sci.math.num-analysis SUNA08, Discretization at Triangles =================================== It was shown in the sci.math.research article "SUNA, 2-D Skew Upwind F.E.D." how the equation for pure steady uniform convection in two dimensions can be discretized for arbitrary curvilinear coordinates, applying a transformation to the velocity components of the corresponding standard F.D. upwind scheme. The analytical formulation reads as follows: u.dT/dx + v.dT/dy = 0 ("d" = partial derivatives) Where: (x,y) = global coordinates, (u,v) = velocity field, T = temperature. At hand of the above equation, we want to do essentially the same thing, namely discretize it for arbitrary curvilinear coordinates. But we will do it now in a different manner. As a first step, the terms dT/d(x,y) will be handled. In order to accomplish their discretization, transformation to isoparametric local coordinates (h,k) at a Finite Element, being a general F.E. technique, seems to be advantageous. Derivatives to the global coordinates must then be expressed in derivatives to the local coordinates. Which is done as follows: dT/dh = dx/dh.dT/dx + dy/dh.dT/dy dT/dk = dx/dk.dT/dx + dy/dk.dT/dy Solving for dT/d(x,y) by Cramer's rule, we find the general formulas: ^^^^^^^ dT/dx = (dy/dk.dT/dh - dy/dh.dT/dk)/J dT/dy = (dx/dh.dT/dk - dx/dk.dT/dh)/J where: J = dx/dh.dy/dk - dx/dk.dy/dh Let's see how such Finite Element discretization works out for the simplest element in 2-D, a linear triangle: 3 o x = x1 + (x2 - x1).h + (x3 - x1).k | \ y = y1 + (y2 - y1).h + (y3 - y1).k | \ T = T1 + (T2 - T1).h + (T3 - T1).k | \ 1 o-------o 2 By differentiation and substitution: J.dT/dx = (y3 - y1).(T2 - T1) - (y2 - y1).(T3 - T1) J.dT/dy = (x2 - x1).(T3 - T1) - (x3 - x1).(T2 - T1) Where: J = (x2 - x1).(y3 - y1) - (x3 - x1).(y2 - y1) Consequently: J.[ u.dT/dx + v.dT/dy ] = [ + (y3 - y1).u - (x3 - x1).v ]/J . (T2 - T1) + [ - (y2 - y1).u + (x2 - x1).v ]/J . (T3 - T1) = 0 This is exactly the same expression as (5) in the abovementioned SUNA article, apart from renumbering issues. Well, ... not entirely: information about (F.D.) upwinding is _no_ longer available in the F.E. formulation. To be continued ... - * Han de Bruijn; Applications&Graphics | "A little bit of Physics * No * TUD Computing Centre; P.O. Box 354 | would be NO idleness in * Oil * 2600 AJ Delft; The Netherlands. | Mathematics" (HdB). * for * E-mail: Han.deBruijn@RC.TUDelft.NL --| Fax: +31 15 78 37 87 ----* Blood