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We propose a new formulation for integrating over smooth curves and surfaces that are described by their closest point mappings. Our method is designed for curves and surfaces that are not defined by any explicit parameterization and is intended to be used in combination with level set techniques. However, contrary to the common practice with level set methods, the volume integrals derived from our formulation coincide exactly with the surface or line integrals that one wishes to compute. We study various aspects of this formulation and provide a geometric interpretation of this formulation in terms of the singular values of the Jacobian matrix of the closest point mapping. Additionally, we extend the formulation-initially derived to integrate over manifolds of codimension one-to include integration along curves in three dimensions. Some numerical examples using very simple discretizations are presented to demonstrate the efficacy of the formulation.

We propose a new formulation for integrating over smooth curves and surfaces that are described by their closest point mappings. Our method is designed for curves and surfaces that are not defined by any explicit parameterization and is intended to be used in combination with level set techniques. However, contrary to the common practice with level set methods, the volume integrals derived from our formulation coincide exactly with the surface or line integrals that one wishes to compute. We study various aspects of this formulation and provide a geometric interpretation of this formulation in terms of the singular values of the Jacobian matrix of the closest point mapping. Additionally, we extend the formulation -initially derived to integrate over manifolds of codimension one -to include integration along curves in three dimensions. Some numerical examples using very simple discretizations are presented to demonstrate the efficacy of the formulation.

This paper proposes an algorithm for simulating interfacial motions in Mullins-Sekerka dynamics using a boundary integral equations formulated on implicit surfaces. The proposed method is able to simulate the dynamics on unbounded domains, and as such will provide a tool to better understand the dynamics obtained in these situations. As pointed out in Gurtin's paper [23], the behavior of the dynamics are far more interesting on domains that are unbounded. However, it seems that there is no other computational method that can simulate the nonlinear interfacial motion of Mullins-Sekerka dynamics in three dimensions on unbounded domains, allowing the interfaces to merge, break up, without the need of finding explicit representations.The Mullins-Sekerka flow is a Stefan-type free boundary problem involving an nonlocal interfacial motion dynamically controlled by the solution of Laplace's equation with appropriate boundary conditions obtained on both sides of the interface. As a result, it is reasonable to consider boundary integral methods, especially for exterior domains, combined with level set methods [40][20], for easy tracking and for being able to handle topological changes.Various numerical methods have been proposed to solve elliptic problems on irregular domains. We mention here some of the few "usual suspects" that use fi-

We propose efficient and accurate algorithms for computing certain area preserving geometric motions of curves in the plane, such as area preserving motion by curvature. These schemes are based on a new class of diffusion generated motion algorithms using signed distance functions. In particular, they alternate two very simple and fast operations, namely convolution with the Gaussian kernel and construction of the distance function, to generate the desired geometric flow in an unconditionally stable manner. We present applications of these area preserving flows to large scale simulations of coarsening.