Thompson, Warsi, Mastin - Numerical Grid Generation, страница 3

Rather these sides are to be considered re-entrant oneach other with points adjacent to one, outside the square, being equivalent to points adjacentto the other, inside the square.Conceptually, the physical region can be considered to have been opened at the cut= 0 and 2 and then deformed into a rectangle to form the transformed region:Here, point correspondence across the re-entrant boundaries (indicated by the dashedconnecting line) in the transformed region is illustrated by the coincidence of the pair ofcircled points. This conceptual device and mode of illustration for the the pointcorrespondence across re-entrant boundaries will serve later for more general configurations.These simple concepts extend to more complicated two-dimensional configurations,the central feature being that one of the curvilinear coordinates is made to be constant on aboundary curve (as was r above), while the other varies monotonically along that boundarycurve (as does ).

The transformation to the rectangle is achieved by making the range anddirection of variation of the varying coordinate the same on each of two opposing boundaries(as varies from 0 to 2 on each circle above).The physical space thus transforms to the rectangle shown above regardless of the shape ofthe physical region.

(It is not necessary to normalize the curvilinear coordinates to theinterval [0,1], and in fact, any normalization can be used. In computational applications thenormalization is more conveniently done to different intervals for each coordinate. The fieldin the transformed space is then rectangular, rather than square.) Familiar examples of thisare elliptical coordinates for the region between two confocal ellipses, spherical coordinatesfor two spheres, parabolic coordinates for two parabolas, etc.These, same concepts will be extended later to completely general configurationsinvolving any number of boundary curves and branch outs.

The extension to threedimensions follows directly, using boundary surfaces instead of curves, i.e., one curvilinearcoordinate will be made constant on a boundary surface, with the other two forming atwo-dimensional coordinate system on the surface.Returning to the concentric circles, if the functional dependence of on , and/or that of ron , had been made more general than the simple linear normalizations given by Eq. (4),the corresponding coordinate lines would have become unequally spaced in the physicalspace, while remaining as radial lines and concentric circles:The transformation, from Eq.

(1), is now given by(6a)(6b)In this case the points on the inner and outer circular boundaries are not equally spacedaround the circles in the physical space for equal increments of , although they remainequally spaced on the top and bottom of the unit square in the transformed space byconstruction. The spacing around these circles is determined by the functional dependence ofon , and, since the points are located at equal increments of by construction, thisfunctional relationship is defined by the placement of these points around the circles. Thispoint, that the coordinate system in the field is determined from the boundary pointdistribution, will be central to the discussion of grid generation to follow.

The distribution ofcircumferential lines is controlled here by the functional relationship between r and , whichis not related to any boundary point distribution. Thus factors other than the boundary pointdistribution may be expected to be involved in grid generation, as well. That the pointdistribution on the boundaries may be controlled by direct placement of the points, while thecoordinate line distribution in the field must be controlled by other means will also continueto appear in the developments to follow.The one-dimensional functional relationship between and in Eq. (6) requires thatthe relative distributions of boundary points around the inner and outer circles be the same.This restriction can be removed by making a function of , as well as of , whileretaining the periodic nature of the dependence on . In this case the ooordinate lines ofconstant will no longer be straight radial lines, although they will continue to connectcorresponding points on the inner and outer circular boundaries.

Similarly thecircumferential coordinate lines (lines of constant here) can be made to depart from circlesby making r dependent on both and , but with the restriction that the dependencevanishes on the inner and outer circular boundaries (wherehere).= 0 and= 1, respectively,Obviously certain constraints will have to be placed on the functions ( , ) and r( , ) tokeep the mapping one-to-one. All of these considerations will reappear in the generaldevelopments that follow.Finally, it should be realized that the intermediate use here of the cylindricalcoordinates (r, ) in defining the transformation between the curvilinear coordinates ( , )and the cartesian coordinates (x,y) has been only in deference to the familiarity of thecylindrical coordinates, and such intermediary coordinates will not appear in general. Thegeneralized statement for the simple configuration under consideration here is as follows:Find (x,y) and (x,y) in the annular region bounded by the curves x2 + y2 = and x2 + y2= , subject to the boundary conditionsSpecified monotonic variation of over [0,1] on x2 + y2 =with same sense of direction on each of these two curves.and on x2 + y2 =It is the inverse problem that will be treated in fact, however, i.e., find x( , ) andy( , ) on the unit square in the transformed space (01, 01), subject to theboundary conditionsx( ,0) and y( ,0) specified on = 0 such that x2( = 0) + y2 ,0) =x( ,1) and y( ,1) specified on = 1 such that x2( = 1) + y2( ,1) =Periodicity in : x(1 + , ) = x( , ) y(1 + , ) = y( , )The simple form for the transformation given by Eq.

(6) is made possible by choosing thesame functional dependence of x and y on on the boundaries, = 0 and = 1. Thefamiliar cylindrical coordinate system is thus a special case of the general grid generationproblem for this simple configuration applicable to the region between two concentriccircles, as is the elliptical coordinate system for two ellipses, etc.2. GeneralizationGeneralizing from the above consideration of cylindrical coordinates, the basic idea ofa boundary-conforming curvilinear coordinate system is to have some coordinate line (in 2D,surface in 3D) coincident with each boundary segment, analogous to the way in which linesof constant radial coordinate coincide with circles in the cylindrical coordinate system.

Theother curvilinear coordinate, analogous to the angular coordinate in the cylindrical system,will vary along the boundary segment and clearly must do so monotonically, else the samepair of values of the curvilinear coordinates will occur at two different physical points. (Itshould be clear that the curvilinear coordinate that varies along a boundary segment musthave the same direction and range of variation over some opposing segment, e.g., as theangular variable varies from 0 to 2 over both of two concentric circles in cylindricalcoordinates).With the values of the curvilinear coordinates thus specified on the boundary, it thenremains to generate values of these coordinates in the field from these boundary values.There must, or course, be a unique correspondence between the cartesian (or other basissystem) and the curvilinear coordinates, i.e., the mapping of the physical region onto thetransformed region must be one-to-one, so that every point in the physical field correspondsto one, and only one, point in the transformed field, and vice versa.

Coordinate lines of thesame family must not cross, and lines of different families must not cross more than once.In this chapter a two-dimensional region will be considered in most of the discussionsin the interest of economy of presentation. Generalization to three dimensions will be evidentin most cases and will be mentioned specifically only when necessary. As noted above, thecurvilinear coordinates may be normalized to any intervals, just as the radial and angularcoordinates of the cylindrical coordinate system can be expressed in many different units.Since the interest of the present discussion is numerical application, it will be generallyconvenient to define the increments of all the curvilinear coordinates to be uniformly unity,and then to normalize these coordinates to the interval [1,N(i)], where N(i) is the total numberof grid points to be used in the i direction.

(The three curvilinear coordinates will beindicated as i, i = 1,2,3, in general. In two dimensions, however, the notation ( , ) willoften be used for the two coordinates 1 and 2 .) The computational field, i.e., the field inthe transformed space, thus will have rectangular boundaries and will be covered by a squaregrid. (It will become clear later that the actual values of the increments in the curvilinearcoordinates are immaterial since they do not appear in the final numer1cal expressions.Therefore no generality is lost in making the grid square and of unit increment in thetransformed field.)A.

Boundary-value Problem -- Physical RegionThe generation of the curvilinear coordinate system may be treated as follows: withthe curvilinear coordinates specified on the boundaries, e.g., (x,y) and (x,y) on aboundary curve (this specification amounting to a constant value for either or on eachsegment of , with a specified monotonic variation of the other over the segment), generatethe values, (x,y) and (x,y), in the field bounded by . This is thus a boundary valueproblem on the physical field with the curvilinear coordinates ( , ) as the dependentvariables and the cartesian coordinates (x,y) as the independent variables, with boundaryconditions specified on curved boundaries:(In these discussions, the transformation is assumed to be from cartesian coordinates in thephysical space.