Applications of Symbolic Computation to the Calculus of Moving Surfaces

In the physical world, objects change shape over time. A soap bubble blowing in the wind changes shape and density as it floats through the air. Red blood cells change shape to carry oxygen through our veins. Modeling these problems requires deforming manifolds. The Calculus of Moving Surfaces (CMS) is an analytical framework for studying deforming manifolds. The CMS is an extension of tensor calculus. Both approach problems from a geometric perspective, without reference to specific coordinate systems. To evaluate a specific realization of a problem, a coordinate system is chosen and a CMS expression is converted to a series of n-dimensional array calculations using standard calculus. This generality has many costs. The length of expressions grows quickly, in many cases exponentially. Although it is applicable to a wide range of problems, calculations quickly become intractable. The expressions generated are not only long and difficult to work with, evaluating them on a specific coordinate system introduces an entirely different set of challenges. We present the first compute algebra system designed specifically for the CMS. Our system, the Symbolic Computation of Moving Surfaces (SCMS) supports the derivation of CMS expressions and the evaluation of expressions on specific coordinate systems. Although large expressions are inherent in the framework, computer automation allows for the application of the CMS to significantly larger problems then can be done by hand and allows the CMS to be applied in an error free way to non-trivial problems. We have developed two libraries making up the SCMS. The first is a term rewrite system, CMSTRS, developed in Java. This library automates the analytic framework of the CMS. Expressions are kept at a high level, retaining the generality of the CMS. The second, CMSTensor, is for evaluation on specific coordinate systems. It is implemented using the Maple computer algebra system. It leverages the power of this computer algebra system to evaluate CMS expressions as a combination of n-dimensional array manipulations and standard calculus operations. We have applied our system to a non-trivial boundary variation problem: the symbolic series expansion of the Laplace Eigenvalues on the N-sided regular polygon under Dirichlet boundary conditions. This series is computed up to N^(-6), two orders higher then previous results. Our calculations confirm previous hand calculations and extend the series beyond what was previously known.Ph.D., Computer Science -- Drexel University, 201