A variational arbitrary Lagrangian-Eulerian finite element formulation for standard dissipative solids at finite strains

A novel Arbitrary Lagrangian-Eulerian (ALE) finite element formulation for standard dissipative media at finite strains is presented. In contrast to previously published ALE approaches accounting for dissipative phenomena, the proposed scheme is fully variational. Consequently, no error estimates are necessary and thus, linearity of the problem and the corresponding Hilbert-space are not required. Hence, the resulting Variational Arbitrary Lagrangian-Eulerian (VALE) finite element method can be applied to highly nonlinear phenomena as well. In case of standard dissipative solids, so-called variational constitutive updates provide a variational principle. Based on these updates, the deformation mapping follows from minimizing an incrementally defined (pseudo) potential, i.e., energy minimization is the overriding criterion that governs every aspect of the system. Therefore, it is natural to allow the variational principle to drive mesh adaption as well. Thus, in the present paper, the discretizations of the deformed as well as the undeformed configuration are optimized jointly by minimizing the respective incremental energy of the considered mechanical system. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

An implicit–explicit residual error estimator for the coupling of dual‐mixed finite elements and boundary elements in elastostatics

We consider the coupling of dual‐mixed finite elements and boundary elements to solve a mixed Dirichlet–Neumann problem of plane elasticity. We derive an a‐posteriori error estimate that is based on the solution of local Dirichlet problems and on a residual term defined on the coupling interface. The general error estimate does not make use of any special finite element or boundary element spaces. Here the residual term is given in a negative order Sobolev norm. In practical applications, where a certain boundary element subspace is used, this norm can be estimated by weighted local L²‐norms. Copyright © 2001 John Wiley & Sons, Ltd.     

Adaptive refinement procedure for the finite difference method

An adaptive refinement procedure consisting of a localized error estimator and a physically based approach to mesh refinement is developed for the finite difference method. The error estimator is a variation of a successful finite element error estimator. The errors are estimated by computing an error energy norm in terms of discontinuous and continuous stress fields formed from the finite difference results for plane stress problems. The error measure identifies regions of high error which are subsequently refined to improve the result. The local refinement procedure utilizes a recently developed approach for developing finite difference templates to produce a graduated mesh. The adaptive refinement procedure is demonstrated with a problem that contains a well-defined singularity. The results are compared to finite element and uniformly refined finite difference results.     

特征值问题协调/非协调有限元后验误差分析

本文研究对称椭圆特征值问题的有限元后验误差估计,包括协调元和非协调元,具有下列特色:(1)对协调/非协调元建立了有限元特征函数uh的误差与相应的边值问题有限元解的误差在局部能量模意义下的恒等关系式,该边值问题的右端为有限元特征值λh与uh的乘积,有限元解恰好为uh.从而边值问题有限元解在能量模意义下的局部后验误差指示子,包括残差型和重构型后验误差指示子,成为有限元特征函数在能量模意义下的局部后验误差指示子.(2)讨论了协调有限元特征函数的基于插值后处理的梯度重构型后验误差估计,对有限元特征函数的导数得到了最大模意义下的渐近准确局部后验误差指示子.     

<Emphasis Type="Italic">L</Emphasis><Superscript>2</Superscript>-error estimates for Dirichlet and Neumann problems on anisotropic finite element meshes

An L ²-estimate of the finite element error is proved for a Dirichlet and a Neumann boundary value problem on a three-dimensional, prismatic and non-convex domain that is discretized by an anisotropic tetrahedral mesh. To this end, an approximation error estimate for an interpolation operator that is preserving the Dirichlet boundary conditions is given. The challenge for the Neumann problem is the proof of a local interpolation error estimate for functions from a weighted Sobolev space.     

An adaptive finite element method for a time‐dependent Stokes problem

In this article, we conduct an a posteriori error analysis of the two‐dimensional time‐dependent Stokes problem with homogeneous Dirichlet boundary conditions, which can be extended to mixed boundary conditions. We present a full time–space discretization using the discontinuous Galerkin method with polynomials of any degree in time and the ? ₂ ? ?₁ Taylor–Hood finite elements in space, and propose an a posteriori residual‐type error estimator. The upper bounds involve residuals, which are global in space and local in time, and an L ²‐error term evaluated on the left‐end point of time step. From the error estimate, we compute local error indicators to develop an adaptive space/time mesh refinement strategy. Numerical experiments verify our theoretical results and the proposed adaptive strategy.     

Galerkin‐type space‐time finite elements for volumetrically coupled problems

The study focuses on error estimation techniques for a coupled problem with two constituents based on the Theory of Porous Media. After developing space‐time finite elements for this mixed problem, we extend the numerical scheme to a coupled space‐time adaptive strategy. Therefore, an adjoint or dual problem is formulated and discussed, which is solved lateron numerically. One advantage of the presented technique is the high flexibility of the error indicator with respect to the error measure.     

Numerical experiments in finite element grid optimization by direct energy search

A series of numerical experiments is conducted to assess the feasibility and practical value of finite element grid optimization based on direct minimization of the total potential energy of the discrete model with respect to the node locations. An implementation relying upon non-linear programming techniques is found to be numerically reliable and to lead to improved grids in accord with engineering intuition. This rigorous approach is hampered, however, by the excessive computational effort required by the energy minimization process. A combination of related techniques is therefore proposed to make dynamic node distribution a useful tool within the framework of large-scale finite element analysis. The combined strategy involves use of substructuring methods, application of a local energy-balancing optimality criterion for fast node distribution, and automatic refinement of previously-improved coarse grids.     

An experimental survey of a posteriori Courant finite element error control for the Poisson equation

This comparison of some a posteriori error estimators aims at empirical evidence for a ranking of their performance for a Poisson model problem with conforming lowest order finite element discretizations. Modified residual-based error estimates compete with averaging techniques and two estimators based on local problem solving. Multiplicative constants are involved to achieve guaranteed upper and lower energy error bounds up to higher order terms. The optimal strategy combines various estimators.     

Adaptive FEM with Stabilized Elements for Parameter Identification of Incompressible Hyperelastic Materials

This work is concerned with the identification of material parameters for isotropic, incompressible hyperelastic material models. Besides the principal stretch-based strain-energy function by Ogden an invariant-based strain-energy function by Rivlin/Saunders is considered for which parameter sensitivities are derived. The identification is formulated as a least-squares minimization problem based on the finite element method to account for inhomogeneous states of stresses and strains in the experimental data which is obtained by optical measurements. For the finite element method low-order tetrahedral elements in a mixed displacement-pressure formulation with stabilization are considered. Special attention is payed to an adaptive mesh-refinement based on a goal-oriented a posteriori error indicator to gain reliable material parameters. To approximate error terms an element-wise recovery technique based on enhanced gradients is introduced. (© 2012 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Sensitivities for topological graph changes in finite element meshes and application to adaptive refinement

We propose a novel approach to adaptivity in FEM based on local sensitivities for topological mesh changes. To this end, we consider refinement as a continuous operation on the edge graph of the finite element discretization, for instance by splitting nodes along edges and expanding edges to elements. Thereby, we introduce the concept of a topological mesh derivative for a given objective function that depends on the discrete solution of the underlying PDE. These sensitivities may in turn be used as refinement indicators within an adaptive algorithm. For their calculation, we rely on the first-order asymptotic expansion of the Galerkin solution with respect to the topological mesh change. As a proof of concept, we consider the total potential energy of a linear symmetric second-order elliptic PDE, minimization of which is known to decrease the approximation error in the energy norm. In this case, our approach yields local sensitivities that are closely related to the reduction of the energy error upon refinement and may therefore be used as refinement indicators in an adaptive algorithm. (© 2014 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Localized pointwise error estimates for mixed finite element methods

In this paper we give weighted, or localized, pointwise error estimates which are valid for two different mixed finite element methods for a general second-order linear elliptic problem and for general choices of mixed elements for simplicial meshes. These estimates, similar in spirit to those recently proved by Schatz for the basic Galerkin finite element method for elliptic problems, show that the dependence of the pointwise errors in both the scalar and vector variables on the derivative of the solution is mostly local in character or conversely that the global dependence of the pointwise errors is weak. This localization is more pronounced for higher order elements. Our estimates indicate that localization occurs except when the lowest order Brezzi-Douglas-Marini elements are used, and we provide computational examples showing that the error is indeed not localized when these elements are employed.

     

Asymptotic expansions of finite element solutions to Robin problems in H 3 and their application in extrapolation cascadic multigrid method

For the Poisson equation with Robin boundary conditions,by using a few techniques such as orthogonal expansion(M-type),separation of the main part and the finite element projection,we prove for the first time that the asymptotic error expansions of bilinear finite element have the accuracy of O(h3)for u∈H3.Based on the obtained asymptotic error expansions for linear finite elements,extrapolation cascadic multigrid method(EXCMG)can be used to solve Robin problems effectively.Furthermore,by virtue of Richardson not only the accuracy of the approximation is improved,but also a posteriori error estimation is obtained.Finally,some numerical experiments that confirm the theoretical analysis are presented.     

New locally conservative finite element methods on a rectangular mesh

A new family of locally conservative, finite element methods for a rectangular mesh is introduced to solve second-order elliptic equations. Our approach is composed of generating PDE-adapted local basis and solving a global matrix system arising from a flux continuity equation. Quadratic and cubic elements are analyzed and optimal order error estimates measured in the energy norm are provided for elliptic equations. Next, this approach is exploited to approximate Stokes equations. Numerical results are presented for various examples including the lid driven-cavity problem.     

An adaptive stabilized finite element method for the generalized Stokes problem

In this work we present an adaptive strategy (based on an a posteriori error estimator) for a stabilized finite element method for the Stokes problem, with and without a reaction term. The hierarchical type estimator is based on the solution of local problems posed on appropriate finite dimensional spaces of bubble-like functions. An equivalence result between the norm of the finite element error and the estimator is given, where the dependence of the constants on the physics of the problem is explicited. Several numerical results confirming both the theoretical results and the good performance of the estimator are given.     

A new mixed finite element method for the Stokes problem

We propose a new mixed formulation of the Stokes problem where the extra stress tensor is considered. Based on such a formulation, a mixed finite element is constructed and analyzed. This new finite element has properties analogous to the finite volume methods, namely, the local conservation of the momentum and the mass. Optimal error estimates are derived. For the numerical implementation of this finite element, a hybrid form is presented. This work is a first step towards the treatment of viscoelastic fluid flows by mixed finite element methods.     

A mixed formulation for 3D magnetostatic problems: theoretical analysis and face-edge finite element approximation

Summary. A mixed field-based variational formulation for the solution of threedimensional magnetostatic problems is presented and analyzed. This method is based upon the minimization of a functional related to the error in the constitutive magnetic relationship, while constraints represented by Maxwell's equations are imposed by means of Lagrange multipliers. In this way, both the magnetic field and the magnetic induction field can be approximated by using the most appropriate family of vector finite elements, and boundary conditions can be imposed in a natural way. Moreover, this method is more suitable than classical approaches for the approximation of problems featuring strong discontinuities of the magnetic permeability, as is usually the case. A finite element discretization involving face and edge elements is also proposed, performing stability analysis and giving error estimates. Received January 23, 1998 / Revised version received July 23, 1998 / Published online September 24, 1999     

A-posteriori error analysis and adaptive finite element methods for singularly perturbed convection-diffusion equations

For singularly perturbed one-dimensional convection-diffusion equations, finite element approximations are constructed based on a so-called approximate symmetrization of the given unsymmetric problem. Local a-posteriori error estimates are established with respect to an appropriate energy norm where the bounds are proved to be realistic. The local bounds, called error indicators, provide a basis for a self-adaptive mesh refinement. For a model problem numerical results are presented showing that the adaptive method detects and resolves the boundary layer.     

An adaptive least squares mixed finite element method for the stress‐displacement formulation of linear elasticity

A least‐squares mixed finite element method for linear elasticity, based on a stress‐displacement formulation, is investigated in terms of computational efficiency. For the stress approximation quadratic Raviart‐Thomas elements are used and these are coupled with the quadratic nonconforming finite element spaces of Fortin and Soulie for approximating the displacement. The local evaluation of the least‐squares functional serves as an a posteriori error estimator to be used in an adaptive refinement algorithm. We present computational results for a benchmark test problem of planar elasticity including nearly incompressible material parameters in order to verify the effectiveness of our adaptive strategy. For comparison, conforming quadratic finite elements are also used for the displacement approximation showing convergence orders similar to the nonconforming case, which are, however, not independent of the Lamé parameters. © 2004 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2005     

Mixed Variational Principles and Robust Finite Element Design of Gradient Plasticity at Finite Strains

The modeling of size effects in elastic-plastic solids, such as the width of shear bands or the grain size dependence in polycrystals, must be based on non-standard theories which incorporate length-scales. This is achieved by models of strain gradient plasticity, incorporating spatial gradients of selected micro-structural fields which describe the evolving dissipative mechanisms. The key aspect of this work is to provide a rigorous incremental variational formulation and mixed finite element design of additive finite gradient plasticity in the logarithmic strain space. We start from a mixed saddle point principle for metric-type plasticity, which is specified for the important model problem of isochoric plasticity with gradient-extended hardening/softening response. To this end, we propose a novel finite element design of the coupled problem incorporating a local-global solution strategy of short- and long-range fields. This includes several new aspects, such as extended Q1P0-type and MINI-type finite elements for gradient plasticity [4]. (© 2013 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)