PSI - Issue 52

Vinit Vijay Deshpande et al. / Procedia Structural Integrity 52 (2024) 391–400 Author name / Structural Integrity Procedia 00 (2019) 000 – 000

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microstructure in the form of another neural network trained on a smaller volume element. Predictions of such a model are much better than the one trained directly on a larger volume element. Since the computational expense to generate data on smaller volume element is significantly lesser than the one on larger volume element, this strategy can lead to surrogate models that predict much better and require less expensive training data. 2. Microstructure reconstruction The real foam microstructure (refer Fig. 1a) resembled a homogeneous distribution of interpenetrating pores. The solid volume fraction of the foam was 0.245. Since the microstructure was random in nature, it was imperative to generate an ensemble of microstructures to understand its statistical behavior. In order to do that, a novel artificial microstructure reconstruction algorithm was developed in Deshpande et al. (2021). It is an optimization algorithm based on Yeong-Torquato (YT) method proposed in Jiao et al. (2008). The YT method is a simulated annealing method to reconstruct the microstructure image of 2 phase systems. In each iteration, random pairs of pixels belonging to opposite phases are selected and switched till the statistical correlation functions of the iterated microstructure matched to that of the target microstructure. In Deshpande et al. (2021), this method was modified this method to take into account the peculiar nature of the present microstructure. Instead of choosing an initial microstructure image as a random distribution of pixels, an initial image with a random distribution of pores whose sizes follow that of the target microstructure. In each iteration, a certain number of pores are randomly selected and their position changed till the statistical correlation functions of the iterated microstructure match to that of the target microstructure. This modification results in the number of required iterations to reduce to hundreds instead of thousands which would have otherwise required in a YT algorithm. The final reconstructed microstructure is shown in Fig. 1b. The objective of the optimization algorithm is defined as an energy functional, E such that E=∑ ∑ [ ( ) − ̅ ( )] 2 , where ( ) and ̅ ( ) are statistical correlation functions of the ℎ order and type α. They are a function of distance in iterated and target microstructures respectively. α indicates number of corr elation functions defined in the energy functional and indicates weight assigned to each function. Three correlation functions namely 2-point correlation function, 2-point cluster correlation function and lineal path function are selected to be used in the objective function. These functions calculated for the target microstructure are given in Fig. 1c.

Fig. 1. 3D binary image of (a) target and (b) reconstructed microstructure; (c) statistical correlation functions of target microstructure.

3. Biaxial compression failure simulation Our recent article Deshpande and Piat (2022) explored in detail the numerical methodology to study uniaxial compression failure of alumina foam taking into consideration the domain size effect and the statistical variation in