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3 Model Construction and Mesh Generation     continued...

A second method of mesh generation is the Delaunay triangulation strategy. Given a two- or three-dimensional set of points which define the boundaries and interior regions of the domain to be modeled, one tessellates the point cloud into an optimal mesh of triangles or tetrahedra. For bioelectric field problems, the advantages and disadvantages tend to be exactly contrary to those arising from the divide and conquer strategy. The primary advantage is that the mesh can be developed to fit any predefined geometry, including subsurfaces, by starting with points which define all the necessary surfaces and subsurfaces and then adding additional interior points to minimize the aspect ratio. For triangles, the aspect ratio is defined to be either the ratio of the maximum horizontal length of the element to the maximum vertical length of the element, or the ratio of the diameter of a circumscribed circle to the maximal distance between vertices. For tetrahedra, the aspect ratio can be defined as where denotes the diameter of the sphere circumscribed about the tetrahedron and is the maximum distance between two vertices. These formulations yield a value of 1 for an ``isosceles'' tetrahedron (triangle) and a value of 0 for a degenerate (flat) element [22]. The closer to the value of 1, the better (it should be noted that this is not a general rule for all problems: in certain problems where there are very high gradients in a uniform direction, as in high speed fluid flow, for example, it is often better to distort the aspect ratio of the element to take into account this phenomena).