A locking-free weak Galerkin finite element method for elasticity problems in the primal formulation

Wang, Chunmei; Wang, Junping; Wang, Ruishu; Zhang, Ran

doi:10.1016/j.cam.2015.12.015

Cited by 83 publications

(40 citation statements)

References 23 publications

(57 reference statements)

Supporting

Mentioning

Contrasting

Order By: Relevance

“…HDG methods for linear elasticity have been coined in [38] (see also [13] for incompressible Stokes flows), and extensions to nonlinear elasticity can be found in [29,34,37]. Other recent developments in the last few years include, among others, Gradient Schemes for nonlinear elasticity with small deformations [22], the Virtual Element Method (VEM) for linear and nonlinear elasticity with small [3] and finite deformations [8,43], the (low-order) hybrid dG method with conforming traces for nonlinear elasticity [44], the hybridizable weakly conforming Galerkin method with nonconforming traces for linear elasticity [30], the Weak Galerkin method for linear elasticity [42], and the discontinuous Petrov-Galerkin method for linear elasticity [7].…”

Section: Introductionmentioning

confidence: 99%

Hybrid High-Order methods for finite deformations of hyperelastic materials

2018

View full text Add to dashboard Cite

We devise and evaluate numerically Hybrid High-Order (HHO) methods for hyperelastic materials undergoing finite deformations. The HHO methods use as discrete unknowns piecewise polynomials of order k ≥ 1 on the mesh skeleton, together with cell-based polynomials that can be eliminated locally by static condensation. The discrete problem is written as the minimization of a broken nonlinear elastic energy where a local reconstruction of the displacement gradient is used. Two HHO methods are considered: a stabilized method where the gradient is reconstructed as a tensor-valued polynomial of order k and a stabilization is added to the discrete energy functional, and an unstabilized method which reconstructs a stable higher-order gradient and circumvents the need for stabilization. Both methods satisfy the principle of virtual work locally with equilibrated tractions. We present a numerical study of the two HHO methods on test cases with known solution and on more challenging three-dimensional test cases including finite deformations with strong shear layers and cavitating voids. We assess the computational efficiency of both methods, and we compare our results to those obtained with an industrial software using conforming finite elements and to results from the literature. The two HHO methods exhibit robust behavior in the quasi-incompressible regime.

show abstract

Section: Introductionmentioning

confidence: 99%

Hybrid High-Order methods for finite deformations of hyperelastic materials

2018

View full text Add to dashboard Cite

show abstract

“…Numerical implementation of WGFEM was discussed in [20]. This method successfully applied to some PDEs such as parabolic equation [21,22], Helmholtz equation [23,24], biharmonic equation [25][26][27][28], Brinkman equation [29,30], elliptic interface problem [31], linear elasticity problems [32,33], Oseen equation [34], Darcy equation [35,36], Darcy-stokes equation [37][38][39], Maxwell equation [40,41], and so on. An important feature of WG methods is applying totally discontinuous piecewise polynomials on the finite element partition.…”

Section: Introductionmentioning

confidence: 99%

Numerical and theoretical study of weak Galerkin finite element solutions of Turing patterns in reaction–diffusion systems

Khaled-Abad

Salehi

2020

Numerical Methods Partial

View full text Add to dashboard Cite

In this paper, we introduce numerical schemes and their analysis based on weak Galerkin finite element framework for solving 2-D reaction-diffusion systems. Weak Galerkin finite element method (WGFEM) for partial differential equations relies on the concept of weak functions and weak gradients, in which differential operators are approximated by weak forms through the Green's theorem. This method allows the use of totally discontinuous functions in the approximation space. In the current work, the WGFEM solves reaction-diffusion systems to find unknown concentrations (u, v) in element interiors and boundaries in the weak Galerkin finite element space WG(P 0 , P 0 , RT 0). The WGFEM is used to approximate the spatial variables and the time discretization is made by the backward Euler method. For reaction-diffusion systems, stability analysis and error bounds for semi-discrete and fully discrete schemes are proved. Accuracy and efficiency of the proposed method successfully tested on several numerical examples and obtained results satisfy the well-known result that for small values of diffusion coefficient, the steady state solution converges to equilibrium point. Acquired numerical results asserted the efficiency of the proposed scheme.

show abstract

“…The weak Galerkin mixed method provides accurate numerical approximations to both primary variable and its flux. A series of model problems are studied by the weak Galerkin method, such as the Stokes equation , the Brinkman equation , the biharmonic equation , and the linear elasticity equation .…”

Section: Introductionmentioning

confidence: 99%

A weak Galerkin mixed finite element method for the Helmholtz equation with large wave numbers

Wang

Zhai

et al. 2018

Numerical Methods Partial

Self Cite

View full text Add to dashboard Cite

In this article, a new weak Galerkin mixed finite element method is introduced and analyzed for the Helmholtz equation with large wave numbers. The stability and well‐posedness of the method are established for any wave number k without mesh size constraint. Allowing the use of discontinuous approximating functions makes weak Galerkin mixed method highly flexible in term of little restrictions on approximations and meshes. In the weak Galerkin mixed finite element formulation, approximation functions can be piecewise polynomials with different degrees on different elements and meshes can consist elements with different shapes. Suboptimal order error estimates in both discrete H1 and L2 norms are established for the weak Galerkin mixed finite element solutions. Numerical examples are tested to support the theory.

show abstract

A locking-free weak Galerkin finite element method for elasticity problems in the primal formulation

Cited by 83 publications

References 23 publications

Hybrid High-Order methods for finite deformations of hyperelastic materials

Hybrid High-Order methods for finite deformations of hyperelastic materials

Numerical and theoretical study of weak Galerkin finite element solutions of Turing patterns in reaction–diffusion systems

A weak Galerkin mixed finite element method for the Helmholtz equation with large wave numbers

Contact Info

Product

Resources

About