Calculation of effective conductivity of 2D and 3D composite materials with anisotropic constituents and different inclusion shapes in Mathematica

Abstract

The study of the effective properties of composite materials with anisotropic constituents and different inclusion shapes has motivated the development of the Mathematica 6.0 package “CompositeMaterials”. This package can be used to calculate the effective anisotropic conductivity tensor of two-phase composites. Any fiber cross section, even percolating ones, can be studied in the 2D composites. “Rectangular Prism” and “Ellipsoidal” inclusion shapes with arbitrary orientations can be investigated in the 3D composites. This package combines the Asymptotic Homogenization Method and the Finite Element Method in order to obtain the effective conductivity tensor. The commands and options of the package are illustrated with two sample applications for two- and three-dimensional composites. Program summary Program title:CompositeMaterials Catalogue identifier:AEAU_v1_0 Program summary URL: http://cpc.cs.qub.ac.uk/summaries/AEAU_v1_0.html Program obtainable from:CPC Program Library, Queen's University, Belfast, N. Ireland Licensing provisions:Standard CPC licence, http://cpc.cs.qub.ac.uk/licence/licence.html No. of lines in distributed program, including test data, etc.:132 183 No. of bytes in distributed program, including test data, etc.:1 334 908 Distribution format:tar.gz Programming language:Mathematica 6.0 Computer:Any that can run Mathematica 6.0 and where the open-source free C-programs Triangle ( http://www.cs.cmu.edu/~quake/triangle.html) and TetGen ( http://tetgen.berlios.de/) can be compiled and executed. Tested in Intel Pentium computers. Operating system:Any that can run Mathematica 6.0 and where the open-source free C-programs Triangle ( http://www.cs.cmu.edu/~quake/triangle.html) and TetGen ( http://tetgen.berlios.de/) can be compiled and executed. Tested in Windows XP. RAM:Small two-dimensional calculations require less than 100 MB. Large three-dimensional calculations require 500 MB or more. Classification:7.9 External routines:One Mathematica Add-on and two external programs: The free Mathematica Add-On IMS ( http://www.imtek.uni-freiburg.de/simulation/Mathematica/IMSweb/), The open-source free C-program Triangle ( http://www.cs.cmu.edu/~quake/triangle.html). The open-source free C-program TetGen ( http://tetgen.berlios.de/). The distribution file contains Windows executables for Triangle and TetGen. Nature of problem:The calculation of effective thermal conductivity tensor for two-dimensional and three-dimensional composite materials with anisotropic constituents and different inclusion shapes. Solution method:Asymptotic Homogenization Method, with the Cell Problems solved with Finite Element Method. Unusual features:Different inclusion shapes can be easily created. The constituents can be anisotropic. The intermediate stages and the final results can be graphed and analyzed with all the power of Mathematica 6.0. The use of the external meshing programs Triangle and TetGen is totally transparent for the end user. A typical calculation requires the use of only four special commands that follow standard Mathematica syntax. Additional comments:The executable binary files for Triangle and TetGen must be accessible from the directory specified by Mathematica's variable $HomeDirectory. The IMS add-on and the CompositeMaterials package, which is the package presented in this work, must be installed in the directory specified by Mathematica's variable $BaseDirectory or in the variable $UserBaseDirectory. The 2D calculations of Composite Materials will run successfully in Mathematica 5.2 and 6.0 but for the 3D calculations it is necessary to use Mathematica 6.0 or higher. Running time:Simple two-dimensional calculations can be done in less than a minute. Complex three-dimensional calculations can take an hour or more.