Thompson, Warsi, Mastin - Numerical Grid Generation, страница 2

The curvilinear coordinate system covers the field and hascoordinate lines (surfaces) coincident with all boundaries. The distribution of lines should besmooth, with concentration in regions of strong solution variation, and the system shouldultimately be capable of sensing these variations and dynamically adjusting itself to resolvethem.A numerically-generated grid is understood here to be the organized set of pointsformed by the intersections of the lines of a boundary-conforming curvilinear coordinatesystem. The cardinal feature of such a system is that some coordinate line (surface in 3D) iscoincident with each segment of the boundary of the physical region. The use of coordinateline intersections to define the grid points provides an organizational structure which allowsall computation to be done on a fixed square grid when the partial differential equations ofinterest have been transformed so that the curvilinear coordinates replace the cartesiancoordinates as the independent variables.This grid frees the computational simulation from restriction to certain boundaryshapes and allows general codes to be written in which the boundary shape is specifiedsimply by input.

The boundaries may also be in motion, either as specified externally or inresponse to the developing physical solution. Similarly, the coordinate system may adjust tofollow variations developing in the evolving physical solution. In any case, thenumerically-generated grid allows all computation to be done on a fixed square grid in thecomputational field which is always rectangular by construction.In the sections which follow, various configurations for the curvilinear coordinatesystem are discussed in Chapter II.

In general, the computational field will be maderectangular, or composed of rectangular sub-regions, and a wide variety of configurations ispossible. Coordinate systems may also be generated separately for sub-regions in thephysical plane and patched together to form a complete system for complex configurations.The basic transformation relations applicable to the use of general curvilinear coordinatesystems are developed in Chapter III; the construction of numerical solutions of partialdifferential equations on those systems is discussed in Chapter IV; and consideration is givenin Chapter V to the evaluation and control of truncation error in the numericalrepresentations.Basically, the procedures for the generation of curvilinear coordinate systems are oftwo general types: (1) numerical solution of partial differential equations and (2)construction by algebraic interpolation.

In the former, the partial differential system may beelliptic (Chapter VI), parabolic or hyperbolic (Chapter VII). Included in the elliptic systemsare both the conformal (Chapter X), and the quasi-conformal mappings, the former beingorthogonal. Orthogonal systems (Chapter IX) do not have to be conformal, and may begenerated from hyperbolic systems as well as from elliptic systems. Some proceduresdesigned to produce coordinates that are nearly orthogonal are also discussed.

The algebraicprocedures, discussed in Chapter VIII, include simple normalization of boundary curves,transfinite interpolation from boundary surfaces, the use of intermediate interpolatingsurfaces, and various other related techniques.Coordinate systems that are orthogonal, or at least nearly orthogonal near theboundary, make the application of boundary conditions more straightforward. Althoughstrict orthogonality is not necessary, and conditions involving normal derivatives cancertainly be represented by difference expressions that combine one-sided differences alongthe line emerging from the boundary with central expressions along the boundary, theaccuracy deteriorates if the departure from orthogonality is too large.

It may also be moredesirable in some cases not to involve adjacent boundary points strongly in therepresentation, e.g., on extrapolation boundaries. The implementation of algebraic turbulencemodels is more reliable with near-orthogonality at the boundary, since information on localboundary normals is usually required in such models. The formulation of boundary-layerequations is also much more straightforward and unambiguous in such systems. Similarly,algorithms based on the parabolic Navier-Stokes equations require that coordinate linesapproximate the flow streamlines, and the lines normal thereto, especially near solidboundaries. It is thus better in general, other considerations being equal, for coordinate linesto be nearly normal to boundaries.Finally, dynamically-adaptive grids are discussed in Chapter XI.

These gridscontinually adapt during the course of the solution in order to follow developing gradients inthe physical solution. This topic is at the frontier of numerical grid generation and may wellprove to be one of its most important aspects.The emphasis throughout is on grids formed by the intersections of coordinate lines ofa curvilinear coordinate system, as opposed to the covering of a field with triangularelements or a random distribution of points.

Neither of these latter collections of points issuitable for really efficient numerical solutions (although numerical representations can beconstructed on each, of course) because of the cumbersome process of identification ofneighbors of a point and the lack of banded structure in the matrices. Thus the subject oftriangular mesh generators, per se, is not addressed here. (Obviously a triangular mesh canbe produced by construction rectangular mesh diagonals.)Considerable progress is being made toward the development of the techniques ofnumerical grid generation and toward casting them in forms that can be readily applied.

Acomprehensive survey of numerical grid generation procedures and applications thereofthrough 1981 was given by Thompson, Warsi, and Mastin in Ref. [1], and the conferenceproceedings published as Ref. [2] contains a number of expository papers on the area, aswell as current results.

Other collections of papers on the area have also appeared (Ref. [3]and [4]), and a later review through 1983 has been given by Thompson in Ref. [5]. Someother earlier surveys are noted in Ref. [1]. A later survey by Eiseman is given in Ref. [37].The present text is meant to be a developmental treatment of the techniques of gridgeneration and its applications, not a survey of results, and therefore no attempt is made hereto cite all related references, rather only those needed to illustrate particular points are noted.The surveys mentioned above should be consulted directly for references to examples ofvarious applications and related contributions.

(Ref [l] gives a short historical developmentof the ideas of grid generation.) Other surveys of particular areas of grid generation are citedlater as topics are introduced.Finally, in regard to implementation, a configuration for the transformed(computational) field is first established as discussed in Chapter II. The grid is generatedfrom a generation system constructed as discussed in Chapters VI -- X. (If the grid is to beadaptive, i.e., coupled with the physical solution done thereon, then the gr1d must becontinually updated as discussed in Chapter XI.) In the construction of the grid, due accountmust be taken of the truncation error induced by the grid discussed in Chapter V.

The partialdifferential equations of the physical problem of interest are transformed according to therelations given in Chapter III. These transformed equations are then discretized, cf. ChapterIV, and the resulting set of algebraic equations is solved on the fixed square grid in therectangular transformed field.II. BOUNDARY-CONFORMING COORDINATE SYSTEMS1. Basic ConceptsTo provide a familiar ground from which to view the general development to follow,consider first a two-dimensional cylindrical coordinate system covering the annular regionbetween two concentric circles:Here the curvilinear coordinates (r, ) vary on the intervals [r1,r2] and [0,2 ], respectively.These curvilinear coordinates are related to the cartesian coordinates (x,y) by thetransformation equations(1)The inverse transformation is given by(2)Note that one of the curvilinear coordinates, r, is constant on each of the phys1calboundaries, while the other coordinate, , varies monotonically over the same range aroundeach of the boundaries.

Note also that the system can be represented as a rectangle on whichthe two physical boundaries correspond to the top and bottom sides:The transformed region, i.e., where the curvilinear coordinates, r and the independentvariables, thus can be thought of as being rectangular, and can be treated as such from acoding standpoint. These points will be central to what follows.The curvilinear coordinates (r, ) can be normalized to the interval [0,1] by introducingthe new curvilinear coordinates ( , ), where(3)or(4)The transformation then may be written(5a)(5b)where now and both vary on the interval [0,1].

This is thus a mapping of the annularregion between the two circles in the physical space onto the unit square in the transformedspace, i.e., each point (x,y) on the annulus corresponds to one, and only one, point ( , ) onthe unit square:The bottom ( = 0) and top ( = 1) of the square correspond, respectively, to the inner andouter circles, r = r1, and r = r2. The sides of the square, = 0 and = 1 correspond to = 0and = 2 , respectively, and hence to the two coincident sides of a branch cut in thephysical space. Therefore, boundary conditions are not to be specified on these sides of theunit square in the transformed space.