Thompson, Warsi, Mastin - Numerical Grid Generation (523190), страница 21

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Section 1A).With generation systems of the Poisson type, negative values of the control function Pi> in Eq. (4), or Pi in Eq. (8) (since gii > 0), will cause the i coordinate lines to concentratein the direction of decreasing i (cf. Section 1B). Several approaches to the determination ofthese control functions are discussed below.A. Attraction to coordinate lines/pointsThis effect can be utilized to achieve attraction of coordinate lines to other coordinatelines and/or points by taking the form of the control functions to be, in 2D, (again with 1=, 2= , P1=P, P2=Q)(30)and an analogous form for Q( , ) with and interchanged. (Here the subscripts identifyparticular -lines and are not to be confused with the superscripts used to refer to thecurvilinear coordinates in general.) In this form, the control functions are functions only ofthe curvilinear coordinates.In the P function, the effect of the amplitude ai is to attract -lines toward the i-line:while the effect of the amplitude bi is to attract-lines toward the single point ( i, i):Note that this attraction to a point is actually attraction of -lines to a point on another-line, and as such acts normal to the -line through the point.

There is no attraction of-lines to this point via the P function. In each case the effect of the attraction decays withdistance in - space from the attraction site according to the decay factors, ci and di. Thisdecay depends only on the distance from the i-line in the first term, so that entire -linesare attracted to the entire i line. In the second term, however, the decay depends on boththe and distances from the attraction point ( i, i), so that the effect is limited toportions of the -lines. With the inclusion of the sign changing function, the attractionoccurs on both sides of the -line, or the ( i, i) point, as the case may be. Without thisfunction, attraction occurs only on the side toward increasing , with repulsion occuring onthe other side. A negative amplitude simply reverses all of these effects, i.e., attractionbecomes repulsion, and vice versa.

The effect of the Q function on -lines followsanalogously.In the case of a boundary that is an -line, positive amplitudes in the Q function willcause -lines off the boundary to move closer to the boundary, assuming that increases offthe boundary. The effect of the P function will be to alter the angle at which the -linesintersect the boundary, if the points on the boundary are fixed, with the -lines tending tolean in the direction of decreasing .

These effects have been noted in figures above, andfurther examples are given below:The first two figures here show the result of attraction to the two circled points, incomparison with the case with no control function. The last figure illustrates strong attractionto the coordinate line coincident with the inner boundary and the branch cut in this C-typesystem. If the boundary is such that decreases off the boundary, then the amplitudes in theQ function must be negative to achieve attraction to the boundary. In any case, theamplitudes ai cause the effects to occur all along the boundary (as in the last figure above),while the effects of the amplitudes bi occur only near selected points on the boundary(second figure above).In configurations involving branch cuts, the attraction lines and/or points in this typeof evaluation of the control function strictly should be considered to exist on all sheets.

Inthe 0-type configuration shown on p. 29, where the two sides of the cut are on opposite sidesof the transformed region, the control function P for attraction to the i-line must beconstructed as follows: In the figure below,when the attraction line is the i=2 line, the =I-1 line experiences a counterclockwiseattraction to this line at a distance of (I-1)-2. However, the i=2 attraction line also appearsas a I+( i-1)=I+(2-1)=I+1 attraction line on the next sheet as the cut is crossed.

Therefore,the =I-1 line also experiences a clockwise attraction to this I+1 line at a distance of(I+1)-(I-1)=2, and this attraction is, of course, stronger than the first mentioned. In fact, sincethe attraction line is repeated on all sheets there strictly must be a summation over all sheetsin Eq. (30), i.e., a summation over k, with i replaced by i+kwhereis the jump inat the cut (=I-1 in the above figure).

Thus i in Eq. (30) would be replaced by the, and the rightside would be summed from k=- to += . However, because of thei+kexponential decay, the terms decrease rapidly as k increases, so that only the term with thesmallest distance in the k summation really needs to be included, i.e., only the term givingclockwise attraction at a distance of 2 from the attraction line for the =I-1 line in the abovefigure.

Since there is no jump in across the cut in this configuration, the evaluation of Q isaffected by this out only through the replacement of i as above in the term for the pointattraction, with summation over k of only this part of the right side. Again only the term withthe smallest distance need actually be included.For the C-type configuration on p. 30, with the two sides of the out on the same side ofthe transformed region, is reflected in the cut, and the construction of the control functionQ is as follows. With reference to the figure below,the attraction line, i=2, is located on both sides of the cut in this configuration.

Now the=3 line above the cut experiences a downward attraction toward the i=2 attraction line ata distance of 3-2=1. Strictly speaking, this line above the cut should also experience adownward attraction toward the portion of the i=2 attraction line below the cut as it appearson the next sheet (and, in fact, on all other sheets), i.e., at cut-( i- cut), where cut is thevalue of on the cut ( cut=1 here). This attraction line on the next sheet is at a distance -[cut-( i- cut)] from the -line of interest, i.e., at 3-[1-(2-1)]=3 from the =3 line above thecut. This attraction line on the next sheet is therefore farther away and hence its effect canperhaps be neglected.

However, for lines between the attraction line and the cut, the effect ofthe attraction line on the next sheet should be considered. In any case it is necessary to takeinto account the attraction lines appearing on the next sheet, those on all other sheets beingtoo far away to be of consequence. Here the evaluation of the control function P is affectedby the cut only through the point attraction part, with i replaced as above.The third type of cut, illustrated on p. 40, for which the two sides of the cut face acrossin both the controla void of the transformed region, is treated by replacing with ifunctions, where-1 is the number of -lines in the void.

There is no additionalsummation in this case.The case on p.52, where the coordinate species changes sign at the cut, requiresindividual attention at each cut. For example, the contribution to the control functions inregion A at a point ( , ) from an attraction site ( i, i) in region B would be evaluatedusing distances of ( - max)+( max- i) and ( - i) in place of - i and - i, respectively.B. Attraction to lines/points in spaceIf the attraction line and/or points are in the field, rather than on a boundary, then theabove attraction is not to a fixed line or point in space, since the attraction line or points arethemselves determined by the solution of the generation system and hence are free to move.It is, of course, also possible to take the control functions to be funtions of x and y instead ofand , and thus achieve attraction to fixed lines and/or points in the physical field.

Thiscase becomes somewhat more complicated, since it must be ensured that coordinate lines arenot attracted parallel to themselves.With the attraction discussed in the previous section, -lines are attracted to other-lines, and -lines are attracted to other -lines. It is unreasonable, of course, to attempt toattract -lines to -lines, since that would have the effect of collapsing the coordinatesystem. When, however, the attraction is to be to certain fixed lines in the physical region,defined by curves y=f(x), care must be exercised to avoid attempting to attract coordinatelines to specified curves that cut the coordinate lines at large angles. Thus, in the figurebelow,it is unreasonable to attract g-lines to the curve y f(x), while it is natural to attract the q-linesto this curve.However, in the general situation, the specified line y=f(x) will not necessarily bealigned with either a or line along its entire length.

Since it is unreasonable to attract aline tangentially to itself, some provision is necessary to decrease the attraction to zero as theangle between the coordinate line and the given line y=f(x) approaches 90°. This can beaccomplished by multiplying the attraction function by the cosine of the angle between thecoordinate line and the line y=f(x). It is also necessary to change the sign on the attractionfunction on either side of the line y=f(x).

This can be done by multiplying by the sine of theangle between the line y=f(x) and the vector to the point on the coordinate line.These two purposes can be accomplished as follows. Let a general point on the -linebe located by the vector (x,y), and let the attraction line y=f(x) be specified by thecollection of points S(xi,yi) i=1,2,...N. Let the unit tangent to the attraction line be (xi,yi),and the unit tangent to a -line be ( ).

Then, with the unit vector normal to thetwo-dimensional plane, and with reference to the following figure,the control functions, P(x,y) and Q(x,y), may logically be taken as(31)The equation for Q simply hasreplaced by( ) orin the above. These functions depend on x( ),and thus must be recalculated at each point as theand y through both anditerative solution proceeds. This form of coordinate control will therefore be more expensiveto implement than that based on attraction to other coordinate lines.There is no real distinction between "line" and "point" attraction with this type ofattraction.

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