Research Papers

A 149 Line Homogenization Code for Three-Dimensional Cellular Materials Written in matlab

[+] Author and Article Information
Guoying Dong

Department of Mechanical Engineering,
McGill University,
817 Rue Sherbrooke Ouest, Room G53,
Montréal, QC H3A 0C3, Canada
e-mail: guoying.dong@mail.mcgill.ca

Yunlong Tang

Department of Mechanical Engineering,
McGill University,
817 Rue Sherbrooke Ouest, Room G53,
Montréal, QC H3A 0C3, Canada
e-mail: tang.yunlong@mail.mcgill.ca

Yaoyao Fiona Zhao

Department of Mechanical Engineering,
McGill University,
817 Rue Sherbrooke Ouest, Room 148,
Montréal, QC H3A 0C3, Canada
e-mail: yaoyao.zhao@mcgill.ca

1Corresponding author.

Contributed by the Materials Division of ASME for publication in the JOURNAL OF ENGINEERING MATERIALS AND TECHNOLOGY. Manuscript received October 25, 2017; final manuscript received May 24, 2018; published online July 10, 2018. Assoc. Editor: Toshio Nakamura.

J. Eng. Mater. Technol 141(1), 011005 (Jul 10, 2018) (11 pages) Paper No: MATS-17-1316; doi: 10.1115/1.4040555 History: Received October 25, 2017; Revised May 24, 2018

Cellular architectures are promising in a variety of engineering applications due to attractive material properties. Additive manufacturing has reduced the difficulty in the fabrication of three-dimensional (3D) cellular materials. In this paper, the numerical homogenization method for 3D cellular materials is provided based on a short, self-contained matlab code. It is an educational description that shows how the homogenized constitutive matrix is computed by a voxel model with one material to be void and another material to be solid. A voxel generation algorithm is proposed to generate the voxel model easily by the wireframe scripts of unit cell topologies. The format of the wireframe script is defined so that the topology can be customized. The homogenization code is then extended to multimaterial cellular structures and thermal conductivity problems. The result of the numerical homogenization shows that different topologies exhibit anisotropic elastic properties to a different extent. It is also found that the anisotropy of cellular materials can be controlled by adjusting the combination of materials.

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Fig. 1

Wireframe of seven topologies

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Fig. 2

An example of the voxel model of a lattice structure with cubic topology: (a) perspective view, (b) top view of each layers, and (c) 3D logical matrix

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Fig. 3

Two cases when calculating the minimum distance from voxel center to the line segment of lattice wireframe

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Fig. 4

Voxel models of seven topologies

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Fig. 5

Local node numbers and the natural coordinate of hexahedron element

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Fig. 6

An example of hexahedron mesh used to model the unit cell: (a) the ID of each element, (b) the nonperiodic node number, (c) global node number of element 1 compared to local node number, and (d) periodic node numbers

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Fig. 7

An example of periodic cellular materials with randomized unit cell: (a) periodic Voronoi structure, (b) one unit cell, and (c) voxel model

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Fig. 8

Effective Young's modulus of lattice structures with 10% relative density

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Fig. 9

Effective Young's modulus of lattice structures with 30% relative density

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Fig. 10

Effective Young's modulus of lattice structures with 50% relative density

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Fig. 11

The relationship between the resolution of the voxel and the axial Young's modulus

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Fig. 12

The convergence plot for PCG method with different preconditioners

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Fig. 13

A three-material lattice cellular structure with cubic-center topologies

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Fig. 14

The Young's modulus surface of cubic-center lattice with (a) multimaterials and (b) single material




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