A Coarea Formulation for Grid-Based Evaluation of Volume Integrals

Abstract

Abstract We present a new method for computing volume integrals based on data sampled on a regular cartesian grid. We treat the case where the domain is defined implicitly by a trivariate inequality f(x,y,z) < 0, and the input data includes sampled values of the defining function f and the integrand. The method employs Federer’s coarea formula to convert the volume integral to an integral over level set values where the integrand is an integral over the level sets. Application of a standard quadrature method produces an approximation of the integral over the continuous range of f in the form of a sum of integrals on level sets corresponding to a discrete set of values of f. The integrals on the discrete collection of level sets are evaluated using the grid-based approach presented by Yurtoglu et al. [1]. We describe how the implementation relates to the implementation described in [1], and we present results for sample problems with known exact results to support a discussion of accuracy and convergence along with a comparison with a traditional Monte Carlo method.