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A methodology for adaptive finite element analysis:Towards an integrated computational environmentG. H. Paulino, I. F. M. Menezes, J. B. Cavalcante Neto, L. F. MarthaAbstract This work introduces a methodology for self-adaptive numerical procedures, which relies on the variouscomponents of an integrated, object-oriented, computa-tional environment involving pre-, analysis, andpost-processing modules. A basic platform for numericalexperiments and further development is provided, whichallows implementation of new elements/error estimatorsand sensitivity analysis. A general implementation of theSuperconvergent Patch Recovery (SPR) and the recentlyproposed Recovery by Equilibrium in Patches (REP) ispresented. Both SPR and REP are compared and used forerror estimation and for guiding the adaptive remeshingprocess. Moreover, the SPR is extended for calculatingsensitivity quantities of first and higher orders. The mesh(re-)generation process is accomplished by means ofmodern methods combining quadtree and Delaunay tri-angulation techniques. Surface mesh generation in arbi-trary domains is performed automatically (i.e. with nouser intervention) during the self-adaptive analysis usingeither quadrilateral or triangular elements. These ideas areimplemented in the Finite Element System Technology inAdaptivity (FESTA) software. The effectiveness and ver-satility of FESTA are demonstrated by representative nu-merical examples illustrating the interconnections amongfinite element analysis, recovery procedures, error esti-mation/adaptivity and automatic mesh generation.Key words finite element analysis, error estimation, ad-aptivity, h-refinement, sensitivity, superconvergent patchrecovery (SPR), recovery by equilibrium in patches (REP),object oriented programming (OOP), interactive computergraphics.1IntroductionThis work presents an integrated (object-oriented) com-putational environment for self-adaptive analyses of ge-neric two-dimensional (2D) problems. This environmentincludes analysis procedures to insure a given level of ac-curacy according to certain criteria, and also the proce-dures to generate and modify the finite elementdiscretization. This computational system, called FESTA(Finite Element System Technology in Adaptivity), involvesfive main components (see shaded boxes in Figure 1):A graphical preprocessor, for defining the geometry ofthe problem, the initial finite element mesh (togetherwith boundary conditions), and the main parametersused in a self-adaptive analysis. Here the geometricalmodel is dissociated from the finite element model.A finite element module for solving the current boun-dary value problem. The code has been developed sothat it is highly modular, expandable, and user-friendly.Thus, it can be properly maintained and continued.Moreover, other users/developers should be able tomodify the basic programming system to fit their spe-cific applications.An error estimation and sensitivity module. Discreti-zation errors are estimated according to available re-covery procedures, e.g. Zienkiewicz and Zhu (ZZ),superconvergent patch recovery (SPR) and recovery byequilibrium in patches (REP). Sensitivities of variousorders (1st., 2nd. or higher) are calculated by means of aprocedure analogous to the SPR. The user chooses thedesired error estimator and sensitivity order.A mesh (re-)generation (rather than mesh enrichment)procedure, based on the combination of quadtree andDelaunay triangulation techniques. According to themagnitude of the error, calculated in the previousmodule, a new finite element mesh is automaticallygenerated (i.e. with no user intervention), using eithertriangular or quadrilateral elements (h-refinement).Computational Mechanics 23 (1999) 361388 Springer-Verlag 1999361G.H. PaulinoDepartment of Civil and Environmental Engineering,University of Illinois at urbana-champaign2209 Newmark Laboratory,205 North Mathews Avenue,Urbana, IL 61801-2352, U.S.A.I.F.M. Menezes, J.B. Cavalcante Neto, L.F. MarthaTeCGraf (Computer Graphics Technology Group), PUC-Rio,Rio de Janeiro, R.J., 22453-900, BrazilJ.B. Cavalcante Neto, L.F. MarthaDepartment of Civil Engineering, PUC-Rio,Rio de Janeiro, R.J., 22453-900, BrazilCorrespondence to: G.H. PaulinoG.H. Paulino acknowledges the support from the United StatesNational Science Foundation (NSF) under Grant No. CMS-9713798. I.F.M. Menezes acknowledges the financial supportprovided by the FAPERJ, which is a Brazilian agency for researchand development in the state of Rio de Janeiro. G.H. Paulino andI.F.M. Menezes also acknowledge the Department of Civil andEnvironmental Engineering at UC-Davis for hospitality while partof this work was performed. J.B. Cavalcante Neto and L.F. Marthaacknowledge the financial support provided by the Brazilianagency CNPq. The authors also thank an anonymous reviewer forproviding relevant suggestions to this work. Finally, a postprocessor module, where all the analysisresults (e.g. deformed shape, sensitivity and stresscontours) can be visualized.Essentially, FESTA is a computational laboratory whichoffers a basic platform for numerical analysis and furtherdevelopment, e.g. implementation of new error estimators,elements, or material models (Cavalcante Neto et al. 1998).Object-oriented programming and integration of pre-,analysis, and post-processing modules make FESTA asoftware well-suited for both practical engineering appli-cations and further research development.The remainder of this paper is organized as follows.A motivation to the work and a brief literature revieware provided in Sect. 2. Afterwards, Sect. 3 presents sometheoretical background on self-adaptive simulations andan overview of the graphical interface used in the FESTAsoftware. Section 4 introduces the mathematical formula-tion of the SPR (using weighted least square systems), theREP, and the sensitivity method. A discussion about theautomatic mesh generation techniques used in this work isgiven in Sect. 5. Relevant information regarding the im-plementation of FESTA is presented in Sect. 6, especiallyaspects related to the SPR and REP recoveries. In order toassess the effectiveness of the proposed computationalsystem, representative numerical examples are given inSect. 7. Finally, in Sect. 8, conclusions are inferred anddirections for future research are discussed.2Motivation and related workNormal practice to solve engineering problems by meansof the Finite Element Method (FEM) or the BoundaryElement Method (BEM) involves increasing the number ofdiscretization points in the computational domain andresolving the resulting system of equations to examine therelative change in the numerical solution. In general, thisprocedure is time consuming, it depends on the experienceof the analyst, and it can be misleading if the solution hasnot entered an asymptotic range.Ideally, with a robust and reliable self-adaptive scheme,one would be able to specify an initial discrete modelwhich is sufficient to describe the geometry/topology ofthe domain and the boundary conditions (BCs), and tospecify a desired error tolerance, according to an appro-priate criterion. Then, the system would automaticallyrefine the model until the error measure falls below theprescribed tolerance. The process should be fully auto-matic and without any user intervention. This is the maingoal which motivated the development of FESTA. Thisapproach increases the overall reliability of the analysisprocedure since it does not depend on the experience, orinexperience, of the analyst.The need for developing better pre-processing tech-niques for the FEM, for performing automated analysis,and for obtaining self-adaptive solutions (which is be-coming a trend for commercial FEM software) have driventhe development of automatic mesh generation algo-rithms, i.e. algorithms which are capable of discretizing anarbitrary geometry into a consistent finite element meshwithout any user intervention. Several algorithms for 2Dgeometries have been developed (e.g. Baehmannet al. 1987; Blacker and Stephenson 1991; Zhu et al. 1991;Potyondy et al. 1995b; Borouchaki and Frey 1998), andapproaches for three-dimensional (3D) geometries haveappeared more recently (e.g. Cass et al. 1996; Escobar andMontenegro 1996; Beall et al. 1997; Lo 1998). The presentwork focus on automatic 2D mesh generation in connec-tion with adaptive solutions. Efficient techniques for gen-erating all-quadrilateral and all-triangular meshes areconsidered in detail. Although the algorithms presentedherein could be extended to mixed meshes, i.e. mesheswith both triangular and quadrilateral elements (see, forexample, Borouchaki and Frey 1998), this topic is notwithin the scope of this work.There exist a vast literature on error estimation andadaptivity, and the reader is directed to the appropriatereferences1. The volumes edited by Brebbia and Alia-badi (1993) and Babuska et al. (1986) review adaptivetechniques for the FEM and the BEM. The book edited byLadeveze and Oden (1998) presents a compilation of pa-pers from the workshop of New Advances in AdaptiveComputational Mechanics, held at Cachan, France, 1719September 1997, which dealt with the latest advances inadaptive methods in mechanics and their impact onsolving engineering problems. Several issues of journalshave also been dedicated to adaptivity, e.g. volume 12(1996), number 2 of Engineering with Computers, vol-ume 15 (1992), numbers 3/4 of Advances in EngineeringSoftware, and volume 36 (1991), number 1 of the Journalof Computational and Applied Mathematics. Surveys ofthe literature in FEM include articles by Noor and Ba-buska (1987), Oden and Demkovicz (1989), Strouboulisand Haque (1992a, b), Babuska and Suri (1994), andAinsworth and Oden (1997). Mackerle (1993, 1994) hascompiled a long list of references on mesh generation,refinement, error analysis and adaptive techniques forFEM and BEM that were published from 1990 to 1993. The(ZZ, SPR, REP, .)Visualization(Postprocessor)Final DiscretizationMeshRegenerationGraphicalPreprocessor(Geometry,Topology, BCs)FiniteElement SolverConver-gence?FESTA ITERATIVE MESH DESIGN CYCLENYError EstimatorFig. 1. Simplified diagram of the FESTA interactive meshing1The list of papers referred here is just a small sampling of theliterature, considering articles of particular interest to the presentwork, and is not intended to be a representative survey of theliterature in the field.362volume edited by Babuska et al. (1983) presents adaptivetechniques for the FEM and the Finite Difference Method(FDM). Relatively recent textbooks in the FEM emphasizethe field of adaptive solution techniques. For example, thebook by Zienkiewicz and Taylor (1989) includes a Chapteron Error Estimation and Adaptivity (Chapter 14), whichis supplemented by the papers by Zienkiewicz andZhu (1992a, b, 1994). Moreover, the book by Szabo andBabuska (1991) is primarily dedicated to this subject.The first papers on adaptive finite elements appearedin the early seventies. Since then, an explosive numberof papers on the subject have been published in thetechnical literature. Babuska and Rheinboldt (1978)presented a pioneering paper about error estimates byevaluating the residuals of the approximate solution andusing them to obtain local, more accurate answers. Theydeveloped the mathematical basis of self-adaptive tech-niques. With the concept of a posteriori error estimates,one can develop a self-adaptive strategy for the FEMsuch that only certain elements should be refined.Zienkiewicz et al. (1982) presented a hierarchical ap-proach for self-adaptive methods. In the early 1980s,computer graphics techniques started to be used asstandard tools by mesh generation programs. Shep-hard (1986) published a paper where geometric model-ing and automatic mesh generation techniques wereused in conjunction with self-adaptive methods. Zien-kiewicz and Zhu (1987) introduced an error estimatorbased on obtaining improved values of gradients(stresses) using some available recovery processes. Easyto be implemented in any finite element code, this typeof technique, based on averaging and on the so called L2projection, has been used to recover the gradients, andreasonable estimators were achieved. In 1992, thistechnique was corrected/improved by the same authors,leading to the so called Superconvergent Patch Recovery:SPR (Zienkiewicz and Zhu 1992a, b, 1994). This methodis a stress-smoothing procedure over local patches ofelements and is based on a discrete least-squares fit of ahigher-order local polynomial stress field to the stressesat the superconvergent sampling points obtained fromthe finite element calculation. Attempts to improve fur-ther the recovery process can be found in various ref-erences, e.g. Wiberg et al. (1994), Wiberg andAbdulwahab (1993), Blacker and Belytschko (1994),Tabbara et al. (1994), and Lee et al. (1997). Essentiallythese improved techniques incorporate equilibrium andboundary conditions on the recovery process. Anexhaustive study by Babuska et al. (1994a, b) showed,through numerous examples, the excellent performanceand superiority of the SPR over residual-type approaches.Recently, Boroomand and Zienkiewicz (1997) have pre-sented a new super-convergent method satisfying theequilibrium condition in a weak form, which does notrequire any knowledge of superconvergent points. Thenew recovery technique has been called Recovery byEquilibrium in Patches: REP. Both SPR and REP are ofparticular interest to the present work.As indicated above, the general field of adaptivity isbroad and has advanced significantly in recent years. Forinstance, Paulino et al. (1997) have proposed a new classof error estimates based on the concept of nodal sensi-tivities, which can be used in conjunction with generalpurpose computational methods such as FEM, BEM orFDM. Rannacher and Suttmeier (1997) have suggested afeedback approach for error control in the FEM. Mahomedand Kekana (1998) have presented an adaptive procedurebased on strain energy equalisation. Moreover, a summaryof recent advances in adaptive computational mechanicscan be found in the book edited by Ladeveze andOden (1998).Quantification of the quality of a model with respect toanother one, taken as the reference, is of primary impor-tance in numerical analysis. This is the case with well es-tablished methods, such as the FEM, or emerging methods,such as the element free Galerkin: EFG (Chung and Bel-ytschko 1998), the symmetric Galerkin BEM (Paulino andGray 1999), or the boundary node method (Mukherjee andMukherjee 1997). Integration of concepts regarding errorestimation and adaptivity in the FEM, within a moderncomputational environment, is the focus of the presentwork.3Theoretical and computational aspectsWhenever a numerical method is used to solve the gov-erning differential equations of a boundary value problem,error is introduced by the discretization process whichreduces the continuous mathematical model to one havinga finite number of degrees of freedom. The discretizationerrors are defined as the difference between the actualsolution and its numerical approximation. By definition,the local errore / / 1 is a measure of the difference between the exact (/) and anapproximate solution (/). Here, / is analogous to a re-sponse quantity (e.g. displacements) in a typical numericalsolution procedure.Self-adaptive methods are numerical schemes whichautomatically adjust themselves in order to improve thesolution of the problem to a specified accuracy. The twobasic components in adaptive methods are error estima-tion and adaptive strategy. These components are discus-sed below.In general, there are two types of discretization errorestimates: a priori and a posteriori. Although a priori es-timates are accurate for the worst case in a particular classof solutions of a problem, they usually do not provideinformation about the actual error for a given model.A posteriori estimates use information obtained during thesolution process, in addition to some a priori assumptionsabout the solution. A posteriori estimates, which canprovide quantitatively accurate measures of the discreti-zation error, have been adopted here.In the context of adaptive strategies, extension methodshave been preferred over others approaches (e.g. dual orcomplementary methods) and are the focus of this work.These methods include h-, p-, and r-extensions. Thecomputer implementation is referred to as the h-, p-, andr-versions, respectively. In the h-extension, the mesh isautomatically refined when the local error indicator ex-363ceeds a preassigned tolerance. The p-extension generallyemploys a fixed mesh. If the error in an element exceeds apreassigned tolerance, the local order of the approxima-tion is increased to reduce the error. The r-extension(node-redistribution) employs a fixed number of nodesand attempts to dynamically move the grid points to areasof high error in the mesh. Any of these extensions can alsobe combined in a special strategy, for example, h-p-, r-h-,among others.3.1Error estimation and adaptive refinementAs pointed out by several authors (e.g. Zienkiewicz andTaylor 1989), the specification of local error in the mannergiven in Eq. (1) is generally not convenient and occa-sionally misleading. Thus mathematical norms are intro-duced to measure the discretization error. The exactdiscretization error in the finite element solution is oftenquantified on the basis of the energy norm for the dis-placement error, jjejj, which can be expressed, in terms ofstresses, asjjejj2ZXrexr TD1rexr dX 2 where rexand r are the exact and the finite element stressfields, D is the constitutive matrix, and X is the problemdefinition domain.The basic idea of error estimators is to substitute thefield rex, which is generally unknown, by the field r, ob-tained by means of recovery procedures (e.g. ZZ, SPR orREP). Therefore, the expression for computing the ap-proximate (estimated) relative error