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A R T k − P k approximation for linear elasticity yielding a broken H ( div ) convergent postprocessed stress
Abstract
We present a non-standard mixed finite element method for the linear elasticity problem in with non-homogeneous Dirichlet boundary conditions. More precisely, our approach is based on a simplified interpretation of the pseudostress–displacement formulation originally proposed in Arnold and Falk (1988), which does not require symmetric tensor spaces in the finite element discretization. We apply the classical Babuška–Brezzi theory to prove that the corresponding continuous and discrete schemes are well-posed. In particular, Raviart–Thomas spaces of order for the pseudostress and piecewise polynomials of degree for the displacement can be utilized. In addition, complementing the results in the aforementioned reference, we introduce a new postprocessing formula for the stress recovering the optimally convergent approximation of the broken -norm. Numerical results confirm our theoretical findings.
... In the literature, an important number of contributions related to mixed formulations for elasticity are available such as [3,4,7,12,15,14], just to mention some of them. These references have the particularity that the results are focused in the load problems. ...
The present paper introduces the analysis of the eigenvalue problem for the elasticity equations when the so called Navier-Lam\'e system is considered. Such a system introduces the displacement, rotation and pressure of some linear and elastic structure. The analysis of the spectral problem is based in the compact operators theory. A finite element method based in polynomials of degree are considered in order to approximate the eigenfrequencies and eigenfunctions of the system. Convergence and error estimate are presented. An a posteriori error analysis is performed, where the reliability and efficiency of the proposed estimator is proved. We end this contribution reporting a series of numerical tests in order to assess the performance of the proposed numerical method, for the a priori and a posteriori estimates.
The present paper introduces the analysis of the eigenvalue problem for the elasticity equations when the so-called Navier-Lam\'e system is considered. This system incorporates the displacement, rotation, and pressure of a linear elastic structure. The analysis of the spectral problem is based on the compact operator theory. A finite element method using polynomials of degree is employed to approximate the eigenfrequencies and eigenfunctions of the system. Convergence and error estimates are presented. An a posteriori error analysis is also performed, where the reliability and efficiency of the proposed estimator are proven. We conclude this contribution by reporting a series of numerical tests to assess the performance of the proposed numerical method for both a priori and a posteriori estimates.
In this paper we extend the utilization of the Banach spaces-based formulations, usually employed for solving diverse nonlinear problems in continuum mechanics via primal and mixed finite element methods, to the virtual element method (VEM) framework and its respective applications. More precisely, we propose and analyze an [Formula: see text] spaces-based mixed virtual element method for a pseudostress-velocity formulation of the two-dimensional Navier–Stokes equations with Dirichlet boundary conditions. To this end, a dual-mixed approach determined by the introduction of a nonlinear tensor linking the usual pseudostress for the Stokes equations with the convective term is employed. As a consequence, this new tensor, say [Formula: see text], and the velocity [Formula: see text] of the fluid constitute the unknowns of the formulation, whereas the pressure is computed via a post-processing formula. The simplicity of the resulting VEM scheme is reflected by the absence of augmented terms, on the contrary to previous works on this and related models, and by the incorporation in it of only the projector onto the piecewise polynomial tensors and the usual stabilizer depending on the degrees of freedom of the virtual element subspace approximating [Formula: see text]. In turn, the non-virtual but explicit subspace given by the piecewise polynomial vectors of degree [Formula: see text] is employed to approximate [Formula: see text]. The corresponding solvability analysis is carried out by using appropriate fixed-point arguments, along with the discrete versions of the Babuška–Brezzi theory and the Banach–Nečas–Babuška theorem, both in subspaces of Banach spaces. A Strang-type lemma is applied to derive the a priori error estimates for the virtual element solution as well as for the fully computable approximation of [Formula: see text], the post-processed pressure, and a second post-processed approximation of [Formula: see text]. Finally, several numerical results illustrating the performance of the mixed-VEM scheme and confirming the rates of convergence predicted by the theory are reported.
In this paper we introduce and analyze a mixed virtual element method (mixed-VEM) for a pseudostress-displacement formulation of the linear elasticity problem with non-homogeneous Dirichlet boundary conditions. We follow a previous work by some of the authors, and employ a mixed formulation that does not require symmetric tensor spaces in the finite element discretization. More precisely, the main unknowns here are given by the pseudostress and the displacement, whereas other physical quantities such as the stress, the strain tensor of small deformations, and the rotation, are computed through simple postprocessing formulae in terms of the pseudostress variable. We first recall the corresponding variational formulation, and then summarize the main mixed-VEM ingredients that are required for our discrete analysis. In particular, we utilize a well-known local projector onto a suitable polynomial subspace to define a calculable version of our discrete bilinear form, whose continuous version requires information of the variables on the interior of each element. Next, we show that the global discrete bilinear form satisfies the hypotheses required by the Babuška–Brezzi theory. In this way, we conclude the well-posedness of our mixed-VEM scheme and derive the associated a priori error estimates for the virtual solutions as well as for the fully computable projections of them. Furthermore, we also introduce a second element-by-element postprocessing formula for the pseudostress, which yields an optimally convergent approximation of this unknown with respect to the broken H(div)-norm. In addition, this postprocessing formula can also be applied to the postprocessed stress tensor. Finally, several numerical results illustrating the good performance of the method and confirming the theoretical rates of convergence are presented.
In this work we introduce and analyze a mixed virtual element method for the two-dimensional nonlinear Brinkman model of porous media flow with non-homogeneous Dirichlet boundary conditions. For the continuous formulation we consider a dual-mixed approach in which the main unknowns are given by the gradient of the velocity and the pseudostress, whereas the velocity itself and the pressure are computed via simple postprocessing formulae. In addition, because of analysis reasons we add a redundant term arising from the constitutive equation relating the pseudostress and the velocity, so that the well-posedness of the resulting augmented formulation is established by using known results from nonlinear functional analysis. Then, we introduce the main features of the mixed virtual element method, which employs an explicit piecewise polynomial subspace and a virtual element subspace for approximating the aforementioned main unknowns, respectively. In turn, the associated computable discrete nonlinear operator is defined in terms of the -orthogonal projector onto a suitable space of polynomials, which allows the explicit integration of the terms involving deviatoric tensors that appear in the original setting. Next, we show the well-posedness of the discrete scheme and derive the associated a priori error estimates for the virtual element solution as well as for the fully computable projection of it. Furthermore, we also introduce a second element-by-element postprocessing formula for the pseudostress, which yields an optimally convergent approximation of this unknown with respect to the broken -norm. Finally, several numerical results illustrating the good performance of the method and confirming the theoretical rates of convergence are presented.
In this paper, we introduce and analyze a mixed virtual element method (mixed-VEM) for the two-dimensional Brinkman model of porous media flow with non-homogeneous Dirichlet boundary conditions. More precisely, we employ a dual-mixed formulation in which the only unknown is given by the pseudostress, whereas the velocity and pressure are computed via postprocessing formulae. We first recall the corresponding variational formulation, and then summarize the main mixed-VEM ingredients that are required for our discrete analysis. In particular, in order to define a calculable discrete bilinear form, whose continuous version involves deviatoric tensors, we propose two well-known alternatives for the local projector onto a suitable polynomial subspace, which allows the explicit integration of these terms. Next, we show that the global discrete bilinear form satisfies the hypotheses required by the Lax-Milgram lemma. In this way, we conclude the well-posedness of our mixed-VEM scheme and derive the associated a priori error estimates for the virtual solution as well as for the fully computable projection of it. Furthermore, we also introduce a second element-by-element postprocessing formula for the pseudostress, which yields an optimally convergent approximation of this unknown with respect to the broken H(div)-norm. Finally, several numerical results illustrating the good performance of the method and confirming the theoretical rates of convergence are presented.
In this paper we present the a priori and a posteriori error analyses of a non-standard mixed finite element method for the linear elasticity problem with non-homogeneous Dirichlet boundary conditions. More precisely, the approach introduced here is based on a simplified interpretation of the pseudostress–displacement formulation originally proposed in Arnold and Falk (1988), which does not require symmetric tensor spaces in the finite element discretization. In addition, physical quantities such as the stress, the strain tensor of small deformations, and the rotation, are computed through a simple postprocessing in terms of the pseudostress variable. Furthermore, we also introduce a second element-by-element postprocessing formula for the stress, which yields an optimally convergent approximation of this unknown with respect to the broken -norm. We apply the classical Babuška–Brezzi theory to prove that the corresponding continuous and discrete schemes are well-posed. In particular, Raviart–Thomas spaces of order for the pseudostress and piecewise polynomials of degree for the displacement can be utilized. Moreover, we remark that in the 3D case the number of unknowns behaves approximately as 9 times the number of elements (tetrahedra) of the triangulation when . This factor increases to 12.5 when one uses the classical PEERS. Next, we derive a reliable and efficient residual-based a posteriori error estimator for the mixed finite element scheme. Finally, several numerical results illustrating the performance of the method, confirming the theoretical properties of the estimator, and showing the expected behaviour of the associated adaptive algorithm, are provided.
We present a new stabilized mixed finite element method for the linear elasticity problem in
. The
approach is based on the introduction of Galerkin least-squares terms arising from the constitutive and
equilibrium equations, and from the relation defining the rotation in terms of the displacement. We show that
the resulting augmented variational formulation and the associated Galerkin scheme are well posed, and that
the latter becomes locking-free and asymptotically locking-free for Dirichlet and mixed boundary conditions,
respectively. In particular, the discrete scheme allows the utilization of Raviart–Thomas spaces of lowest
order for the stress tensor, piecewise linear elements for the displacement, and piecewise constants for the
rotation. In the case of mixed boundary conditions, the essential one (Neumann) is imposed weakly, which
yields the introduction of the trace of the displacement as a suitable Lagrange multiplier. This trace is
then approximated by piecewise linear elements on an independent partition of the Neumann boundary whose mesh
size needs to satisfy a compatibility condition with the mesh size associated to the triangulation of the
domain. Several numerical results illustrating the good performance of the augmented mixed finite element
scheme in the case of Dirichlet boundary conditions are also reported.
We propose a new mixed variational formulation for the equations of linear elasticity. It does not require symmetric tensors and consequently is easy to discretize by adapting mixed finite elements developed for scalar second order elliptic equations.
In this paper we introduce and analyze two augmented mixed finite element methods for a velocity-pressure-stress formulation of the stationary Stokes equations. Our approach, which extends analog results for linear elasticity problems, is based on the introduction of the Galerkin least-squares-type terms arising from the constitutive and equilibrium equations and the Dirichlet boundary condition for the velocity, all of them multiplied by suitable stabilization parameters. We show that these parameters can be chosen so that the resulting augmented variational formulations are defined by strongly coercive bilinear forms, whence the associated Galerkin schemes become well-posed for any choice of finite element subspaces. In particular, we can use continuous piecewise linear velocities, piecewise constant pressures, and Raviart-Thomas elements for the stresses, thus yielding a number of unknowns behaving asymptotically as 5 times the number of triangles of the triangulation. Alternatively, the above factor reduces to 4 when a second augmented variational formulation, involving only the velocity and the stress as unknowns, is employed. Next, we derive a reliable and efficient residual-based a posteriori error estimator for the augmented mixed finite element schemes. Finally, extensive numerical experiments illustrating the performance of the augmented mixed finite element methods, confirming the properties of the a posteriori estimators, and showing the behavior of the associated adaptive algorithms are reported.
In this paper we present the a priori and a posteriori error analyses of a non-standard mixed finite element method for the linear elasticity problem with non-homogeneous Dirichlet boundary conditions. More precisely, the approach introduced here is based on a simplified interpretation of the pseudostress–displacement formulation originally proposed in Arnold and Falk (1988), which does not require symmetric tensor spaces in the finite element discretization. In addition, physical quantities such as the stress, the strain tensor of small deformations, and the rotation, are computed through a simple postprocessing in terms of the pseudostress variable. Furthermore, we also introduce a second element-by-element postprocessing formula for the stress, which yields an optimally convergent approximation of this unknown with respect to the broken -norm. We apply the classical Babuška–Brezzi theory to prove that the corresponding continuous and discrete schemes are well-posed. In particular, Raviart–Thomas spaces of order for the pseudostress and piecewise polynomials of degree for the displacement can be utilized. Moreover, we remark that in the 3D case the number of unknowns behaves approximately as 9 times the number of elements (tetrahedra) of the triangulation when . This factor increases to 12.5 when one uses the classical PEERS. Next, we derive a reliable and efficient residual-based a posteriori error estimator for the mixed finite element scheme. Finally, several numerical results illustrating the performance of the method, confirming the theoretical properties of the estimator, and showing the expected behaviour of the associated adaptive algorithm, are provided.
We consider a non-standard mixed approach for the Stokes problem in which the velocity, the pressure, and the pseudostress are the main unknowns. Alternatively, the pressure can be eliminated from the original equations, thus yielding an equivalent formulation with only two unknowns. In this paper we develop a priori and a posteriori error analyses of both approaches. We first apply the Babuška–Brezzi theory to prove the well-posedness of the continuous and discrete formulations. In particular, we show that Raviart–Thomas elements of order k⩾0 for the pseudostresses, and piecewise polynomials of degree k for the velocities and the pressures, ensure unique solvability and stability of the associated Galerkin schemes. Then, we derive reliable and efficient residual-based a posteriori error estimators for both schemes, without and with pressure unknown. Finally, we provide several numerical results illustrating the good performance of the resulting mixed finite element methods, confirming the theoretical properties of the estimators, and showing the behaviour of the associated adaptive algorithms.
Using Fortin operators we give a new proof of stability for Stenbergs family of BDMS elements in linear elasticity. Our approach yields the inf-sup condition with respect to the standard norms, which is indispensable for a posteriori error analysis. Furthermore our technique allows the construction of another family of finite elements strongly related to the classical PEERS element. The given estimates are robust for nearly incompressible materials.
In this paper we introduce and analyze a new augmented mixed finite element method for linear elasticity problems in 3D. Our approach is an extension of a technique developed recently for plane elasticity, which is based on the introduction of consistent terms of Galerkin least-squares type. We consider non-homogeneous and homogeneous Dirichlet boundary conditions and prove that the resulting augmented variational formulations lead to strongly coercive bilinear forms. In this way, the associated Galerkin schemes become well posed for arbitrary choices of the corresponding finite element subspaces. In particular, Raviart–Thomas spaces of order 0 for the stress tensor, continuous piecewise linear elements for the displacement, and piecewise constants for the rotation can be utilized. Moreover, we show that in this case the number of unknowns behaves approximately as 9.5 times the number of elements (tetrahedrons) of the triangulation, which is cheaper, by a factor of 3, than the classical PEERS in 3D. Several numerical results illustrating the good performance of the augmented schemes are provided.