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hp FEM for ReactionDiffusion Equations I: Robust Exponential Convergence
, 1997
"... A singularly perturbed reactiondiffusion equation in two dimensions is considered. We assume analyticity of the input data, i.e., the boundary of the domain is an analytic curve and the right hand side is analytic. We show that the hp version of the finite element method leads to robust exponential ..."
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Cited by 14 (6 self)
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A singularly perturbed reactiondiffusion equation in two dimensions is considered. We assume analyticity of the input data, i.e., the boundary of the domain is an analytic curve and the right hand side is analytic. We show that the hp version of the finite element method leads to robust exponential convergence provided that one layer of needle elements of width O(p") is inserted near the domain boundary, that is, the rate of convergence is O \Gamma exp(\Gammabp) \Delta and independent of the perturbation parameter ". Additionally, we show that the use of numerical quadrature for the evaluation of the stiffness matrix and the load vector retains the exponential rate of convergence. In particular, the Spectral Element Method based on the use of a GaussLobatto quadrature rule with (p+1) \Theta (p+1) points yields robust exponential convergence.
A.: NURBSenhanced finite element method (NEFEM) for Euler equations
 Internat. J. Numer. Methods Fluids
, 2008
"... The development of NURBSEnhanced Finite Element Method (NEFEM) is revisited. This technique allows a seamless integration of the CAD boundary representation of the domain and the finite element method (FEM). The importance of the geometrical model in finite element simulations is addressed and the ..."
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Cited by 13 (8 self)
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The development of NURBSEnhanced Finite Element Method (NEFEM) is revisited. This technique allows a seamless integration of the CAD boundary representation of the domain and the finite element method (FEM). The importance of the geometrical model in finite element simulations is addressed and the benefits and potential of NEFEM are discussed and compared with respect to other curved finite element techniques.
A.: Comparison of highorder curved finite elements
 Internat. J. Numer. Methods Engrg
, 2011
"... Several finite element techniques used in domains with curved boundaries are discussed and compared, with particular emphasis in two issues: the exact boundary representation of the domain and the consistency of the approximation. The influence of the number of integration points in the accuracy of ..."
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Cited by 7 (4 self)
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Several finite element techniques used in domains with curved boundaries are discussed and compared, with particular emphasis in two issues: the exact boundary representation of the domain and the consistency of the approximation. The influence of the number of integration points in the accuracy of the computation is also studied. Two dimensional numerical examples, solved with continuous and discontinuous Galerkin formulations, are used to test and compare all these methodologies. In every example shown, the recently proposed NURBSenhanced finite element method (NEFEM) provides the maximum accuracy for a given spatial discretization, at least one order of magnitude more accurate than classical isoparametric finite element methods (FEM). Moreover, NEFEM outperforms Cartesian FEM and pFEM, stressing the importance of the geometrical model as well as the relevance of a consistent approximation in finite element simulations.
Abstract pFEM for finite deformation powder compaction
, 2007
"... The simulation of powder compaction problems (diecompaction and cold isostatic pressing) is considered herein by an implicit highorder (pversion) finite element method. In this class of problems use is made of a finite strain viscoplasticity model with evolution equations for internal variables de ..."
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The simulation of powder compaction problems (diecompaction and cold isostatic pressing) is considered herein by an implicit highorder (pversion) finite element method. In this class of problems use is made of a finite strain viscoplasticity model with evolution equations for internal variables developed for the highly compressible behavior in powder compaction processes. The classical approach of implicit finite elements applies the combination of BackwardEuler integration scheme and the MultilevelNewton algorithm to solve the system of differentialalgebraic equations resulting from the spacediscretized weak formulation by means of pversion finite elements. This approach requires on Gausspoint level a robust stressalgorithm. The challenging investigations are the incorporation of the applied highly nonlinear viscoplasticity model into a pversion finite element formulation using follower load applications. Several axisymmetric numerical examples show the feasibility and good performance of this pversion approach. Ó 2007 Elsevier B.V. All rights reserved.