Dual-mixed Finite Element Research Articles

An extended crystal plasticity model for latent hardening in polycrystals

In this contribution, a computational approach to modeling size-dependent self- and latent hardening in polycrystals is presented. Latent hardening is the hardening of inactive slip systems due to active slip systems. We focus attention on the investigation of glide system interaction, latent hardening and excess dislocation development. In particular, latent hardening results in a transition to patchy slip as a first indication and expression of the development of dislocation microstructures. To this end, following Nye (Acta Metall 1:153-162, 1953), Kondo (in Proceedings of the second Japan national congress for applied mechanics. Science Council of Japan, Tokyo, pp. 41-47, 1953), and many others, local deformation incompatibility in the material is adopted as a measure of the density of geometrically necessary dislocations. Their development results in additional energy being stored in the material, leading to additional kinematic-like hardening effects. A large-deformation model for latent hardening is introduced. This approach is based on direct exploitation of the dissipation principle to derive all field relations and (sufficient) forms of the constitutive relations as based on the free energy density and dissipation potential. The numerical implementation is done via a dual-mixed finite element method. A numerical example for polycrystals is presented.

A Dual Mixed Formulation for Non-isothermal Oldroyd–Stokes Problem

We propose a mixed formulation for non-isothermal Oldroyd-Stokes problem where the both extra stress and the heat flux's vector are considered. Based on such a formulation, a dual mixed finite element is constructed and analyzed. This finite element method enables us to obtain precise approximations of the dual variable which are, for the non-isothermal fluid flow problems, the viscous and polymeric components of the extra-stress tensor, as well as the heat flux. Furthermore, it has properties analogous to the finite volume methods, namely, the local conservation of the momentum and the mass.

A priori error estimation for the dual mixed finite element method of the elastodynamic problem in a polygonal domain, II

In this paper we analyze a new dual mixed formulation of the elastodynamic system in polygonal domains by using an implicit scheme for the time discretization. After the analysis of stability of the fully discrete scheme, L ∞ in time, L 2 in space a priori error estimates for the approximation of the displacement, the strain, the pressure and the rotational are derived. Numerical tests are presented which confirm our theoretical results.

Modeling of polycrystals with gradient crystal plasticity: A comparison of strategies

This paper treats the computational modeling of size dependence in microstructure models of metals. Different gradient crystal plasticity strategies are analyzed and compared. For the numerical implementation, a dual-mixed finite element formulation which is suitable for parallelization is suggested. The paper ends with a representative numerical example for polycrystals.

A residual-based a posteriori error estimator for a two-dimensional fluid–solid interaction problem

In this paper, we develop an a posteriori error analysis of a mixed finite element method for a fluid–solid interaction problem posed in the plane. The media are governed by the acoustic and elastodynamic equations in time-harmonic regime, respectively, and the transmission conditions are given by the equilibrium of forces and the equality of the normal displacements of the solid and the fluid. The coupling of primal and dual-mixed finite element methods is applied to compute both the pressure of the scattered wave in the linearized fluid and the elastic vibrations that take place in the elastic body. The finite element subspaces consider continuous piecewise linear elements for the pressure and a Lagrange multiplier defined on the interface, and PEERS for the stress and rotation in the solid domain. We derive a reliable and efficient residual-based a posteriori error estimator for this coupled problem. Suitable auxiliary problems, the continuous inf-sup conditions satisfied by the bilinear forms involved, a discrete Helmholtz decomposition, and the local approximation properties of the Clement interpolant and Raviart–Thomas operator are the main tools for proving the reliability of the estimator. Then, Helmholtz decomposition, inverse inequalities, and the localization technique based on triangle-bubble and edge-bubble functions are employed to show the efficiency. Finally, some numerical results confirming the reliability and efficiency of the estimator are reported.

Dual-mixed finite element approximation of Stokes and nonlinear Stokes problems using trace-free velocity gradients

In this work a finite element method for a dual-mixed approximation of Stokes and nonlinear Stokes problems is studied. The dual-mixed structure, which yields a twofold saddle point problem, arises in a formulation of this problem through the introduction of unknown variables with relevant physical meaning. The method approximates the velocity, its gradient, and the total stress tensor, but avoids the explicit computation of the pressure, which can be recovered through a simple postprocessing technique. This method improves an existing approach for these problems and uses Raviart–Thomas elements and discontinuous piecewise polynomials for approximating the unknowns. Existence, uniqueness, and error results for the method are given, and numerical experiments that exhibit the reduced computational cost of this approach are presented.

A priori error estimation for the dual mixed finite element method of the elastodynamic problem in a polygonal domain, I

In this paper we analyze a new dual mixed formulation of the elastodynamic system in polygonal domains. In this formulation the symmetry of the strain tensor is relaxed by the rotation of the displacement. For the time discretization of this new dual mixed formulation, we use an explicit scheme. After the analysis of stability of the fully discrete scheme, L ∞ in time, L 2 in space a priori error estimates are derived for the approximation of the displacement, the strain, the pressure and the rotation. Numerical experiments confirm our theoretical predictions.

Dual mixed finite element methods for the elasticity problem with Lagrange multipliers

We study a dual mixed formulation of the elasticity system in a polygonal domain of the plane with mixed boundary conditions and its numerical approximation. The (essential) Neumann boundary conditions (or traction boundary condition) are imposed using a discontinuous Lagrange multiplier corresponding to the trace of the displacement field. Moreover, a strain tensor is introduced as a new unknown and its symmetry is relaxed, also by the use of a Lagrange multiplier (the rotation). The singular behaviour of the solution requires us to use refined meshes to restore optimal rates of convergence. Uniform error estimates in the Lamé coefficient λ are obtained for large λ . The hybridization of the problem is performed and numerical tests are presented confirming our theoretical results.

A new dual-mixed finite element method for the plane linear elasticity problem with pure traction boundary conditions

In this paper we consider the stress–displacement–rotation formulation of the plane linear elasticity problem with pure traction boundary conditions and develop a new dual-mixed finite element method for approximating its solution. The main novelty of our approach lies on the weak enforcement of the non-homogeneous Neumann boundary condition through the introduction of the boundary trace of the displacement as a Lagrange multiplier. A suitable combination of PEERS and continuous piecewise linear functions on the boundary are employed to define the dual-mixed finite element scheme. We apply the classical Babuška–Brezzi theory to show the well-posedness of the continuous and discrete formulations. Then, we derive a priori rates of convergence of the method, including an estimate for the global error when the stresses are measured with the L 2-norm. Finally, several numerical results illustrating the good performance of the mixed finite element scheme are reported.

A Priori and a Posteriori Error Analysis of a Wavelet-Based Stabilization for the Mixed Finite Element Method

We use Galerkin least-squares terms and biorthogonal wavelet bases to develop a new stabilized dual-mixed finite element method for second-order elliptic equations in divergence form with Neumann boundary conditions. The approach introduces the trace of the solution on the boundary as a new unknown that acts also as a Lagrange multiplier. We show that the resulting stabilized dual-mixed variational formulation and the associated discrete scheme defined with Raviart–Thomas spaces are well-posed and derive the usual a priori error estimates and the corresponding rate of convergence. Furthermore, a reliable and efficient residual-based a posteriori error estimator and a reliable and quasi-efficient one are provided.

A dual-mixed finite element method for nonlinear incompressible elasticity with mixed boundary conditions

In this paper we consider the Hu-Washizu principle and propose a new dual-mixed finite element method for nonlinear incompressible plane elasticity with mixed boundary conditions. The approach extends a related previous work on the Dirichlet problem and imposes the Neumann (essential) boundary condition in a weak sense by means of an additional Lagrange multiplier. The resulting variational formulation becomes a twofold saddle point operator equation which, for convenience of the subsequent analysis, is shown to be equivalent to a nonlinear threefold saddle point problem. In this way, a slight generalization of the classical Babuška–Brezzi theory is applied to show the well-posedness of the continuous and discrete formulations, and to derive the corresponding a priori error estimates. In particular, the classical PEERS space is suitably enriched to define the associated Galerkin scheme. Next, we develop a local problems-based a posteriori error analysis and derive an implicit reliable and quasi-efficient estimate, and a fully explicit reliable one. Finally, several numerical results illustrating the good performance of the explicit a posteriori estimate for the adaptive computation of the discrete solutions are provided.

Analysis of the Coupling of Primal and Dual-Mixed Finite Element Methods for a Two-Dimensional Fluid-Solid Interaction Problem

This paper deals with a time-harmonic fluid-solid interaction problem posed in the plane. More precisely, we apply the coupling of primal and dual-mixed finite element methods to compute both the pressure of the scattered wave in the linearized fluid and the elastic vibrations that take place in the solid elastic body. To this end, we solve a transmission problem holding between the cross-section of the infinitely long cylinder representing the obstacle and an annular region surrounding it. The novelty of our method lies in the use of a dual-mixed variational formulation in the obstacle, while maintaining the usual primal formulation in the fluid. In other words, we introduce a stress-pressure formulation of the problem instead of the traditional displacement-pressure encountered in the literature. As a consequence, one of the transmission conditions becomes essential, and hence we enforce it weakly by means of a Lagrange multiplier. Next, we apply the abstract framework developed in a recent work by A. Buffa, prove that our coupled variational formulation is well posed, and define the corresponding discrete scheme by using PEERS in the solid domain and standard Lagrange finite elements in the fluid domain. Then we show that the resulting Galerkin scheme is uniquely solvable and convergent and derive optimal error estimates. Finally, we illustrate our analysis with some results from computational experiments.

A Posteriori Error Estimation for the Dual Mixed Finite Element Method of the Stokes Problem

The paper presents an a posteriori error estimator of residual type for the stationary Stokes problem in a possibly multiply-connected polygonal domain of the plane, using the dual mixed finite element method (FEM). We prove lower and upper error bounds with the explicit dependence of the viscosity parameter and without any regularity assumption on the solution.

Functional-type <I>a posteriori</I> error estimates for mixed finite element methods

This paper concerns a posteriori error estimation for the primal and dual mixed finite element methods applied to the diffusion problem. The problem is considered in a general setting with inhomogeneous mixed Dirichlet–Neumann boundary conditions. New functional-type a posteriori error estimators are proposed that exhibit the ability both to indicate the local error distribution and to ensure upper bounds for discretization errors in primal and dual (flux) variables. The latter property is a direct consequence of the absence in the estimators of any mesh-dependent constants; the only constants present in the estimates stem from the Friedrichs and trace inequalities and, thus, are global and dependent solely on the domain geometry and the bounds of the diffusion matrix. The estimators are computationally cheap and require only the projections of piecewise constant functions onto the spaces of the lowest-order Raviart–Thomas or continuous piecewise linear elements. It is shown how these projections can be easily realized by simple local averaging.

Mixed and hybrid finite element methods for convection-diffusion equations and their relationships with finite volume

We study the relationship between finite volume and mixed finite element methods for the the hyperbolic conservation laws, and the closely related convection-diffusion equations. A general framework is proposed for the derivation and a functional framework is developed which could allow the analysis of relating finite volume (FV) schemes. We show via two non- standard formulations, that numerous FV schemes, including centred, up- wind, Lax-Friedrichs, Roe, Engquist-Osher, the central Nessyahu-Tadmor schemes, etc., can be recovered in the unique dual mixed and hybrid (DMH) finite element framework. That makes possible a better understanding of these FV schemes. Moreover, the large number of DMH finite element re- sults available can then give the analysis of these FV methods in a unified fashion. Furthermore, stabilized methods are proposed. In particular, inter- pretation in terms of the Lagrange multiplier of flux-limiter is given. We end by presenting numerical results to validate the newly proposed stabilized schemes.

A posteriori error estimation for the dual mixed finite element method of the elasticity problem in a polygonal domain

AbstractIn this article, we propose a residual based reliable and efficient error estimator for the new dual mixed finite element method of the elasticity problem in a polygonal domain, introduced by M. Farhloul and M. Fortin. With the help of a specific generalized Helmholtz decomposition of the error on the strain tensor and the classical decomposition of the error on the gradient of the displacements, we show that our global error estimator is reliable. Efficiency of our estimator follows by using classical inverse estimates. The lower and upper error bounds obtained are uniform with respect to the Lamé coefficient λ, in particular avoiding locking phenomena. © 2005 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2005.

Functional-type a posteriori error estimates for mixed finite element methods

This paper concerns a posteriori error estimation for the primal and dual mixed finite element methods applied to the diffusion problem. The problem is considered in a general setting with inhomogeneous mixed Dirichlet–Neumann boundary conditions. New functional-type a posteriori error estimators are proposed that exhibit the ability both to indicate the local error distribution and to ensure upper bounds for discretization errors in primal and dual (flux) variables. The latter property is a direct consequence of the absence in the estimators of any mesh-dependent constants; the only constants present in the estimates stem from the Friedrichs and trace inequalities and, thus, are global and dependent solely on the domain geometry and the bounds of the diffusion matrix. The estimators are computationally cheap and require only the projections of piecewise constant functions onto the spaces of the lowest-order Raviart–Thomas or continuous piecewise linear elements. It is shown how these projections can be easily realized by simple local averaging.

An a posteriori error estimate for a linear-nonlinear transmission problem in plane elastostatics

We consider the coupling of dual-mixed finite element and boundary element methods to solve a linear-nonlinear transmission problem in plane hyperelasticity with mixed boundary conditions. Besides the displacement and the stress tensor, we introduce the strain tensor as an additional unknown, which yields a two-fold saddle point operator equation as the corresponding variational formulation. We derive a reliable a posteriori error estimate that depends on the solution of local Dirichlet problems and on residual terms on the transmission and Neumann boundaries, which are given in a negative order Sobolev norm. Our approach does not need the exact Galerkin solution, but any reasonable approximation of it. In addition, the analysis does not depend on special finite element or boundary element subspaces. However, for certain specific subspaces we are able to provide two fully local a posteriori error estimates, in which the residual terms are bounded by weighted local L2-norms. Further, one of the error estimates does not require the explicit solution of the local problems.

A posteriorierror estimates for linear exterior problemsviamixed-FEM and DtN mappings

In this paper we combine the dual-mixed finite element method with a Dirichlet-to-Neumann mapping (given in terms of a boundary integral operator) to solve linear exterior transmission problems in the plane. As a model we consider a second order elliptic equation in divergence form coupled with the Laplace equation in the exterior unbounded region. We show that the resulting mixed variational formulation and an associated discrete scheme using Raviart-Thomas spaces are well posed, and derive the usual Cea error estimate and the corresponding rate of convergence. In addition, we develop two different a-posteriori error analyses yielding explicit residual and implicit Bank-Weiser type reliable estimates, respectively. Several numerical results illustrate the suitability of these estimators for the adaptive computation of the discrete solutions.

Temperature-driven fluid flow in porous media using a mixed finite element method and a finite volume method

We present a numerical scheme for the computation of conservative fluid velocity, pressure and temperature fields in a porous medium. For the velocity and pressure we use the primal–dual mixed finite element method of Trujillo and Thomas while for the temperature we use a cell-centered finite volume method. The motivation for this choice of discretization is to compute accurate conservative quantities. Since the variant of the mixed finite element method we use is not commonly used, the numerical schemes are presented in detail. We sketch the computational details and present numerical experiments that justify the accuracy predicted by the theory.