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A dual-mixed finite element method for nonlinear incompressible elasticity with mixed boundary conditions
Abstract
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.
... On the other hand, in the recent thesis [24] the results from [22] were extended to the case of mixed boundary conditions, in which no additional uniqueness constraint is needed for well posedness. However, differently from [22], the analysis in [24] was performed through an equivalent three-fold saddle point operator equation, whence a further generalization of the theory from [16] was required there. In addition, a local problems-based a-posteriori error analysis, following the same lines of [22], is also developed in [24]. ...
... However, differently from [22], the analysis in [24] was performed through an equivalent three-fold saddle point operator equation, whence a further generalization of the theory from [16] was required there. In addition, a local problems-based a-posteriori error analysis, following the same lines of [22], is also developed in [24]. ...
... In addition, we remark that γ and ξ play the role of the Lagrange multipliers associated to the symmetry of σ and the Neumann condition, respectively. Then, proceeding as in [24] (see also [22]), we arrive to the following variational formulation of the boundary value problem (2. ...
In this paper, we reconsider the a priori and a posteriori error analysis of a new mixed finite element method for nonlinear incompressible elasticity with mixed boundary conditions. The approach, being based only on the fact that the resulting variational formulation becomes a two-fold saddle-point operator equation, simplifies the analysis and improves the results provided recently in a previous work. Thus, a well-known 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 estimate. In particular, enriched PEERS subspaces are required for the solvability and stability of the associated Galerkin scheme. In addition, we use the Ritz projection operator to obtain a new reliable and quasi-efficient a posteriori error estimate. Finally, several numerical results illustrating the good performance of the associated adaptive algorithm are presented. Copyright © 2006 John Wiley & Sons, Ltd.
... And this method produces direct approximations to both stress and displacement. The most popular methods include mixed finite element methods [2,3,21,23,28], dual-mixed methods [17][18][19], and hybrid discontinuous Galerkin methods [12,35]. Compared to volume locking, gradient-robustness is a new concept. ...
... The scheme (S1) satisfies both locking-free and gradient-robust properties. The reconstruction operator in CR scheme (18) is used to map discretely divergencefree functions to divergence-free functions, while the scheme (S1) does not require reconstructing. ...
In this paper, we investigate a low-order robust numerical method for the linear elasticity problem. The method is based on a Bernardi–Raugel-like -conforming method proposed first for the Stokes flows in [Li and Rui, IMA J. Numer. Anal. 42 (2022) 3711–3734]. Therein, the lowest-order -conforming Raviart–Thomas space () was added to the classical conforming pair to meet the inf-sup condition, while preserving the divergence constraint and some important features of conforming methods. Due to the inf-sup stability of the pair, a locking-free elasticity discretization with respect to the Lamé constant can be naturally obtained. Moreover, our scheme is gradient-robust for the pure and homogeneous displacement boundary problem, that is, the discrete -norm of the displacement is when the external body force is a gradient field. We also consider the mixed displacement and stress boundary problem, whose discretization should be carefully designed due to a consistency error arising from the part. We propose both symmetric and nonsymmetric schemes to approximate the mixed boundary case. The optimal error estimates are derived for the energy norm and/or -norm. Numerical experiments demonstrate the accuracy and robustness of our schemes.
... Mixed finite element methods are a natural remedy for schemes based on primal formulations that are prone to poor performance and numerical artefacts for bending dominated and nearly incompressible regimes (in the sense that the computational results can differ substantially from what is expected from the physical and mathematical structure of the problem [10]), but other approaches also are available to improve the approximation of large deformation problems. These alternatives include high-order pure displacement methods designed to be robust in the nearly incompressible limit [11], reduced and selective integration [12], nonconforming displacementpressure [13], generalised schemes [14], projection methods [15], enhanced strain [16,17], methods based on biorthogonal systems [18] or extended finite element approaches [19] (see also the general monographs and reviews in [1,20,21]). ...
... For this we could explore different formulations using Kirchhoff stress, for instance applying integration by parts differently in the derivation of the variational formulation, and leading to divergence-conforming approximations of stress. This has been used successfully in a number of three-field formulations for linear elasticity using the Cauchy stress [21,25,79,80], but the principle is not straightforwardly carried over to the hyperelastic framework. Some formulations are available [3][4][5] in that context, but they consider the first Piola-Kirchhoff stress and displacement or adequate stress reconstructions from displacement and pressure [26]. ...
The numerical approximation of hyperelasticity must address nonlinear constitutive laws, geometric nonlinearities associated with large strains and deformations, the imposition of the incompressibility of the solid, and the solution of large linear systems arising from the discretisation of 3D problems in complex geometries. We adapt the three-field formulation for nearly incompressible hyperelasticity introduced in Chavan et al. (2007) to the fully incompressible case. The mixed formulation is of Hu–Washizu type and it differs from other approaches in that we use the Kirchhoff stress, displacement, and pressure as principal unknowns. We also discuss the solvability of the linearised problem restricted to neo-Hookean materials, illustrating the interplay between the coupling blocks. We construct a family of mixed finite element schemes (with different polynomial degrees) for simplicial meshes and verify its error decay through computational tests. We also propose a new augmented Lagrangian preconditioner that improves convergence properties of iterative solvers. The numerical performance of the family of mixed methods is assessed with benchmark solutions, and the applicability of the formulation is further tested in a model of cardiac biomechanics using orthotropic strain energy densities. The proposed methods are advantageous in terms of physical fidelity (as the Kirchhoff stress can be approximated with arbitrary accuracy and no locking is observed) and convergence (the discretisation and the preconditioners are robust and computationally efficient, and they compare favourably at least with respect to classical displacement–pressure schemes).
... For an example of a triple saddle point problem, see, e.g., [11]. Other sources of multiple saddle point problems can be found, e.g., in [7], [5], [6], [1] and the references therein. ...
... See, in particular, among many others contributions [16], for Schur complement based approaches. For other results on the analysis and the construction of preconditioners for double/twofold and triple/threefold saddle point problems see, e.g., [7], [5], [6], [11], [17], [1]. ...
The importance of Schur complement based preconditioners are well-established for classical saddle point problems in . In this paper we extend these results to multiple saddle point problems in Hilbert spaces . For such problems with a block tridiagonal Hessian and a well-defined sequence of associated Schur complements, sharp bounds for the condition number of the problem are derived which do not depend on the involved operators. These bounds can be expressed in terms of the roots of the difference of two Chebyshev polynomials of the second kind. If applied to specific classes of optimal control problems the abstract analysis leads to new existence results as well as to the construction of efficient preconditioners for the associated discretized optimality systems.
... A critical aspect in the process was to avoid blocking the elements under a complex configuration on the movement of the mesh in the F-S interaction (Gatica et al., 2007;Dauge and Schwab, 2002). As a strategy, the distortion of 3D elements is oriented with reference to dynamics of a biological bicuspid valve, as well as the left ventricle. ...
... In this way, a greater increase in numerical and mathematical process was avoided. This consideration was appropriate in the F-S simulation by promoting the stability of isoparametric elements, defined by the 'parcel criterion' (Gatica et al., 2007;Dauge and Schwab, 2002;Deparis et al., 2003). ...
Simulated experience was performed through a computational haemodynamic fluid-structure solution, with the implementation of a mechanical bio-pump as a functional element of a ventricular assistance device, in which the mechanical dynamics of latissimus dorsi muscle is represented. The influence of chamber geometry and constituent material parameters is evaluated upon the pumping functionality. By analysis of haemodynamic variables, it is found that geometry and elasticity are parameters influencing the determination of the pump outflow. In conclusion, the computational haemodynamic model helps evaluate the mechanical behaviour of the functional cardiac assist device through haemodynamic variables.
... In [22], an elementwise Hu-Washizu variational principle is used to derive the numerical method when restricted to discrete sets of displacements, strains and stresses that are possibly discontinuous across element boundaries. The method shares a number of features with others also stemming from the Hu-Washizu variational principle, e.g., [32,34], most prominent among them are assumed strain methods, see, e.g., [35,33,14]. The obvious difference is that the methods we present here use discontinuous approximations for the displacements. ...
... Solutions for this problem in the linear elasticity case are well known, such as reduced integration, mixed methods, enhanced strain methods, assumed strain methods and discontinuous Galerkin methods, among others. Some of these ideas remain valid when nonlinear constitutive equations are adopted with either linearized [14] or exact [36] kinematic descriptions. The numerical examples in section 6.6 suggest that when the Bassi and Rebay numerical fluxes are chosen, together with piecewise linear elements, the resulting discontinuous Galerkin method is locking-free for nearly incompressible situations with the body undergoing large deformations, a notable advantage over mixed methods that need to carry the extra pressure field. ...
This paper presents the formulation and a partial analysis of a class of discontinuous Galerkin methods for quasistatic non-linear elasticity problems. These methods are endowed with several salient features. The equations that define the numerical scheme are the Euler–Lagrange equations of a one-field variational principle, a trait that provides an elegant and simple derivation of the method. In consonance with general discontinuous Galerkin formulations, it is possible within this framework to choose different numerical fluxes. Numerical evidence suggests the absence of locking at near-incompressible conditions in the finite deformations regime when piecewise linear elements are adopted. Finally, a conceivable surprising characteristic is that, as demonstrated with numerical examples, these methods provide a given accuracy level for a comparable, and often lower, computational cost than conforming formulations.
Stabilization is occasionally needed for discontinuous Galerkin methods in linear elliptic problems. In this paper we propose a sufficient condition for the stability of each linearized non-linear elastic problem that naturally includes material and geometric parameters; the latter needed to account for buckling. We then prove that when a similar condition is satisfied by the discrete problem, the method provides stable linearized deformed configurations upon the addition of a standard stabilization term.
We conclude by discussing the complexity of the implementation, and propose a computationally efficient approach that avoids looping over both elements and element faces. Several numerical examples are then presented in two and three dimensions that illustrate the performance of a selected discontinuous Galerkin method within the class. Copyright
... Twofold saddle point problems arise ubiquitously when mixed finite element formulations are used to approximate the stress in an incompressible fluid or solid [2, 9, 27, 29, 24, 15, 22, 23,6, 26]. When the usual linear relation S 0 = νD(u) for the viscous stress is replaced with a more general relation S 0 = νA(D(u)) the equations for the creeping (Stokes) flow of a fluid in a domain Ω ⊂ R d become, ...
... Sufficient conditions for solvability and abstract error analysis are also given in [21] and [25]. This theory has been applied to many problems in elasticity and fluids as well, see [27, 29, 24, 15, 22, 23, 26] for some examples. Problems of type (1.2) were also studied in [16, 17]. ...
Necessary and sufficient conditions for existence and uniqueness of solutions are developed for twofold saddle point problems
which arise in mixed formulations of problems in continuum mechanics. This work extends the classical saddle point theory
to accommodate nonlinear constitutive relations and the twofold saddle structure. Application to problems in incompressible
fluid mechanics employing symmetric tensor finite elements for the stress approximation is presented.
Mathematics Subject Classification (2000)65N30
... In particular, nonlinear poroelasticity equations arise, for example, in models of geomechanics and in the study of deformable soft tissues (such as filtration of aqueous humor through cartilage-like structures in the eye and with application in glaucoma formation). In the present work we focus on the case of fully saturated deformable porous media (the Apart from the application of Hu-Washizu formulations in many works for linear elasticity (see, e.g., [21][22][23][24][25] and the references therein), the solvability analysis of the continuous and discrete twofold saddle-point mixed problems (including also error estimates) has been carried out in [26,27] for Hencky-strain nonlinear elasticity, as well as for more recent models for stress-assisted diffusion coupled with poroelasticity [28]. There, one tests the constitutive equation for stress against the test function associated with the space of infinitesimal strains. ...
We propose four-field and five-field Hu–Washizu-type mixed formulations for nonlinear poroelasticity – a coupled fluid diffusion and solid deformation process – considering that the permeability depends on a linear combination between fluid pressure and dilation. As the determination of the physical strains is necessary, the first formulation is written in terms of the primal unknowns of solid displacement and pore fluid pressure as well as the poroelastic stress and the infinitesimal strain, and it considers strongly symmetric Cauchy stresses. The second formulation imposes stress symmetry in a weak sense and it requires the additional unknown of solid rotation tensor. We study the unique solvability of the problem using the Banach fixed-point theory, properties of twofold saddle-point problems, and the Banach–Nečas–Babuška theory. We propose monolithic Galerkin discretisations based on conforming Arnold–Winther for poroelastic stress and displacement, and either PEERS or Arnold–Falk–Winther finite element families for the stress–displacement-rotation field variables. The wellposedness of the discrete problem is established as well, and we show a priori error estimates in the natural norms. Some numerical examples are provided to confirm the rates of convergence predicted by the theory, and we also illustrate the use of the formulation in some typical tests in Biot poroelasticity.
... Apart from the application of Hu-Washizu formulations in many works for linear elasticity (see, e.g., [22,23,37,38,51] and the references therein), the solvability analysis of the continuous and discrete twofold saddle-point mixed problems (including also error estimates) has been carried out in [30,32] for Hencky-strain nonlinear elasticity, as well as for more recent models for stress-assisted diffusion coupled with poroelasticity [33]. There, one tests the constitutive equation for stress against the test function associated with the space of infinitesimal strains. ...
We propose four-field and five-field Hu--Washizu-type mixed formulations for nonlinear poroelasticity -- a coupled fluid diffusion and solid deformation process -- considering that the permeability depends on a linear combination between fluid pressure and dilation. As the determination of the physical strains is necessary, the first formulation is written in terms of the primal unknowns of solid displacement and pore fluid pressure as well as the poroelastic stress and the infinitesimal strain, and it considers strongly symmetric Cauchy stresses. The second formulation imposes stress symmetry in a weak sense and it requires the additional unknown of solid rotation tensor. We study the unique solvability of the problem using the Banach fixed-point theory, properties of twofold saddle-point problems, and the Banach--Ne\v{c}as--Babu\v{s}ka theory. We propose monolithic Galerkin discretisations based on conforming Arnold--Winther for poroelastic stress and displacement, and either PEERS or Arnold--Falk--Winther finite element families for the stress-displacement-rotation field variables. The wellposedness of the discrete problem is established as well, and we show a priori error estimates in the natural norms. Some numerical examples are provided to confirm the rates of convergence predicted by the theory, and we also illustrate the use of the formulation in some typical tests in Biot poroelasticity.
... In addition, it is well known that for low-order finite element approximations of elasticity problems subject to the volume preserving constraint, the undesirable so-called volume locking phenomenon, in which the displacement decreases severely in a non-physical fashion may show up [12]. Several remedies have been proposed in the literature, including for instance resorting to different kinds of mixed and double-mixed formulations [4,6,10,15]. Herein, following the framework in [5,27], we formulate a suitable discontinuous Galerkin (DG) method. ...
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.
... The Herrmann principle has also been extended to nonlinear hyperelastic problems in Reissner & Atluri (1989); van den Bogert et al. (1991);de Souza et al. (1996); Piltner & Taylor (1999). Other mixed formulations for nonlinear elasticity are introduced in Gatica & Stephan (2002); Gatica et al. (2007), see also Carstensen & Funken (2001); Brink & Stephan (2001); Carstensen et al. (2005). The approaches based on the bilinear or trilinear displacement and piecewise constant pressure are often used to overcome the volumetric locking in nearly incompressible elasticity, see Hughes (1987); Brezzi & Fortin (1991); Braess (2001). ...
We consider finite-element methods based on simplices to solve the problem of nearly incompressible elasticity. Two different
approaches based, respectively, on dual meshes and dual bases are presented, where in both approaches pressure is discontinuous
and can be statically condensed out from the system. These novel approaches lead to displacement-based low-order finite-element
methods for nearly incompressible elasticity based on rigorous mathematical framework. Numerical results are provided to demonstrate
the efficiency of the approach.
... In [12], the pseudostress was introduced as an additional variable and used in the modeling equations instead of the symmetric stress tensor. A similar approach that, in the context of fluid problems, introduces the pseudostress and velocity gradient as additional variables, has been employed in several recent works (see [13, 14, 15, 16, 17, 18] ). Both approaches yield saddle-point problems which require certain infsup conditions to be satisfied by the chosen finite element approximation spaces. ...
In this work a finite element method for a dual-mixed approximation of Stokes and generalized Stokes problems in two or three space dimensions is studied. A variational formulation of the generalized Stokes problems is accomplished through the introduction of the pseudostress and the trace-free velocity gradient as unknowns, yielding a twofold saddle point problem. The method avoids the explicit computation of the pressure, which can be recovered through a simple post-processing technique. Compared to an existing approach for the same problem, the method presented here reduces the global number of degrees of freedom by up to one third in two space dimensions. The method here also represents a connection between existing dual-mixed and pseudostress methods for Stokes problems. Existence, uniqueness, and error results for the generalized Stokes problem are given, and two new penalty methods for approximating the steady Stokes problem are derived. Numerical experiments that illustrate the theoretical results are presented.
In this paper we recast the analysis of twofold saddle point variational formulations for several nonlinear
boundary value problems arising in continuum mechanics, and derive reliable and efficient residual-based
a posteriori error estimators for the associated Galerkin schemes.We illustrate the main results with nonlinear
elliptic equations modelling heat conduction and hyperelasticity. The main tools of our analysis
include a global inf–sup condition for a linearization of the problem, Helmholtz’s decompositions, local
approximation properties of the Raviart–Thomas and Clément interpolation operators, inverse inequalities,
and the localization technique based on triangle-bubble and edge-bubble functions. Finally, several
numerical results confirming the theoretical properties of the estimator and showing the behaviour of the
associated adaptive algorithms are provided.
SUMMARYA Galerkin FEM is developed for nonlinear, incompressible (hyper) elasticity that takes account of nonlinearities in both the strain tensor and the relationship between the strain tensor and the stress tensor. By using suitably defined linearised dual problems with appropriate boundary conditions, a posteriori error estimates are then derived for both linear functionals of the solution and linear functionals of the stress on a boundary, where Dirichlet boundary conditions are applied. A second, higher order method for calculating a linear functional of the stress on a Dirichlet boundary is also presented together with an a posteriori error estimator for this approach. An implementation for a 2D model problem with known solution, where the entries of the strain tensor exhibit large, rapid variations, demonstrates the accuracy and sharpness of the error estimators. Finally, using a selection of model problems, the a posteriori error estimate is shown to provide a basis for effective mesh adaptivity. Copyright © 2014 John Wiley & Sons, Ltd.
In this paper, we present a geometric discretization scheme for incompressible linearized elasticity. We use ideas from discrete exterior calculus (DEC) to write the action for a discretized elastic body modeled by a simplicial complex. After characterizing the configuration manifold of volume-preserving discrete deformations, we use Hamilton’s principle on this configuration manifold. The discrete Euler–Lagrange equations are obtained without using Lagrange multipliers. The main difference between our approach and the mixed finite element formulations is that we simultaneously use three different discrete spaces for the displacement field. We explicitly derive the governing equations for the two-dimensional case, where the discrete spaces for the displacement field are constructed by P1P1 polynomials over primal meshes for incompressibility constraint, P0P0 polynomials over dual meshes for the kinetic energy, and P1P1 polynomials over support volumes for the elastic energy, and the discrete space of the pressure field is constructed by P0P0 polynomials over primal meshes. We test the efficiency and robustness of this geometric scheme using some numerical examples. In particular, we do not see any volume locking and/or checkerboarding of pressure in our numerical examples. This suggests that our choice of discrete solution spaces is compatible.
We combine a dual-mixed finite element method with the boundary integral equation method, to study the solvability and Galerkin approximations of a linear-nonlinear transmission problem arising in plane elastostatics. Our approach is based on the introduction of the strain tensor as an auxiliary unknown in the finite element domain, and it leads to what we call a dual-dual mixed formulation since the corresponding operator equation becomes of a twofold saddle-point structure. We derive existence and uniqueness of solution for the continuous and discrete formulations and provide the associated error analysis. In particular, we extend the finite element space for plane elasticity given by the PEERS element and define an explicit finite element/boundary element subspace satisfying all the required compatibility conditions. Most of our analysis makes use of an extension of the classical Babuska--Brezzi theory to a class of nonlinear saddle-point problems.
A mixed finite element procedure for plane elasticity is introduced and analyzed. The symmetry of the stress tensor is enforced
through the introduction of a Lagrange multiplier. An additional Lagrange multiplier is introduced to simplify the linear
algebraic system. Applications are made to incompressible elastic problems and to plasticity problems.
The Dirichler problem for the equations of plane elasticity is approximated by a mixed finite element method using a new family of composite finite elements having properties analogous to those possessed by the Raviart-Thomas mixed finite elements for a scalar, second-order elliptic equation. Estimates of optimal order and minimal regularity are derived for the errors in the displacement vector and the stress tensor inL
2(), and optimal order negative norm estimates are obtained inH
s
() for a range ofs depending on the index of the finite element space. An optimal order estimate inL
() for the displacement error is given. Also, a quasioptimal estimate is derived in an appropriate space. All estimates are valid uniformly with respect to the compressibility and apply in the incompressible case. The formulation of the elements is presented in detail.
We combine a dual-mixed nite element method with a Dirichlet- to-Neumann mapping (derived by the boundary integral equation method) to study the solvability and Galerkin approximations of a class of exterior non- linear transmission problems in the plane. As a model problem, we consider a nonlinear elliptic equation in divergence form coupled with the Laplace equa- tion in an unbounded region of the plane. Our combined approach leads to what we call a dual-dual mixed variational formulation since the main operator involved has itself a dual-type structure. We establish existence and unique- ness of solution for the continuous and discrete formulations, and provide the corresponding error analysis by using Raviart-Thomas elements. The main tool of our analysis is given by a generalization of the usual Babuska-Brezzi theory to a class of nonlinear variational problems with constraints.
This article discusses nite element Galerkin schemes for a number of lin- ear model problems in electromagnetism. The nite element schemes are in- troduced as discrete dierential forms, matching the coordinate-independent statement of Maxwell's equations in the calculus of dierential forms. The asymptotic convergence of discrete solutions is investigated theoretically. As discrete dierential forms represent a genuine generalization of conventional Lagrangian nite elements, the analysis is based upon a judicious adaptation of established techniques in the theory of nite elements. Risks and diculties haunting nite element schemes that do not t the framework of discrete dif- ferential forms are highlighted.
We extend the applicability of stable mixed finite elements for linear plane elasticity, such as PEERS, to a mixed variational formulation of hyperelasticity. The present approach is based on the introduction of the strain tensor as a further unknown, which yields a two-fold saddle point nonlinear operator equation for the corresponding weak formulation. We provide the uniqueness of solution for the continuous and discrete schemes, and derive the usual Cea estimate for the associated error. Finally, a reliable a-posteriori error estimate, based on the solution of local Dirichlet problems, and well suited for adaptive computations. is also given.
. We present three new a posteriori error estimators in the energy norm for finite element solutions to elliptic partial differential equations. The estimators are based on solving local Neumann problems in each element. The estimators differ in how they enforce consistency of the Neumann problems. We prove that as the mesh size decreases, under suitable assumptions, two of the estimators approach upper bounds on the norm of the true error, and all three estimators are within multiplicative constants of the norm of the true error. We present numerical results in which one of the estimators appears to converge to the norm of the true error. 1. Introduction In this work, we will describe several methods for computing a posteriori error estimates for finite element calculations. That is, given some piecewise polynomial approximation U to uH , the solution of an elliptic partial differential equation, we seek some practical method for computing an estimate of jjju H Gamma U jjj for an ap...
. Computable a posteriori error bounds and related adaptive mesh-refining algorithms are provided for the numerical treatment of monotone stationary flow problems with a quite general class of conforming and nonconforming finite element methods. A refined residual-based error estimate generalises the works of Verfurth, Dari, Duran & Padra, Bao & Barrett. As a consequence, reliable and efficient averaging estimates can be established on unstructured grids. The symmetric formulation of the incompressible flow problem models certain non-Newtonian flow problems and the Stokes problem with mixed boundary conditions. A Helmholtz decomposition avoids any regularity or saturation assumption in the mathematical error analysis. Numerical experiments for the partly nonconforming method analysed by Kouhia and Stenberg indicate efficiency of related adaptive mesh-refining algorithms. 1. Introduction Adaptive finite element methods play an important practical role in computational fluid dynamics. T...
We present three new a posteriori error estimators in the energy norm for finite element solutions to elliptic partial differential equations. The estimators are based on solving local Neumann problems in each element. The estimators differ in how they enforce consistency of the Neumann problems. We prove that as the mesh size decreases, under suitable assumptions, two of the error estimators approach upper bounds on the norm of the true error, and all three error estimators are within multiplicative constants of the norm of the true error. We present numerical results in which one of the error estimators appears to converge to the norm of the true error.
We generalize the usual Babuska-Brezzi theory to a class of nonlinear variational problems with constraints. The corresponding operator equation has a dual-dual type structure since the nonlinear operator involved has itself a dual structure with a strongly monotone and Lipschitz-continuous main operator. We provide sufficient conditions for the existence and uniqueness of solution of the continuous and Galerkin formulations, and derive a Strang-type estimate for the associated error. An application to the coupling of mixed-FEM and BEM for a nonlinear transmission problem in potential theory is also described.
In this paper, we provide a priori and a posteriori error analyses of an augmented mixed finite element method with Lagrange multipliers applied to elliptic equations in divergence form with mixed boundary conditions. The augmented scheme is obtained by including the Galerkin least-squares terms arising from the constitutive and equilibrium equations. We use the classical Babuska-Brezzi theory to show that the resulting dual-mixed variational formulation and its Galerkin scheme defined with Raviart-Thomas spaces are well posed, and also to derive the corresponding a priori error estimates and rates of convergence. Then, we develop a reliable and efficient residual-based a posteriori error estimate and a reliable and quasi-efficient Ritz projection-based one, as well. Finally, several numerical results illustrating the performance of the augmented scheme and the associated adaptive algorithms are reported.
The numerical approximation of the mixed velocity-pressure-stress formulation of the Stokes problem using spectral methods is considered. In addition to the compatibility condition between the discrete velocity and pressure spaces, a second condition between the discrete velocity and stress spaces must also be satisfied in order to have a well-posed problem. The theory is developed by considering a doubly constrained minimization problem in which the viscous stress tensor is minimized subject to the constraint that the viscous forces are irrotational. The discrete problem is analyzed and error estimates are derived. A comparison between the mixed approach and the standard velocity-pressure formulation is made in terms of the condition number of the resulting systems.
The mathematical basis of FEMs for incompressible steady interior flow
problems is examined in a text (a revised and expanded version of the
work of Girault and Raviart, 1979) intended for a postgraduate course in
numerical analysis. Chapters are devoted to the mathematical foundation
of the Stokes problem, results obtained with a standard FEM
approximation, the numerical solution of the Stokes problem in primitive
variables, incompressible mixed FEMs for solving the Stokes problem, and
the theory and approximation of the Navier-Stokes problem.
In this paper we apply the Babuska-Brezzi theory to obtain new results on the solvability and Galerkin approximations of some operator equations arising in linear and nonlinear variational problems with linear constraints. This constitutes an extension of the usual theory to a class of variational formulations that can be written in a dual-dual form
The coupling of finite elements and boundary elements is analyzed, where in the FEM domain we assume an incompressible elastic material governed by a uniformly monotone operator and use a Stokes-type mixed FEM. In the BEM domain, linear elasticity is considered. We prove existence and uniqueness of the solution and quasi-optimal convergence of a Galerkin method. We derive an a posteriori error estimator of explicit residual type. © 2001 John Wiley & Sons, Inc. Numer Methods Partial Differential Eq 17: 79–92, 2001
Three a posteriori error estimators for PEERS and BDMS elements in linear elasticity are presented: one residual error estimator and two estimators based on the solution of auxiliary local problems with different boundary conditions. All of them are reliable and efficient with respect to the standard norm and furthermore robust for nearly incompressible materials.
A residue based reliable and efficient error estimator is established for finite element solutions of mixed boundary value
problems in linear, planar elasticity. The proof of the reliability of the estimator is based on Helmholtz type decompositions
of the error in the stress variable and a duality argument for the error in the displacements. The efficiency follows from
inverse estimates. The constants in both estimates are independent of the Lamé constant , and so locking phenomena for are properly indicated. The analysis justifies a new adaptive algorithm for automatic mesh–refinement.
We prove a discrete extension theorem for the equilibrium finite elements spaces. The most important part of the work consists
of establishing the basic properties used for proving the extension theorem.
A new mixed finite element has allowed us to calculate flows of Maxwell-B and Oldroyd-B fluids at very high values of the Deborah number, De. The element is divided into several bilinear sub-elements for the stresses, while streamline-upwinding is used for discretizing the constitutive equation. The method is applied to the stick-slip problem, the flow through a tapered contraction and the flow through four-to-one abrupt plane and circular contractions. Important corner vortices develop at high values of De in the circular contraction. We have not encountered upper limits for the Deborah number in our calculations with Oldroyd-B fluids.
We present a mixed finite element method for a class of non-linear Stokes models arising in quasi-Newtonian fluids. Our results include, as a by-product, a new mixed scheme for the linear Stokes equation. The approach is based on the introduction of both the flux and the tensor gradient of the velocity as further unknowns, which yields a twofold saddle point operator equation as the resulting variational formulation. We prove that the continuous and discrete formulations are well posed, and derive the associated a priori error analysis. The corresponding Galerkin scheme is defined by using piecewise constant functions and Raviart–Thomas spaces of lowest order.
We analyse three different a posteriori error estimators for elliptic partial differential equations. They are based on the evaluation of local residuals with respect to the strong form of the differential equation, on the solution of local problems with Neumann boundary conditions, and on the solution of local problems with Dirichlet boundary conditions. We prove that all three are equivalent and yield global upper and local lower bounds for the true error. Thus adaptive mesh-refinement techniques based on these estimators are capable to detect local singularities of the solution and to appropriately refine the grid near these singularities. Some numerical examples prove the efficiency of the error estimators and the mesh-refinement techniques.
A method to obtain a posteriori an accurate measure of the error in finite element computations is to solve a Neumann problem with equilibrated tractions for each finite element. We extend this approach to large-strain elasticity and mixed finite elements, which are well suited for nearly incompressible rubber-like materials. For an L2-projection of the error in the first Piola-Kirchhoff stress tensor we prove that the estimator provides a guaranteed upper bound without multiplicative constants. Numerical examples confirm the effectiveness for both linear and Ogden material.
This is the second part of a work dealing with a low-order mixed finite element method for a class of nonlinear Stokes models arising in quasi-Newtonian fluids. In the first part we showed that the resulting variational formulation is given by a twofold saddle point operator equation, and that the corresponding Galerkin scheme becomes well posed with piecewise constant functions and Raviart–Thomas spaces of lowest order as the associated finite element subspaces. In this paper we develop a Bank–Weiser type a posteriori error analysis yielding a reliable estimate and propose the corresponding adaptive algorithm to compute the mixed finite element solutions. Several numerical results illustrating the efficiency of the method are also provided.
In this paper we present a new a posteriori error estimate for the boundary element method applied to an integral equation of the rst kind. The estimate is local and sharp for quasi-uniform meshes and so improves earlier work of ours. The mesh-dependence of the constants is analyzed and shown to be weaker than expected from our previous work. Besides the Galerkin bound- ary element method, the collocation method and the qualocation method are considered. A numerical example is given involving an adaptive feedback al- gorithm.
We provide a general abstract theory for the solvability and Galerkin approximation of nonlinear twofold saddle point problems. In particular, a Strang error estimate containing the consistency terms arising from the approximation of the continuous operators involved is deduced. Then we apply these results to analyse a fully discrete Galerkin scheme for a twofold saddle point formulation of a nonlinear elliptic boundary value problem in divergence form. Some numerical results are also presented.
Introduction The classical Babuska-Brezzi theory for variational problems with constraints is a very well established tool for studying the weak solutions and Galerkin approximations, via mixed formulations, of a wide class of linear boundary value problems in physics and engineering sciences (see, e.g. [2], [9] and the references therein). ? This research was partially supported by Fondecyt-Chile through the project No. 1980122 and by the Program A on Numerical Analysis of the FONDAP in Applied Mathematics. 2 Gabriel N. Gatica However, the corresponding theory for nonlinear problems has not been fully developed and only a few fundamental references can be found in the literature. For the case of nonlinear problems with nonlinear constraints, we refer to [13] and [14], where the abstract theory and its applications to some problems in nonlinear incompressible elasticity are presented. Also, we may mention the wor
Mixed finite element methods such as PEERS or the BDMS methods are designed to avoid locking for nearly incompressible materials in plane elasticity. In this paper, we establish a robust adaptive mesh-refining algorithm that is rigorously based on a reliable and efficient a posteriori error estimate. Numerical evidence is provided for the -independence of the constants in the a posteriori error bounds and for the efficiency of the adaptive mesh-refining algorithm proposed. Key words: a posteriori error estimates, adaptive algorithm, reliability, mixed finite element method, locking 1 Introduction In this paper we investigate finite element solutions of the Lam'e system in linear elasticity and consider a plane elastic body with reference configuration Omega ae R 2 and boundary @Omega = Gamma = Gamma D [ Gamma N , Gamma D 6= ;, Gamma N = GammanGamma D . Given a volume force f :Omega ! R 2 and a traction g : Gamma N ! R 2 , we seek (an approximation to) the displace...
This paper deals with an expanded mixed nite element formulation, based on the HuWashizu principle, for a nonlinear incompressible material in the plane. We follow our related previous works and introduce both the stress and the strain tensors as further unknowns, which yields a two-fold saddle point operator equation as the corresponding variational formulation. A slight generalization of the classical Babu^ska-Brezzi's theory is applied to prove unique solvability of the continuous and discrete formulations, and to derive the corresponding a-priori error analysis. An extension of the well known PEERS space is used to dene an stable associated Galerkin scheme. Finally, we provide an a-posteriori error analysis based on the classical Bank-Weiser approach. Key words. Mixed nite elements, PEERS, hyperelasticity, a-posteriori error estimate Mathematics subject classications (1991). 65N30, 74B20, 65N15 1 Introduction It is well known that the displacement-based variational formulat...