Animal Shapes, Modal Analysis, and Visualization of Motion (IV): Geometric Constructions and Implementation

We address computer implementation and technology issues in geometric constructions for mathematical and computational modeling purposes, especially regarding 3D finite-element mesh generation for large scale scientific computation of partial differential equations in applications. This mesh generation process for irregular shapes is time-consuming and cannot be done by hand, and has long been recognized as a major bottle-neck in large-scale scientific computing. Significant computer skills and proficiency are required for the applied mathematicians and numerical analysts engaged in the process of 3D-geometry mesh generations. But these days specialty software packages are available that can expedite geometric constructions. Their applications are now the canonical way to carry out the needed numerics of grids. In fact, new technologies such as laser scan are now available which generate a point cloud that, through further processing to remedy some common types of defects, can further expedite the generation of mesh of high quality for complex geometries. In this paper, we use Geomagic Design X as the basic platform to construct CAD containing the underlying geometry from a point cloud. From this CAD, we can then generate finite element mesh by HyperMesh for solving partial differential equations. This paper is mostly survey and tutorial in nature in the hope of providing a somewhat self-contained account for the researcher working on shape analysis and geometric designs. Step-by-step procedures of computer work are explained, with four instances of concrete examples. Trouble-shooting for common mesh generation problems, whether from a point cloud or CAD such as the repair of overlapping regions, mutually penetrating finite elements, gaps and holes, are also indicated.