Keywords: neural network, multiscale, homogenization, concrete.

Many phenomena in computational mechanics are characterized by multiscale systems. Properties on the macroscopic level are induced by the mesoscopic structure, e.g. the nonlinear behaviour of concrete on the macroscale is caused by microcracks on the mesoscale. For the numerical approach a model on the macroscopic level is often used, which does not consider the underlying physical processes on the mesoscale. On the contrary a multiscale approach offers the possibility to couple mesoscale and macroscale formulation. This approach requires a homogenization technique that connects the mesoscale with the macroscale. In general, the properties on the mesocale are nonlinear so that for each evaluation for a point on the macroscale a calculation on the mesoscale has to be performed. A problem is the complexity of the mesomodel. In order to capture the underlying mesostructure with its actual physical phenomena a high resolution of the mesomodel is required. As a result this multiscale approach is computationally very expensive for large macroscopic structures.

In this paper a multiscale approach based on artificial neural networks is proposed. Mesoscale simulations, which explicitly take into account the physical phenomena on the lower scale, are performed to model the response in a representative volume element (RVE). In a second step the mesomodel is approximated by a multilayer feedforward network and describes the material behaviour of the macroscopic level. The advantage is that no material model for the macroscale has to be specified. As a result the evaluation of the homogenized mesocale response for a macroscopic point using the neural network is much faster than the evaluation of the mesomodel itself. This approach requires a training of the neural network with training data from mesoscale simulations. On the one hand the number of training samples should be reduced to a minimum due to the complexity of a mesoscale simulation, and on the other hand the training samples should capture all the phenomena on the mesoscale. Within certain regions the nonlinearity of the mesoscale model is more pronounced than in others, requiring a varying density of training samples within the parameter range.

As an example the pullout of a reinforcement bar in reinforced concrete is analyzed. Within the mesomodel of concrete, the aggregates, the matrix and the interfacial transition zone, as well as the ribbed bar are explicitly considered. The input parameters of the neural network are the normal and tangential relative displacements of the opposite boundaries of the mesomodel, whereas the output parameters is a homogenized tangential stress. In the macromodel, the interface zone is represented by zero-thickness interface elements. Due to the incremental increase of the displacements in the mesoscale simulation, the input parameters are arranged along several linear paths. At the beginning the undamaged material has a high stiffness, and small changes in the input parameters lead to large variations of the output parameters, thus requiring a high density of training samples within that region. After the peak load has been reached, a gradual decrease requires only a low density of training samples. However, the simulation on the mesoscale requires shorter displacement increments due to numerical reasons close to the peak point. A procedure to select or interpolate the training samples as input data to the neural network from the mesoscale simulations is presented. In order to obtain more accurate results within the training procedure, a transformation of the training samples is performed so that the transformed distribution for each coordinate of the training samples is a uniform distribution. This leads to a considerable increase in the accuracy of the neural network approximation. Finally a calculation on the macroscale is presented to show the applicability of the method.

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