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

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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
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