Using Self Organizing Maps for 3D surface and volume adaptive mesh generation

Adaptive mesh methods are commonly used to improve the accuracy of numerical solution of problems without essential increase in the number of mesh nodes (Lebedev et al., 2002). Within the scope of all adaptive mesh methods, there is an important class of methods in which the mesh is an image under an appropriate mapping of a fixed mesh over a computational domain (Bern & Plassmann, 1999). Most of widely used conventional methods from the above class, such as equidistribution method (Shokina, 2001), Thompson’s method (Thompson et al., 1985), elliptic method (Liseikin, 1999), etc. determine the mapping by solving a complicated system of nonlinear partial differential equations (PDEs). This often leads to significant difficulties. First, the convergence of numerical solution of these PDEs highly depends on an initial mesh, requires fixing boundary mesh nodes beforehand and imposes quite strong limitations on the properties of mesh density function (Khakimzyanov et al., 2001). Second, efficient parallelization of solvers for the PDEs meets overwhelming difficulties. Finally, the PDEs for mesh construction are not universal and need to be proposed for 1D, 2D or 3D spaces specifically. The complexity of numerical solution of these PDEs essentially grows with increasing the dimensionalities (Khakimzyanov et al., 2001). Moreover, there is no methods and techniques in the above mentioned class that can provide a fully automatic adaptive mesh construction in 3D case. This chapter demonstrates the great ability of the Kohonen’s Self Organizing Maps (SOM) (Kohonen, 2001) to perform high quality adaptive mesh construction. Since the SOM model provides a topology preserving mapping of high-dimensional data onto a low-dimensional space with approximation of input data distribution, the proposed mesh construction method uses the same algorithms for different dimensionalities of a physical domain that proves its universality. In our investigation, the classical SOM model has been studied and modified in order to overcome border effect and provide topology preservation. Based on the ideas in (Nechaeva, 2006), the composition of SOM models of different dimensionalities has been proposed which alternates mesh construction on the border and inside a physical domain. It has been shown that the SOM learning algorithm can be used as a mesh smoothing tool. All the algorithms has been implemented using the GeomBox (Bessmeltsev, 2009) and 9