Grid Equations Research Articles

Two-Dimensional Elliptic Grid Solver Using Boundary Grid Control and Curvature Correction

The Thompson method of explicitly evaluating control function of the elliptic grid solver is employed to improve local grid quality around convex and concave boundaries. The curvature correction is principally added to the grid lines around a boundary and is quickly attenuated as the grid points move inward. The algebraic advancing front method is employed to estimate two levels of orthogonal grids for evaluating the control functions at the boundary. If the grid equation based on the Cauchy-Riemann relation with ξ and η as independent variables is employed, numerical examinations show that a lengthy trial-and-error procedure is required to get a satisfactory grid distribution. In contrast, the method based on the Cauchy-Riemann relation with x and y as independent variables, which benefits from the maximum principle, not only removes the undesired grid clustering and diluting easily but also effectively improves overall grid smoothness and slightly enhances boundary grid orthogonality

Transient eddy-current calculation with the FI-method

This paper introduces time-domain formulations based on the Maxwell grid equations of the consistent finite integration (FI) method which allows calculation of slowly-varying electromagnetic fields with various implicit time-stepping techniques. The resultant large sparse linear systems of equations have to be solved by iterative methods for each time step. The preconditioned conjugate gradient methods allow the calculation of nongauged (quasi-)static problems efficiently. Numerical results are shown for the TEAM workshop problem 7 and an SRC high density magnetic head problem, which could be calculated with more than one million degrees of freedom.

A study of incorporating the multigrid method into the three-dimensional finite element discretization: a modular setting and application

Increasing the efficiency of solving linear/linearized matrix equations is a key point to save computer time in numerical simulation, especially for three-dimensional problems. The multigrid method has been determined to be efficient in solving boundary-value problems. However, this method is mostly linked to the finite difference discretization, rather than to the finite element discretization. This is because the grid relationship between fine and coarse grids was not achieved effectively for the latter case. Consequently, not only is the coding complicated but also the performance is not satisfactory when incorporating the multigrid method into the finite element discretization. Here we present an approach to systematically prepare necessary information to relate fine and coarse grids regarding the three-dimensional finite element discretization, such that we can take advantage of using the multigrid method. To achieve a consistent approximation at each grid, we use A2h=I2hhAhIh2h and b2h=I2hh bh, starting from the composed matrix equation of the finest grid, to prepare the matrix equations for coarse grids. Such a process is implemented on an element level to reduce the computation to its minimum. To demonstrate the performance, this approach has been used to adapt two existing three-dimensional finite element subsurface flow and transport models, 3DFEMWATER and 3DLEWASTE, to their multigrid version, 3DMGWATER and 3DMGWASTE, respectively. Two example problems, one for each model, are considered for illustration. The computational result shows that the multigrid method can help solve the example problems very efficiently with our presented modular setting. © 1998 John Wiley & Sons, Ltd.

Development of an error-minimizing adaptive grid method

We present the first step in the development of a moving adaptive grid technique for flow and transport problems that is based on the approximate minimization in the L 1 norm of the modeling error due to discretization. The technique consists of a second-order accurate discretization scheme designed for error analysis, a filter technique to ensure sufficiently smooth solutions, and an iterative solver for the nonlinear adaptive grid equations. We describe how these elements have been used for the numerical approximation of a given one-dimensional function. This application has been chosen to test and optimize the error-minimizing adaptive grid method. Results are presented for a function having a smooth part, a discontinuity and a uniform part. In comparison with a uniform grid, a gain in accuracy of a factor 10 to almost 1000 was obtained using 40 to 200 grid points.

Sensitivity analysis of the aeroacoustic response of turbomachinery blade rows

A method for computing the change or sensitivity of the aeroacoustic response of a cascade to small changes in airfoil and cascade geometry is presented. The steady flow is modeled by the full potential equation, which is discretized using a variational finite element technique. A streamline computational grid is generated as part of the steady solution. Newton iteration is used to solve for the nonlinear steady flow and grid equations with lower-upper (LU) matrix decomposition plus one forward and one back substitution used to solve the resulting matrix equations. Similarly, the unsteady small disturbance flow about the nonlinear mean flow is modeled by the linearized potential equation together with rapid distortion theory to account for vortical gusts. These linearized equations are discretized using finite elements and solved with a single LU decomposition. The sensitivities of the steady and unsteady flowfields to small changes in geometry are computed by perturbing the discretized equations about the nominal solutions. The resulting linear system of equations can be solved very efficiently because the LU factors of the resulting matrix equations are computed as part of the nominal steady and unsteady solution. Results are presented in this paper to show the accuracy and efficiency of the method, and the implications for aeroacoustic design of turbomachinery blades are discussed.

Solution-adaptive grid generation for compressible flow

A new efficient method of generating solution-adaptive grids for compressible flow problems is introduced. This scheme uses the Laplace equations, which are then transformed by using stretching functions to the final computational domain so that the generated grid can be clustered in the desired regions. Thus, the resulting generating equations are uncoupled. To adapt the grid to solutions, the control functions are chosen to depend upon the curvature and the gradient of solution at each grid point and the grid spacing is controlled by these values. This adaptive grid equations are employed for compressible flow calculations. The compressible Euler equations are solved using a second-order, symmetric TVD, finite volume scheme.

Fourier analysis of a robust multigrid method for convection-diffusion equations

We consider a two-grid method for solving 2D convection-diffusion problems. The coarse grid correction is based on approximation of the Schur complement. As a preconditioner of the Schur complement we use the exact Schur complement of modified fine grid equations. We assume constant coefficients and periodic boundary conditions and apply Fourier analysis. We prove an upper bound for the spectral radius of the two-grid iteration matrix that is smaller than one and independent of the mesh size, the convection/diffusion ratio and the flow direction; i.e. we have a (strong) robustness result. Numerical results illustrating the robustness of the corresponding multigrid $W$ -cycle are given.

Using the Maxwell grid equations to solve large problems

The capabilities of a set of finite-difference codes, based on the finite-integration technique (FIT), for solving Maxwell's equations have been examined. Each of the Maxwell equations is separately discretized to produce an equivalent matrix equation. In this way, problems from a very broad area of physics and engineering can be solved by a method which not only produces unique, physical solutions, but in which the divergence equations are also satisfied. With only 60-204 bytes per mesh cell and six unknowns per cell, it is possible to calculate very large problems with up to 12 million unknowns on a modern workstation with a 128-MB memory. To indicate the range and complexity of the problems which can be solved using this method, a series of large (approaching a million unknowns) problems is presented-in particular, in the frequency domain for both resonant and eddy current problems and in the time domain for the broadband calculation of filter structures.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

A local adaptive grid procedure for incompressible flows with multigridding and equidistribution concepts

AbstractAn adaptive grid solution procedure is developed for incompressible flow problems in which grid refinement based on an equidistribution law is performed in high‐error‐estimate regions that are flagged from a preliminary coarse grid solution. Solutions on the locally refined and equidistributed meshes are obtained using boundary conditions interpolated from the preliminary coarse grid solution, and solutions on both the refined and coarse grid regions are successively improved using a multigrid approach. For this purpose, suitable correction terms for the coarse grid equations are derived for all variables in the flagged regions. This procedure with Local Adaptation, Multigridding and Equidistribution (LAME) concepts is applied to various flow problems to demonstrate the accuracy improvements obtained using this method.

Numerical Methods for Grid Equations: Vol. I, Direct Methods

Numerical Methods for Grid Equations: Vol. I, Direct Methods

Maxwell's Grid Equations

A numerical approach for the solution of Maxwell's equation is presented. Based on a finite difference Yee lattice the method transforms each of the four Maxwell equations into an equivalent matrix expression that can be subsequently treated by matrix mathematics and suitable numerical methods for solving matrix problems. The algorithm, although derived from integral equations, can be considered to be a special case of finite difference formalisms

Multigrid method for stability problems

The problem of calculating the stability of steady state solutions of differential equations is treated. Leading eigenvalues (i.e., having maximal real part) of large matrices that arise from discretization are to be calculated. An efficient multigrid method for solving these problems is presented. The method begins by obtaining an initial approximation for the dominant subspace on a coarse level using a damped Jacobi relaxation. This proceeds until enough accuracy for the dominant subspace has been obtained. The resulting grid functions are then used as an initial approximation for appropriate eigenvalue problems. These problems are solved first on coarse levels, followed by refinement until a desired accuracy for the eigenvalues has been achieved. The method employs local relaxation on all levels together with a global change on the coarsest level only, which is designed to separate the different eigenfunctions as well as to update their corresponding eigenvalues. Coarsening is done using the FAS formulation in a nonstandard way in which the right-hand side of the coarse grid equations involves unknown parameters to be solved for on the coarse grid. This in particular leads to a new multigrid method for calculating the eigenvalues of symmetric problems. Numerical experiments with a model problem that are presented demonstrate the effectiveness of the method proposed. Using an FMG algorithm a solution to the level of discretization errors is obtained in just a few work units (less than 10), where a work unit is the work involved in one Jacobi relaxation on the finest level.

Multigrid method for the equilibrium equations of elasticity using a compact scheme

A compact difference scheme derived in (Phillips and Rose, SIAM J. Statist. Comput. 7, 288 (1986)) for treating the equilibrium equations of elasticity is studied. The scheme truns out to be inconsistent and unstable. A multigrid method which takes into account these properties is described. The solution of the discrete equations, up to the level of discretization errors, is obtained by this method in just 16 work units, where a work unit is the work involved in relaxing the finest grid equations once.

Numerical solution of a problem of pressurized filtration under a hydraulic engineering structure

The general problem of fluid filtration under a hydraulic engineering structure is examined. A computing algorithm based on the method of fictitious regions and the iterative alternate triangular method of solving grid equations is proposed, and examples of calculations are presented.

Fictitious component method for solving grid equations used to approximate higher-order elliptic equations with natural boundary conditions

Article Fictitious component method for solving grid equations used to approximate higher-order elliptic equations with natural boundary conditions was published on January 1, 1986 in the journal Russian Journal of Numerical Analysis and Mathematical Modelling (volume 1, issue 1).

An analysis of the modified exclusion method in certain problems

A quantitative estimate is given of the error of a modified exclusion method with an increased stability to rounding errors when solving symmetric grid equations generated by a second-order scalar elliptic operator.

The problem of constructing variational-difference schemes for elliptic equations

In [l] variational difference schemes were constructed and the question of the rate of convergence of these schemes was studied, for boundary value problems for elliptic equations of order 2m with natural boundary conditions. However, for domains of arbitrary form the problem of the method of solving the system of grid equations obtained remains open, since the matrix of the system may be arbitrarily ill conditioned (it is easy to be convinced of this by the example of a onedimensional boundary value problem on the segment [a, bl, if the grid is so arranged that the size of the intersection of the segment [XC,, ri] with [a, b] is sufficiently small in comparison with h = xi zi-+; here x,, x1, . . . are the nodes of the grid).

Grid equations of the axial-symmetric problem of the elasticity theory in curvilinear coordinates applied to turbine elements report no. 1

Grid equations of the axial-symmetric problem of the elasticity theory in curvilinear coordinates applied to turbine elements report no. 1