چكيده به لاتين
Considering the necessity of implementation and computation from continuous structure (continuous differential geometry) to discrete structure, based on the data presented in reference [17], this thesis is trying to lay a foundation to come up with geometry-based computational modeling purposes. However, as we know, abstract geometric spaces could not be described practically. Therefore, this thesis intends to introduce concepts and methods to solve this problem. So, Discrete Exterior Calculus (DEC) introduce a scheme preserving significant structures and properties of continuous spaces. Using this method and required devises, first we transform the domain into triangulation meshes by creating the replication of discrete continuous manifold. Furthermore, by introducing their duality in the next phases, we are going to examine the discretization of forms and their operators which lead these differential forms to be considered as volumes on triangulated meshes. To work toward the main goal, concepts of wedge product and Hodge star will be examined in the discrete space. Finally, introducing the replication of hodge decomposition discrete will lead us to get a glimpse at the main goal. It should be noted that the idea of discrete hodge decomposition for computational purposes, could be applicable in various domains. Despite of the recent significant and extensive developments, this approach to computations is still fairly new. As a result, there are a significant amount of details to examine in order to show the idea of forms as a basis of discretizational elements of differential equations can be used for various purposes.
