Numerische Mathematik

, Volume 124, Issue 1, pp 73–97 | Cite as

Finite element network approximation of conductivity in particle composites

Article

Abstract

A new finite element method computes conductivity in some unstructured particle-reinforced composite material. The 2-phase material under consideration is composed of a poorly conducting matrix material filled by highly conducting circular inclusions which are randomly dispersed. The mathematical model is a Poisson-type problem with discontinuous coefficients. The discontinuities are huge in contrast and quantity. The proposed method generalizes classical continuous piecewise affine finite elements to special computational meshes which encode the particles in a network structure. Important geometric parameters such as the volume fraction are preserved exactly. The computational complexity of the method is (almost) proportional to the number of inclusions. This is minimal in the sense that the representation of the underlying geometry via the positions and radii of the inclusions is of the same complexity. The discretization error is proportional to the distance of neighboring inclusions and independent of the conductivity contrast in the medium.

Mathematics Subject Classification (2000)

65N15 65N30 74Q20 

References

  1. 1.
    Bebendorf, M.: Why finite element discretizations can be factored by triangular hierarchical matrices. SIAM J. Numer. Anal. 45(4), 1472–1494 (2007)MathSciNetMATHCrossRefGoogle Scholar
  2. 2.
    Börm, S.: Approximation of solution operators of elliptic partial differential equations by \({\cal {H}}\)- and \({\cal {H}}^{2}\)-matrices. Numer. Math. 115(2), 165–193 (2010)MathSciNetMATHCrossRefGoogle Scholar
  3. 3.
    Berlyand, L., Kolpakov, A.: Network approximation in the limit of small interparticle distance of the effective properties of a high-contrast random dispersed composite. Arch. Ration. Mech. Anal. 159(3), 179–227 (2001)MathSciNetMATHCrossRefGoogle Scholar
  4. 4.
    Berlyand, L., Novikov, A.: Error of the network approximation for densely packed composites with irregular geometry. SIAM J. Math. Anal. 34(2), 385–408 (2002) (electronic)Google Scholar
  5. 5.
    Borcea, L., Papanicolaou, G.C.: Network approximation for transport properties of high contrast materials. SIAM J. Appl. Math 58, 501–539 (1998)Google Scholar
  6. 6.
    Brenner, S.C., Scott, L.R.: The mathematical theory of finite element methods. In: Texts in Applied Mathematics, 3rd edn, vol. 15. Springer, New York (2008)Google Scholar
  7. 7.
    Chu, C.-C., Graham, I.G., Hou, T.Y.: A new multiscale finite element method for high-contrast elliptic interface problems. Math. Comput. 79, 1915–1955 (2010)MathSciNetMATHCrossRefGoogle Scholar
  8. 8.
    Ciarlet, P.: The Finite Element Method for Elliptic Problems. North Holland, Amsterdam (1978)MATHGoogle Scholar
  9. 9.
    Davis, T.A.: Direct methods for sparse linear systems. In: Fundamentals of Algorithms, vol. 2. Society for Industrial and Applied Mathematics (SIAM), Philadelphia (2006)Google Scholar
  10. 10.
    Delaunay, B.: Sur la sphère vide. Izvestia Akademii Nauk SSSR, Otdelenie Matematicheskikh i Estestvennykh Nauk 7, 793–800 (1934)Google Scholar
  11. 11.
    Weinan, E., Engquist, B.: The heterogeneous multiscale methods. Commun. Math. Sci. 1(1), 87–132 (2003)MathSciNetMATHGoogle Scholar
  12. 12.
    Eigel, M., Peterseim, D.: Network FEM for Composite Materials with A Posteriori Control DFG Research Center Matheon Berlin, Preprint Series, vol. 985 (2012)Google Scholar
  13. 13.
    Fortune, S.: A sweepline algorithm for Voronoĭ diagrams. Algorithmica 2(2), 153–174 (1987)MathSciNetMATHCrossRefGoogle Scholar
  14. 14.
    Gavrilova, M., Rokne, J.: Swap conditions for dynamic Voronoi diagrams for circles and line segments. Comput. Aided Geom. Design 16(2), 89–106 (1999)MathSciNetMATHCrossRefGoogle Scholar
  15. 15.
    George, A., Liu, J.: Computer Solution of Large Sparse Positive Definite Systems. Prentice-Hall, Englewood Cliffs (1981)MATHGoogle Scholar
  16. 16.
    Hou, T.Y., Wu, X.-H.: A multiscale finite element method for elliptic problems in composite materials and porous media. J. Comput. Phys. 134, 169–189 (1997)MathSciNetMATHCrossRefGoogle Scholar
  17. 17.
    Hughes, T.J.R., Feijóo, G.R., Mazzei, L., Quincy, J.-B.: The variational multiscale method—a paradigm for computational mechanics. Comput. Methods Appl. Mech. Eng. 166(1–2), 3–24 (1998)Google Scholar
  18. 18.
    Kim, D.-S., Kim, D., Sugihara, K.: Voronoi diagram of a circle set from Voronoi diagram of a point set. I. Topology. Comput. Aided Geom. Design 18(6), 541–562 (2001)MathSciNetMATHCrossRefGoogle Scholar
  19. 19.
    Kolpakov, A.A., Kolpakov, A.G.: Capacity and transport in contrast composite structures. CRC Press, Boca Raton (2010)Google Scholar
  20. 20.
    Larson, M.G., Målqvist, A.: Adaptive variational multiscale methods based on a posteriori error estimation: energy norm estimates for elliptic problems. Comput. Methods Appl. Mech. Eng. 196(21–24), 2313–2324 (2007)MATHCrossRefGoogle Scholar
  21. 21.
    Målqvist, A., Peterseim, D.: Localization of Elliptic Multiscale Problems. ArXiv e-prints, 1110.0692 (2011)Google Scholar
  22. 22.
    Mao, S., Nicaise, S., Shi, Z.-C.: On the interpolation error estimates for \(Q_1\) quadrilateral finite elements. SIAM J. Numer. Anal. 47(1), 467–486 (2008)MathSciNetCrossRefGoogle Scholar
  23. 23.
    Peterseim, D.: Generalized Delaunay partitions and composite material modeling. DFG Research Center Matheon Berlin, Preprint Series, vol. 690 (2010)Google Scholar
  24. 24.
    Peterseim, D.: Triangulating a system of disks. In: Proceedings of the EuroCG 2010. Dortmund, Germany (2010)Google Scholar
  25. 25.
    Peterseim, D.: Robustness of Finite Element Simulations in Densely Packed Random Particle Composites. Netw. Heterog Media 7(1), 113–126 (2012)Google Scholar
  26. 26.
    Stein, E.M.: Singular Integrals and Differentiablity Properties of Function. Priceton Univ. Press, New York (1970)Google Scholar
  27. 27.
    Voronoi, G.F.: Nouvelles applications des paramètres continus à la théorie des formes quadratiques. Journal für die Reine und Angewandte Mathematik 133, 97–178 (1907)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2012

Authors and Affiliations

  1. 1.Institut für MathematikHumboldt-Universität zu BerlinBerlinGermany
  2. 2.Department of CSEYonsei UniversitySeoulKorea

Personalised recommendations