Convergence of a stabilized discontinuous Galerkin method for incompressible nonlinear elasticity

In this paper we present a discontinuous Galerkin method applied to incompressible nonlinear elastostatics in a total Lagrangian deformation-pressure formulation, for which a suitable interior penalty stabilization is applied. We prove that the proposed discrete formulation for the linearized problem is well-posed, asymptotically consistent and that it converges to the corresponding weak solution. The derived convergence rates are optimal and further confirmed by a set of numerical examples in two and three spatial dimension

Convergence of a stabilized discontinuous Galerkin method for incompressible nonlinear elasticity

In this paper we present a discontinuous Galerkin method applied to incompressible nonlinear elastostatics in a total Lagrangian deformation-pressure formulation, for which a suitable interior penalty stabilization is applied. We prove that the proposed discrete formulation for the linearized problem is well-posed, asymptotically consistent and that it converges to the corresponding weak solution. The derived convergence rates are optimal and further confirmed by a set of numerical examples in two and three spatial dimensions

Stabilized lowest order finite element approximation for linear three-field poroelasticity

A stabilized conforming mixed finite element method for the three-field (displacement, fluid flux and pressure) poroelasticity problem is developed and analyzed. We use the lowest possible approximation order, namely piecewise constant approximation for the pressure and piecewise linear continuous elements for the displacements and fluid flux. By applying a local pressure jump stabilization term to the mass conservation equation we ensure stability and avoid pressure oscillations. Importantly, the discretization leads to a symmetric linear system. For the fully discretized problem we prove existence and uniqueness, an energy estimate and an optimal a-priori error estimate, including an error estimate for the divergence of the fluid flux. Numerical experiments in 2D and 3D illustrate the convergence of the method, show the effectiveness of the method to overcome spurious pressure oscillations, and evaluate the added mass effect of the stabilization term.Comment: 25 page

Explicit mixed strain–displacement finite elements for compressible and quasi-incompressible elasticity and plasticity

The final publication is available at Springer via http://dx.doi.org/ 10.1007/s00466-016-1305-zThis paper presents an explicit mixed finite element formulation to address compressible and quasi-incompressible problems in elasticity and plasticity. This implies that the numerical solution only involves diagonal systems of equations. The formulation uses independent and equal interpolation of displacements and strains, stabilized by variational subscales. A displacement sub-scale is introduced in order to stabilize the mean-stress field. Compared to the standard irreducible formulation, the proposed mixed formulation yields improved strain and stress fields. The paper investigates the effect of this enhancement on the accuracy in problems involving strain softening and localization leading to failure, using low order finite elements with linear continuous strain and displacement fields (P1P1 triangles in 2D and tetrahedra in 3D) in conjunction with associative frictional Mohr–Coulomb and Drucker–Prager plastic models. The performance of the strain/displacement formulation under compressible and nearly incompressible deformation patterns is assessed and compared to analytical solutions for plane stress and plane strain situations. Benchmark numerical examples show the capacity of the mixed formulation to predict correctly failure mechanisms with localized patterns of strain, virtually free from any dependence of the mesh directional bias. No auxiliary crack tracking technique is necessary.Peer ReviewedPostprint (author's final draft

Mixed stabilized finite element methods in nonlinear solid mechanics: Part I: Formulation

This paper exploits the concept of stabilized finite element methods to formulate stable mixed stress/displacement and strain/displacement finite elements for the solution of nonlinear solid mechanics problems. The different assumptions and approximations used to derive the methods are exposed. The proposed procedure is very general, applicable to 2D and 3D problems. Implementation and computational aspects are also discussed, showing that a robust application of the proposed formulation is feasible. Numerical examples show that the results obtained compare favorably with those obtained with the corresponding irreducible formulation

A Hybrid High-Order method for nonlinear elasticity

In this work we propose and analyze a novel Hybrid High-Order discretization of a class of (linear and) nonlinear elasticity models in the small deformation regime which are of common use in solid mechanics. The proposed method is valid in two and three space dimensions, it supports general meshes including polyhedral elements and nonmatching interfaces, enables arbitrary approximation order, and the resolution cost can be reduced by statically condensing a large subset of the unknowns for linearized versions of the problem. Additionally, the method satisfies a local principle of virtual work inside each mesh element, with interface tractions that obey the law of action and reaction. A complete analysis covering very general stress-strain laws is carried out, and optimal error estimates are proved. Extensive numerical validation on model test problems is also provided on two types of nonlinear models.Comment: 29 pages, 7 figures, 4 table