Numerical algorithm of multipole expansion method for conductivity of ellipsoidal particle composite

Abstract The paper addresses the numerical implementation of the multipole expansion method in application to the conductivity problem for the ellipsoidal particle composite with isotropic constituents and imperfect interfaces. The main steps of the numerical algorithm are considered in detail including generation of the geometry model, evaluating the re-expansion coefficients, fulfilling the periodicity and interface conditions and determination of the series expansion coefficients by iterative solving a set of linear equations. For each step, the numerical tests are performed which illustrate an accuracy and efficiency of the method. The rational computational strategies and the ways of boosting the computer code performance are discussed. The algorithm-based code provides a fast and accurate analysis of the local potential fields and the effective conductivity of composite with an adequate account for the arrangement and orientation of inhomogeneities. The imperfect thermal contact between the matrix and inhomogeneities is taken into account accurately. The application area covers the polydisperse and multiphase composites of matrix type. Its extension to the ellipsoidal particle composites with anisotropic constituents is straightforward. Incorporation of this feature yields possibly the most general model of composite that may be considered in the framework of analytical approach, and the proposed numerical algorithm is by far the most efficient method for studying these models.

[1]  Martin Lévesque,et al.  A fully automated numerical tool for a comprehensive validation of homogenization models and its application to spherical particles reinforced composites , 2012 .

[2]  J. J. Alvarado-Gil,et al.  On the thermal conductivity of particulate nanocomposites , 2011 .

[3]  S. Torquato,et al.  Random Heterogeneous Materials: Microstructure and Macroscopic Properties , 2005 .

[4]  E. Hobson.  The Theory of Spherical and Ellipsoidal Harmonics , 1955 .

[5]  Salvatore Torquato,et al.  Determining elastic behavior of composites by the boundary element method , 1993 .

[6]  I. Sevostianov,et al.  Local fields and effective conductivity tensor of ellipsoidal particle composite with anisotropic constituents , 2017, Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences.

[7]  Jan Drewes Achenbach,et al.  Effect of interphases on micro and macromechanical behavior of hexagonal-array fiber composites , 1990 .

[8]  Aleksandar Donev,et al.  Neighbor list collision-driven molecular dynamics simulation for nonspherical hard particles. , 2005 .

[9]  L. Kantorovich,et al.  Approximate methods of higher analysis , 1960 .

[10]  Martin Lévesque,et al.  Random generation of periodic hard ellipsoids based on molecular dynamics: A computationally-efficient algorithm , 2013, J. Comput. Phys..

[11]  Touvia Miloh,et al.  On the effective conductivity of composites with ellipsoidal inhomogeneities and highly conducting interfaces , 1999, Proceedings of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences.

[12]  L Greengard,et al.  Fast Algorithms for Classical Physics , 1994, Science.

[13]  Hervé Moulinec,et al.  A numerical method for computing the overall response of nonlinear composites with complex microstructure , 1998, ArXiv.

[14]  Edward J. Garboczi,et al.  An algorithm for computing the effective linear elastic properties of heterogeneous materials: Three-dimensional results for composites with equal phase poisson ratios , 1995 .

[15]  Peter Wriggers,et al.  An Introduction to Computational Micromechanics , 2004 .

[16]  F. Stillinger,et al.  Neighbor list collision-driven molecular dynamics simulation for nonspherical hard particles. II. Applications to ellipses and ellipsoids , 2005 .

[17]  Robert Lipton,et al.  Variational methods, bounds and size effects for composites with highly conducting interface , 1997 .

[18]  I. Sevostianov,et al.  Maxwell homogenization scheme as a rigorous method of micromechanics: Application to effective conductivity of a composite with spheroidal particles , 2016 .

[19]  R. Brenner,et al.  Random distribution of polydisperse ellipsoidal inclusions and homogenization estimates for porous elastic materials , 2018, Computers & Structures.

[20]  F. Pavanello,et al.  Generalized interface models for transport phenomena: unusual scale effects in composite nanomaterials , 2012, 1212.4133.

[21]  Anhua Lin,et al.  On the Distance between Two Ellipsoids , 2002, SIAM J. Optim..

[22]  H. Hasimoto.  On the periodic fundamental solutions of the Stokes equations and their application to viscous flow past a cubic array of spheres , 1959, Journal of Fluid Mechanics.

[23]  B. Klusemann,et al.  Generation of 3D representative volume elements for heterogeneous materials: A review , 2018, Progress in Materials Science.

[24]  A. Stroud,et al.  Approximate Calculation of Integrals , 1962 .

[25]  Javier Segurado,et al.  Computational micromechanics of composites: The effect of particle spatial distribution , 2006 .

[26]  I. Sevostianov,et al.  Maxwell's methodology of estimating effective properties: Alive and well. , 2019, International journal of engineering science.

[27]  Leslie Greengard,et al.  A renormalization method for the evaluation of lattice sums , 1994 .

[28]  V. Kushch.  Interacting ellipsoidal inhomogeneities by multipole expansion method and effective conductivity of particulate composite , 2017 .

[29]  P. P. Ewald.  Die Berechnung optischer und elektrostatischer Gitterpotentiale , 1921 .

[30]  Toshio Mura,et al.  Micromechanics of defects in solids , 1982 .

[31]  George Dassios,et al.  Ellipsoidal Harmonics: Theory and Applications , 2012 .

[32]  L. Dagum,et al.  OpenMP: an industry standard API for shared-memory programming , 1998 .

[33]  Y. Saad,et al.  GMRES: a generalized minimal residual algorithm for solving nonsymmetric linear systems , 1986 .

[34]  Volodymyr I. Kushch,et al.  Elastic ellipsoidal inhomogeneity with imperfect interface: Complete displacement solution in terms of ellipsoidal harmonics , 2019, International Journal of Solids and Structures.

[35]  D. Hasselman,et al.  Effective Thermal Conductivity of Composites with Interfacial Thermal Barrier Resistance , 1987 .

[36]  Touvia Miloh,et al.  The effective conductivity of composites with imperfect thermal contact at constituent interfaces , 1986 .

[37]  D. Jeulin,et al.  Determination of the size of the representative volume element for random composites: statistical and numerical approach , 2003 .

[38]  J. Banavar,et al.  Computer Simulation of Liquids , 1988 .