Keywords: aggregate particle, concrete specimen, spherical harmonic analysis, surface triangulation, region triangulation, advancing front technique.

The design of concrete with specified properties became of increasing importance with the wide use of high-performance concretes (HPCs), such as pumpable concrete or self compacting concrete (SCC). Many concrete properties, starting from the mechanical properties as the compressive strength and modulus of elasticity, over the rheological properties influencing the workability of fresh concrete, up to physical properties such as diffusivity and thermal and electric conductivity, for example, can be assessed by an appropriate computational model representing concrete as a multiscale random composite material with realistically described aggregates. However, incorporation of three dimensional aggregate particles into a computational code requires their proper discretization. This is not straightforward due to rather difficult mathematical characterization of aggregate particles of random shape.

Modern technologies 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 (without important features being captured) or too fine (with unimportant features captured by excessive number of elements). To make the fine discretization appropriate for numerical analysis, it has to be further processed. One alternative [2] is to adapt the triangulation by successive modifications using a set of geometrical and topological operators according to the desired resolution. Another way is to employ the original triangulation only as the control grid from which a smooth surface is reconstructed (using an appropriate subdivision technique) and then subjected to retriangulation [3] complying with the desired resolution.

In the present work, the digital representation is first used to derive a smooth representation of aggregate particle using the expansion into spherical harmonic functions [4]. Although this representation is not universal it is suitable for almost all aggregates used in structural concrete. The significant advantage of this approach is that the resolution of the smooth representation can be flexibly controlled by the number of terms in the expansion. In the next phase, the surface of aggregate particle is subjected to discretization using the advancing front technique (AFT). Although the representation of the surface is parameterized (by two spherical angles), the actual triangulation is performed directly on the surface in the real space [5] and not in 2D parametric space with subsequent mapping to the real space. The advantage of this procedure consists in the fact that the anisotropic meshing of the parametric space as well as the demanding calculations related to the reparameterization or the inverse mapping are avoided. The singular points (at the top and bottom of the aggregate particle) and their immediate surroundings are covered by a patch of elements sharing a single node coinciding with the singular point. The edges on the perimeter of the patch form the initial front for the startup of the AFT. Finally, the obtained surface triangulation is used as the initial boundary triangulation for subsequent tetrahedrization of the particle volume using a truly three-dimensional AFT [6].

The performance of the presented discretization algorithm is demonstrated on the example of a cylindrical concrete specimen with a few different aggregate particles. This example reveals that the adopted algorithms are capable to produce high quality uniform and graded tetrahedral meshes that can be incorporated into various computational models.

1

W.E. Lorensen and 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

D. Rypl and 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

4

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

5

D. Rypl, "Approaches to Discretization of 3D Surfaces", CTU Reports, 7 (2), 2003.

6

D. Rypl, "Sequential and Parallel Generation of Unstructured 3D Meshes", CTU Reports, 2 (3), 1998.

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