Discrete Exterior Calculus

The language of modern mechanics is calculus on manifolds, and exterior calculus is an important part of that. It consists of objects like differential forms, general tensors and vector fields on manifolds, and operators that act on these. While the smooth exterior calculus has a long history going back to Cartan, Lie, Grassmann, Hodge, de Rham and many others, the need for a discrete calculus has been spurred on recently by the need to do computations. This thesis presents the beginnings of a theory of discrete exterior calculus (DEC). This is motivated by potential applications in computational methods for field theories (elasticity, fluids, electromagnetism) and in areas of computer vision and computer graphics. One approach to approximating a smooth exterior calculus is to consider the given mesh as approximating some smooth manifold at least locally, and then defining the discrete operators by truncating the smooth ones. Another approach is to consider the discrete mesh as the only given thing and developing an entire calculus using only discrete combinatorial and geometric operations. The derivations may require that the objects on the discrete mesh, but not the mesh itself, are interpolated. It is this latter route that we have taken and this leads to a discrete exterior calculus. Our theory includes not only discrete equivalents of differential forms, but also discrete vector fields and the operators acting on these objects. General tensors are not developed, though we suggest a possible way to do that towards the end. The presence of forms and vector fields allows us to address the various interactions between forms and vector fields which are important in applications. With a few exceptions, most previous attempts at discrete exterior calculus have addressed only differential forms, or vector fields as proxies for forms. We also show that the circumcentric dual of a simplicial complex plays a useful role in the metric dependent part of this theory. The importance of dual complexes in this field has been well understood, but with a few exceptions previous researchers have used barycentric duals. The use of duals is reminiscent of the use of staggered meshes in computational mechanics. The appearance of dual complexes leads to a proliferation of the operators in the discrete theory. For example there are primal-primal, primal-dual etc. versions of many operators. This is of course unique to the discrete side. In many examples we find that the formulas derived from our discrete exterior calculus are identitical to the existing formulas in literature. We define discrete differential forms in the usual way, as cochains on a simplicial complex. The discrete vector fields are defined as vector valued 0-forms, and they live either on the primal, or on the dual vertices.