Finite-element mesh generation using self-organizing neural networks

Neural networks are applied to the problem of mesh placement for the finite-element method. When the finite-element method is used to numerically solve a partial differential equation with boundary conditions over a domain, the domain must be divided into "elements." The precise placement of the nodes of the elements has a major affect on the accuracy of the numeric method. In this paper the self-organizing algorithm of Kohonen is adapted to solve the problem of automatically assigning (in a near-optimal way) coordinates from a two-dimensional domain to a given topologic grid (or mesh) of nodes in order to apply the finite-element method effectively when solving a partial differential equation with boundary conditions over that domain. One novelty of the method is the interweaving of versions of the Kohonen algorithm in different dimensions simultaneously in order to handle the boundary of the domain properly. Our method allows for the use of arbitrary types of two-dimensional elements (in particular, quadrilaterals or mixed shapes as opposed to just triangles) and for varying desired densities over the domain. (Thus more elements can be placed automatically near "areas of interest.") The methods and experiments developed here are for two-dimensional domains but seem naturally extendable to higher-dimensional problems. The method uses a mixture of both one- and two-dimensional versions of the Kohonen algorithm, with an improvement suggested by Tabakman and Exman, and further adapted to the particular problem here. Experimental results comparing this algorithm with a well-known two-dimensional grid-generating system (PLTMG) are presented.