4 and 45 we could store information about a face

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Unformatted text preview: o use variable-order approximations (p-re nement). Without edge data, we need a way of determining those edges that are on @ . This can be done by adopting a convention that the edge between the rst and second vertices of each face is Edge 1. Remaining edges are numbered in counterclockwise order. A sample boundary data table for the mesh of Figure 5.3.4 is shown on the right of Table 5.3.4. The rst row of the table identi es Edge 1 of Face 1 as being on a boundary of the domain. Similarly, the second row of the table identi es Edge 4 of Face 1 as being a boundary edge, etc. Regions with curved edges would need pointers back to the geometric database. 5.4. Coordinate Transformations Vertex 1 2 3 4 5 6 7 8 15 Coordinates 0.00 0.00 1.00 0.00 2.00 0.00 0.00 1.00 1.00 1.00 2.00 1.00 0.50 2.00 1.50 2.00 Face Edge 1 1 1 4 2 1 2 2 3 3 4 2 5 2 9 0.50 0.00 10 1.50 0.00 11 0.00 0.50 12 1.00 0.50 13 2.00 0.50 14 0.50 1.00 15 1.50 1.00 16 0.25 1.50 17 0.75 1.50 18 1.25 1.50 19 1.75 1.50 21 0.50 0.50 22 1.50 0.50 Table 5.3.4: Vertex and coordinate data (left) and boundary data (right) for the nite element mesh shown in Figure 5.3.4. 5.4 Coordinate Transformations Coordinate transformations enable us to develop element sti ness and mass matrices and load vectors on canonical triangular, square, tetrahedral, and cubic elements in a computational domain and map these to actual elements in the physical domain. Useful transformations must (i) be simple to evaluate, (ii) preserve continuity of the nite element solution and geometry, and (iii) be invertible. The latter requirement ensures that each point within the actual element corresponds to one and only one point in the canonical element. Focusing on two dimensions, this requires the Jacobian Je := x x (5.4.1) yy of the transformation of Element e in the physical (x y)-plane to the canonical element in the computational ( )-plane to be nonsingular. The most popular coordinate transformations are, naturally, piecewise-polynomial 16 Mesh Generation and Assembly functions. These mappings are called subparametric,...
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This document was uploaded on 03/16/2014 for the course CSCI 6860 at Rensselaer Polytechnic Institute.

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