Research Papers

Treat All Integrals as Volume Integrals: A Unified, Parallel, Grid-Based Method for Evaluation of Volume, Surface, and Path Integrals on Implicitly Defined Domains

[+] Author and Article Information
Mete Yurtoglu

747 6th Street South,
Kirkland, WA 98033
e-mail: myurtoglu@me.com

Molly Carton

Mechanical Engineering,
University of Washington,
Seattle, WA 98195-2600
e-mail: mcarton@uw.edu

Duane Storti

Mechanical Engineering,
University of Washington,
P.O. Box: 352600,
Seattle, WA 98195-2600
e-mail: storti@uw.edu

Manuscript received September 8, 2017; final manuscript received March 9, 2018; published online April 26, 2018. Assoc. Editor: Yong Chen.

J. Comput. Inf. Sci. Eng 18(2), 021013 (Apr 26, 2018) (9 pages) Paper No: JCISE-17-1179; doi: 10.1115/1.4039639 History: Received September 08, 2017; Revised March 09, 2018

We present a unified method for numerical evaluation of volume, surface, and path integrals of smooth, bounded functions on implicitly defined bounded domains. The method avoids both the stochastic nature (and slow convergence) of Monte Carlo methods and problem-specific domain decompositions required by most traditional numerical integration techniques. Our approach operates on a uniform grid over an axis-aligned box containing the region of interest, so we refer to it as a grid-based method. All grid-based integrals are computed as a sum of contributions from a stencil computation on the grid points. Each class of integrals (path, surface, or volume) involves a different 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 defined over the continuous domain so that grid refinement 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 numerical simulation of fluid flow, or any other method that produces data corresponding to values of a function sampled on a regular grid. Every step of a grid-based integral computation (including evaluating a function on a grid, application of stencils on a grid, and reduction of the contributions from the grid points to a single sum) is well suited for parallelization. We present results from a parallelized CUDA implementation of grid-based integrals that faithfully reproduces the output of a serial implementation but with significant reductions in computing time. We also present example grid-based integral results to quantify convergence rates associated with grid refinement and dependence of the convergence rate on the specific choice of difference stencil (corresponding to a particular genus of Daubechies wavelet).

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


Edwards, H. M. , 1993, Advanced Calculus: A Differential Forms Approach, Birkhäuser, Boston, MA.
Sarraga, R. , 1982, “ Computation of Surface Areas in GMSolid,” IEEE Comput. Graph. Appl., 2(7), pp. 65–72. [CrossRef]
Lee, Y. T. , and Requicha, A. A. , 1982, “ Algorithms for Computing the Volume and Other Integral Properties of Solids—I: Known Methods and Open Issues,” Commun. ACM, 25(9), pp. 635–641. [CrossRef]
Lee, Y. T. , and Requicha, A. A. , 1982, “ Algorithms for Computing the Volume and Other Integral Properties of Solids—II: A Family of Algorithms Based on Representation Conversion and Cellular Approximation,” Commun. ACM, 25(9), pp. 642–650. [CrossRef]
Ellis, J. L. , Kedem, G. , Lyerly, T. , Thielman, D. , Marisa, R. J. , Menon, J. , and Voelcker, H. B. , 1991, “ The Ray Casting Engine and Ray Representatives,” First ACM Symposium on Solid Modeling Foundations and CAD/CAM Applications, Austin, TX, June 5–7, pp. 255–267.
Wang, C. C. , Leung, Y.-S. , and Chen, Y. , 2010, “ Solid Modeling of Polyhedral Objects by Layered Depth-Normal Images on the GPU,” Comput.-Aided Des., 42(6), pp. 535–544. [CrossRef]
Wu, J. , Kramer, L. , and Westermann, R. , 2016, “ Shape Interior Modeling and Mass Property Optimization Using Ray-Reps,” Comput. Graph., 58, pp. 66–72. [CrossRef]
Gonzalez-Ochoa, C. , McCammon, S. , and Peters, J. , 1998, “ Computing Moments of Objects Enclosed by Piecewise Polynomial Surfaces,” ACM Trans. Graph., 17(3), pp. 143–157. [CrossRef]
Krishnamurthy, A. , and McMains, S. , 2010, “ Accurate Moment Computation Using the GPU,” 14th ACM Symposium on Solid and Physical Modeling (SPM'10), Haifa, Israel, Sept. 1–3, pp. 81–90.
Krishnamurthy, A. , and Mcmains, S. , 2011, “ Accurate GPU-Accelerated Surface Integrals for Moment Computation,” Comput.-Aided Des., 43(10), pp. 1284–1295. [CrossRef]
Luft, B. , Shapiro, V. , and Tsukanov, I. , 2008, “ Geometrically Adaptive Numerical Integration,” ACM Symposium on Solid and Physical Modeling, Stony Brook, NY, June 2–4, pp. 147–157.
Lynch, P. , 2014, “ The High Power Hypar,” 50 Visions of Mathematics, S. Parc, ed., Oxford University Press, Oxford, UK, pp. 110–113. [PubMed] [PubMed]
Shapiro, V. , Tsukanov, I. , and Grishin, A. , 2011, “ Geometric Issues in Computer Aided Design/Computer Aided Engineering Integration,” ASME J. Comput. Inf. Sci. Eng., 11(2), p. 021005. [CrossRef]
Storti, D. , Ganter, M. , Ledoux, W. , Ching, R. , Hu, Y. , and Haynor, D. , 2009, “ Wavelet SDF-Reps: Solid Modeling With Volumetric Scans,” ASME J. Comput. Inf. Sci. Eng., 9(3), p. 031006. [CrossRef]
Lorensen, W. E. , and Cline, H. E. , 1987, “ Marching Cubes: A High Resolution 3D Surface Construction Algorithm,” ACM Siggraph Computer Graphics, Vol. 21, Association for Computing Machinery, New York, pp. 163–169.
Zames, F. , 1977, “ Surface Area and the Cylinder Area Paradox,” Two-Year Coll. Math. J., 8(4), pp. 207–211. [CrossRef]
Koenderink, J. J. , 1990, Solid Shape, MIT Press, Cambridge, MA, p. 714.
Yurtoglu, M. , 2017, “GPU-Based Parallel Computation of Integral Properties of Volumetrically Digitized Objects,” Ph.D. thesis, University of Washington, Seattle, WA. https://digital.lib.washington.edu/researchworks/handle/1773/38661
Resnikoff, H. L. , and Raymond, O., Jr. , 2012, Wavelet Analysis: The Scalable Structure of Information, Springer Science & Business Media, New York.
Storti, D. , 2010, “Using Lattice Data to Compute Surface Integral Properties of Digitized Objects,” Integrated Design and Manufacturing in Mechanical Engineering—Virtual Concept, Bordeaux, France, pp. 1–6.
Romine, C. , and Peyton, B. , 1997, “Computing Connection Coefficients of Compactly Supported Wavelets on Bounded Intervals,” Oak Ridge National Laboratory, Oak Ridge, TN, Report No. ORNL/TM-13413. https://www.osti.gov/servlets/purl/661583
Bulut, F. , 2016, “ An Alternative Approach to Compute Wavelet Connection Coefficients,” Appl. Math. Lett., 53, pp. 1–9. [CrossRef]
Latto, A. , Resnikoff, H. , and Tenenbaum, E. , 1991, “ The Evaluation of Connection Coefficients of Compactly Supported Wavelets,” French-USA Workshop on Wavelets and Turbulence, Princeton, NJ, June, pp. 76–89.
Latto, A. , and Tenenbaum, E. , 1990, “ Les ondelletes à support compact et la solution numérique de l'équation de burgers,” C. R. Acad. Sci. France, 311(903), pp. 303–309.
Storti, D. , and Yurtoglu, M. , 2015, CUDA for Engineers: An Introduction to High-Performance Parallel Computing, Addison-Wesley Professional, New York.
McCool, M. D. , Robison, A. D. , and Reinders, J. , 2012, Structured Parallel Programming: Patterns for Efficient Computation, Elsevier, Waltham, MA.
NVIDIA, 2015, “CUDA C Programming Guide, v7.5,” NVIDIA Corporation, accessed Mar. 23, 2018, https://developer.nvidia.com/cuda-75-downloads-archive.
Myers, J. A. , 1962, “ Handbook of Equations for Mass and Area Properties of Various Geometrical Shapes,” U.S. Naval Ordinance Test Station Publication 2838, China Lake, CA, NAVWEPS Report No. 7827.
Wang, W. , 2013, “Kinematic Study of the Evolution and Properties of Flame Surfaces in Turbulent Nonpremixed Combustion With Local Extinction and Reignition,” Ph.D. thesis, University of Washington, Seattle, WA. https://digital.lib.washington.edu/researchworks/handle/1773/24157
Blakeley, B. C. , Riley, J. J. , Storti, D. W. , and Wang, W. , 2017, “ On the Kinematics of Scalar ISO-Surfaces in Turbulent Flow,” 70th Annual Meeting of the APS Division of Fluid Dynamics, Denver, CO, Nov. 19–21.
Newman, T. S. , and Yi, H. , 2006, “ A Survey of the Marching Cubes Algorithm,” Comput. Graph., 30(5), pp. 854–879. [CrossRef]
Liu, Y.-S. , Yi, J. , Zhang, H. , Zheng, G.-Q. , and Paul, J.-C. , 2010, “ Surface Area Estimation of Digitized 3D Objects Using Quasi-Monte Carlo Methods,” Pattern Recognit., 43(11), pp. 3900–3909. [CrossRef]


Grahic Jump Location
Fig. 1

Data from convergence study for grid-based evaluation of surface area of a torus. (a) Raw data and best fit log–log plots of relative error versus grid refinement are shown for genus 1 and 2. The genus 2 stencil (with radius 2) produces a significant reduction in relative error across the full range of refinement. (b) Best-fit lines for log–log plots of relative error versus refinement for genus 1–7. Increasing genus beyond G = 2 does not consistently provide decreases in relative error to make up for increased computational cost associated longer stencils.

Grahic Jump Location
Fig. 2

Results of convergence study for genus 1 grid-based evaluation of moment of inertia of a toroidal shell

Grahic Jump Location
Fig. 3

Results of convergence study for genus 1 grid-based evaluation of the volume of a solid torus. Log–log plot of relative error versus grid refinement shows raw data and best-fit line.

Grahic Jump Location
Fig. 4

Results of convergence study for genus 1 grid-based evaluation of the moment of inertia of a solid torus. Log–log plot of relative error versus grid refinement shows raw data and best-fit line.

Grahic Jump Location
Fig. 5

Results of convergence study for genus 1 grid-based evaluation of the path length of a sphere-sphere intersection curve. Log–log plot of relative error versus grid refinement shows raw data and best-fit line.

Grahic Jump Location
Fig. 6

Visualization of stoichiometric surface implicitly defined by a 512 × 512 grid of direct numerical simulation data. Area and area density were successfully computed using the grid-based surface integral method [30].

Grahic Jump Location
Fig. 7

Computed volume/actual volume versus number of points for a the calculation of the volume of a torus, comparing a Monte Carlo method with direct integration. Direct integration results are within ±0.0005 in under 1 million points, where some Monte Carlo runs remain outside that range at 7 million points. (a) Volume error versus number of points and (b) detail of (a).

Grahic Jump Location
Fig. 8

Timing comparison of calculation of torus volume with wavelet genus 1 in number of points evaluated versus execution time (ms). Indicated are timings for direct integration on the CPU and GPU (including full execution time and kernel time) as well as Monte Carlo timing.




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