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Stabilization Parameter (stabilization + parameter)
Selected AbstractsAnisotropic meshes and streamline-diffusion stabilization for convection,diffusion problemsINTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN BIOMEDICAL ENGINEERING, Issue 10 2005Torsten Linß Abstract We study a convection-diffusion problem with dominant convection. Anisotropic streamline aligned meshes with high aspect ratios are recommended to resolve characteristic interior and boundary layers and to achieve high accuracy. We address the question of how the stabilization parameter in the streamline-diffusion FEM (SDFEM) and the Galerkin least-squares FEM (GLSFEM) should be chosen inside the layers. Using a residual free bubbles approach, we show that within the layers the stabilization must be drastically reduced. Copyright © 2005 John Wiley & Sons, Ltd. [source] Compressible flow SUPG parameters computed from element matricesINTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN BIOMEDICAL ENGINEERING, Issue 9 2005L. Catabriga Abstract We present, for the SUPG formulation of inviscid compressible flows with shocks, stabilization parameters defined based on the element-level matrices. These definitions are expressed in terms of the ratios of the norms of the matrices and take into account the flow field, the local length scales, and the time step size. Calculations of these stabilization parameters are straightforward and do not require explicit expressions for length or velocity scales. We compare the performance of these stabilization parameters, accompanied by a shock-capturing parameter introduced earlier, with the performance of a stabilization parameter introduced earlier, accompanied by the same shock-capturing parameter. We investigate the performance difference between updating the stabilization and shock-capturing parameters at the end of every time step and at the end of every non-linear iteration within a time step. We also investigate the influence of activating an algorithmic option that was introduced earlier, which is based on freezing the shock-capturing parameter at its current value when a convergence stagnation is detected. Copyright © 2005 John Wiley & Sons, Ltd. [source] CBS versus GLS stabilization of the incompressible Navier,Stokes equations and the role of the time step as stabilization parameterINTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN BIOMEDICAL ENGINEERING, Issue 2 2002R. Codina Abstract In this work we compare two apparently different stabilization procedures for the finite element approximation of the incompressible Navier,Stokes equations. The first is the characteristic-based split (CBS). It combines the characteristic Galerkin method to deal with convection dominated flows with a classical splitting technique, which in some cases allows us to use equal velocity,pressure interpolations. The second approach is the Galerkin-least-squares (GLS) method, in which a least-squares form of the element residual is added to the basic Galerkin equations. It is shown that both formulations display similar stabilization mechanisms, provided the stabilization parameter of the GLS method is identified with the time step of the CBS approach. This identification can be understood from a formal Fourier analysis of the linearized problem. Copyright © 2001 John Wiley & Sons, Ltd. [source] An optimally convergent discontinuous Galerkin-based extended finite element method for fracture mechanicsINTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN ENGINEERING, Issue 6 2010Yongxing Shen Abstract The extended finite element method (XFEM) enables the representation of cracks in arbitrary locations of a mesh. We introduce here a variant of the XFEM rendering an optimally convergent scheme. Its distinguishing features are as follows: (a) the introduction of singular asymptotic crack tip fields with support on only a small region around the crack tip (the enrichment region), (b) only one and two enrichment functions are added for anti-plane shear and planar problems, respectively and (c) the relaxation of the continuity between the enrichment region and the rest of the domain, and the adoption of a discontinuous Galerkin (DG) method therein. The method is provably stable for any positive value of a stabilization parameter, and by weakly enforcing the continuity between the two regions it eliminates ,blending elements' partly responsible for the suboptimal convergence of some early XFEMs. Moreover, the particular choice of enrichment functions results in a surprisingly sparse stiffness matrix that remains reasonably conditioned as the mesh is refined. More importantly, the stress intensity factors can be extracted with a satisfactory accuracy as primary unknowns. Quadrature strategies required for the optimal convergence are also discussed. Finally, the DG method was modified to retain stability based on an inf-sup condition. Copyright © 2009 John Wiley & Sons, Ltd. [source] A variational multiscale model for the advection,diffusion,reaction equationINTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN BIOMEDICAL ENGINEERING, Issue 7 2009Guillaume Houzeaux Abstract The variational multiscale (VMS) method sets a general framework for stabilization methods. By splitting the exact solution into coarse (grid) and fine (subgrid) scales, one can obtain a system of two equations for these unknowns. The grid scale equation is solved using the Galerkin method and contains an additional term involving the subgrid scale. At this stage, several options are usually considered to deal with the subgrid scale equation: this includes the choice of the space where the subgrid scale would be defined as well as the simplifications leading to compute the subgrid scale analytically or numerically. The present study proposes to develop a two-scale variational method for the advection,diffusion,reaction equation. On the one hand, a family of weak forms are obtained by integrating by parts a fraction of the advection term. On the other hand, the solution of the subgrid scale equation is found using the following. First, a two-scale variational method is applied to the one-dimensional problem. Then, a series of approximations are assumed to solve the subgrid space equation analytically. This allows to devise expressions for the ,stabilization parameter' ,, in the context of VMS (two-scale) method. The proposed method is equivalent to the traditional Green's method used in the literature to solve residual-free bubbles, although it offers another point of view, as the strong form of the subgrid scale equation is solved explicitly. In addition, the authors apply the methodology to high-order elements, namely quadratic and cubic elements. The proposed model consists in assuming that the subgrid scale vanishes also on interior nodes of the element and applying the strategy used for linear element in the segment between these interior nodes. The proposed scheme is compared with existing ones through the solution of a one-dimensional numerical example for linear, quadratic and cubic elements. In addition, the mesh convergence is checked for high-order elements through the solution of an exact solution in two dimensions. Copyright © 2008 John Wiley & Sons, Ltd. [source] Compressible flow SUPG parameters computed from element matricesINTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN BIOMEDICAL ENGINEERING, Issue 9 2005L. Catabriga Abstract We present, for the SUPG formulation of inviscid compressible flows with shocks, stabilization parameters defined based on the element-level matrices. These definitions are expressed in terms of the ratios of the norms of the matrices and take into account the flow field, the local length scales, and the time step size. Calculations of these stabilization parameters are straightforward and do not require explicit expressions for length or velocity scales. We compare the performance of these stabilization parameters, accompanied by a shock-capturing parameter introduced earlier, with the performance of a stabilization parameter introduced earlier, accompanied by the same shock-capturing parameter. We investigate the performance difference between updating the stabilization and shock-capturing parameters at the end of every time step and at the end of every non-linear iteration within a time step. We also investigate the influence of activating an algorithmic option that was introduced earlier, which is based on freezing the shock-capturing parameter at its current value when a convergence stagnation is detected. Copyright © 2005 John Wiley & Sons, Ltd. [source] An 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] | ||||||||||