‐regular elements of arbitrary shape and show that, whenever piecewise‐polynomial approximations of degree are used and the exact solution is smooth enough, the antisymmetric part of the gradient of the displacement converges with order k, the stress and the symmetric part of the gradient of the displacement converge with order k + 1/2, and the displacement converges with order k + 1. We also provide numerical results showing that the orders of convergence are actually sharp. Copyright © 2014 John Wiley & Sons, Ltd.

A locking-free discontinuous Galerkin method for linear elasticity in locally nearly incompressible heterogeneous media

In this work we consider the problem of numerical locking in composite materials featuring quasi-incompressible and compressible sections. More specifically, we start by extending a classical regularity estimate for the H1-norm of the divergence of the displacement field to the heterogeneous case. The proof is based on a reformulation of the elasticity problem as a Stokes system with nonzero divergence constraint. This result is then used to design a locking-free discontinuous Galerkin method. The key point is to make sure that the multiplicative constant in the estimate of the convergence rate uniquely depends on this bounded quantity. Thanks to a fine tuning of the penalty term, the lower bound for the penalty parameter appearing in the method is simply expressed in terms of the space dimension. To conclude, numerical validation of the theoretical results is provided.

On the Local Discontinuous Galerkin Method for Linear Elasticity

Following Castillo et al. (2000) and Cockburn (2003), a general framework of constructing discontinuous Galerkin (DG) methods is developed for solving the linear elasticity problem. The numerical traces are determined in view of a discrete stability identity, leading to a class of stable DG methods. A particular method, called the LDG method for linear elasticity, is studied in depth, which can be viewed as an extension of the LDG method discussed by Castillo et al. (2000) and Cockburn (2003). The error bounds inL2-norm,H1-norm, and a certain broken energy norm are obtained. Some numerical results are provided to confirm the convergence theory established.

A hybridizable discontinuous Galerkin method for linear elasticity

AbstractThis paper describes the application of the so‐called hybridizable discontinuous Galerkin (HDG) method to linear elasticity problems. The method has three significant features. The first is that the only globally coupled degrees of freedom are those of an approximation of the displacement defined solely on the faces of the elements. The corresponding stiffness matrix is symmetric, positive definite, and possesses a block‐wise sparse structure that allows for a very efficient implementation of the method. The second feature is that, when polynomials of degree k are used to approximate the displacement and the stress, both variables converge with the optimal order of k+1 for any k⩾0. The third feature is that, by using an element‐by‐element post‐processing, a new approximate displacement can be obtained that converges at the order of k+2, whenever k⩾2. Numerical experiments are provided to compare the performance of the HDG method with that of the continuous Galerkin (CG) method for problems with smooth solutions, and to assess its performance in situations where the CG method is not adequate, that is, when the material is nearly incompressible and when there is a crack. Copyright © 2009 John Wiley & Sons, Ltd.

Discontinuous Galerkin methods for incompressible elastic materials

In this paper, we introduce and analyze a local discontinuous Galerkin method for linear elasticity. A simple post-processing is introduced which takes advantage of the special structure of the method. It allows us to construct an approximation to the displacement which is H(div)-conforming and to enforce the equation that links the pressure to the divergence of the displacement strongly inside each element. As a consequence, when the material is exactly incompressible, the displacement is also exactly incompressible. This is achieved without having to deal with the almost impossible task of constructing finite dimensional subspaces of incompressible displacements. We provide an error analysis of the method that holds uniformly with respect to the Poisson ratio. In particular, we show that the displacement converges in L 2 with order k + 1 when polynomials of degree k > 0 are used. We also display numerical experiments confirming that the theoretical orders of convergence are actually achieved and that they do not deteriorate when the material becomes incompressible.

Optimal BV estimates for a discontinuous Galerkin method for linear elasticity

We analyze a discontinuous Galerkin method for linear elasticity. The discrete formulation derives from the Hellinger-Reissner variational principle with the addition of stabilization terms analogous to those previously considered by others for the Navier-Stokes equations and a scalar Poisson equation. For our formulation, we first obtain convergence in a mesh-dependent norm and in the natural mesh-independent BD norm. We then prove a generalization of Korn's second inequality which allows us to strengthen our results to an optimal, mesh-independent BV estimate for the error.

Discontinuous Galerkin and the Crouzeix–Raviart element: Application to elasticity

We propose a discontinuous Galerkin method for linear elasticity, based on discontinuous piecewise linear approximation of the displacements. We show optimal order a priori error estimates, uniform in the incompressible limit, and thus locking is avoided. The discontinuous Galerkin method is closely related to the non-conforming Crouzeix-Raviart (CR) element, which in fact is obtained when one of the stabilizing parameters tends to infinity. In the case of the elasticity operator, for which the CR element is not stable in that it does not fulfill a discrete Korn's inequality, the discontinuous framework naturally suggests the appearance of (weakly consistent) stabilization terms. Thus, a stabilized version of the CR element, which does not lock, can be used for both compressible and (nearly) incompressible elasticity. Numerical results supporting these assertions are included. The analysis directly extends to higher order elements and three spatial dimensions.