sci.math.research SUNA07, Specializations of 3-D SUFED ==================================== This article is part of the "Series on Unified Numerical Approximations". SKEW UPWIND FINITE ELEMENT DIFFERENCES (SUFED) FOR PURE STEADY UNIFORM CONVECTION IN THREE DIMENSIONS PART II : SPECIAL CASES In order to make the meaning of the equations clear, a unit cube is subdivided into tetrahedra. These tetrahedra act like "funnels", which conduct the flow towards the vertex with index (0). The funnels must be such that the whole flow in the first octant is "catched up" by them; nothing "leaks through". At first, a proper local numbering of the unit cube vertices should be defined. As follows. The vertices are numbered in binary, with 1 bit per dimension: Numbering bit pattern x y z 0 000 (0,0,0) 1 001 (1,0,0) 2 010 (0,1,0) 3 011 (1,1,0) (1) 4 100 (0,0,1) 5 101 (1,0,1) 6 110 (0,1,1) 7 111 (1,1,1) Note that the unit coordinates conform to the reversed bit pattern of the node numbering. Also note that the numbering starts with 0 instead of 1. Then there exist several possibilities for arranging funnels in a cube: four to be precise. All four arrangements correspond to a special SUFED scheme. 0. Unskewed scheme The zero'th possibility is denoted as such, because it is nothing new. In the first octant, it involves 4 vertices and 1 tetrahedron, namely the unit tetrahedron. Its base vectors, according to table (1), are not transformed: (0,1,2,4) : (1,0,0) (0,1,0) (0,0,1) The upwind scheme corresponding to this is the well known equation (Part I,3): U = u , V = v , W = w . The complete scheme has a 7 node computational molecule. 1. First skewed scheme The first skewed scheme involves 5 nodes and 3 tetrahedra in the first octant. (0,1,2,7) , (0,1,4,7) , (0,2,4,7) The base vectors are transformed according to table (1): Unit tetrahedron: (1,0,0) (0,1,0) (0,0,1) (0,1,2,7) : (1,0,0) (0,1,0) (1,1,1) (0,1,4,7) : (1,0,0) (0,0,1) (1,1,1) (0,2,4,7) : (0,1,0) (0,0,1) (1,1,1) Which induces a transformation of the velocities, according to (Part I,4): (0,1,2,7) : u = U + W , v = V + W , w = W (0,1,4,7) : u = U + W , v = W , w = V + W (0,2,4,7) : u = W , v = U + W , w = V + W Determine the inverses of these transformations: 1 : U = u - w , V = v - w , W = w 2 : U = u - v , V = w - v , W = v 3 : U = v - u , V = w - u , W = u Substitute this into the classical upwind scheme (Part I,3). Also the upwind conditions U <= 0 , V <= 0 , W <= 0 must be transformed. This finally results in: 1 : (u - w).(T0 - T1) + (v - w).(T0 - T2) + w.(T0 - T7) = 0 2 : (u - v).(T0 - T1) + (w - v).(T0 - T4) + v.(T0 - T7) = 0 3 : (v - u).(T0 - T2) + (w - u).(T0 - T4) + u.(T0 - T7) = 0 1 : for u,v <= w <= 0 2 : for u,w <= v <= 0 3 : for v,w <= u <= 0 Similar equations can be derived for the other octants. The completed scheme has a 15 node computational molecule, and covers all possible (u,v,w) cases. 2. Second skewed scheme The second skewed scheme involves 7 nodes and 4 tetrahedra in the first octant: (0,3,5,6) , (0,1,3,5) , (0,2,3,6) , (0,4,5,6) The base vectors are transformed according to table (1): Unit tetrahedron: (1,0,0) (0,1,0) (0,0,1) (0,3,5,6) : (1,1,0) (1,0,1) (0,1,1) (0,1,3,5) : (1,0,0) (1,1,0) (1,0,1) (0,2,3,6) : (0,1,0) (1,1,0) (0,1,1) (0,4,5,6) : (0,0,1) (1,0,1) (0,1,1) Which induces a transformation of the velocities, according to (Part I,4): (0,3,5,6) : u = U + V , v = U + V , w = V + W (0,1,3,5) : u = U + V + W , v = V , w = W (0,2,3,6) : u = V , v = U + V + W , w = W (0,4,5,6) : u = V , v = W , w = U + V + W Determine the inverses of these transformations: 1 : U = (u + v - w)/2 , V = (u + w - v)/2 , W = (v + w - u)/2 2 : U = u - v - w , V = v , W = w 3 : U = v - u - w , V = u , W = w 4 : U = w - u - v , V = u , W = v Substitute this into the classical upwind scheme (Part I,3). Also the upwind conditions U <= 0 , V <= 0 , W <= 0 must be transformed. This finally results in: 1 : (u + v - w).(T0 - T3) + (u + w - v).(T0 - T5) + (v + w - u).(T0 - T6) = 0 2 : (u - v - w).(T0 - T1) + v.(T0 - T3) + w.(T0 - T5) = 0 3 : (v - u - w).(T0 - T2) + u.(T0 - T3) + w.(T0 - T6) = 0 4 : (w - u - v).(T0 - T4) + u.(T0 - T5) + v.(T0 - T6) = 0 1 : for u + v <= w , u + w <= v , v + w <= u 2 : for u <= v + w , v <= 0 , w <= 0 3 : for v <= u + w , u <= 0 , w <= 0 4 : for w <= u + v , u <= 0 , v <= 0 Similar equations can be derived for the other octants. The completed scheme has a 19 node computational molecule, and covers all possible (u,v,w) cases. 3. Third skewed scheme The third skewed scheme involves 8 nodes and 6 tetrahedra in the first octant: (0,1,3,7) , (0,1,5,7) , (0,2,3,7) , (0,2,6,7) , (0,4,5,7) , (0,4,6,7) The base vectors are transformed according to table (1): Unit tetrahedron: (1,0,0) (0,1,0) (0,0,1) (0,1,3,7) : (1,0,0) (1,1,0) (1,1,1) (0,1,5,7) : (1,0,0) (1,0,1) (1,1,1) (0,2,3,7) : (0,1,0) (1,1,0) (1,1,1) (0,2,6,7) : (0,1,0) (0,1,1) (1,1,1) (0,4,5,7) : (0,0,1) (1,0,1) (1,1,1) (0,4,6,7) : (0,0,1) (0,1,1) (1,1,1) Substitute the vertex coordinates into the velocity-transformations (Part I,4): (1,3,7): u = U + V + W , v = V + W , w = W (1,5,7): u = U + V + W , v = W , w = V + W (2,3,7): u = V + W , v = U + V + W , w = W (2,6,7): u = W , v = U + V + W , w = V + W (4,5,7): u = V + W , v = W , w = U + V + W (4,6,7): u = W , v = V + W , w = U + V + W And determine the inverse transformations: 1 : U = u - v , V = v - w , W = w 2 : U = u - w , V = w - v , W = v 3 : U = v - u , V = u - w , W = w 4 : U = v - w , V = w - u , W = u 5 : U = w - u , V = u - v , W = v 6 : U = w - v , V = v - u , W = u Substitute this into the classical upwind scheme (Part I,3) and simplify. Also the upwind conditions U < 0 , V < 0 , W < 0 must be transformed. This finally results in: u.(T0 - T1) + v.(T1 - T3) + w.(T3 - T7) = 0 for u <= v <= w <= 0 u.(T0 - T1) + w.(T1 - T5) + v.(T5 - T7) = 0 for u <= w <= v <= 0 v.(T0 - T2) + u.(T2 - T3) + w.(T3 - T7) = 0 for v <= u <= w <= 0 v.(T0 - T2) + w.(T2 - T6) + u.(T6 - T7) = 0 for v <= w <= u <= 0 w.(T0 - T4) + u.(T4 - T5) + v.(T5 - T7) = 0 for w <= u <= v <= 0 w.(T0 - T4) + v.(T4 - T6) + u.(T6 - T7) = 0 for w <= v <= u <= 0 In fact, this is the "skewest" SUFED scheme that exists in 3-D. It is remarked that special cases occur for u = v , v = w , u = w , assuring a smooth transition between the different parts of the scheme. For u = v = w all parts of the scheme are identical, simply resulting in: T0 = T7 Similar equations can be derived for the other octants. The completed scheme has a 27 node computational molecule, and covers all possible (u,v,w) cases. Acknowledgement: The skewed schemes (1.) and (2.) were discovered by David Paterson. To be continued later. - * 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