0
research-article

Treat All Integrals as Volume Integrals: A uni?ed, parallel, grid-based method for evaluation of volume, surface, and path integrals on implicitly de?ned domains

[+] Author and Article Information
Mete Yurtoglu

Google, 747 6th St S., Kirkland, WA
myurtoglu@me.com

Molly Carton

Student Mem. ASME, University of Washington - Seattle, Mechanical Engineering, Seattle, WA 98195-2600
mcarton@uw.edu

Duane W. Storti

University of Washington - Seattle, Mechanical Engineering, Box 352600, Seattle, WA 98195-2600
storti@uw.edu

1Corresponding author.

ASME doi:10.1115/1.4039639 History: Received September 08, 2017; Revised March 09, 2018

Abstract

Wepresentauni?edmethodfornumericalevaluationofvolume,surface,andpathintegralsofsmooth,bounded functions on implicitly de?ned bounded domains. The method avoids both the stochastic nature (and slow convergence) of Monte Carlo methods and problem-speci?c domain decompositions required by most traditional numericalintegrationtechniques. Ourapproachoperatesonauniformgridoveranaxis-alignedboxcontainingtheregion ofinterest,sowerefertoitasagrid-basedmethod. Allgrid-basedintegralsarecomputedasasumofcontributions fromastencilcomputationonthegridpoints. Eachclassofintegrals(path,surface,orvolume)involvesadifferent stencil formulation, but grid-based integrals of a given class can be evaluated by applying the same stencil on the same set of grid points; only the data on the grid points changes. When functions are de?ned over the continuous domain so that grid re?nement is possible, grid-based integration is supported by a convergence proof based on wavelet analysis. Given the foundation of function values on a uniform grid, grid-based integration methods apply directly to data produced by volumetric imaging (including computed tomography and magnetic resonance), direct numericalsimulation(DNS)of?uid?ow,oranyothermethodthatproducesdatacorrespondingtovaluesofafunctionsampledonaregulargrid. Everystepofagrid-basedintegralcomputation(includingevaluatingafunctionon a grid, application of stencils on a grid, and reduction of the contributions from the grid points to a single sum) is well-suitedforparallelization. WepresentresultsfromaparallelizedCUDAimplementationofgrid-basedintegrals that faithfully reproduces the output of a serial implementation but with signi?cant reductions in computing time. We also present example grid-based integral results to quantify convergence rates associated with grid re?nement and dependence of the convergence rate on the speci?c choice of difference stencil (corresponding to a particular genus of Daubechies wavelet).

Copyright (c) 2018 by ASME
Your Session has timed out. Please sign back in to continue.

References

Figures

Tables

Errata

Discussions

Some tools below are only available to our subscribers or users with an online account.

Related Content

Customize your page view by dragging and repositioning the boxes below.

Related Journal Articles
Related eBook Content
Topic Collections

Sorry! You do not have access to this content. For assistance or to subscribe, please contact us:

  • TELEPHONE: 1-800-843-2763 (Toll-free in the USA)
  • EMAIL: asmedigitalcollection@asme.org
Sign In