Elliptic Grid Research Articles

Elliptic Grid Generation Based on Laplace Equations and Algebraic Transformations

An elliptic grid generation method is presented to generate boundary conforming grids in domains in 2D and 3D physical space and on minimal surfaces and parametrized surfaces in 3D physical space. The elliptic grid generation method is based on the use of a composite mapping. This composite mapping consists of a nonlinear transfinite algebraic transformation and an elliptic transformation. The elliptic transformation is based on the Laplace equations for domains, or on the Laplace-Beltrami equations for surfaces. The algebraic transformation maps the computational space one-to-one onto a parameter space. The elliptic transformation maps the parameter space one-to-one onto the domains or surfaces. The composition of these two mappings is a differentiable one-to-one mapping from computational space onto the domains or surfaces and has a nonvanishing Jacobian. This composite mapping defines the grid point distribution in the interior of the domains or surfaces. For domains and minimal surfaces, the composite mapping obeys a nonlinear elliptic Poisson system with control functions completely defined by the algebraic transformation. The solution of the Poisson systems is obtained by Picard iteration and black-box multigrid solvers. For parametrized curved surfaces, it is not necessary to define and solve a nonlinear elliptic Poisson system. Instead a linear elliptic system and an inversion problem is solved to generate the grid in the interior of the surface.

Transient Three-Dimensional Fire-Induced Air Flow in a Full Scale Ventilated Tunnel

Abstract A modified three-dimensional version of the turbulence k—∊ model is applied to simulate the transient fire-induced air flow and smoke movement in a full scale ventilated tunnel. The mode! adopts the general curvilinear coordinate system to preserve the actual geometry of the tunnel. A modified elliptic grid generation method is introduced to specify all of the boundaries in terms of grid distribution, grid size and angle of intersection. The effects of the buoyancy and mean streamline curvature on turbulence are accounted for in the model. The smoke concentration is expressed in terms of smoke obscuration and is calculated by a conservation equation. The QUICK discretization scheme is used. The predictions are shown to be in reasonable agreement with reported experiment data. Application of the mixed convection parameter Gr∗/Re5/2 H has been extended and this is confirmed by the observed temperature stratification of the heated air flow in the full scale tunnel.

An approximate factorization scheme for elliptic grid generation with control functions

AbstractAn Alternating Direction Implicit (ADI), Approximate Factorization (AF) scheme is presented here for the solution of the two‐dimensional elliptic partial differential equations, with control functions as source terms, used for grid generation. This scheme requires significantly less computational effort than a Successive Over Relaxation (SOR) scheme. The dependence of the choice of the acceleration parameter on the rate of convergence of the AF scheme has been studied. As an example, grids generated by this method are shown, along with a comparison of the convergence history for the present AF and SOR schemes. © 1994 John Wiley & Sons, Inc.

Surface grid generation based on elliptic PDE models

Elliptic grid generation methods for the construction of surface grids are discussed. An improved elliptic surface grid generation method for parametric surfaces will be presented. Two methods of imposing grid orthogonality have been implemented. Several control function options are available for maintaining a desired distribution of grid points on the surface. The overall effect of these improvements is a reliable and versatile elliptic method for generating and improving surface grids. Examples will be presented to demonstrate the application of the method in generating practical grids.

Laminar flow and heat tranfer from multiple impinging slot jets with an inclined confinement surface

Results of numerical simulation of two-dimensional flow field and heat transfer due to laminar heated multiple slot jets discharging normally into a converging confined channel are presented. A control volume-based finite difference method was employed to solve the governing mass, momentum and energy equations. To handle the complexity of the geometry a program was developed using a non-orthogonal body-fitted coordinate system. A fully staggered grid system was generated using the solution to a set of elliptic grid generation equations. The parameters studied were: the jet Reynolds number (600 < Re < 1000), and the angle of inclination of the upper surface (0° < θ < 20°). Inclination of the confined surface so as to accelerate the exhaust flow was found to level the Nusselt number distribution on the impingement surface.

A boundary-fitted grid model for tidal motions: Orthogonal coordinates generation in 2-D embodying Singapore coastal waters

This paper proposes a computational procedure for the generation of an orthogonal coordinate system based on the existing curvilinear coordinates generation techniques. The paper is divided into three sections. Firstly a brief review of the curvilinear coordinates generation by the elliptic grid generation technique along with the curvature constraint that governs the metric of a coordinate system is presented. A procedure for the generation of an orthogonal coordinate system is also outlined. Secondly numerical solution for the grid generation system suggested in the present grid generation procedure is described. The proposed grid generation procedure is adopted for the generation of an orthogonal coordinate system for the 2-D domain of Singapore coastal waters. The generated grid demonstrates very satisfactory results.

Hybrid-CVFE Method for Flexible-Grid Reservoir Simulation

Summary The flexible control-volume finite-element (CVFE) method is used to construct hybrid grids. The method involves use of a local cylindrical or elliptical grid to represent near-well flow accurately while honoring complex reservoir boundaries. The grid transition is smooth without any special discretization approximation, which eliminates the grid transition problem experienced with Cartesian local grid refinement and hybrid Cartesian gridding techniques.

Elliptic grid generation system: control functions revisited—I

A methodology for the development of control functions associated with the elliptic grid generation system is presented. This development is based on the optimal blending of the best characteristics of hyperbolic generation system and algebraic techniques. The algorithm is readily applicable to grid adaptation. Computational examples are presented to demonstrate the success of this methodology.

Nonstaggered Calculation of Laminar and Turbulent Flows Using Curvilinear Nonorthogonal Coordinates

A solution algorithm for the Navier-Stokes equations is tested by calculating laminar and turbulent flows over backward facing steps and laminar flow over obstacles. The solution algorithm is based on the finite volume method, curvilinear nonorthogonal coordinates, and a nonstaggered grid arrangement, and the connective fluxes are described by the power law scheme or the second-order upwind scheme. The Rhie and Chow interpolation method is selected for calculating the convective velocities on the cell faces to get good coupling between the velocity and pressure fields. When the flow is turbulent, the k-e turbulence model with wall Junctions is employed. A highly deformed grid generated with an elliptic grid generator is used around the corners of the backward facing steps and obstacles; this represents a severe test of the algorithm. The results are compared with measurements and other calculations.

A Tensor Product B-Spline Method for Numerical Grid Generation

We present a numerical grid generation method in which the Cartesian coordinate functions are expanded in tensor product B-spline basis functions and collocation is used to solve the elliptic grid generation equations. The efficiency of the method derives from the fact that the smoothness of the basis functions is exploited to compute fine grids in the physical domain over a coarse set of knots in the computational domain. We formulate the tensor product B-spline method, investigate its computational complexity and compare its performance to the finite difference method for several 2D grids. We show that for comparable grids the computational cost of the tensor product B-spline method is less than the cost of the finite difference method.

Study of supersonic intersection flowfield at modified wing-body junctions

The problem of supersonic flow control using fillets and sweep for a wing-body junction has been investigated numerically using a three-dimensional Navier-Stokes code, which employs the MacCormack's time-split finite volume technique. An elliptic grid generation technique with direct control over spacing has been developed for constructing the grid at a filleted wing-body junction. The computed results for pressure distribution, particle paths, and limiting streamlines on the flat plate and fin surface for a swept fin show a decrease in the peak pressure on the fin leading edge and in the extent of the separated flow region. Moreover, the results for filleted juncture clearly show that the flow streamline patterns lose much of their vortical character with proper filleting. It has been demonstrated that fillets with a radius of three-and-one-half times the fin leading-edge diameter are required to weaken the vorticity in the horseshoe vortex by a factor of three for the Mach number and Reynolds number considered in the present study.

Structured background grids for generation of unstructured grids by advancing-front method

A new method of background grid construction is introduced for generation of unstructured grids using the advancing-front technique. Unlike the conventional triangular/tetrahedral background grids that are difficult to construct and usually inadequate in performance, the new method exploits the simplicity of uniform Cartesian meshes and the superiority of the elliptic grid point distribution for unstructured grid generation. The approach is analogous to solving a steady-state heat conduction problem with discrete heat sources. The spacing parameters of grid points are distributed over the nodes of a Cartesian background grid by solving a Poisson equation. To increase the control over the grid point distribution, a directional clustering approach is developed. The new method is convenient to use and provides better grid quality and flexibility. Some sample two-dimensional results are presented to demonstrate the power of the method.

Numerical Mapping of Arbitrary Domains Using Spectral Methods

In order to maintain spectral accuracy, the grids on which a physical problem is to be solved must also be obtained by spectrally accurate techniques. The purpose of this paper is to describe a method of solving the quasilinear elliptic grid generation equations by spectral techniques both in Euclidean (E2) and Riemannian (R2) spaces. A parametric continuation method is used to generate grids in completely arbitrary domains.

Analysis of fluid flow in constricted tubes and ducts using body‐fitted non‐staggered grids

AbstractA finite volume method for the calculation of laminar and turbulent fluid flows inside constricted tubes and ducts is described. The selected finite volume method is based on curvilinear non‐orthogonal co‐ordinates (body‐fitted co‐ordinates) and a non‐staggered grid arrangement. The grids are either generated by transfinite interpolation or an elliptic grid generator. The method is employed for calculation of laminar flows through a tube, a converging‐diverging duct and different constricted tubes by both a two‐ and a three‐dimensional computer program. In addition, turbulent flow through an axisymmetric constricted tube is calculated. Both the power law scheme and the second‐order upwind scheme are used. The calculated results are compared with the experimental data and with other numerical solutions.

Enhancements of a three-dimensional hyperbolic grid generation scheme

A hyperbolic grid generation scheme formulated from grid orthogonality and cell volume specification has been significantly enhanced so that high quality three-dimensional grids can be obtained for a wide variety of geometries. While the speed of the scheme remains one to two orders of magnitude faster than typical elliptic grid generation methods, the robustness of the scheme has been greatly improved over previous applications of the three-dimensional hyperbolic grid generation procedure. Enhancements included the use of spatially-variable smoothing coefficient, metric correction procedures, local treatment of severe convex corners, and new extrapolation treatments of floating and axis boundaries. The versatility of the new hyperbolic grid generation scheme is demonstrated by three-dimensional grids generated for external components of the integrated Space Shuttle vehicle and the SOFIA telescope.

Grid generation for internal flow configurations

This paper is concerned with analysis and applications of geometry-grid generation methodologies for internal flow configurations. Weighted transfinite interpolation technique is combined with cubic splines and Bezier curves to accomplish well-distributed zonal grids. The concept of surface distribution mesh and volume distribution mesh is presented and its natural application to adaptive grids is developed. Grid refinement algorithms are developed using applications of elliptic grid systems. Various computational examples of practical interest are presented.

TWO MODIFIED VERSIONS OF HSU-LEE'S ELLIPTIC SOLVER OF GRID GENERATION

Two modifications of the Hsu and Lee elliptic grid generation method (which is, in turn, a modified Sorenson method) are developed. The first modification specifies the control functions along all of the boundaries in terms of grid distribution, grid size, and angle of intersection. The interior control functions are interpolated by transfinite interpolation. The second modification successfully employs one of the control functions in order to specify the grid size distribution line by line between two opposite boundaries, the other control function being retained for orthogonality control. By linking the arc length specification with the weighting function of Dwyer et al., the second method is extended to become an adaptive grid scheme.

A robust elliptic grid generator

A variational principle is proposed that results in a robust elliptic grid generator having many of the strengths of original Winslow or homogeneous Thompson-Thames-Mastin method (hTTM). The new grid generator places grid lines more uniformly over the domain than does hTTM, without loss of orthogonality. Numerically generated examples are given to demonstrate these effects. Grid quality measures are introduced to quantify differences between discrete grids. Both the hTTM and the new grid generator can generate folded grids on certain pathological regions, but overall they are very robust. Grid weighting for solution-adaptive calculations is briefly considered. Generalization of the new method to surface and volume grid generation is straightforward.

Application of Bubnov-Galerkin formulation to orthogonal grid generation

A Bubnov-Galerkin formulation is used to solve an elliptic grid generation system by using linear and quadratic isoparametric elements. Good orthogonality characteristics are obtained for symmetric and non-symmetric physical domains using both complete boundary correspondence or a combination of Dirichlet and Neuman boundary conditions. The method exhibits excellent stability and requires a low number of iterations to attain convergence. Results are compared with those presented in (E. D. Chikliwala and Y. C. Yortsos, J. Comput. Phys., 57, 391, 1985).

Grid control for non‐orthogonal elliptic grids

AbstractA relation is given between grid properties such as grid angle and aspect ratio and the metric coefficients of the transformation from the physical to the computational domain. A transformation is proposed in which these grid properties occur explicitly, such that the resulting grid will have the required grid angle and aspect ratio. This leads to an elliptic grid‐generation procedure with non‐linear coefficients. Some examples are given which illustrate that the properties can be predicted.