In this contribution we propose a data-driven surrogate model for the prediction of permeabilities and laminar flow through two-dimensional random micro-heterogeneous materials; here Darcy’s law is used. The philosophy of the proposed scheme is to provide a large number of training sets through a numerically “cheap” (stochastic) model instead of using an “expensive” (FEM) one. In order to achieve an efficient computational tool for the generation of the database (up to $$10^3$$ 10 3 and much more realizations), needed for the training of the neural networks, we apply a stochastic model based on the Brownian motion. An efficient algebraic algorithm compared to a classical Monte Carlo approach is based on the evaluation of stochastic transition matrices. For the encoding of the microstructure and the optimization of the surrogate model, we compare two architectures, the so-called UResNet model and the Fourier Convolutional Neural Network (FCNN). Here we analyze two FCNNs, one based on the discrete cosine transformation and one based on the complex-valued discrete Fourier transformation. Finally, we compare the flux fields and the permeabilities for independent microstructures (not used in the training set) with results from the $$\hbox {FE}^2$$ FE 2 method, a numerical homogenization scheme, in order to demonstrate the efficiency of the proposed surrogate model.
In this contribution we present an approach to generate a data driven surrogate model for the prediction of permeabilities and flow through two dimensional random micro-heterogeneous materials. The laminar flow is well described by Darcy's law. In order to achieve an efficient computational tool for the generation of the database (up to 10 3 realizations), needed for the training of the neural networks, we apply a stochastic model based on the Brownian motion. The stationary state of the resulting stochastic model solves the Darcy equation and can be iteratively solved by a Monte Carlo approach applied to a particle simulation. Improved numerical efficiency can be yield by usage of the related transition matrix.
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