A subdivision approach to the construction of smooth differential forms

Abstract

Vertex- and face-based subdivision schemes are now routinely used in geometric modeling and computational science, and their primal/dual relationships are well studied. In this thesis we interpret these schemes as defining bases for discrete differential 0- resp. 2 -forms, and present a novel subdivision-based method of constructing smooth differential forms on simplicial surfaces. It completes the picture of classic primal/dual subdivision by introducing a new concept named r-cochain subdivision. Such subdivision schemes map scalar coefficients on r-simplexes from the coarse to the refined mesh and converge to r-forms on the mesh. We perform convergence and smoothness analysis in an arbitrary topology setting by utilizing the techniques of matrix subdivision and the subdivision differential structure. The other significance of our method is its preserving exactness of differential forms. We prove that exactness preserving is equivalent to the commutative relations between the subdivision schemes and the topological exterior derivative. Our construction is based on treating r- and (r + 1)-cochain subdivision schemes as a pair and enforcing the commutative relations. As a result, our low-order construction recovers classic Whitney forms, while the high-order construction yields a new class of high order Whitney forms. The 1-form bases are C1, except at irregular vertices where they are C0. We also demonstrate extensions to three-dimensional subdivision schemes and non-simplicial meshes as well, such as quadrilaterals and octahedra. Our construction is seamlessly integrated with surface subdivision. Once a metric is supplied, the scalar 1-form coefficients define a smooth tangent vector filed on the underlying subdivision surface. Design of tangent vector fields is made particularly easy with this machinery as we demonstrate. The subdivision r-forms can also be used as finite element bases for physical simulations on curved surfaces. We demonstrate the optimal rate of convergence in solving the Laplace and bi-Laplace equations of 1-forms.