Local refinement of 3-D triangulations using object-oriented methods

Abstract

The data structures used to model meshes for solving problems by finite element methods is based on different arrays. In these arrays information is stored related to, among other components, nodes, edges, faces, tetrahedra and connectivities. These structures provide optimum results but, in many cases, they need additional programming to be maintained. In adaptive simulation, the meshes undergo refinement/derefinement processes to improve the numerical solution at each step. These processes produce new elements and eliminate others, so the arrays should reflect the state of the mesh in each of these steps. Using traditional language, memory should be pre-assigned at the outset of the program, so it is only required to estimate the changes taking place in the mesh. In the same respect, it was necessary to compact the arrays to recover space from erased elements. With the advent of languages such as C, memory can be assigned dynamically, resolving most of the problem. However, arrays are costly to maintain, as they require adapting the mesh treatment to the data model, and not inversely. The object-oriented programming suggests a new focus in implementing data structures to work with meshes. The classes create data types that may be adjusted to the needs of each case, allowing each element to be modelled independently. Inheritance and encapsulation enable us to simplify the programming tasks and increase code reuse. We propose a data structure based on meshes-treating objects. Finally, we present an implementation of a local refinement algorithm based on 8-subtetrahedron subdivision and some experiments.