Uma formulação de volumes finitos 3D utilizando malhas OcTree

Abstract

In order to better handle problems with complex domain, a numerical scheme with OcTree meshes based on the Finite Volume Method is developed in the present work. The main feature of this mesh with cube form is that the ratio between the edges of two volumes retains a power of two. The possibility of having small volumes adjacent to irregular interfaces leads to a more precise capture of interfaces. Regions with constant property can be represented by larger volumes, resulting in saving both memory and CPU time. On the other hand, in the case of the study of seismic waves, the use of larger volumes in regions with higher wave propagation velocity reduces significantly data storage and processing time. Differential discrete gradient and divergent operators are constructed through the assembly of matrices that connect the volumes to their faces. In addition, two new approaches to the differential gradient operator are developed, aiming at raising the original approximation order and minimizing numerical reflections by virtue of the nonconforming mesh. The proposed technique is applied and discussed to phenomena governed by the heat and wave equations with substantially great practical interest, including a modeling of a treatment for cancer by hyperthermia and of the propagation of acoustic waves in complex geological environments.