Level-Set Minimization of Potential Controlled Hadwiger Valuations for Molecular Solvation

A level-set method is developed for the numerical minimization of a class of Had-wiger valuations with a potential on a set of three-dimensional bodies. Such valuations are linear combinations of the volume, surface area, and surface integral of mean curvature. The potential increases rapidly as the body shrinks beyond a critical size. The combination of the Hadwiger valuation and the potential is the mean-field free-energy functional of the solvation of non-polar molecules in the recently developed variational implicit-solvent model. This functional of surfaces is minimized by the level-set evolution in the steepest decent of the free energy. The normal velocity of this surface evolution consists of both the mean and Gaussian curvatures, and a lower-order, "forcing" term arising from the potential. The forward Euler method is used to discretize the time derivative with a dynamic time stepping that satisfies a CFL condition. The normal velocity is decomposed into two parts. The first part consists of both the mean and Gaussian curvature terms. It is of parabolic type with parameter correction, and is discretized by central differencing. The second part has all the lower-order terms. It is of hyperbolic type, and is discretized by an upwinding scheme. New techniques of local level-set method and numerical integration are developed. Numerical tests demonstrate a second-order convergence of the method. Examples of application to the modeling of molecular solvation are presented.