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Convergence Behavior (convergence + behavior)
Selected AbstractsOn the C1 continuous discretization of non-linear gradient elasticity: A comparison of NEM and FEM based on Bernstein,Bézier patchesINTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN ENGINEERING, Issue 10 2010P. Fischer Abstract In gradient elasticity, the appearance of strain gradients in the free energy density leads to the need of C1 continuous discretization methods. In the present work, the performances of C1 finite elements and the C1 Natural Element Method (NEM) are compared. The triangular Argyris and Hsieh,Clough,Tocher finite elements are reparametrized in terms of the Bernstein polynomials. The quadrilateral Bogner,Fox,Schmidt element is used in an isoparametric framework, for which a preprocessing algorithm is presented. Additionally, the C1 -NEM is applied to non-linear gradient elasticity. Several numerical examples are analyzed to compare the convergence behavior of the different methods. It will be illustrated that the isoparametric elements and the NEM show a significantly better performance than the triangular elements. Copyright © 2009 John Wiley & Sons, Ltd. [source] Numerical simulation of bubble and droplet deformation by a level set approach with surface tension in three dimensionsINTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN FLUIDS, Issue 9 2010Roberto Croce Abstract In this paper we present a three-dimensional Navier,Stokes solver for incompressible two-phase flow problems with surface tension and apply the proposed scheme to the simulation of bubble and droplet deformation. One of the main concerns of this study is the impact of surface tension and its discretization on the overall convergence behavior and conservation properties. Our approach employs a standard finite difference/finite volume discretization on uniform Cartesian staggered grids and uses Chorin's projection approach. The free surface between the two fluid phases is tracked with a level set (LS) technique. Here, the interface conditions are implicitly incorporated into the momentum equations by the continuum surface force method. Surface tension is evaluated using a smoothed delta function and a third-order interpolation. The problem of mass conservation for the two phases is treated by a reinitialization of the LS function employing a regularized signum function and a global fixed point iteration. All convective terms are discretized by a WENO scheme of fifth order. Altogether, our approach exhibits a second-order convergence away from the free surface. The discretization of surface tension requires a smoothing scheme near the free surface, which leads to a first-order convergence in the smoothing region. We discuss the details of the proposed numerical scheme and present the results of several numerical experiments concerning mass conservation, convergence of curvature, and the application of our solver to the simulation of two rising bubble problems, one with small and one with large jumps in material parameters, and the simulation of a droplet deformation due to a shear flow in three space dimensions. Furthermore, we compare our three-dimensional results with those of quasi-two-dimensional and two-dimensional simulations. This comparison clearly shows the need for full three-dimensional simulations of droplet and bubble deformation to capture the correct physical behavior. Copyright © 2009 John Wiley & Sons, Ltd. [source] Some results on the accuracy of an edge-based finite volume formulation for the solution of elliptic problems in non-homogeneous and non-isotropic mediaINTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN FLUIDS, Issue 3 2009Darlan Karlo Elisiário de Carvalho Abstract The numerical simulation of elliptic type problems in strongly heterogeneous and anisotropic media represents a great challenge from mathematical and numerical point of views. The simulation of flows in non-homogeneous and non-isotropic porous media with full tensor diffusion coefficients, which is a common situation associated with the miscible displacement of contaminants in aquifers and the immiscible and incompressible two-phase flow of oil and water in petroleum reservoirs, involves the numerical solution of an elliptic type equation in which the diffusion coefficient can be discontinuous, varying orders of magnitude within short distances. In the present work, we present a vertex-centered edge-based finite volume method (EBFV) with median dual control volumes built over a primal mesh. This formulation is capable of handling the heterogeneous and anisotropic media using structured or unstructured, triangular or quadrilateral meshes. In the EBFV method, the discretization of the diffusion term is performed using a node-centered discretization implemented in two loops over the edges of the primary mesh. This formulation guarantees local conservation for problems with discontinuous coefficients, keeping second-order accuracy for smooth solutions on general triangular and orthogonal quadrilateral meshes. In order to show the convergence behavior of the proposed EBFV procedure, we solve three benchmark problems including full tensor, material heterogeneity and distributed source terms. For these three examples, numerical results compare favorably with others found in literature. A fourth problem, with highly non-smooth solution, has been included showing that the EBFV needs further improvement to formally guarantee monotonic solutions in such cases. Copyright © 2008 John Wiley & Sons, Ltd. [source] On CFL evolution strategies for implicit upwind methods in linearized Euler equationsINTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN FLUIDS, Issue 1 2009H. M. Bücker Abstract In implicit upwind methods for the solution of linearized Euler equations, one of the key issues is to balance large time steps, leading to a fast convergence behavior, and small time steps, needed to sufficiently resolve relevant flow features. A time step is determined by choosing a Courant,Friedrichs,Levy (CFL) number in every iteration. A novel CFL evolution strategy is introduced and compared with two existing strategies. Numerical experiments using the adaptive multiscale finite volume solver QUADFLOW demonstrate that all three CFL evolution strategies have their advantages and disadvantages. A fourth strategy aiming at reducing the residual as much as possible in every time step is also examined. Using automatic differentiation, a sensitivity analysis investigating the influence of the CFL number on the residual is carried out confirming that, today, CFL control is still a difficult and open problem. Copyright © 2008 John Wiley & Sons, Ltd. [source] Convergence acceleration by self-adjusted time stepsize using Bi-CGSTAB method for turbulent separated flow computationINTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN FLUIDS, Issue 2 2002W. B. Tsai Abstract Poor convergence behavior is usually encountered when numerical computations on turbulent separated flow are performed. A design of self-adjusted stepsize concept both in time span and spatial coordinate systems to achieve faster convergence is demonstrated in this study. The determination of the time stepsize based on the concept of minimization of residuals using the Bi-CGSTAB algorithm is proposed. The numerical results show that the time stepsize adjusted by the proposed method indeed improves the convergence rate for turbulent separated flow computations using advanced turbulence models in low-Reynolds number forms. Copyright © 2002 John Wiley & Sons, Ltd. [source] An accurate total energy density functionalINTERNATIONAL JOURNAL OF QUANTUM CHEMISTRY, Issue 15 2007Baojing Zhou Abstract We propose a new density functional for the evaluation of the total electronic energy by subtracting the Roothaan energy, i.e. the Hartree energy of the density residual, from the Hohenberg,Kohn,Sham (HKS) functional, which is normally used in self-consistent Kohn,Sham (KS) density functional theory (DFT) calculations. Because of the positive semi-definite nature of the Roothaan energy, the resulting Wang,Zhou (WZ) functional always produces a total energy lower than that from the HKS functional and usually converges to the exact total energy from below. Following the same spirit of the Zhou,Wang-, (ZW,) functional in the recently proposed orbital-corrected orbital-free (OO) DFT method (Zhou and Wang, J Chem Phys 2006, 124, 081107), we linearly mix the WZ functional with the HKS functional to allow further systematic error cancellations. The resulting Wang,Zhou-, (WZ,) functional is compared with the ZW, functional in OO-DFT calculations for systems within different chemical environment. We find that the optimal value of , for the WZ, functional is more stable than that of , for the ZW, functional. This is because the WZ functional remedies the oscillatory convergence behavior of the Harris functional and renders the direct evaluation of , for the WZ, functional more plausible in the application of the linear-scaling OO-DFT method for large systems. © 2007 Wiley Periodicals, Inc. Int J Quantum Chem, 2007 [source] Thermophoresis of axisymmetric aerosol particles along their axes of revolutionAICHE JOURNAL, Issue 1 2009Yu C. Chang Abstract The axisymmetric thermophoretic motion of an aerosol particle of revolution in a uniformly prescribed temperature gradient is studied theoretically. The Knudsen number is assumed to be small so that the fluid flow is described by a continuum model. A method of distribution of a set of spherical singularities along the axis of revolution within a prolate particle or on the fundamental plane within an oblate particle is used to find the general solutions for the temperature distribution and fluid velocity field. The jump/slip conditions on the particle surface are satisfied by applying a boundary-collocation technique to these general solutions. Numerical results for the thermophoretic velocity of the particle are obtained with good convergence behavior for various cases. For the axisymmetric thermophoresis of an aerosol spheroid with no temperature jump and frictional slip at its surface, the agreement between our results and the available analytical solutions is very good. The thermophoretic velocity of a spheroid along its axis of revolution in general increases with an increase in its axial-to-radial aspect ratio, but there are exceptions. For most practical cases of a spheroid with a specified aspect ratio, its thermophoretic mobility is not a monotonic function of its relative jump/slip coefficients and thermal conductivity. © 2008 American Institute of Chemical Engineers AIChE J, 2009 [source] A comparison of abstract versions of deflation, balancing and additive coarse grid correction preconditionersNUMERICAL LINEAR ALGEBRA WITH APPLICATIONS, Issue 4 2008R. Nabben Abstract In this paper we consider various preconditioners for the conjugate gradient (CG) method to solve large linear systems of equations with symmetric positive definite system matrix. We continue the comparison between abstract versions of the deflation, balancing and additive coarse grid correction preconditioning techniques started in (SIAM J. Numer. Anal. 2004; 42:1631,1647; SIAM J. Sci. Comput. 2006; 27:1742,1759). There the deflation method is compared with the abstract additive coarse grid correction preconditioner and the abstract balancing preconditioner. Here, we close the triangle between these three methods. First of all, we show that a theoretical comparison of the condition numbers of the abstract additive coarse grid correction and the condition number of the system preconditioned by the abstract balancing preconditioner is not possible. We present a counter example, for which the condition number of the abstract additive coarse grid correction preconditioned system is below the condition number of the system preconditioned with the abstract balancing preconditioner. However, if the CG method is preconditioned by the abstract balancing preconditioner and is started with a special starting vector, the asymptotic convergence behavior of the CG method can be described by the so-called effective condition number with respect to the starting vector. We prove that this effective condition number of the system preconditioned by the abstract balancing preconditioner is less than or equal to the condition number of the system preconditioned by the abstract additive coarse grid correction method. We also provide a short proof of the relationship between the effective condition number and the convergence of CG. Moreover, we compare the A -norm of the errors of the iterates given by the different preconditioners and establish the orthogonal invariants of all three types of preconditioners. Copyright © 2008 John Wiley & Sons, Ltd. [source] INCOME THRESHOLDS AND GROWTH CONVERGENCE: A PANEL DATA APPROACH,THE MANCHESTER SCHOOL, Issue 2 2006TSUNG-WU HO This paper applies a dynamic panel model to explore whether the low-income countries ,catch up' with the rich ones by examining the threshold effects of per capita income on the convergence behavior of growth rates. Empirical evidence from 121 Penn World Table economies and 48 US states indicates that income levels have substantial impacts on the convergence behavior. First, convergence is insignificantly found in the lowest-income regimes, which is interpreted that these poor countries persist at their income levels, which cause possible income barriers-to-growth. That is, the poor countries may not be able to catch up with the rich ones easily, unless an income threshold is overcome. Second, convergence is significantly found beyond the lowest-income regime, implying that the low-income countries catch up with the rich. We conclude that when a certain income threshold is overcome, the poor countries catch up with the rich ones; hence a subsidiary income policy can be helpful. [source] | |||||||||||||