Adaptive computational methods for variational inequalities based on mixed formulations

This work describes concepts for a posteriori error estimation and adaptive mesh design for finite element models where the solution is subjected to inequality constraints. These methods are developed here for several model problems. Based on these examples, unified frameworks are proposed, which provide a systematic way of adaptive error control for problems stated in form of variational inequalities. Copyright © 2006 John Wiley & Sons, Ltd.     

Space adaptive finite element methods for dynamic Signorini problems

Space adaptive techniques for dynamic Signorini problems are discussed. For discretisation, the Newmark method in time and low order finite elements in space are used. For the global discretisation error in space, an a posteriori error estimate is derived on the basis of the semi-discrete problem in mixed form. This approach relies on an auxiliary problem, which takes the form of a variational equation. An adaptive method based on the estimate is applied to improve the finite element approximation. Numerical results illustrate the performance of the presented method. Electronic supplementary material  The online version of this article (doi:) contains supplementary material, which is available to authorized users.     

Numerical inclusion methods of solutions for variational inequalities

We consider a numerical method that enables us to verify the existence of solutions for variational inequalities. This method is based on the infinite dimensional fixed point theorems and explicit error estimates for finite element approximations. Using the finite element approximations and explicit a priori error estimates, we present an effective verification procedure that through numerical computation generates a set which includes the exact solution. Copyright © 2002 John Wiley & Sons, Ltd.     

自然对流问题两重网格算法的残量型后验误差估计(英文)

本文得到了自然对流问题基于牛顿迭代两重网格算法的残量型后验误差估计.相对于标准有限元一层方法的后验误差估计,牛顿迭代两重网格算法的后验误差估计多了一些额外项.通过研究这些额外项的渐近行为,本文得到了这些额外项在误差估计中所起的作用.对于牛顿迭代两重网格方法的最优粗细网格匹配尺寸,这些额外项的收敛阶不高于离散解的收敛阶.数值算例验证了理论分析结论.     

On the duality of finite element discretization error control in computational Newtonian and Eshelbian mechanics

In this paper, goal-oriented a posteriori error estimators of the averaging type are presented for the error obtained while approximately evaluating theJ-integral in nonlinear elastic fracture mechanics. Since the value of the J-integral is one component of the material force acting on the crack tip of a pre-cracked elastic body, the appropriate mechanical framework to be chosen is the one named after Eshelby rather than classical Newtonian mechanics. However, in a finite element setting, the discretized Eshelby problem is generally not solved explicitly. Rather, its solution is approximated by the finite element solution of the corresponding discretized dual Newton problem. As a consequence, discrete material forces arise not only at the crack tip but also at other nodes of the current finite element mesh. It is the objective of this paper to establish goal-oriented a posteriori error estimators in both the framework of Eshelbian and Newtonian mechanics and to elaborate their dual relations. This allows to control the error of the J-integral while, at the same time, no further discrete material forces arise during the adaptive mesh refinement process which could lead to misleading mechanical interpretations of the results obtained by the finite element method. The paper is concluded by numerical examples that illustrate our theoretical results. Dedicated to the memory of the esteemed colleague Professor Karl Popp, University of Hannover, who unexpectedly passed away on April 24, 2005.     

Adaptive hp-versions of boundary element methods for elastic contact problems

The variational formulation of elastic contact problems leads to variational inequalities on convex subsets. These variational inequalities are solved with the boundary element method (BEM) by making use of the Poincaré–Steklov operator. This operator can be represented in its discretized form by the Schur-complement of the dense Galerkin-matrices for the single layer potential operator, the double layer potential operator and the hypersingular integral operator. Due to the difficulties in discretizing the convex subsets involved, traditionally only the h-version is used for discretization. Recently, p- and hp-versions have been introduced for Signorini contact problems in Maischak and Stephan (Appl Numer Math, 2005) . In this paper we show convergence for the quasi-uniform hp-version of BEM for elastic contact problems, and derive a-posteriori error estimates together with error indicators for adaptive hp-algorithms. We present corresponding numerical experiments.     

A posteriori error estimates of mixed methods for miscible displacement problems

The miscible displacement of one incompressible fluid by another in a porous medium is governed by a system of two equations. One is an elliptic equation of the pressure and the other is a parabolic equation of the concentration of one of the fluids. Since the pressure appears in the concentration only through its velocity field, we choose a mixed finite element method to approximate the pressure equation and for the concentration we use the standard Galerkin method. We shall obtain an explicit a posteriori error estimator in L²(L²) for the semi‐discrete scheme applied to the non‐linear coupled system. Copyright © 2007 John Wiley & Sons, Ltd.     

A Type of Finite Element Gradient Recovery Method Based on Vertex-Edge-Face Interpolation: The Recovery Technique and Superconvergence Property

In this paper, a new type of gradient recovery method based on vertex-edge-face interpolation is introduced and analyzed. This method gives a new way to recover gradient approximations and has the same simplicity, efficiency, and superconvergence properties as those of superconvergence patch recovery method and polynomial preserving recovery method. Here, we introduce the recovery technique and analyze its superconvergence properties. We also show a simple application in the a posteriori error estimates. Some numerical examples illustrate the effectiveness of this recovery method.     

Partition of unity for localization in implicit a posteriori finite element error control for linear elasticity

The partition of unity for localization in adaptive finite element method (FEM) for elliptic partial differential equations has been proposed in Carstensen and Funken (SIAM J. Sci. Comput. 2000; 21 : 1465–1484) and is applied therein to the Laplace problem. A direct adaptation to linear elasticity in this paper yields a first estimator η_L based on patch‐oriented local‐weighted interface problems. The global Korn inequality with a constant C_Korn yields reliability for any finite element approximation u_h to the exact displacement u. In order to localize this inequality further and so to involve the global constant C_Korn directly in the local computations, we deduce a new error estimator µ_L. The latter estimator is based on local‐weighted interface problems with rigid body motions (RBM) as a kernel and so leads to effective estimates only if RBM are included in the local FE test functions. Therefore, the excluded first‐order FEM has to be enlarged by RBM, which leads to a partition of unit method (PUM) with RBM, called P₁+RBM or to second‐order FEMs, called P₂ FEM. For P₁+RBM and P₂ FEM (or even higher‐order schemes) one obtains the sharper reliability estimate . Efficiency holds in the strict sense of . The local‐weighted interface problems behind the implicit error estimators η_L and µ_L are usually not exactly solvable and are rather approximated by some FEM on a refined mesh and/or with a higher‐order FEM. The computable approximations are shown to be reliable in the sense of . The oscillations are known functions of the given data and higher‐order terms if the data are smooth for first‐order FEM. The mathematical proofs are based on weighted Korn inequalities and inverse estimates combined with standard arguments. The numerical experiments for uniform and adapted FEM on benchmarks such as an L‐shape problem, Cook's membrane, or a slit problem validate the theoretical estimates and also concern numerical bounds for C_Korn and the locking phenomena. Copyright © 2007 John Wiley & Sons, Ltd.     

Error-controlled adaptive mixed finite element methods for second-order elliptic equations

In this contribution, we deal with a posteriori error estimates and adaptivity for mixed finite element discretizations of second-order elliptic equations, which are applied to the Poisson equation. The method proposed is an extension to the one recently introduced in [10] to the case of inhomogeneous Dirichlet and Neumann boundary conditions. The residual-type a posteriori error estimator presented in this paper relies on a postprocessed and therefore improved solution for the displacement field which can be computed locally, i.e. on the element level. Furthermore, it is shown that this discontinuous postprocessed solution can be further improved by an averaging technique. With these improved solutions at hand, both upper and lower bounds on the finite element discretization error can be obtained. Emphasis is placed in this paper on the numerical examples that illustrate our theoretical results.     

A posteriori error estimates and adaptive procedures for the meshless Galerkin boundary node method for 3D potential problems

The Galerkin boundary node method (GBNM) is a boundary only meshless method that combines variational formulations of boundary integral equations with the moving least-squares approximations. This paper presents the mathematical derivation of a posteriori error estimates and adaptive refinement procedures for the GBNM for 3D potential problems. Two types of error estimators are developed in detail. One is a perturbation error estimator that is formulated based on the difference between numerical solutions obtained using two successive nodal arrangements. The other is a projection error estimator that is formulated based on the difference between the GBNM solution itself and its L²-orthogonal projection. The reliability and efficiency of both types of error estimators is established. That is, these error estimators are proven to have an upper and a lower bound by the constant multiples of the exact error in the energy norm. A localization technique is introduced to accommodate the non-local property of integral operators for the needed local and computable a posteriori error indicators. Convergence analysis results of corresponding adaptive meshless procedures are also given. Numerical examples with high singularities illustrate the theoretical results and show that the proposed adaptive procedures are simple, effective and efficient.     

Improved Discontinuity-capturing Finite Element Techniques for Reaction Effects in Turbulence Computation

Recent advances in turbulence modeling brought more and more sophisticated turbulence closures (e.g. k-ɛ, k-ɛ -v ²-f, Second Moment Closures), where the governing equations for the model parameters involve advection, diffusion and reaction terms. Numerical instabilities can be generated by the dominant advection or reaction terms. Classical stabilized formulations such as the Streamline–Upwind/Petrov–Galerkin (SUPG) formulation (Brook and Hughes, comput methods Appl Mech Eng 32:199–255, 1982; Hughes and Tezduyar, comput methods Appl Mech Eng 45: 217–284, 1984) are very well suited for preventing the numerical instabilities generated by the dominant advection terms. A different stabilization however is needed for instabilities due to the dominant reaction terms. An additional stabilization term, called the diffusion for reaction-dominated (DRD) term, was introduced by Tezduyar and Park (comput methods Appl Mech Eng 59:307–325, 1986) for that purpose and improves the SUPG performance. In recent years a new class of variational multi-scale (VMS) stabilization (Hughes, comput methods Appl Mech Eng 127:387–401, 1995) has been introduced, and this approach, in principle, can deal with advection–diffusion–reaction equations. However, it was pointed out in Hanke (comput methods Appl Mech Eng 191:2925–2947) that this class of methods also need some improvement in the presence of high reaction rates. In this work we show the benefits of using the DRD operator to enhance the core stabilization techniques such as the SUPG and VMS formulations. We also propose a new operator called the DRDJ (DRD with the local variation jump) term, targeting the reduction of numerical oscillations in the presence of both high reaction rates and sharp solution gradients. The methods are evaluated in the context of two stabilized methods: the classical SUPG formulation and a recently-developed VMS formulation called the V-SGS (Corsini et al. comput methods Appl Mech Eng 194:4797–4823, 2005). Model problems and industrial test cases are computed to show the potential of the proposed methods in simulation of turbulent flows.     

H(curl)空间中椭圆最优控制问题的自适应有限元算法

针对H(curl)空间椭圆型最优控制问题提出了一种自适应有限元方法。首先将H(curl)空间Maxwell方程的最优控制模型转化为偏微分方程组,给出了偏微分方程组的解得正则性。其次利用自适应有限元方法求解此偏微分方程组,同时讨论了方法的后验误差及收敛性。最后通过数值算例给出了该方法的数值结果,验证了有限元方法的有效性和可靠性。这一方法可以应用于更复杂的最优控制问题的求解。     

Goal-Oriented Error Estimates Based on Different FE-Spaces for the Primal and the Dual Problem with Applications to Fracture Mechanics

The objective of this paper is to derive goal-oriented a posteriori error estimators for the error obtained while approximately evaluating the nonlinear J-integral as a fracture criterion in linear elastic fracture mechanics (LEFM) using the finite element method. Such error estimators are based on the well-established technique of solving an auxiliary dual problem. In a straightforward fashion, the solution to the discretized dual problem is sought in the same FE-space as the solution to the original (primal) problem, i.e. on the same mesh, although it merely acts as a weight of the discretization error only. In this paper, we follow the strategy recently proposed by Korotov et al. [J Numer Math 11:33–59, 2003; Comp Lett (in press)] and derive goal-oriented error estimators of the averaging type, where the discrete dual solution is computed on a different mesh than the primal solution. On doing so, the FE-solutions to the primal and the dual problems need to be transferred from one mesh to the other. The necessary algorithms are briefly explained and finally some illustrative numerical examples are presented.     

A numerical method for unilateral problems

Variational inequalities connected with Signorini's problem have appeared as a natural generalization of the minimum potential-energy theorem for bodies with unilateral constraints. In this paper, we describe numerical experience on the use of variational inequalities and Pade approximants to obtain approximate solutions to a class of unilateral boundary value problems of elasticity, like those describing the equilibrium configuration of an elastic membrane stretched over an elastic obstacle. These problems have the peculiar feature of being alternatively formulated as nonlinear boundary value problems without constraints for which the technique of Pade approximants can be successfully employed. The variational inequality formulation is used to discuss the problem of uniqueness and existence of the solution.     

大坝有限元分析应力取值的研究

提出了基于误差控制下的自适应网格的有限元应力取值标准:即给定一全局误差限作为自适应有限元网格剖分的准则,以此网格计算所得应力即为有限元应力取值。应用适用于工程计算的Z2后验误差估计方法以及h-型自适应策略,对一个典型的重力坝剖面进行了线弹性自适应有限元计算。计算结果表明:给定一个全局误差限,网格剖分调整若干次后即可满足误差要求,不会出现因角缘应力集中出现剖分不收敛的情况;存在一个全局误差限,使得当继续降低误差限时,坝踵和坝趾的角缘应力趋于稳定值。     

A variational constitutive model for porous metal plasticity

This paper presents a variational formulation of viscoplastic constitutive updates for porous elastoplastic materials. The material model combines von Mises plasticity with volumetric plastic expansion as induced, e.g., by the growth of voids and defects in metals. The finite deformation theory is based on the multiplicative decomposition of the deformation gradient and an internal variable formulation of continuum thermodynamics. By the use of logarithmic and exponential mappings the stress update algorithms are extended from small strains to finite deformations. Thus the time-discretized version of the porous-viscoplastic constitutive updates is described in a fully variational manner. The range of behavior predicted by the model and the performance of the variational update are demonstrated by its application to the forced expansion and fragmentation of U-6%Nb rings.     

On the elastodynamic solution of frictional contact problems using variational inequalities

This article is concerned with the development, implementation and application of variational inequalities to treat the general elastodynamic contact problem. The solution strategy is based upon the iterative use of two subproblems. Quadratic programming and Lagrange multipliers are used to solve the respective first and second subproblems and to identify the candidate contact surface and contact stresses. This approach guarantees the imposition of the active kinematic contact constraints, avoids the use of special contact elements and the interference of the user in dictating the accuracy of the solution. A modified Newmark formulation is developed to integrate the resulting time‐dependent variational inequality. This newly devised implicit time integration scheme is unconditionally stable, second‐order accurate, avoids numerical oscillations present in the traditional Newmark method, and does not cause numerical dissipation. To demonstrate the versatility and accuracy of the newly proposed algorithm, several examples are examined and compared with existing solutions where the penalty method has been employed. Copyright © 2001 John Wiley & Sons, Ltd.     

Nodally exact Ritz discretizations of 1D diffusion–absorption and Helmholtz equations by variational FIC and modified equation methods

This article presents the first application of the Finite Calculus (FIC) in a Ritz-FEM variational framework. FIC provides a steplength parametrization of mesh dimensions, which is used to modify the shape functions. This approach is applied to the FEM discretization of the steady-state, one-dimensional, diffusion–absorption and Helmholtz equations. Parametrized linear shape functions are directly inserted into a FIC functional. The resulting Ritz-FIC equations are symmetric and carry a element-level free parameter coming from the function modification process. Both constant- and variable-coefficient cases are studied. It is shown that the parameter can be used to produce nodally exact solutions for the constant coefficient case. The optimal value is found by matching the finite-order modified differential equation (FOMoDE) of the Ritz-FIC equations with the original field equation. The inclusion of the Ritz-FIC models in the context of templates is examined. This inclusion shows that there is an infinite number of nodally exact models for the constant coefficient case. The ingredients of these methods (FIC, Ritz, MoDE and templates) can be extended to multiple dimensions     

A convergent adaptive finite element method for the primal problem of elastoplasticity

The boundary value problem representing one time step of the primal formulation of elastoplasticity with positive hardening leads to a variational inequality of the second kind with some nondifferentiable functional. This paper establishes an adaptive finite element algorithm for the solution of this variational inequality that yields the energy reduction and, up to higher order terms, the R‐linear convergence of the stresses with respect to the number of loops. Applications include several plasticity models: linear isotropic‐kinematic hardening, linear kinematic hardening, and multisurface plasticity as model for nonlinear hardening laws. For perfect plasticity, the adaptive algorithm yields strong convergence of the stresses. Numerical examples confirm an improved linear convergence rate and study the performance of the algorithm in comparison with the more frequently applied maximum refinement rule. Copyright © 2006 John Wiley & Sons, Ltd.