Thompson, Warsi, Mastin - Numerical Grid Generation (523190), страница 26
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(107) or (104) is used togenerate a new surface coordinate system (,) by generating values of the parametriccoordinates (u,v) as functions of the curvilinear coordinates (, ), analogous to theplane generation systems which generate values of the cartesian coordinates as functions ofthe curvilinear coordinates. In fact, as noted above, the surface generation systemdegenerates to the plane system when the surface curvature vanishes.With (u,v) now available, as described above, the metric elements with overbars canbe calculated from the definitions in Eq. (103), using second-order central differences for allderivatives as in the plane case. The quantities 2u and 2v are then calculated in the samemanner from Eq. (106).
Also the control fucntions are evaluated from the same relationsgiven above for the plane case. All derivatives in the system (107) or (104) are representedby second-order central difference epxressions, and the resulting nonlinear differenceequations are solved as in the plane case.Exercises1. Demonstrate the validity of Eq.
(4) -- (6).2. For plane polar coordinates (r, ) defined asshow that the curvilinear coordinatesare solutions of the Laplace equations2=0,2=0.3. Show that the one-dimensional control function in Eq. (13) that is equivalent to the use ofa subsequent exponential stretching transformation by the function given by Eq. (VIII-26) isP = - /I. Hint: x/L = ( /I)4. Show that the one-dimensional control function in Eq. (13) that corresponds to ahyperbolic tangent stretching transformation by the function given by Eq. (VIII-32) iswhere u is given by Eq. (VIII-33) and5. Show that the one-dimensional control function for the generation system given by Eq.(23) corresponding to a distribution x( ) isNote that this control function will be considerably larger than that for Eq.
(12) because ofthe higher inverse power of x6. Show that a solution of Eq. (20), with P = P( ) and Q = Q( ), for a rectangular regionwithand,and, is given bywhere7. Show that a solution of Eq. (20), with P = P( ) and Q( ), for an annular region betweentwo concentric circles of radius r1 and r2 is given, withand, bywherewith F( ) and G( ) given in the preceding exercise. Show also that for P = p/q , R( ) and ( ) become=q/pand Q8. From the result of the preceding exercise show that the control function Q( ) required toproduce a specified radial distribution r( ) is given by9.
Show that the first term in the control function Q( ) given in the preceding exercise arisesfrom the first term in Eq. (45), and that the second term arises from curvature term in (45).10. Consider the generating system (23) for plane curvilinear coordinates. Let the controlfunctions P and Q be defined as follows:where k > 0 is a constant. Let it be desired to solve Eq. (23) for the generation of coordinatesin the region of a circular annulus with = i(r = 1) as the inner circle and = o(r = R) asthe outer circle.
Considering the clockwise traverse in the -direction as positive, setwherein Eq. (23), and show thatwhere11. Show that the control function P in Exercise 4 has the following values at the boundaries:Note that an iterative procedure could be set up in the manner of Section 2F in which andA are determined from P(0) and P(I) and then these are used in the P( ) of Exercise 4 todefine the control function in the field, rather than interpolating from the boundary values.12.
Show that the Beltrami operator reduces to the Laplacian for a plane surface.13. Verify Eq. (71).14. Verify Eq. (77a).15. Consider a sphere of unit radius in which it is desired to introduce a coordinate system (, ) in such a way that (i) is orthogonal, and (ii) the resulting metric coefficients g33 andg11 are equal.
(Such systems are known to be isothermic.)(a) Verify by inspection that for isothermic coordinates Eq. (80) is identically satisfied.(b) To obtain the isothermic coordinates on a sphere setand show that(c) Show that the relation between the standard longitude and latitude surface coordinatesand where 0< 2 and 0 < < , is16. Using Eq.
(15), (20) and (21) of Appendix A, show that the sum of the principalcurvatures,+, of a prolate ellipsoid defined asis17. Verify the correspondence between Eq. (90) and (91).18. Verify Eq. (92).19. Let ( , ) be the surface coordinates in the surface on whichshown in Appendix A, Eq. (21),= constant. Then asLet a new coordinate system ( , ) be introduced in the same surface such that( , ), and = ( , ) are admissible transformation functions.(a) Use the chain rule of differentiation to show that the components of the normalsurface are coordinate invariants, i.e.,(b) Also show that on coordinate transformation=to the20.
Let it be desired to obtain the 3D curvilinear coordinates in the region bounded by aprolate ellipsoid as an inner boundary ( = i) and a sphere as an outer boundary ( = o).The (x,y,z) for both the inner and outer bodies are given below in which and i are theparameters of the ellipsoid:(a) First write Eq. (91) as three equations in x, y and z for the generation of those surfaces onwhich = constant. Also set P = Q = 0 and transform the three equations mentioned abovefrom to , where = i + ( ).(b) Assume the solution isand compute all the needed derivatives to find g11, g12, g22 while keeping fixed. Alsousing Eq.
(15), (20) and (21) of Appendix A obtain the expressions for the components ofi(3) and R(3).(c) Use all the quantities obtained in (b) in the equations written in (a), and show thatwhere21. Let a surface in the xyz-space by given asShow the following:(a) The components of the unit normal vector to the surface area are(b) The element of area dA on the surface is(c) The element of length ds of a surface curve is given by(d) The sum of the principal curvatures is given by22.
(a) Show that the unit tangent vectorF(x,y,z) = 0, G(x,y,z) = 0 isto the curve of intersection of two surfaceswhereand m,n,k are in the cyclic per mutations of 1,2,3.(b) Using the formula for the normal vectorfind the Cartesian components of23. Verify Eq. (104)..to F constant, .i,e.,VII. PARABOLIC AND HYPERBOLIC GENERATION SYSTEMSIt is also possible to base a grid generation system on hyperbolic or parabolic partialdifferential equations, rather than elliptic equations.
In each of these eases the grid isgenerated by numerically solving the partial differential equations, marching in the directionof one curvilinear coordinate between two boundary curves in two dimensions, or betweentwo boundary surfaces in three dimensions. In neither case can the entire boundaries of ageneral region be specified -- only the elliptic equations allow that.The parabolic system can be applied to generate the grid between the two boundariesof a doubly-connected region with each of these boundaries specified.
The hyperbolic case,however, allows only one boundary to be specified, and is therefore of interest only for usein calculation on physically unbounded regions where the precise location of acomputational outer boundary is not important. Both parabolic and hyperbolic gridgeneration systems have the advantage of being generally faster than elliptic generationsystems, but, as just noted, are applicable only to certain configurations. Hyperbolicgeneration systems can be used to generate orthogonal grids.1.
Hyperbolic Grid GenerationIn two dimensions the condition of orthogonality is simply(1)If either the cell area,or the cell diagonal length (squared), g11 + g22, is a specifiedfunction of the curvilinear coordinates, i.e.,(2a)or(2b)then the system consisting of Eq. (1) and either (2a) or (2b), as appropriate, is hyperbolic.A hyperbolic generation system based on Eq. (1) and (2a) is constructed as follows (cf.Ref. [28-29]). Eq. (1) and (2a) become, with 1 = , 2 = , x1 = x, x2 = y,(3a)(3b)where the cell volume distribution, V( , ), is specified.
This system is hyperbolic andtherefore a non-iterative marching solution can be constructed proceeding in one coordinatedirection, say , away from a specified boundary.The equations are first locally linearized about a known solution denoted x°, y°. Thus(4)whereThen with second-order central differences for the -derivatives and first-order backwarddifferences for the derivatives we have, with = i and = j,(5)withiij=i+1,j -ij andj=ij -i,j-1 andwhere A and B, and V° in, are evaluatedat j, and the last term is an added fourth-order dissipation term for stability. Withandevaluated using central differences at j,andcan be evaluated by simulatenous solutionof Eq.
(3a) and (3b). Eq. (5) then is a 2x2 block tridiagonal equation which is solved on eachsuccessive -line, proceeding away from the specified boundary, to generate the grid.The cell volume distribution in the field is controlled by the specified function,V( , ). One form of this specification is as follows. Let points be distributed on a circlehaving a perimeter equal to that of the specified boundary at the same are length distributionas on that boundary. Then specify a radial distribution of concentric circles about this circleaccording to some distribution function, e.g., the hyperbolic tangent discussed in ChapterVIII.
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