# American Institute of Mathematical Sciences

March  2012, 7(1): 113-126. doi: 10.3934/nhm.2012.7.113

## Robustness of finite element simulations in densely packed random particle composites

 1 Humboldt-Universität zu Berlin, Institut für Mathematik, Unter den Linden 6, 10099 Berlin, Germany

Received  July 2011 Revised  October 2011 Published  February 2012

This paper presents some weighted $H^2$-regularity estimates for a model Poisson problem with discontinuous coefficient at high contrast. The coefficient represents a random particle reinforced composite material, i.e., perfectly conducting circular particles are randomly distributed in some background material with low conductivity. Based on these regularity results we study the percolation of thermal conductivity of the material as the volume fraction of the particles is close to the jammed state. We prove that the characteristic percolation behavior of the material is well captured by standard conforming finite element models.
Citation: Daniel Peterseim. Robustness of finite element simulations in densely packed random particle composites. Networks & Heterogeneous Media, 2012, 7 (1) : 113-126. doi: 10.3934/nhm.2012.7.113
##### References:
 [1] I. Babuška and B. Q. Guo, Regularity of the solution of elliptic problems with piecewise analytic data. II: The trace spaces and application to the boundary value problems with nonhomogeneous boundary conditions, SIAM J. Math. Anal., 20 (1989), 763-781. doi: 10.1137/0520054.  Google Scholar [2] L. Berlyand and A. Kolpakov, 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 (2001), 179-227. doi: 10.1007/s002050100142.  Google Scholar [3] L. Berlyand and A. Novikov, Error of the network approximation for densely packed composites with irregular geometry, SIAM J. Math. Anal., 34 (2002), 385-408. doi: 10.1137/S0036141001397144.  Google Scholar [4] L. Borcea and G. C. Papanicolaou, Network approximation for transport properties of high contrast materials, SIAM J. Appl. Math., 58 (1998), 501-539. doi: 10.1137/S0036139996301891.  Google Scholar [5] G. A. Chechkin, Yu. O. Koroleva and L.-E. Persson, On the precise asymptotics of the constant in Friedrich's inequality for functions vanishing on the part of the boundary with microinhomogeneous structure, J. Inequal. Appl., 2007, Art. ID 34138, 13 pp.  Google Scholar [6] L. C. Evans, "Partial Differential Equations,'' 2nd edition, Graduate Studies in Mathematics, 19, American Mathematical Society, Providence, RI, 2010.  Google Scholar [7] J. M. Melenk, "$hp$-Finite Element Methods for Singular Perturbations,'' Lecture Notes in Mathematics, 1796, Springer-Verlag, Berlin, 2002.  Google Scholar [8] D. Peterseim, Generalized delaunay partitions and composite material modeling, preprint, DFG Research Center Matheon Berlin, 690 (2010). Google Scholar [9] D. Peterseim, Triangulating a system of disks, in "Proc. 26th European Workshop on Computational Geometry,'' (2010), 241-244. Google Scholar [10] D. Peterseim and C. Carstensen, Finite element network approximation of conductivity in particle composites, preprint, DFG Research Center Matheon Berlin, 807 (2010). Google Scholar

show all references

##### References:
 [1] I. Babuška and B. Q. Guo, Regularity of the solution of elliptic problems with piecewise analytic data. II: The trace spaces and application to the boundary value problems with nonhomogeneous boundary conditions, SIAM J. Math. Anal., 20 (1989), 763-781. doi: 10.1137/0520054.  Google Scholar [2] L. Berlyand and A. Kolpakov, 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 (2001), 179-227. doi: 10.1007/s002050100142.  Google Scholar [3] L. Berlyand and A. Novikov, Error of the network approximation for densely packed composites with irregular geometry, SIAM J. Math. Anal., 34 (2002), 385-408. doi: 10.1137/S0036141001397144.  Google Scholar [4] L. Borcea and G. C. Papanicolaou, Network approximation for transport properties of high contrast materials, SIAM J. Appl. Math., 58 (1998), 501-539. doi: 10.1137/S0036139996301891.  Google Scholar [5] G. A. Chechkin, Yu. O. Koroleva and L.-E. Persson, On the precise asymptotics of the constant in Friedrich's inequality for functions vanishing on the part of the boundary with microinhomogeneous structure, J. Inequal. Appl., 2007, Art. ID 34138, 13 pp.  Google Scholar [6] L. C. Evans, "Partial Differential Equations,'' 2nd edition, Graduate Studies in Mathematics, 19, American Mathematical Society, Providence, RI, 2010.  Google Scholar [7] J. M. Melenk, "$hp$-Finite Element Methods for Singular Perturbations,'' Lecture Notes in Mathematics, 1796, Springer-Verlag, Berlin, 2002.  Google Scholar [8] D. Peterseim, Generalized delaunay partitions and composite material modeling, preprint, DFG Research Center Matheon Berlin, 690 (2010). Google Scholar [9] D. Peterseim, Triangulating a system of disks, in "Proc. 26th European Workshop on Computational Geometry,'' (2010), 241-244. Google Scholar [10] D. Peterseim and C. Carstensen, Finite element network approximation of conductivity in particle composites, preprint, DFG Research Center Matheon Berlin, 807 (2010). Google Scholar
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