Integration Over Curves and Surfaces Defined by the Closest Point Mapping

Abstract

We propose a new formulation using the closest point mapping for integrating over smooth curves and surfaces with boundaries that are described by their closest point mappings. 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 wish 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 are presented.

Open PDF

Document Details

Document Type
Technical Report
Publication Date
Apr 01, 2015
Accession Number
ADA623636

Entities

People

  • Catherine Kublik
  • Richard Tsai

Organizations

  • University of Dayton

Tags

Communities of Interest

  • Air Platforms
  • C4I

DTIC Thesaurus Topics

  • Boundaries
  • Cartesian Coordinates
  • Computations
  • Convergence
  • Coordinate Systems
  • Curvature
  • Differential Equations
  • Eigenvalues
  • Equations
  • Grids
  • Integrals
  • Mathematics
  • Numerical Integration
  • Partial Differential Equations
  • Standards
  • Three Dimensional
  • Two Dimensional

Fields of Study

  • Mathematics

Readers

  • Computational Fluid Dynamics (CFD)
  • Graph Algorithms and Convex Optimization.
  • Joint Military Operations and Doctrine.