Geometric Processing of Cad Data and Meshes as Input of Integral Equation Solvers

Among the presently known numerical solvers of integral equations, two main categories of approaches can be traced: • Mesh-free approaches • Mesh-based approaches. We will propose some techniques to process geometric data so that they can be efficiently used in subsequent numerical treatments of integral equations. In order to prepare geometric information so that the above two approaches can be automatically applied, we need the following items. • Splitting a given surface Γ into several four-sided patches Γi. • Generating a diffeomorphism γi from the unit square to Γi. • Generating a mesh M on a given surface. • Patching of a given triangulation. In order to have a splitting Γ = ∪i=1Γi, we need to approximate the surfaces first by polygonal regions. We use afterwards quadrangulation techniques by removing quadrilaterals repeatedly. We will generate the diffeomorphisms by means of transfinite interpolations of Coons and Gordon types. The generation of a mesh M from a piecewise Riemannian surface will use some generalized Delaunay techniques in which the mesh size will be determined with the help of the Laplace-Beltrami operator. We will describe our experiences with the IGES format because of two reasons. First, most of our implementations have been done with it. Next, some of the proposed methodologies assume that the curve and surface representations are similar to those of IGES. Patching a mesh consists in approximating or interpolating it by a set of practical surfaces such as B-spline patches. That approach proves useful when we want to utilize a mesh-free integral equation solver but the input geometry is represented as a mesh.