[...]... as Transfinite Interpolation and the marching algorithm (a grid generator for hyperbolic partial differential equations) The programming language is the standard FORTRAN 77/90 Our objective in this book is to give an introduction to the most important aspects of grid generation Our coverage of the literature is rather selective, and by no means complete For further information and a much wider range... set of curvilinear co-ordinates {x i } with covariant base vectors gi and contravariant base vectors gi , we can define the covariant and contravariant metric tensors respectively as the scalar products gij = gi · gj (1.14) g ij = gi · gj , (1.15) where i and j can take any values from 1 to 3 From eqns (1.5), (1.10), for the background cartesian components of gi and gi , it follows that gij = ∂yk ∂yk... to time to assist the reader In addition, the companion website (www.bh.com/companions/ 0750650583) provides a series of easy -to- follow, clearly annotated numerical codes, closely associated with Chapters 4, 5, 6, and 8 The aim has been to show the application of the theory to the generation of numerical grids in fairly simple two-dimensional domains, varying from rectangles, circles and ellipses to. .. parallelepiped (Fig 1.2) with sides g1 , g2 , g3 ) The fact that g1 is perpendicular to g2 and g3 , which are tangential to the co-ordinate curves on which x 2 and x 3 , respectively, vary, implies that g1 must be perpendicular to the plane which contains these tangential directions; this is just the tangent plane to the co-ordinate surface at P on which x 1 is constant Thus gi must be normal to the co-ordinate... presented by the authors in Farrashkhalvat and Miles (1990); this book started at an elementary level, and had the restricted aim, compared with many of the more wide-ranging books on tensor calculus, of showing how to use tensor methods to transform partial differential equations of physics and engineering from one co-ordinate system to another (an aim which remains relevant in the present context) There are... the transformation, with i-j element equal to ∂x i /∂x j , has a determinant not equal to zero, so that eqn (1.58) may be inverted We define the Jacobian J of the transformation as J = det A (1.59) 9 10 Basic Structured Grid Generation Exercise 4 Show that if we define the matrix B as that whose i-j element is equal to ∂x j /∂x i , then (1.60) AB T = I and det B = J −1 (1.61) We obtain new covariant base... tensor properties of the covariant derivative and the fact that in cartesian co-ordinate systems covariant derivatives reduce to straightforward partial derivatives Since gij takes constant values in a cartesian system, the partial derivatives of these values are all zero, and these will transform to zero under tensor transformation to any other co-ordinate system It can be shown similarly that ij g,k... index’ and ‘lowering the index’, respectively It is straightforward to show that the scalar product of vectors u and v is given by u · v = ui vi = ui v i = gij ui v j = g ij ui vj (1.54) and hence that the magnitude of a vector u is given by |u| = gij ui uj = g ij ui uj (1.55) It is important to note the special transformation properties of covariant and contravariant components under a change of... LINUX Service Manager at Imperial College of Science, Technology and Medicine for help with computer administration M Farrashkhalvat J.P Miles 1 Mathematical preliminaries – vector and tensor analysis 1.1 Introduction In this chapter we review the fundamental results of vector and tensor calculus which form the basis of the mathematics of structured grid generation We do not feel it necessary to give derivations... (1.13) can be said to constitute a covariant vector, since by the usual chain rule they transform according to eqn (1.70), i.e ∂ϕ ∂x j ∂ϕ = ∂x i ∂x i ∂x j Exercise 5 Show that the transformation rule for contravariant components of a vector is ∂x i ui = j uj , (1.72) ∂x or  1   1  u u  u2  = A  u2  (1.73) u3 u3 Note the important consequence that the scalar product (1.54) is an invariant quantity . w0 h0" alt="" Basic Structured Grid Generation with an introduction to unstructured grid generation Basic Structured Grid Generation with an introduction to unstructured grid generation M. Farrashkhalvat. contain an introduction to the basic techniques (mainly in two dimensions) of structured grid generation, involving algebraic methods and dif- ferential models. Again, in an attempt to be reasonably. equations). The program- ming language is the standard FORTRAN 77/90. Our objective in this book is to give an introduction to the most important aspects of grid generation. Our coverage of the