Thompson, Warsi, Mastin - Numerical Grid Generation (523190), страница 19
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(14) thereby verifying Eq. (16). Certain identitieswill be useful, such as11. Use the identitycos to verify the inequality in (17):12. Use the following relations to write the truncation error in Eq. (18) in the form of Eq.(19).VI. ELLIPTIC GENERATION SYSTEMSAs noted in Chapter II, the generation of a boundary-conforming coordinate system isaccomplished by the determination of the values of the curvilinear coordinates in the interiorof a physical region from specified values (and/or slopes of the coordinate lines intersectingthe boundary) on the boundary of the region:One coordinate will be constant on each segement of the physical boundary curve (surface in3D), while the other varies monotonically along the segment (cf.
Chapter II).The equivalent problem in the transformed region is the determination of values of thephysical (cartesian or other) coordinates in the interior of the transformed region fromspecified values and/or slopes on the boundary of this region, as discussed in Chapter II:This is a more amenable problem for computation, since the boundary of the transformedregion is comprised of horizontal and vertical segments, so that this region is composed ofrectangular blocks which are contiguous, at least in the sense of being joined by re-entrantboundaries (branch cuts), as described in Chapter II.The generation of field values of a function from boundary values can be done invarious ways, e.g., by interpolation between the boundaries, etc., as is discussed in ChapterVIII.
The solution of such a boundary-value problem, however, is a classic problem ofpartial differential equations, so that it is logical to take the coordinates to be solutions of asystem of partial differential equations. If the coordinate points (and/or slopes) are specifiedon the entire closed boundary of the physical region, the equations must be elliptic, while ifthe specification is on only a portion of the boundary the equations would be parabolic orhyperbolic.
This latter case would occur, for instance, when an inner boundary of a physicalregion is specified, but a surrounding outer boundary is arbitrary. The present chapter,however, treats the general case of a completely specified boundary, which requires anelliptic partial differential system. Hyperbolic and parabolic generation systems arediscussed in Chapter VII.Some general discussion of elliptic generation systems has been given in Ref. [19],and numerous references to the application thereof appear in the surveys given by Ref. [1]and [5].1. Generation EquationsThe extremum principles, i.e., that extrema of solutions cannot occur within thefield,that are exhibited by some elliptic systems can serve to guarantee a one-to-one mappingbetween the physical and transformed regions (cf.
Ref. [20] and [21]). Thus, since thevariation of the curvilinear coordinate along a physical boundary segment must bemonotonic, and is over the same range along facing boundary segments (cf. Chapter II), itclearly follows that extrema of the curvilinear coordinates cannot be allowed in the interiorof the physical region, else overlapping of the coordinate system will occur. Note that it isthe extremum principles of the partial differential system in the physical space, i.e., with thecurvilinear coordinates as the dependent variables, that is relevant since it is the curvilinearcoordinates, not the cartesian coordinates, that must be constant or monotonic on theboundaries.
Thus it is the form of the partial differential equations in the physical space, i.e.,containing derivatives with respect to the cartesian coordinates, that is important.Another important property in regard to coordinate system generation is the inherentsmoothness that prevails in the solutions of elliptic systems. Furthermore, boundary slopediscontinuities are not propagated into the field. Finally, the smoothing tendencies of ellipticoperators, and the extremum principles, allow grids to be generated for any configurationswithout overlap of grid lines. Some examples appear below:There are thus a number of advantages to using a system of elliptic partial differentialequations as a means of coordinate system generation. A disadvantage, of course, is that asystem of partial differential equations must be solved to generate the coordinate system.The historical progress of the form of elliptic systems used for grid generation hasbeen traced in Ref.
[1]. Consequently, references to all earlier work will not be made here.Numerous examples of the generation and application of coordinate systems generated fromelliptic partial differential equations are covered in the above reference, as well as in Ref.[2].A. Laplace systemThe most simple elliptic partial differential system, and one that does exhibit anextremum principle and considerable smoothness is the Laplace system:(1)This generation system guarantees a one-to-one mapping for boundary-conformingcurvilinear coordinate systems on general closed boundaries.These equations can, in fact, be obtained from the Euler equations for theminimization of the integral(2)as is discussed further in Chapter XI.
Since the coordinate lines are located at equali| can be considered a measure ofincrements of the curvilinear coordinate, the quantity |the grid point density along the coordinate line on which i varies, i.e., i must changerapidly in physical space where grid points are clustered. Minimization of this integral thusleads to the smoothest coordinate line distribution over the field.With this generating system the coordinate lines will tend to be equally spaced in theabsence of boundary curvature because of the strong smoothing effect of the Laplacian, butwill become more closely spaced over convex boundaries, and less so over concaveboundaries, as illustrated below.
(In this and other illustrations and applications in twodimensions, 1 and 2 will be denoted and , respectively, while x and y will be used forx1 and x2.)In the left figure we have xx > 0 because of the convex (to the interior) curvature of thelines of constant ( -lines). Therefore it follows that yy < 0, and hence the spacingbetween the -lines must increase with y. The -lines thus will tend to be more closelyspaced over such a convex boundary segment. For concave segments, illustrated in the rightfigure, we have xx < 0, so that yy must be positive, and hence the spacing of the -linesmust decrease outward from this concave boundary. Some examples of grids generated fromthe Laplace system are shown below.
The inherent smoothness and the behavior nearconcave and convex boundaries are evident in these examples.B. Poisson systemControl of the coordinate line distribution in the field can be exercised by generalizingthe elliptic generating system to Poisson equations:(3)in which the "control functions" Pi can be fashioned to control the spacing and orientation ofthe coordinate lines. The extremum principles may be weakened or lost completely withsuch a system, but the existence of an extremum principle is a sufficient, but not a necessary,condition for a one-to-one mapping, so that some latitude can be taken in the form of thecontrol functions.Considering the equation 2 = Q and the figures above (P1 = P and P2 = Q in theillustrations here), since a negative value of the control function would tend to make yymore negative, it follows that negative values of Q will tend to cause the coordinate linespacing in the cases shown above to increase more rapidly outward from the boundary.Generalizing, negative values of the control function Q will cause the -lines to tend tomove in the direction of decreasing , while negative values of P in 2 = P will cause-lines to tend to move in the direction of decreasing .
These effects are illustrated belowfor an -line boundary:With the boundary values fixed, the -lines here cannot change the intersection with theboundary. The effect of the control function P in this case is to change the angle ofintersection at the boundary, causing the -lines to lean in the direction of decreasing .These effects are illustrated in the following figures:Here the -lines are radial and the -lines are circumferential.
In the left illustration thecontrol function Q is locally non-zero near a portion of the inner boundary as indicated, sothe -lines move closer to that portion of the boundary while in the right figure, P is locallynon-zero, resulting in a change in intersection angle of the -lines with that portion of theboundary. If the intersection angle, instead of the point location, on the boundary isspecified, so that the points are free to move along the boundary, then the -lines wouldmove toward lines with lower values of :In general, a negative value of the Laplacian of one of the curvilinear coordinatescauses the lines on which that coordinate is constant to move in the direction in which thatcoordinate decreases.
Positive values of the Laplacian naturally result in the opposite effect.C. Effect of boundary point distributionBecause of the strong smoothing tendencies that are inherent in the Laplacianoperator, in the absence of the control functions, i.e., with Pi = 0, the coordinate lines willtend to be generally equally spaced away from the boundaries regardless of the boundarypoint distribution. For example, the simple case of a coordinate system comprised ofhorizontal and vertical lines in a rectangular physical region, (cf. the right figure below)cannot be obtained as a solution of Eq. (3) with P=Q=0 unless the boundary points areequally spaced.Withyy= xx = 0, Eq. (3) reduces toand thus P and Q cannot vanish if the point distribution is not uniform on the horizontal andvertical boundaries, respectively.
With P=Q=0 the lines tend to be equally-spaced away fromthe boundary. These effects are illustrated further in the figures below. Here the controlfunctions are zero in the left figure.Although the spacing is not uniform on the semi-circular outer boundary in this figure, theangular spacing is essentially uniform away from the boundary.
By contrast, nonzero controlfunctions in the right figure, evaluated from the boundary point distribution, cause the fieldspacing to follow that on the boundary. Thus, if the coordinate lines in the interior of theregion are to have the same general spacing as the point distributions on the boundarieswhich these lines connect, it is necessary to evaluate the control functions to be compatiblewith the boundary point distribution.
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