Thompson, Warsi, Mastin - Numerical Grid Generation, страница 42

Here, by ’space’ we mean a region in which arbitrary independent coordinates can beintroduced; the number of independent coordinates determines the dimensionsion of thesapce. A space is termed Eulclidean when rectangular cartesian coordinates can beintroduced in it on a global scale. Examples are 2D or 3D regions in a plane or in arectangular box, respectively. It must, however, be pointed out that in an Euclidean space,besides rectangular cartesian coordinates, any general coordinate system can be introducedwithout disturbing the basic nature of the space itself.

Since this book is mainly concernedwith the general coordinate systems in either 2D or 3D Euclidean spaces, or to 2D surfacesembedded in a 3D space, we shall restrict our attention to the Christoffel symbols for spaceand for surfaces only.A. Space Christoffel symbolsFrom the definition of the base vectorstwo indices i and k,i,we first note the following result.

For anyThus(23)We now select any three indices. say i,j,k, and consider the following three equations,Adding the second and third equations, and subtracting the first equation, while using Eq.(23), we get(24)where(25)is called the Christoffel symbol of the first kind.Eq. (24) implies that(26)Taking the dot product on both sides of Eq. (26) byl,we obtain(27)where(28)is called the Christoffel symbol of the second kind.Eq.(27) implies that(29)It must be noted that both kinds of Christofiel symbols are symmetric in the first two indices,viz.,It is also easy to show, based on the definition ofthat(30)The Christoffel symbolscan be computed by using the following expanded formulae:(31)where the indices l,i,j range from 1 to 3 in 3D, or from 1 to 2 in 2D.B. Christoffel symbols in a surfaceThe Christoffel symbols, (25) and (28) are applicable both to 2D and 3D Euclideanspaces.

In fact, if we take (25) and (28) as the definitions of some 3-index symbols withoutany consideration of an Euclidean space, then they are also applicable to an n-dimensionalnon-Euclidean space.The Christoffel symbols for a 2D surface embedded in a 3D Euclidean space aredefined exactly as for any other space. Since in a surface only two independent coordinatescan be introduced, we again use the Greek indices to emphasize this point and write(32)(33)as the Christoffel symbols of the first and second kind respectively, of a surface.

Here theindices assume only two values.An important point to note here is that for a 2D space the metric coefficients gij do notdepend on one of the cartesian coordinate, say z. On the other hand for a 2D space formed bya surface in 3D Euclidean space the metric coefficients appearing in (32) and (33) depend onall three cartesian coordinates.Gauss indirectly introduced the definition of the Christoffel symobls by arguing that ina surface the base vectors,and the unit normal (Eq. (15)) form a triad ofindependent vectors. Thus any other vector in the surface can be presented as a linearcombination of,, .

Following this argument, the second derivative of the positionvectorcan be expressed as(34)which are called the formulae of Gauss. Thus, for a surfaceare the current coordinates, Eq. (34) is written as3= constant in which1,2(35)where Eq. (35) represents the second derivatives11,12,22.APPENDIX BEULER EQUATIONS1.

Variational Principle in Transformed SpaceConsider the integralwhereis the covariant metric tensor, with elements gij defined by Eq. (III-5), and w( ) isa weight function dependent on.A. Grid Generation SystemThe Euler equations then are given by(2)as has been noted. Sinceand F depends on (xi)jonly through the elements of the metric tensor,, we have(3)wherei isi,throughthe unit vector in the xi-direction. Here the operation indicated by the notation,is the simple replacement of, we havej byi inF. Also, since F depends onj onlyorTherefore,Since F depends ononly through the weight function we haveThen the Euler Equations can be written asor as the vector equation(6)(Note that the symmetric elements of the metric tensor, g jk = g kj, are to be left as distinctelements in F until after the differentiation has been performed.)Expanding thej-derivative,we then haveBut alsoso thatThus we have the grid generation system, withwritten as F’,(7)where(8)This is a quasi-linear, second-order partial differential equation for the cartesian coordinates.If the weight function depends directly on, instead of onin Eq.

(1), thenin Eq. (2). Also in his case, thethat appears on p. 439 and in thedevelopment that leads to Eq. (7) is replaced by simply w j . Then Eq. (7) is replaced by(9)for a weight function w( ) in Eq. (1).B. Two-Dimensional ExamplesIn two dimensions, the generation system (7) becomes (with1=and2= )andIf the weight function depends on , rather than on x, the terms(10) become w and w , respectively, and the last term, -- 1/2 F’ w, vanishes.As an example, consider Fw from Eq. (XI-71). Then we havein Eq.Then the generation system based on concentration by Eq. (7) is(11)With F taken to be a measure of orthogonality, i.e., Fo from Eq. (XI-70), we have,The generation system based only on orthogonality then is(12)Finally, for the smoothness integral, Eq, (XI-69), the derivatives needed areThe complete generation system is then obtained as the linear combination of theconcentration system, Eq.

(11), the orthogonality system, Eq. (12), and the smoothnesssystem which is formed by substituting the above relations into the general equations (7).The three-dimensional case follows in an analogous fashion.2. Variational Principle in Physical SpaceWith the variational problem formulated in the physical space, consider the integral(13)where is the contravariant metric tensor, i.e., with elements gij from Eq. (III-37), and theweight function is a function of .A. Grid Generation SystemThen for the Euler equations, we have(14)Now,and F depends on ( i)x only throughjAlso, since F depends oni only.

Thenthrough gik (k = 1,2,3) we haveTherefore,Also, since F depends ononly through the weight function, we haveThen the Euler equations can be writtenorNowandThenor,NowandThen the generation system is, withwritten as F’,(15)where(16)This can also be written as(17)(18)Then(19)where Cik is the signed cofactor of Aki.If the weight function in the integral (13) is a function of , rather thanin the Euler equation (14), and Eq. (15) is replaced by, then(20)In this case Si of Eq. (18) are redefined as(21)APPENDIX CCODE DEVELOPMENT AND COMPUTER EXERCISES1. Code Development Exercises1.

Take the computational region to be a rectangle on which the curvilinear coordinates aredefined to be equal to the indices of the field arrays, i.e., = I and = J, with x = X (I,J)and y = Y (I,J).Make provision for reading in values of x and y on any segments of the boundary of thecomputational region. Generate x and y in the interior by interpolating linearly between thetop and bottom boundaries. Plot the grid.2. Modify the code to allow horizontal (in the computational plane) interpolation, as well,the choice being specified by input.3.

Now add the choice of interpolation from the four corners (tensor product interpolation).4. Finally add the choice of transfinite interpolation.5. Generalize the interpolation to cubic Hermite interpolation, with the grid being orthogonalat the boundary.6. Generalize the interpolation to use the hyperbolic tangent distribution function, rather thanbeing linear, with specified relative spacing on each end.7.

Modify the code to provide for reading in x and y on any segment of any horizontal orvertical line in the computational region. Also provide for the interpolation tion to be doneon any rectangular segment of the computational region (including a segment that is only aline.)8. Add another field array TYPE (I,J) which is a flag to identify each point as one for whichthe x,y values are (1) fixed, e.g., specified points on the physical boundary, (2) out of thecomputation, e.g., points inside a slab, or (3) to be generated.

Provide for the designation (1)and (2) to be made by input for any rectangular segment of the computational region, thedefault being to the designation (3).9. Modify the dimensions of the field arrays so that an extra layer of points surrounds thecomputational region. Also add two more field arrays, ILINK (I,J) and JLINK (I,J).

Providefor any segments of any horizontal or vertical lines to be designated as image points inTYPE by input, i.e., points for which the values of x and y are set equal to those at someother point. Also provide for the indices of these other points to be put in ILINK and JLINKby input.10. Add an elliptic generator, based on Laplace equation, to the code. Use the algebraicgenerator (the interpolation) to provide the initial guess for point SOR iteration.11. Add control functions to the elliptic generator.

Let the control function be evaluated onthe boundaries and interpolated into the field by transfinite interpolation.2. Computer Exercises1. Generate an algebraic grid between two concentric circles. Use linear interpolationbetween the circles.2. Generate an algebraic grid between two ellipses, both of which are centered at the originbut which may have different eccentricities, using interpolation between the ellipses.Compare grids generated using linear and Hermite interpolation, the latter being orthogonalat the boundaries.3. Generate a C-type algebraic grid for an ellipse inside an outer boundary formed by asemicircle replacing one side of a rectangle:Compare (1) vertical interpolation in the computational region boundary, (2) horizontalinterpolation, (3) tensor product interpolation, and (4) transfinite interpolation, using linearinterpolation in each case.