Keywords: Keywords: lineal path function, homogenization, statistically equivalent periodic unit cell, graphics processing unit.

This paper concerns the modelling of heterogeneous materials consisting of more material components or from one material in a different physical state of decay, saturation etc. When we can separate those components of material or its states knowing the properties of the particular components, we can predict the behaviour of a whole heterogeneous material. However, we have to take into account the phase distribution and its influence on both physical and macroscopic properties and such information cannot be obtained from a single sample of the material. To study the material behaviour more generally, we would need a set of testing samples and expensive experimental tests. To avoid this we use a few samples and their images to create a new artificial material with similar structural information.

The unifying theoretical framework is provided by homogenization theories, which are the replacement of the heterogeneous microstructure with an equivalent homogeneous material. It is generally accepted that detailed discretisation techniques, and the finite element method (FEM) in particular, remain the most powerful and flexible tools available. In spite of the tedious computation time, it provides us details of the local stress and strain fields. Moreover, it is convenient to characterize the material heterogeneity by introducing the concept of a periodic unit cell (PUC). When dealing with the complex random microstructures, the unit cell representing exactly the periodic morphology needs to be replaced by a statistically equivalent periodic unit cell (SEPUC) preserving the important material properties in a statistical manner.

Nowadays there are plenty of statistical descriptors that differ from each other by their computational load, mathematical description and the information involved. The output is always a kind of matrix with values corresponding to the probability of finding such a phase at a particular place given by the exact formula of the statistical descriptor. One of the most commonly used group of descriptors embodies a set of general n-point probability functions, applicable to an arbitrary two-phase composite.

Our objective is to describe and reconstruct a medium with considerable phase connection. Therefore we utilize the lineal path descriptor. The SEPUC can be then created by randomly generated modifications of an arbitrary starting geometry within the process minimizing the difference between the lineal path obtained and the one corresponding to the original medium. The lineal path is a low-order statistical descriptor based on a more complex fundamental function able to capture certain information about the phase connectivity. Its main disadvantage is the computational cost. Note that the optimization process needs to evaluate this function in the range of millions of iterations depending on a sample size and complexity. The acceleration of the objective function was done in several software and hardware optimization steps. In this contribution, we present the reformulation of the sequential C++ code for evaluation of the lineal path function into the parallel C++ code with compute unified device architecture (CUDA) extensions enabling the usage of computational potential of the NVIDIA graphics processing unit (GPU).

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