Communications in Mathematics and Statistics

, Volume 7, Issue 4, pp 459–474 | Cite as

Numerical Integration Over Implicitly Defined Domains with Topological Guarantee

  • Tianhui Yang
  • Ammar Qarariyah
  • Hongmei Kang
  • Jiansong DengEmail author


Numerical integration over the implicitly defined domains is challenging due to topological variances of implicit functions. In this paper, we use interval arithmetic to identify the boundary of the integration domain exactly, thus getting the correct topology of the domain. Furthermore, a geometry-based local error estimate is explored to guide the hierarchical subdivision and save the computation cost. Numerical experiments are presented to demonstrate the accuracy and the potential of the proposed method.


Isogeometric analysis Numerical integration Implicitly defined domains Topological guarantee Interval arithmetic Local error estimate Hierarchical subdivision 

Mathematics Subject Classification




We would like to thank the anonymous reviewers and our laboratory group for helpful discussions and comments. The work is supported by the National Natural Science Foundation of China (No. 11771420).


  1. 1.
    Barendrecht, P.J., Bartoň, M., Kosinka, J.: Efficient quadrature rules for subdivision surfaces in isogeometric analysis. Comput. Methods Appl. Mech. Eng. 333, 128–149 (2018)MathSciNetGoogle Scholar
  2. 2.
    Bartoň, M., Calo, V.M.: Gauss-galerkin quadrature rules for quadratic and cubic spline spaces and their application to isogeometric analysis. Comput. Aided Des. 82, 57–67 (2017)MathSciNetGoogle Scholar
  3. 3.
    Berntsen, J., Espelid, T.O., Genz, A.: An adaptive algorithm for the approximate calculation of multiple integrals. ACM Trans. Math. Softw. (TOMS) 17(4), 437–451 (1991)MathSciNetzbMATHGoogle Scholar
  4. 4.
    Caprani, O., Madsen, K., Rall, L.B.: Integration of interval functions. SIAM J. Math. Anal. 12(3), 321–341 (1981)MathSciNetzbMATHGoogle Scholar
  5. 5.
    Cheng, K.W., Fries, T.-P.: Higher-order xfem for curved strong and weak discontinuities. Int. J. Numer. Methods Eng. 82(5), 564–590 (2010)MathSciNetzbMATHGoogle Scholar
  6. 6.
    Cottrell, J.A., Hughes, T.J.R., Bazilevs, Y.: Isogeometric Analysis: Toward Integration of CAD and FEA. Wiley, New York (2009)zbMATHGoogle Scholar
  7. 7.
    Davis, P.J., Rabinowitz, P.: Methods of Numerical Integration. Courier Corporation, North Chelmsford (2007)zbMATHGoogle Scholar
  8. 8.
    Dokken, T., Skytt, V., Barrowclough, O.: Trivariate spline representations for computer aided design and additive manufacturing. arXiv preprint arXiv:1803.05756 (2018)
  9. 9.
    Dolbow, J., Belytschko, T.: Numerical integration of the galerkin weak form in meshfree methods. Comput. Mech. 23(3), 219–230 (1999)MathSciNetzbMATHGoogle Scholar
  10. 10.
    Drescher, L., Heumann, H., Schmidt, K.: A high order method for the approximation of integrals over implicitly defined hypersurfaces. SIAM J. Numer. Anal. 55(6), 2592–2615 (2017)MathSciNetzbMATHGoogle Scholar
  11. 11.
    Edalat, A., Krznaric, M.: Numerical Integration with Exact Real Arithmetic. Springer, Berlin (1999)zbMATHGoogle Scholar
  12. 12.
    Engwer, C., Nüßing, A.: Geometric integration over irregular domains with topologic guarantees. arXiv preprint arXiv:1601.03597 (2016)
  13. 13.
    Farin, G.E.: Curves and Surfaces for CAGD: A Practical Guide. Morgan Kaufmann, Burlington (2002)Google Scholar
  14. 14.
    Gautschi, W., Notaris, S.E.: Gauss–kronrod quadrature formulae for weight functions of bernstein–szegö type. J. Comput. Appl. Math. 25(2), 199–224 (1989)MathSciNetzbMATHGoogle Scholar
  15. 15.
    Gomes, A., Voiculescu, I., Jorge, J., Wyvill, B., Galbraith, C.: Implicit Curves and Surfaces: Mathematics, Data Structures and Algorithms. Springer, Berlin (2009)zbMATHGoogle Scholar
  16. 16.
    Hartshorne, R.: Algebraic Geometry, vol. 52. Springer, Berlin (2013)zbMATHGoogle Scholar
  17. 17.
    Hollig, K.: Finite Element Methods with B-Splines, vol. 26. Siam, New Delhi (2003)zbMATHGoogle Scholar
  18. 18.
    Höllig, K., Hörner, J.: Programming finite element methods with weighted b-splines. Comput. Math. Appl. 70(7), 1441–1456 (2015)MathSciNetGoogle Scholar
  19. 19.
    Huang, Pu, Wang, Charlie CL, Chen, Yong: Intersection-free and topologically faithful slicing of implicit solid. J. Comput. Inf. Sci. Eng. 13(2), 021009 (2013)Google Scholar
  20. 20.
    Li, Z., Ito, K.: The Immersed Interface Method: Numerical Solutions of PDEs Involving Interfaces and Irregular Domains, vol. 33. Siam, New Delhi (2006)zbMATHGoogle Scholar
  21. 21.
    Martin, R., Shou, H., Voiculescu, I., Bowyer, A., Wang, G.: Comparison of interval methods for plotting algebraic curves. Comput. Aided Geom. Des. 19(7), 553–587 (2002)MathSciNetzbMATHGoogle Scholar
  22. 22.
    Mitchell, D.P.: Robust ray intersection with interval arithmetic. In: Proceedings of Graphics Interface, vol. 90, pp. 68–74 (1990)Google Scholar
  23. 23.
    Moës, N., Dolbow, J., Belytschko, T.: A finite element method for crack growth without remeshing. Int. J. Numer. Methods Eng. 46(1), 131–150 (1999)MathSciNetzbMATHGoogle Scholar
  24. 24.
    Moore, R.: Interval Arithmetic. Prentice-Hall, Englewood Cliffs (1966)Google Scholar
  25. 25.
    Moore, R.E.: Methods and Applications of Interval Analysis, vol. 2. Siam, Philadelphia (1979)zbMATHGoogle Scholar
  26. 26.
    Moore, R.E.: Reliability in Computing: The Role of Interval Methods in Scientific Computing, vol. 19. Elsevier, Amsterdam (2014)Google Scholar
  27. 27.
    Müller, B., Kummer, F., Oberlack, M.: Highly accurate surface and volume integration on implicit domains by means of moment-fitting. Int. J. Numer. Methods Eng. 96(8), 512–528 (2013)MathSciNetzbMATHGoogle Scholar
  28. 28.
    Olshanskii, M.A., Safin, D.: Numerical integration over implicitly defined domains for higher order unfitted finite element methods. Lobachevskii J. Math. 37(5), 582–596 (2016)MathSciNetzbMATHGoogle Scholar
  29. 29.
    Peskin, C.S.: The immersed boundary method. Acta Numer. 11, 479–517 (2002)MathSciNetzbMATHGoogle Scholar
  30. 30.
    Press, W.H.: Numerical Recipes: The Art of Scientific Computing, 3rd edn. Cambridge University Press, Cambridge (2007)zbMATHGoogle Scholar
  31. 31.
    Rank, E., Ruess, M., Kollmannsberger, S., Schillinger, D., Düster, A.: Geometric modeling, isogeometric analysis and the finite cell method. Comput. Methods Appl. Mech. Eng. 249, 104–115 (2012)zbMATHGoogle Scholar
  32. 32.
    Rvachev, V.L., Shevchenko, A.N., Veretel’nik, V.V.: Numerical integration software for projection and projection-grid methods. Cybern. Syst. Anal. 30(1), 154–158 (1994)zbMATHGoogle Scholar
  33. 33.
    Saye, R.I.: High-order quadrature methods for implicitly defined surfaces and volumes in hyperrectangles. SIAM J. Sci. Comput. 37(2), A993–A1019 (2015)MathSciNetzbMATHGoogle Scholar
  34. 34.
    Shapiro, V., Tsukanov, I.: The architecture of sage—a meshfree system based on rfm. Eng. Comput. 18(4), 295–311 (2002)Google Scholar
  35. 35.
    Shou, H., Lin, H., Martin, R., Wang, G.: Modified affine arithmetic is more accurate than centered interval arithmetic or affine arithmetic. In: Mathematics of Surfaces, Springer, pp. 355–365 (2003)Google Scholar
  36. 36.
    Sukumar, N., Chopp, D.L., Moës, N., Belytschko, T.: Modeling holes and inclusions by level sets in the extended finite-element method. Comput. Methods Appl. Mech. Eng. 190(46–47), 6183–6200 (2001)MathSciNetzbMATHGoogle Scholar
  37. 37.
    Thiagarajan, V.: Shape Aware Quadratures. The University of Wisconsin-Madison, Madison (2017)zbMATHGoogle Scholar
  38. 38.
    Thiagarajan, V., Shapiro, V.: Adaptively weighted numerical integration over arbitrary domains. Comput. Math. Appl. 67(9), 1682–1702 (2014)MathSciNetzbMATHGoogle Scholar
  39. 39.
    Thiagarajan, V., Shapiro, V.: Adaptively weighted numerical integration in the finite cell method. Comput. Methods Appl. Mech. Eng. 311, 250–279 (2016)MathSciNetGoogle Scholar
  40. 40.
    Thiagarajan, V., Shapiro, V.: Shape aware quadratures. J. Comput. Phys. 374, 1239 (2018)MathSciNetzbMATHGoogle Scholar
  41. 41.
    Tornberg, A.-K., Engquist, B.: Numerical approximations of singular source terms in differential equations. J. Comput. Phys. 200(2), 462–488 (2004)MathSciNetzbMATHGoogle Scholar
  42. 42.
    Upreti, K., Subbarayan, G.: Signed algebraic level sets on nurbs surfaces and implicit boolean compositions for isogeometric cad-cae integration. Comput. Aided Des. 82, 112–126 (2017)MathSciNetGoogle Scholar
  43. 43.
    Wegst, U.G.K., Bai, H., Saiz, E., Tomsia, A.P., Ritchie, R.O.: Bioinspired structural materials. Nat. Mater. 14(1), 23 (2015)Google Scholar
  44. 44.
    Wolfe, M.A.: Interval enclosures for a certain class of multiple integrals. Appl. Math. Comput. 96(2–3), 145–159 (1998)MathSciNetzbMATHGoogle Scholar
  45. 45.
    Xu, G., Kwok, T.-H., Wang, C.C.L.: Isogeometric computation reuse method for complex objects with topology-consistent volumetric parameterization. Comput. Aided Des. 91, 1–13 (2017)Google Scholar
  46. 46.
    Xu, J., Sun, N., Shu, L., Rabczuk, T., Xu, G.: An improved isogeometric analysis method for trimmed geometries. arXiv preprint arXiv:1707.00323 (2017)

Copyright information

© School of Mathematical Sciences, University of Science and Technology of China and Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  • Tianhui Yang
    • 1
  • Ammar Qarariyah
    • 1
  • Hongmei Kang
    • 2
  • Jiansong Deng
    • 1
    Email author
  1. 1.School of Mathematical ScienceUniversity of Science and Technology of ChinaHefeiChina
  2. 2.School of Mathematical SciencesSoochow UniversitySuzhouChina

Personalised recommendations