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Galerkin Formulation (galerkin + formulation)
Selected AbstractsDiscontinuous Galerkin framework for adaptive solution of parabolic problemsINTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN ENGINEERING, Issue 1 2007Deepak V. Kulkarni Abstract Non-conforming meshes are frequently employed in adaptive analyses and simulations of multi-component systems. We develop a discontinuous Galerkin formulation for the discretization of parabolic problems that weakly enforces continuity across non-conforming mesh interfaces. A benefit of the DG scheme is that it does not introduce constraint equations and their resulting Lagrange multiplier fields as done in mixed and mortar methods. The salient features of the formulation are highlighted through an a priori analysis. When coupled with a mesh refinement scheme the DG formulation is able to accommodate multiple hanging nodes per element edge and leads to an effective adaptive framework for the analysis of interface evolution problems. We demonstrate our approach by analysing the Stefan problem of solidification. Copyright © 2006 John Wiley & Sons, Ltd. [source] An assumed-gradient finite element method for the level set equationINTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN ENGINEERING, Issue 8 2005Hashem M. Mourad Abstract The level set equation is a non-linear advection equation, and standard finite-element and finite-difference strategies typically employ spatial stabilization techniques to suppress spurious oscillations in the numerical solution. We recast the level set equation in a simpler form by assuming that the level set function remains a signed distance to the front/interface being captured. As with the original level set equation, the use of an extensional velocity helps maintain this signed-distance function. For some interface-evolution problems, this approach reduces the original level set equation to an ordinary differential equation that is almost trivial to solve. Further, we find that sufficient accuracy is available through a standard Galerkin formulation without any stabilization or discontinuity-capturing terms. Several numerical experiments are conducted to assess the ability of the proposed assumed-gradient level set method to capture the correct solution, particularly in the presence of discontinuities in the extensional velocity or level-set gradient. We examine the convergence properties of the method and its performance in problems where the simplified level set equation takes the form of a Hamilton,Jacobi equation with convex/non-convex Hamiltonian. Importantly, discretizations based on structured and unstructured finite-element meshes of bilinear quadrilateral and linear triangular elements are shown to perform equally well. Copyright © 2005 John Wiley & Sons, Ltd. [source] An enriched meshless method for non-linear fracture mechanicsINTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN ENGINEERING, Issue 2 2004B. N. Rao Abstract This paper presents an enriched meshless method for fracture analysis of cracks in homogeneous, isotropic, non-linear-elastic, two-dimensional solids, subject to mode-I loading conditions. The method involves an element-free Galerkin formulation and two new enriched basis functions (Types I and II) to capture the Hutchinson,Rice,Rosengren singularity field in non-linear fracture mechanics. The Type I enriched basis function can be viewed as a generalized enriched basis function, which degenerates to the linear-elastic basis function when the material hardening exponent is unity. The Type II enriched basis function entails further improvements of the Type I basis function by adding trigonometric functions. Four numerical examples are presented to illustrate the proposed method. The boundary layer analysis indicates that the crack-tip field predicted by using the proposed basis functions matches with the theoretical solution very well in the whole region considered, whether for the near-tip asymptotic field or for the far-tip elastic field. Numerical analyses of standard fracture specimens by the proposed meshless method also yield accurate estimates of the J -integral for the applied load intensities and material properties considered. Also, the crack-mouth opening displacement evaluated by the proposed meshless method is in good agreement with finite element results. Furthermore, the meshless results show excellent agreement with the experimental measurements, indicating that the new basis functions are also capable of capturing elastic,plastic deformations at a stress concentration effectively. Copyright © 2003 John Wiley & Sons, Ltd. [source] A Galerkin/least-squares finite element formulation for consolidationINTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN ENGINEERING, Issue 8 2001A. Truty Abstract A Galerkin/least-squares (GLS) finite element formulation for problem of consolidation of fully saturated two-phase media is presented. The elimination of spurious pressure oscillations appearing at the early stage of consolidation for standard Galerkin finite elements with equal interpolation order for both displacements and pressures is the goal of the approach. It will be shown that the least-squares term, based exclusively on the residuum of the fluid flow continuity equation, added to the standard Galerkin formulation enhances its stability and can fully eliminate pressure oscillations. A reasonably simple framework designed for derivation of one-dimensional as well as multi-dimensional estimates of the stabilization factor is proposed and then verified. The formulation is validated on one-dimensional and then on two-dimensional, linear and non-linear test problems. The effect of the fluid incompressibility as well as compressibility will be taken into account and investigated. Copyright © 2001 John Wiley & Sons Ltd. [source] Variance-reduced Monte Carlo solutions of the Boltzmann equation for low-speed gas flows: A discontinuous Galerkin formulationINTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN FLUIDS, Issue 4 2008Lowell L. Baker Abstract We present and discuss an efficient, high-order numerical solution method for solving the Boltzmann equation for low-speed dilute gas flows. The method's major ingredient is a new Monte Carlo technique for evaluating the weak form of the collision integral necessary for the discontinuous Galerkin formulation used here. The Monte Carlo technique extends the variance reduction ideas first presented in Baker and Hadjiconstantinou (Phys. Fluids 2005; 17, art. no. 051703) and makes evaluation of the weak form of the collision integral not only tractable but also very efficient. The variance reduction, achieved by evaluating only the deviation from equilibrium, results in very low statistical uncertainty and the ability to capture arbitrarily small deviations from equilibrium (e.g. low-flow speed) at a computational cost that is independent of the magnitude of this deviation. As a result, for low-signal flows the proposed method holds a significant computational advantage compared with traditional particle methods such as direct simulation Monte Carlo (DSMC). Copyright © 2008 John Wiley & Sons, Ltd. [source] A numerical approach for groundwater flow in unsaturated porous mediaINTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN FLUIDS, Issue 9-10 2006F. Quintana Abstract In this article, a computational tool to simulate groundwater flow in variably saturated non-deformable fractured porous media is presented, which includes a conceptual model to obtain analytical expressions of water retention and hydraulic conductivity curves for fractured hard rocks and a numerical algorithm to solve the Richards equation. To calculate effective saturation and relative hydraulic conductivity curves we adopt the Brooks,Corey model assuming fractal laws for both aperture and number of fractures. A standard Galerkin formulation was employed to solve the Richards' equation together with a Crank,Nicholson scheme with Richardson extrapolation for the time discretization. The main contribution of this paper is to group an analytical model of the authors with a robust numerical algorithm designed to solve adequately the highly non-linear Richards' equation generating a tool for porous media engineering. Copyright © 2006 John Wiley & Sons, Ltd. [source] Dispersion analysis of the least-squares finite-element shallow-water systemINTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN FLUIDS, Issue 6 2003D. Y. Le Roux Abstract The frequency or dispersion relation for the least-squares mixed formulation of the shallow-water equations is analysed. We consider the use of different approximation spaces corresponding to co-located and staggered meshes, respectively. The study includes the effect of Coriolis, and the dispersion properties are compared analytically and graphically with those of the mixed Galerkin formulation. Numerical solutions of a test problem to simulate slow Rossby modes illustrate the theoretical results. Copyright © 2003 John Wiley & Sons, Ltd. [source] | ||||||||||