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Stabilization Terms (stabilization + term)
Selected AbstractsAn adaptive stabilization strategy for enhanced strain methods in non-linear elasticityINTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN ENGINEERING, Issue 11 2010Alex Ten Eyck Abstract This paper proposes and analyzes an adaptive stabilization strategy for enhanced strain (ES) methods applied to quasistatic non-linear elasticity problems. The approach is formulated for any type of enhancements or material models, and it is distinguished by the fact that the stabilization term is solution dependent. The stabilization strategy is first constructed for general linearized elasticity problems, and then extended to the non-linear elastic regime via an incremental variational principle. A heuristic choice of the stabilization parameters is proposed, which in the numerical examples proved to provide stable approximations for a large range of deformations, different problems and material models. We also provide explicit lower bounds for the stabilization parameters that guarantee that the method will be stable. These are not advocated, since they are generally larger than the ones based on heuristics, and hence prone to deteriorate the locking-free behavior of ES methods. Numerical examples with two different non-linear elastic models in thin geometries and incompressible situations show that the method remains stable and locking free over a large range of deformations. Finally, the method is strongly based on earlier developments for discontinuous Galerkin methods, and hence throughout the paper we offer a perspective about the similarities between the two. Copyright © 2009 John Wiley & Sons, Ltd. [source] A new mixed finite element method for poro-elasticityINTERNATIONAL JOURNAL FOR NUMERICAL AND ANALYTICAL METHODS IN GEOMECHANICS, Issue 6 2008Maria Tchonkova Abstract Development of robust numerical solutions for poro-elasticity is an important and timely issue in modern computational geomechanics. Recently, research in this area has seen a surge in activity, not only because of increased interest in coupled problems relevant to the petroleum industry, but also due to emerging applications of poro-elasticity for modelling problems in biomedical engineering and materials science. In this paper, an original mixed least-squares method for solving Biot consolidation problems is developed. The solution is obtained via minimization of a least-squares functional, based upon the equations of equilibrium, the equations of continuity and weak forms of the constitutive relationships for elasticity and Darcy flow. The formulation involves four separate categories of unknowns: displacements, stresses, fluid pressures and velocities. Each of these unknowns is approximated by linear continuous functions. The mathematical formulation is implemented in an original computer program, written from scratch and using object-oriented logic. The performance of the method is tested on one- and two-dimensional classical problems in poro-elasticity. The numerical experiments suggest the same rates of convergence for all four types of variables, when the same interpolation spaces are used. The continuous linear triangles show the same rates of convergence for both compressible and entirely incompressible elastic solids. This mixed formulation results in non-oscillating fluid pressures over entire domain for different moments of time. The method appears to be naturally stable, without any need of additional stabilization terms with mesh-dependent parameters. Copyright © 2007 John Wiley & Sons, Ltd. [source] Simple modifications for stabilization of the finite point methodINTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN ENGINEERING, Issue 3 2005B. Boroomand Abstract A stabilized version of the finite point method (FPM) is presented. A source of instability due to the evaluation of the base function using a least square procedure is discussed. A suitable mapping is proposed and employed to eliminate the ill-conditioning effect due to directional arrangement of the points. A step by step algorithm is given for finding the local rotated axes and the dimensions of the cloud using local average spacing and inertia moments of the points distribution. It is shown that the conventional version of FPM may lead to wrong results when the proposed mapping algorithm is not used. It is shown that another source for instability and non-monotonic convergence rate in collocation methods lies in the treatment of Neumann boundary conditions. Unlike the conventional FPM, in this work the Neumann boundary conditions and the equilibrium equations appear simultaneously in a weight equation similar to that of weighted residual methods. The stabilization procedure may be considered as an interpretation of the finite calculus (FIC) method. The main difference between the two stabilization procedures lies in choosing the characteristic length in FIC and the weight of the boundary residual in the proposed method. The new approach also provides a unique definition for the sign of the stabilization terms. The reasons for using stabilization terms only at the boundaries is discussed and the two methods are compared. Several numerical examples are presented to demonstrate the performance and convergence of the proposed methods. Copyright © 2005 John Wiley & Sons, Ltd. [source] Stabilized finite element method for viscoplastic flow: formulation with state variable evolutionINTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN ENGINEERING, Issue 2 2003Antoinette M. Maniatty Abstract A stabilized, mixed finite element formulation for modelling viscoplastic flow, which can be used to model approximately steady-state metal-forming processes, is presented. The mixed formulation is expressed in terms of the velocity, pressure and state variable fields, where the state variable is used to describe the evolution of the material's resistance to plastic flow. The resulting system of equations has two sources of well-known instabilities, one due to the incompressibility constraint and one due to the convection-type state variable equation. Both of these instabilities are handled by adding mesh-dependent stabilization terms, which are functions of the Euler,Lagrange equations, to the usual Galerkin method. Linearization of the weak form is derived to enable a Newton,Raphson implementation into an object-oriented finite element framework. A progressive solution strategy is used for improving convergence for highly non-linear material behaviour, typical for metals. Numerical experiments using the stabilization method with hierarchic shape functions for the velocity, pressure and state variable fields in viscoplastic flow and metal-forming problems show that the stabilized finite element method is effective and efficient for non-linear steady forming problems. Finally, the results are discussed and conclusions are inferred. Copyright © 2002 John Wiley & Sons, Ltd. [source] A preconditioner for generalized saddle point problems: Application to 3D stationary Navier-Stokes equationsNUMERICAL METHODS FOR PARTIAL DIFFERENTIAL EQUATIONS, Issue 6 2006C. Calgaro Abstract In this article we consider the stationary Navier-Stokes system discretized by finite element methods which do not satisfy the inf-sup condition. These discretizations typically take the form of a variational problem with stabilization terms. Such a problem may be transformed by iteration methods into a sequence of linear, Oseen-type variational problems. On the algebraic level, these problems belong to a certain class of linear systems with nonsymmetric system matrices ("generalized saddle point problems"). We show that if the underlying finite element spaces satisfy a generalized inf-sup condition, these problems have a unique solution. Moreover, we introduce a block triangular preconditioner and we show how the eigenvalue bounds of the preconditioned system matrix depend on the coercivity constant and continuity bounds of the bilinear forms arising in the variational problem. Finally we prove that the stabilized P1-P1 finite element method proposed by Rebollo is covered by our theory and we show that the condition number of the preconditioned system matrix is independent of the mesh size. Numerical tests with 3D stationary Navier-Stokes flows confirm our results. © 2006 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 2006 [source] | ||||||||||