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.
Similar content being viewed by others
References
Bebendorf, M.: Why finite element discretizations can be factored by triangular hierarchical matrices. SIAM J. Numer. Anal. 45(4), 1472–1494 (2007)
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)
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)
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)
Borcea, L., Papanicolaou, G.C.: Network approximation for transport properties of high contrast materials. SIAM J. Appl. Math 58, 501–539 (1998)
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)
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)
Ciarlet, P.: The Finite Element Method for Elliptic Problems. North Holland, Amsterdam (1978)
Davis, T.A.: Direct methods for sparse linear systems. In: Fundamentals of Algorithms, vol. 2. Society for Industrial and Applied Mathematics (SIAM), Philadelphia (2006)
Delaunay, B.: Sur la sphère vide. Izvestia Akademii Nauk SSSR, Otdelenie Matematicheskikh i Estestvennykh Nauk 7, 793–800 (1934)
Weinan, E., Engquist, B.: The heterogeneous multiscale methods. Commun. Math. Sci. 1(1), 87–132 (2003)
Eigel, M., Peterseim, D.: Network FEM for Composite Materials with A Posteriori Control DFG Research Center Matheon Berlin, Preprint Series, vol. 985 (2012)
Fortune, S.: A sweepline algorithm for Voronoĭ diagrams. Algorithmica 2(2), 153–174 (1987)
Gavrilova, M., Rokne, J.: Swap conditions for dynamic Voronoi diagrams for circles and line segments. Comput. Aided Geom. Design 16(2), 89–106 (1999)
George, A., Liu, J.: Computer Solution of Large Sparse Positive Definite Systems. Prentice-Hall, Englewood Cliffs (1981)
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)
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)
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)
Kolpakov, A.A., Kolpakov, A.G.: Capacity and transport in contrast composite structures. CRC Press, Boca Raton (2010)
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)
Målqvist, A., Peterseim, D.: Localization of Elliptic Multiscale Problems. ArXiv e-prints, 1110.0692 (2011)
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)
Peterseim, D.: Generalized Delaunay partitions and composite material modeling. DFG Research Center Matheon Berlin, Preprint Series, vol. 690 (2010)
Peterseim, D.: Triangulating a system of disks. In: Proceedings of the EuroCG 2010. Dortmund, Germany (2010)
Peterseim, D.: Robustness of Finite Element Simulations in Densely Packed Random Particle Composites. Netw. Heterog Media 7(1), 113–126 (2012)
Stein, E.M.: Singular Integrals and Differentiablity Properties of Function. Priceton Univ. Press, New York (1970)
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)
Author information
Authors and Affiliations
Corresponding author
Additional information
The work of D. Peterseim was supported by the DFG Research Center Matheon Berlin through project C33.
Rights and permissions
About this article
Cite this article
Peterseim, D., Carstensen, C. Finite element network approximation of conductivity in particle composites. Numer. Math. 124, 73–97 (2013). https://doi.org/10.1007/s00211-012-0509-1
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00211-012-0509-1