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Unformatted text preview: Lecture 41 Finite Elements Triangulating a Region A disadvantage of finite difference methods is that they require a very regular grid, and thus a very regular region, either rectangular or a regular part of a rectangle. Finite elements is a method that works for any shape region because it is not built of a grid, but on triangulation of the region, i.e. cutting the region up into triangles as we did in a previous lecture. The following figure shows a triangularization of a region.-2.5-2-1.5-1-0.5 0.5 1 1.5 2 2.5-2.5-2-1.5-1-0.5 0.5 1 1.5 2 2.5 x y Figure 41.1: An annular region with a triangulation. Notice that the nodes and triangles are very evenly spaced. This figure was produced by the script program mywasher.m . Notice that the nodes are evenly distributed. This is good for the finite element process where we will use it. Open the program mywasher.m . This program defines a triangulation by defining the vertices in a matrix V in which each row contains the x and y coordinates of a vertex. Notice that we list the interior nodes first, then the boundary nodes. Triangles are defined in the matrix T . Each row of T has three integer numbers indicating the indices of the nodes that form a triangle. For instance the first row is 43 42 25 , so T 1 is the triangle 140 141 a116 a68 a68 a68 a68 a68 a68 a68 a68 a4 a4 a4 a4 a4 a4 a4 a4 a116 a66 a66 a66 a66 a66 a66 a66 a66 a2 a2 a2 a2 a2 a2 a2 a2 a116 a65 a65 a65 a65 a65 a65 a65 a65 a1 a1 a1 a1 a1 a1 a1 a1 a116 a65 a65 a65 a65 a65 a65 a65 a65 a1 a1 a1 a1 a1 a1 a1 a1 a116 a66 a66...
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- Fall '08
- Piecewise linear function, Linear function, Elliptic boundary value problem, Cj Φj