An adaptive finite element approach for frictionless contact problems

The derivation of an a posteriori error estimator for frictionless contact problems under the hypotheses of linear elastic behaviour and infinitesimal deformation is presented. The approximated solution of this problem is obtained by using the finite element method. A penalization or augmented‐Lagrangian technique is used to deal with the unilateral boundary condition over the contact boundary. An a posteriori error estimator suitable for adaptive mesh refinement in this problem is proposed, together with its mathematical justification. Up to the present time, this mathematical proof is restricted to the penalization approach. Several numerical results are reported in order to corroborate the applicability of this estimator and to compare it with other a posteriori error estimators. Copyright © 2001 John Wiley & Sons, Ltd.     

Towards automatic adaptive geometrically non-linear shell analysis. Part I: Implementation of an h-adaptive mesh refinement procedure

Effective methods leading to automated adaptive numerical solutions to geometrically non-linear shell-type problems are studied in this work. In particular, procedures for improving the accuracy, the reliability and the computational efficiency of the finite element solutions are of primary interest here. This is addressed using h-adaptive mesh refinement based on a posteriori error estimation, self-adaptive methods in global incremental/iterative processes, as well as smart algorithms and heuristic approaches based on methods of knowledge engineering. Seemless integration of h-adaptive finite element methods with adaptive step-length control makes it possible to maintain a prescribed accuracy while maintaining the solution efficiency without user intervention throughout the process of a non-linear analysis. Several examples illustrate the merit and potential of the approach studied herein and confirm the feasibility of developing an automatic adaptive environment for geometrically non-linear analysis of shell structures.     

Adaptive mesh refinement of the boundary element method for potential problems by using mesh sensitivities as error indicators

This paper presents a novel method for error estimation and h-version adaptive mesh refinement for potential problems which are solved by the boundary element method (BEM). Special sensitivities, denoted as mesh sensitivities, are used to evaluate a posteriori error indicators for each element, and a global error estimator. A mesh sensitivity is the sensitivity of a physical quantity at a boundary node with respect to perturbation of the mesh. The element error indicators for all the elements can be evaluated from these mesh sensitivities. Mesh refinement can then be performed by using these element error indicators as guides.The method presented here is suitable for both potential and elastostatics problems, and can be applied for adaptive mesh refinement with either linear or quadratic boundary elements. For potential problems, the physical quantities are potential and/or flux; for elastostatics problems, the physical quantities are tractions/displacements (or tangential derivatives of displacements). In this paper, the focus is on potential problems with linear elements, and the proposed method is validated with two illustrative examples. However, it is easy to extend these ideas to elastostatics problems and to quadratic elements.The computing for this research has been supported by the Cornell National Supercomputer Facility.     

Mesh refinement and iterative solution methods for finite element computations

Global and element residuals are introduced to determine a posteriori, computable, error bounds for finite element computations on a given mesh. The element residuals provide a criterion for determining where a finite element mesh requires refinement. This indicator is implemented in an algorithm in a finite element research program. There it is utilized to automatically refine the mesh for sample two-point problems exhibiting boundary layer and interior layer solutions. Results for both linear and nonlinear problems are presented. An important aspect of this investigation concerns the use of adaptive refinement in conjunction with iterative methods for system solution. As the mesh is being enriched through the refinement process, the solution on a given mesh provides an accurate starting iterate for the next mesh, and so on. A wide range of iterative methods are examined in a feasibility study and strategies for interweaving refinement and iteration are compared.     

Error estimation and mesh adaptivity for the virtual element method based on recovery by compatibility in patches

In this article, a recovery by compatibility in patches (RCP)-based a posteriori error estimator is proposed for the virtual element method (VEM), and it is utilized to drive adaptive mesh refinement processes in two-dimensional elasticity problems. In RCP, recovered stresses are obtained by minimizing the complementary energy of patches of elements over a set of assumed equilibrated stress modes. To this aim, the explicit knowledge of displacements is only needed along the patch boundaries and no knowledge of superconvergent points is required, so making the RCP naturally suitable for the VEM. The a posteriori error estimation is conducted by comparing the stress field of a standard displacement-based VEM solution and the stress field obtained through RCP. The procedure is simple, and it does not require ad hoc modifications for small patches. The capability of this RCP-based error estimator to drive adaptive mesh refinements is successfully demonstrated through various numerical examples.     

An error estimator for adaptive mesh refinement analysis based on strain energy equalisation

The two most widely used error estimators for adaptive mesh refinement are discussed and developed in the context of non-linear elliptic problems. The first is based on the work of Babuska and Rheinboldt (1978) where the error norm is a function of the residual and the inter-element discontinuity of the stress field. The discontinuous stress field arises in the Finite Element formulation where C ₀ continuity of the velocity field is assumed. The second error estimator is based on the work of Zienkiewicz and Zhu (1987). This method also uses the discontinuous stress field to measure the error, but results in a more simplified expression for the error norm. In fact, the equivalence between the two error norms has been shown by Zienkiewicz. Finally, an error estimator which is based on the approximation velocity space only is proposed. This estimator has the advantage in that it does not require the a posteriori calculation of the pressure (or stress) field. The method is applied to non-Newtonian Stokes flow which has a similar formulation to non-linear elasticity problems.     

On error estimates and adaptivity in elastoplastic solids: Applications to the numerical simulation of strain localization in classical and Cosserat continua

The a posteriori error estimates based on the post-processing approach are introduced for elastoplastic solids. The standard energy norm error estimate established for linear elliptic problems is generalized here to account for the presence of internal variables through the norm associated with the complementary free energy. This is known to represent a natural metric for the class of elastoplastic problems of evolution. In addition, the intrinsic dissipation functional is utilized as a basis for a complementary a posteriori error estimates. A posteriori error estimates and adaptive refinement techniques are applied to the finite element analysis of a strain localization problem. As a model problem, the constitutive equations describing a generalization of standard J₂-elastoplasticity within the Cosserat continuum are used to overcome serious limitations exhibited by classical continuum models in the post-instability region. The proposed a posteriori error estimates are appropriately modified to account for the Cosserat continuum model and linked with adaptive techniques in order to simulate strain localization problems. Superior behaviour of the Cosserat continuum model in comparison to the classical continuum model is demonstrated through the finite element simulation of the localization in a plane strain tensile test for an elastopiastic softening material, resulting in convergent solutions with an h-refinement and almost uniform error distribution in all considered error norms.     

An adaptive hp-version of the finite element method applied to flame propagation problems

This paper describes an adaptive hp-version mesh refinement strategy and its application to the finite element solution of one-dimensional flame propagation problems. The aim is to control the spatial and time discretization errors below a prescribed error tolerance at all time levels. In the algorithm, the optimal time step is first determined in an adaptive manner by considering the variation of the computable error in the reaction zone. Later, the method uses a p-version refinement till the computable a posteriori error is brought down below the tolerance. During the p-version, if the maximum allowable degree of approximation is reached in some elements of the mesh without satisfying the global error tolerance criterion, then conversion from p- to h-version is performed. In the conversion procedure, a gradient based non-uniform h-version refinement has been introduced in the elements of higher degree approximation. In this way, p-version and h-version approaches are used alternately till the a posteriori error criteria are satisfied. The mesh refinement is based on the element error indicators, according to a statistical error equi-distribution procedure. Numerical simulations have been carried out for a linear parabolic problem and premixed flame propagation in one-space dimension. © 1997 John Wiley & Sons, Ltd.     

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.     

An hp‐adaptive finite element method for scattering problems in computational electromagnetics

An hp‐adaptive finite element (FE) approach is presented for a reliable, efficient and accurate solution of 3D electromagnetic scattering problems. The radiation condition in the far field is satisfied automatically by approximation with infinite elements (IE). Near optimal discretizations that can effectively resolve local rapid variations in the scattered field are sought adaptively by mesh refinements blended with graded polynomial enrichments. The p‐enrichments need not be spatially isotropic. The discretization error can be controlled by a self‐adaptive process, which is driven by implicit or explicit a posteriori error estimates. The error may be estimated in the energy norm or in a quantity of interest. A radar cross section (RCS) related linear functional is used in the latter case. Adaptively constructed solutions are compared to pure uniform p approximations. Numerical, highly accurate, and fairly converged solutions for a number of generic problems are given and compared to previously published results. Copyright © 2004 John Wiley & Sons, Ltd.     

Spatial mesh adaptation in semidiscrete finite element analysis of linear elastodynamic problems

A spatial mesh adaptation procedure in semidiscrete finite element analysis of 2D linear elastodynamic problems is presented. The procedure updates, through an automatic remeshing scheme, the spatial mesh when found necessary in order to gain control of the spatial discretization error from time to time. An a posteriori error estimate developed by Zienkiewicz and Zhu (1987) for elliptic problems is extended to dynamic analysis to estimate the spatial discretization error at a certain time, which is found to be reasonable by analyzing an a priori error estimate. Numerical examples are used to demonstrate the performance of the procedure. It is indicated that the extended error estimation and the procedure are capable of monitoring the moving of steep stress regions by updating the spatial mesh according to a prescribed error tolerance, thus providing a reliable finite element solution in an efficient manner.     

An algorithm for adaptive refinement of triangular element meshes

A simple algorithm is developed for adaptive and automatic h refinement of two-dimensional triangular finite element meshes. The algorithm is based on an element refinement ratio that can be calculated from an a posteriori error indicator. The element subdivision algorithm is robust and recursive. Smooth transition between large and small elements is achieved without significant degradation of the aspect ratio of the elements in the mesh. Several example problems are presented to illustrate the utility of the approach.     

A postprocessed error estimate and an adaptive procedure for the semidiscrete finite element method in dynamic analysis

In the paper we present a postprocessed type of a posteriori error estimate and a h-version adaptive procedure for the semidiscrete finite element method in dynamic analysis. In space the super-convergent patch recovery technique is used for determining higher-order accurate stresses and, thus, a spatial error estimate. In time a postprocessing technique is developed for obtaining a local error estimate for one step time integration schemes (the HHT-α method). Coupling the error estimate with a mesh generator, a h-version adaptive finite element procedure is presented for two-dimensional dynamic analysis. It updates the spatial mesh and time step automatically so that the discretization errors are controlled within specified tolerances. Numerical studies on different problems are presented for demonstrating the performances of the proposed adaptive procedure.     

Solution of viscoelastic scattering problems in linear acoustics using hp Boundary/Finite Element Method

The interaction of acoustic waves with submerged structures remains one of the most difficult and challenging problems in underwater acoustics. Many techniques such as coupled Boundary Element (BE)/Finite Element (FE) or coupled Infinite Element (IE)/Finite Element approximations have evolved. In the present work, we focus on the steady‐state formulation only, and study a general coupled hp‐adaptive BE/FE method. A particular emphasis is placed on an a posteriori error estimation for the viscoelastic scattering problems. The highlights of the proposed methodology are as follows: (1) The exterior Helmholtz equation and the Sommerfeld radiation condition are replaced with an equivalent Burton–Miller (BM) boundary integral equation on the surface of the scatterer. (2) The BM equation is coupled to the steady‐state form of viscoelasticity equations modelling the behaviour of the structure. (3) The viscoelasticity equations are approximated using hp‐adaptive FE isoparametric discretizations with order of approximation p⩾5 in order to avoid the ‘locking’ phenomenon. (4) A compatible hp superparametric discretization is used to approximate the BM integral equation. (5) Both the FE and BE approximations are based on a weak form of the equations, and the Galerkin method, allowing for a full convergence analysis. (6) An a posteriori error estimate for the coupled problem of a residual type is derived, allowing for estimating the error in pressure on the wet surface of the scatterer. (7) An adaptive scheme, an extension of the Texas Three Step Adaptive Strategy is used to manipulate the mesh size h and the order of approximation p so as to approximately minimize the number of degrees of freedom required to produce a solution with a specified accuracy. The use of this hp‐scheme may exhibit exponential convergence rates. Several numerical experiments illustrate the methodology. These include detailed convergence studies for the problem of scattering of a plane acoustic wave on a viscoelastic sphere, and adaptive solutions of viscoelastic scattering problems for a series of MOCK0 models. Copyright © 1999 John Wiley & Sons, Ltd.     

Adaptive finite element computation of dielectric and mechanical intensity factors in piezoelectrics with impermeable cracks

The paper deals with the application of an adaptive, hierarchic‐iterative finite element technique to solve two‐dimensional electromechanical boundary value problems with impermeable cracks in piezoelectric plates. In order to compute the dielectric and mechanical intensity factors, the interaction integral technique is used. The iterative finite element solver takes advantage of a sequence of solutions on hierarchic discretizations. Based on an a posteriori error estimation, the finite element mesh is locally refined or coarsened in each step. Two crack configurations are investigated in an infinite piezoelectric plate: A finite straight crack and a finite kinked crack. Fast convergence of the numerical intensity factors to the corresponding analytical solution is exemplarily proved during successive adaptive steps for the first configuration. Similar tendency can be observed for the second configuration. Furthermore, the computed intensity factors for the kinks are found to coincide well with the corresponding analytical values. In order to simulate the kinks spreading from a straight crack, the finite element mesh is modified automatically with a specially developed algorithm. This forms the basis for a fully adaptive simulation of crack propagation. Copyright © 2009 John Wiley & Sons, Ltd.     

Adaptive finite element analysis of geometrically non-linear plates and shells,especially buckling

Based on both moderate and finite rotation bending theories of thin elastic shells including shear deformation, adaptive non-linear static finite element analysis is treated within a displacement approach and h-adaptivity. The a posteriori error indicator given by Rheinboldt, gained by linearization, is investigated in order to decide whether the deformations influence the indicator explicitely and how parameter dependent problems (like the Reissner–Mindlin model) behave in the process of adaptation. In order to achieve overall consistency, dimensional adaptivity (to 3-D elasticity) is implemented within disturbed subdomains, especially at supports. Results are that Rheinboldt's error indicator is valid under certain restrictions but not directly at bifurcation points and that robustness is not improved by adaptation. Nested quadrilateral finite elements are used for studying pre- and post-buckling states of plates and shells.     

Adaptive component mode synthesis in linear elasticity

Component mode synthesis (CMS) is a classical method for the reduction of large‐scale finite element models in linear elasticity. In this paper we develop a methodology for adaptive refinement of CMS models. The methodology is based on a posteriori error estimates that determine to what degree each CMS subspace influence the error in the reduced solution. We consider a static model problem and prove a posteriori error estimates for the error in a linear goal quantity as well as in the energy and L² norms. Automatic control of the error in the reduced solution is accomplished through an adaptive algorithm that determines suitable dimensions of each CMS subspace. The results are demonstrated in numerical examples. Copyright © 2010 John Wiley & Sons, Ltd.     

Analysis of transfer procedures in elastoplasticity based on the error in the constitutive equations: Theory and numerical illustration

The aim of this work is to illustrate a methodology for the assessment of adaptive strategies for the solution of associative rate‐independent plasticity problems solved by employing the incremental displacement conforming finite element method. This is the first step towards a more rational definition of transfer operators in terms of the ensuing error. The motivating idea is the observation that change of data and/or finite element mesh from one time interval to the other can be both related to a discontinuity jump of the approximate solution across the time instant t_n. Thus, reliable a posteriori estimates will have to depend not only on the time step and finite element mesh size but also on the value of the jump. A new error estimate based on the error in the constitutive equations is developed which allows characterization of the discontinuity jump. Copyright © 2004 John Wiley & Sons, Ltd.     

Local boundary value problems for the error in FE approximation of non‐linear diffusion systems

In this work we investigate the a posteriori error estimation for a class of non‐linear, multicomponent diffusion operators, which includes the Stefan–Maxwell equations. The local error indicators for the global error are based on local boundary value problems, which are chosen to approximate either the global residual of the finite element approximation or the global linearized error equation. Using representative numerical examples, it is shown that the error indicators based on the latter approach are more accurate for estimating the global error for this problem class as the problem becomes more non‐linear, and can even produce better adaptive mesh refinement (AMR). In addition, we propose a new local error indicator for the error in output functionals that is accurate, inexpensive to compute, and is suitable for AMR, as demonstrated by numerical examples. Copyright © 2007 John Wiley & Sons, Ltd.     

Adaptive Poly-FEM for the analysis of plane elasticity problems

In this work, we present an adaptive polygonal finite element method (Poly-FEM) for the analysis of two-dimensional plane elasticity problems. The generation of meshes consisting of n ? sided polygonal finite elements is based on the generation of a centroidal Voronoi tessellation (CVT). An unstructured tessellation of a scattered point set, that minimally covers the proximal space around each point in the point set, is generated whereby the method also includes tessellation of nonconvex domains. In this work, we propose a region by region adaptive polygonal element mesh generation. A patch recovery type of stress smoothing technique that utilizes polygonal element patches for obtaining smooth stresses is proposed for obtaining the smoothed finite element stresses. A recovery type a ? posteriori error estimator that estimates the energy norm of the error from the recovered solution is then adopted for the Poly-FEM. The refinement of the polygonal elements is then made on an region by region basis through a refinement index. For the numerical integration of the Galerkin weak form over polygonal finite element domains, we resort to classical Gaussian quadrature applied to triangular subdomains of each polygonal element. Numerical examples of two-dimensional plane elasticity problems are presented to demonstrate the efficiency of the proposed adaptive Poly-FEM.