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The purpose of this paper is to introduce efficient methods for solving the 2D biharmonic equation with Dirichlet boundary conditions of second kind. This equation…
The purpose of this paper is to introduce efficient methods for solving the 2D biharmonic equation with Dirichlet boundary conditions of second kind. This equation describes the deflection of loaded plate with boundary conditions of simply supported plate kind. Also it can be derived from the calculus of variations combined with the variational principle of minimum potential energy. Because of existing fourth derivatives in this equation, introducing high‐order accurate methods need to use artificial points. Also solving the resulted linear system of equations suffers from slow convergence when iterative methods are used. This paper aims to introduce efficient methods to overcome these problems.
The paper considers several compact finite difference approximations that are derived on a nine‐point compact stencil using the values of the solution and its second derivatives as the unknowns. In these approximations there is no need to define special formulas near the boundaries and boundary conditions can be incorporated with these techniques. Several iterative linear systems solvers such as Krylov subspace and multigrid methods and their combination (with suitable preconditioner) have been developed to compare the efficiency of each method and to design powerful solvers.
The paper finds that the combination of compact finite difference schemes with multigrid method and Krylov iteration methods preconditioned by multigrid have excellent results for the second biharmonic equation, and that Krylov iteration methods preconditioned by multigrid are the most efficient methods.
The paper is of value in presenting, via some tables and figures, some numerical experiments which resulted from applying new methods on several test problems, and making comparison with conventional methods.
This paper presents some linear adaptive non‐nested multigrid methods which are applied to linear elastic problems discretized with triangular and tetrahedral finite elements using unstructured and Delaunay mesh generators. The Zienkiewicz‐Zhu error estimator and a h‐refinement procedure are used to obtain the non‐nested meshes used by the multigrid methods. We solve problems with a specified percentage error in the energy norm using the optimal performance of multigrid methods.
The steady compressible Navier—Stokes equations coupled to thek—ε turbulence equations are discretized within avertex‐centered finite volume formulation. The convective…
The steady compressible Navier—Stokes equations coupled to the k—ε turbulence equations are discretized within a vertex‐centered finite volume formulation. The convective fluxes are obtained by the polynomial flux‐difference splitting upwind method. The first order accurate part results directly from the splitting. The second order part is obtained by the flux‐extrapolation technique using the minmod limiter. The diffusive fluxes are discretized in the central way and are split into a normal and a tangential contribution. The first order accurate part of the convective fluxes together with the normal contribution of the diffusive fluxes form a positive system which allows solution by classical relaxation methods. The source terms in the low‐Reynolds k‐ε equations are grouped into positive and negative terms. The linearized negative source terms are added to the positive system to increase the diagonal dominance. The resulting positive system forms the left hand side of the equations. The remaining terms are put in the right hand side. A multigrid method based on successive relaxation, full weighting, bilinear interpolation and W‐cycle is used. The multigrid method itself acts on the left hand side of the equations. The right hand side is updated in a defect correction cycle.
Presents an implementation of the algebraic multigrid method. It can work in two ways: as pure multigrid method and as a pre‐conditioner for the conjugate gradient method…
Presents an implementation of the algebraic multigrid method. It can work in two ways: as pure multigrid method and as a pre‐conditioner for the conjugate gradient method. Shows applications of the iterative solvers for problems in linear and non‐linear elasticity. Shows the range of possible applications with different examples with regular and non‐regular meshes and three‐dimensional problems.
A flux‐difference splitting based on the polynomial character of the flux vectors is applied to steady Euler equations, discretized with a vertex‐centred finite volume…
A flux‐difference splitting based on the polynomial character of the flux vectors is applied to steady Euler equations, discretized with a vertex‐centred finite volume method. In first order accurate form, a discrete set of equations is obtained which is both conservative and positive. Due to the positivity, the set of equations can be solved by collective relaxation methods in multigrid form. A full multigrid method based on successive relaxation, full weighting, bilinear interpolation and W‐cycle is used. Second order accuracy is obtained by the Chakravarthy‐Osher flux‐extrapolation technique, using the Roe‐Chakravarthy minmod limiter. In second order form, direct relaxation of the discrete equations is no longer possible due to the loss of positivity. A defect‐correction is used in order to solve the second order system.
The averaged Navier‐Stokes and the k‐e turbulence model equations are used to simulate turbulent flows in some internal flow cases. The discrete equations are solved by…
The averaged Navier‐Stokes and the k‐e turbulence model equations are used to simulate turbulent flows in some internal flow cases. The discrete equations are solved by different variations of Multigrid methods. These include both steady state as well as time dependent solvers. Locally refined grids can be added dynamically in all cases. The Multigrid schemes result in fast convergence rates, whereas local grid refinements allow improved accuracy with rational increase in problem size. The applications of the solver to a 3‐D (cold) furnace model and to the simulation of the flow in a wind tunnel past an object prove the efficiency of the Multigrid scheme with local grid refinement.
This paper discusses the robustness of the algebraic multigrid (AMG) method as well as geometric multigrid (GMG) method against mesh distortion in edge‐based finite…
This paper discusses the robustness of the algebraic multigrid (AMG) method as well as geometric multigrid (GMG) method against mesh distortion in edge‐based finite element analysis.
Analyzes a simple magnetostatic problem, in which the model consists of a cubic iron and the surrounding air region. Prepares three meshes which have same number of elements to evaluate the robustness of multigrid against the distortion of mesh.
The AMG method is shown to be more robust against mesh distortion than the GMG method.
Shows that the AMG is more robust than the GMG. This result is of practical interest to the researchers in this field.
Detailed results of numerical calculations of transient, 2D incompressible flow around and in the wake of a square prism at Re = 100, 200 and 500 are presented. An…
Detailed results of numerical calculations of transient, 2D incompressible flow around and in the wake of a square prism at Re = 100, 200 and 500 are presented. An implicit finite‐difference operator‐splitting method, a version of the known SIMPLEC‐like method on a staggered grid, is described. Appropriate theoretical results are presented. The method has second‐order accuracy in space, conserving mass, momentum and kinetic energy. A new modification of the multigrid method is employed to solve the elliptic pressure problem. Calculations are performed on a sequence of spatial grids with up to 401 × 321 grid points, at sequentially halved time steps to ensure grid‐independent results. Three types of flow are shown to exist at Re = 500: a steady‐state unstable flow and two which are transient, fully periodic and asymmetric about the centre line but mirror symmetric to each other. Discrete frequency spectra of drag and lift coefficients are presented.
Develops a finite element analysis and solution strategy for the augmented drift‐diffusion equations in semiconductors device theory using a multilevel scheme. Decouples…
Develops a finite element analysis and solution strategy for the augmented drift‐diffusion equations in semiconductors device theory using a multilevel scheme. Decouples the drift‐diffusion equations using Gummel iteration with incremental continuation in the applied voltage. Includes augmentation of the carrier mobility to address the issue of non‐local electric field effects (velocity overshoot) within the framework of the drift‐diffusion formulation. Gives comparison results with hydrodynamic and Monte Carlo models and sensitivity studies with respect to the augmentation parameter. Discretizes the equations with a special finite element method using bases of variable polynomial degree. Accomplishes solution of the resulting linear systems with a multilevel method using the basis degree as the grid level. Presents performance results and comparison studies with direct elimination.
The focus of this paper is on the efficient numerical computation of 3D electromagnetic field problems by using the finite element (FE) and multigrid (MG) methods. The…
The focus of this paper is on the efficient numerical computation of 3D electromagnetic field problems by using the finite element (FE) and multigrid (MG) methods. The magnetic vector potential is used as the field variable and the discretization is performed by Lagrange (nodal) as well as Ne´de´lec (edge) finite elements. The resulting system of equations is solved by applying a preconditioned conjugate gradient (PCG) method with an adapted algebraic multigrid (AMG) as well as an appropriate geometric MG preconditioner.