Keywords: microstructure, digital representation, mesh generation, interpolating recursive subdivision, advancing front technique.

The present paper deals with the modelling of multi-phase microstructures. Modern technologies such as computer tomography (CT) or magnetic resonance tomography (MRT) offer a powerful nondestructive technique for digital representation of opaque solid objects. This voxel based representation can be discretized using for example the marching cubes algorithm [1]. The resolution of the resulting triangulation, however, is strongly dependent on the resolution of the digital representation which might be either too coarse or too fine. To make the fine discretization appropriate for numerical analysis, it has to be further processed. One choice [2] is to adapt the triangulation by successive modifications using a set of geometrical and topological operators according to the desired resolution. Alternatively, the digital representation may be first used to derive a smooth representation which is then subjected to triangulation of a variable resolution. In [3], the smooth representation is recovered using the spherical harmonic analysis. However, this method is limited to star-shaped objects without internal voids and cannot be therefore applied to a general microstructure.

In this work a different approach is adopted. The gray scale digital representation is firstly thresholded into voxels of appropriate discrete values of gray corresponding to individual phases of the processed microstructure. In the next phase, the boundary voxels of individual phases are identified. A triangulated boundary representation is then obtained from the boundary voxel representation by replacing the boundary sides of boundary voxels by boundary triangulation derived from the corners and centres of those boundary sides. This triangulation is then subjected to recursive interpolating subdivision [4] yielding a C¹ continuous surface. In the final phase, the smooth boundary surface is triangulated using the advancing front technique [5]. Since there is available no global mapping of the recovered surface, the discretization is performed directly in three-dimensional space on the surface. Note that the resolution of the final triangulation is independent of the resolution of the initial digital representations and is driven mainly by the user specification and properties (curvature) of the recovered smooth representation. Finally, the interior of the microstructure is discretized by the advancing front technique using the microstructure surface triangulation as the initial front. The algorithm is capable of producing high quality uniform and graded tetrahedral meshes that can be incorporated into various computational models.

1

W.E. Lorensen, H.E. Cline, "Marching cubes: A high resolution 3D surface construction algorithm", Computer Graphics, 21, 163-169, 1987. doi:10.1145/37402.37422

2

P. Frey, "About surface remeshing", In "Proceedings of 9th International Meshing Roundtable", 2000.

3

E.J. Garboczi, "Three-dimensional mathematical analysis of particle shape using X-ray tomography and spherical harmonics: Application to aggregate used in concrete", Cement and Concrete Research, 32, 1621-1638, 2002. doi:10.1016/S0008-8846(02)00836-0

4

D. Zorin, P. Schröder, W. Sweldens, "Interpolating subdivision for meshes with arbitrary topology", in "Computer Graphics Proceedings (SIGGRAPH '96)", 189-192, 1996.

5

D. Rypl, Z. Bittnar, "Triangulation of 3D surfaces reconstructed by interpolating subdivision", Computers and Structures, 82 (23-26), 2093-2103, 2004. doi:10.1016/j.compstruc.2004.03.064

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