Uniform and Non Uniform Mesh Generation Problem




The design of a printed circuit board raises a number of interesting geometric problem. One of the problems is known as the mesh generation problem. Many components emit heat during operation. The emission of heat should remain below a certain threshold. In order to acquire an effective design of the circuit chip, the designers often build the simulation of the board. Simulation allows testing during the initial stages of the design phase, so that faulty designs can be rejected as early as possible.
The transfer of heat between different materials on the board is a very complicated method. Therefore, a way to has to simulate the heat processes on the board is to resort to approximation using finite element method. This method divides the board into many small regions, or elements which can be triangles or quadrilaterals. The heat generated by each element is assumed to be known. It is also assumed to be known how neighboring elements influence each other. A series of equations is then generated.
The accuracy of the finite element method depends largely on the complexity of the mesh. The finer the mesh the better the solution, but the computation time for the solution increases drastically as the number of elements increases. The mesh elements should also be contained in only one region. The shape of the mesh elements also plays an important role. Irregular shaped objects are discouraged.
The problem consists of computing a triangular mesh of a square, that is a subdivision of the square into triangles. The mesh should have the following features:
The mesh should be non-uniform, it should be finer near the edges of the components and coarse far away from the edges. This would make our calculations much simpler.
First of all the triangulation of the vertices of the components need not respect the edges of the components, otherwise we will suffer from the problem of having very small angles. Since the Delaunay triangulation maximaizes the minimum angle, it seems impossible to generate a mesh with only well-shaped triangles. But the triangles in a mesh are not required to have their vertices at the input points. We introduce new points called Steiner points, to help us obtain well-shaped triangles. But this produces a mesh which has small triangles everywhere not only near the edges, so it is a uniform mesh. The solution to this problem is the generation of non-uniform mesh.

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Last modified: Sun Mar 5 22:41:09 EST 2000
Spent approximately 5 and half hours on this assignment.