an rp-adaptive finite element discretization applied to physically non ...

VIII International Conference on Computational PlasticityCOMPLAS VIIIE. Oñate and D.R.J. Owen (Eds)c○CIMNE, Barcelona, 2005AN RP -ADAPTIVE FINITE ELEMENT DISCRETIZATIONAPPLIED TO PHYSICALLY NON-LINEAR PROBLEMSVera Nübel, Alexander Düster and Ernst RankLehrstuhl für Bauinformatik, Fakultät für Bauingenieur- und VermessungswesenTechnische Universität München, Munich, Germanye-mail: {nuebel,duester,rank}@bv.tum.de, web page: http://www.inf.bv.tum.de/Key words: Adaptivity, p-version, r-version, deformation theory of plasticity.Summary. This article proposes an efficient discretization strategy for physically nonlinearproblems. It combines a high order finite element formulation which increases thepolynomial degree of the shape functions on a coarse mesh with the r-method moving nodesand edges of an existing FE-mesh. The basic idea of the rp-version is to iteratively adaptthe finite element mesh to the shape of the elastic-plastic interface in order to take intoaccount the loss of regularity which arises along the boundary of the plastic zone. Thisdiscretization technique leads to an exponential rate of convergence.1 IntroductionThe development of adaptive low-order finite element approaches for physically nonlinearproblems has been in the center of research for a long time. In recent years, numericalinvestigations have shown that high-order finite elements enable an improvement ofefficiency for this class of problems (Düster, Rank [1]). In order to enhance the resultsof a uniform p-extension the rp-adaptive approach which leads to an exponential rate ofconvergence even for elastoplastic analysis has been developed and implemented. It isbased on the p-version of the finite element method which — in contrast to the classicalh-version — leaves the mesh unchanged and increases the polynomial degree of the shapefunctions locally or globally to reduce the error of the approximation (see Szabó et al.[3]).2 The rp-discretizationThe idea of a more efficient discretization scheme for physically non-linear problemsis sketched in Figure 1. A singularity arises along the boundary of the plastic zone. Itscontour corresponds to a curve (in the 2D case) or a surface (in the 3D case). Due to thisloss of regularity, only an algebraic rate of convergence can be achieved if the interfaceintersects the interior of elements. To obtain an exponential rate of convergence evenfor this class of problem, we decompose the whole computational domain Ω into twosubdomains Ω e and Ω p (see Figure 1), so that their boundaries coincide with the shapeof the elastic-plastic interface. The exact solution over these subdomains — belongingto either an elastic or a plastic area of the structure — is smooth, assuming that no1

Vera Nübel, Alexander Düster and Ernst Rankadditional singularities occur, such as those due to reentrant corners. Each subdomain isthen discretized by the p-version applied on a coarse mesh, where no refinement towardsthe elastic-plastic interface is necessary. The computational domain with a non-smoothsolution is split into areas for which regular solutions do exist. As the boundary ofthe elastic-plastic interface describes an arbitrarily curved contour, element edges areaccordingly shaped. To describe such arbitrarily bounded elements we use the blendingfunction method first proposed by Gordon and Hall (see [3]). It is an important featureof the p-version that enables a large varity of element shapes.PSfrag replacementsΓ NtΩΩ eΩ puΓ DFigure 1: Domain Ω, decomposed into an elastic region Ω e and a plastic region Ω pAs position and shape of the elastic-plastic interface are not known a priori, an rpadaptivealgorithm iteratively detects the boundary of the plastic zone. Within thisadaptive cycle elements may be arbitrarily curved and strongly distorted. As the p-version is very robust with respect to distortion (see Szabó et al. [3]), these effects can becompensated with an increase of the polynomial degree of the finite element computation.3 Numerical examplesNumerical studies on the rp-adaptive approach for two-dimensional problems haveshown that the proposed discretization strategy enables an improvement of efficiencyand provides accurate results [4]. Furthermore it turned out to be superior to adaptiveh-version approaches when investigating the example of a rectangular domain with acentral hole which has been defined by Stein et al. [2] as a benchmark problem forthe joint German research project ’Adaptive Finite-Element-Methods in ComputationalMechanics’. This can be seen from Figure 2 which shows the relative error in u y in [%]at point 4, located at the upper right corner. Three of the corresponding finite-elementmeshes,arising during the rp-adaptive cycle, are presented in Figure 3.A similar behaviour of the rp-method can be observed in 3D, when investigating the threedimensionalproblem of a thick-walled hollow sphere under internal pressure. Figure 4shows the results for a uniform h-version, a p-version with a priori adjusted interface atthe plastic front and an adaptive rp-version.2

tnb hr1 234567PSfrag replacementsVera Nübel, Alexander Düster and Ernst Rankrelative error of uy,FE, at point 4 [%]1001010.10.010.0010.00011e-05adaptive rp-version64,256,1024,4096 graded Q1-P0 elements875 adapted Q1-P0 elementsuniform h-version, p=2PSfrag replacementsrelative error of u y,FE , at point 4 [%]degrees of freedom Nadaptive rp-version64,256,1024,4096 graded Q1-P0 elements875 adapted Q1-P0 elementsuniform h-version, p=21e-0610 100 1000 10000 100000 1e+06degrees of freedom Nyht n5761 2bx43rFigure 2: Relative error |u y,ref −u y,F E|u y,ref100 [%] at point 4Figure 3: Finite element meshes during the rp-adaptive algorithm (iteration steps 3,4,5)A detailed discussion on the efficiency of the rp-method for the deformation theory ofplasticity is given in [4].4 ConclusionsThe proposed rp-adaptive algorithm for physically non-linear problems iteratively detectsthe boundary of the elastic-plastic interface to subdivide the whole computationaldomain with a non-regular solution into areas where smooth solutions exist. If the shapeof the plastic zone is represented accurately, an exponential rate of convergence can beobtained even for this class of problems.3

1e+02PSfrag replacementsxyzVera Nübel, Alexander Düster and Ernst Rankrira1e+021e+011e+00PSfrag replacementsrelative error η [%]1e+011e+001e-011e-021e-031e-04interface mesh, 6 elementsuniform h-version, p=2adaptive rp-version1e+021e+011e+001e-011e-021e-031e-041e-051e+001e+011e+021e+031e+041e+051e+06relative error η [%]degrees of freedom Ninterface mesh, 6 elementsuniform h-version, p=2adaptive rp-versionyzr ir ax1e-051e+01 1e+02 1e+03 1e+04 1e+05 1e+06degrees of freedom N√|Uref −UFigure 4: Relative error η =F E|∫U ref100 [%] with U = 1 2 Ω εT σ dΩ for 3D-problemAcknowledgementThis study has been sponsored with the help of a grant from the Deutsche Forschungsgemeinschaft(DFG, RA 624/6-3, RA 624/6-4). This support is gratefully acknowledged.REFERENCES[1] A. Düster and E. Rank. The p-version of the finite element method compared toan adaptive h-version for the deformation theory of plasticity, Computer Methods inApplied Mechanics and Engineering, 190, 1925-1935, 2001.[2] E. Stein. Error-Controlled Adaptive Finite Elements in Solid Mechanics, John Wiley& Sons, 2002.[3] B. Szabó, A. Düster and E. Rank. The p-version of the finite element method, In:Encyclopedia of Computational Mechanics, E. Stein, R. de Borst and T. Hughes,editors, John Wiley & Sons, 2004.[4] V. Nübel, A. Düster and E. Rank. An rp-adaptive finite element method for thedeformation theory of plasticity, submitted to: Computational Mechanics, 2004.4