Keywords: artificial neural networks, hierarchical composites, effective properties.

In this paper we describe how an Artificial Neural Network (ANN) can be used to approximate and memorise parameters characterising the effective behaviour of hierarchical composites. The present approach uses the direct results of the homogenisation theory. The asymptotic theory of homogenization permits the deduction of the matrix of the effective material characteristics for the composite from the given properties of the components and their spatial arrangement inside the representative volume of the heterogeneous material (cell of periodicity). To compute the effective material properties, the applied version of the homogenisation technique requires a solution of a boundary value problem (BVP) with the periodic boundary condition posed over the cell of periodicity. The effective material properties are computed point-wise for given data, i.e. for each new geometry of the cell and each variation of the values of the properties of components. It is impossible to obtain a closed form expression for them. In our approach, the neural network is used as an approximator of the functional dependence of the components of the effective constitutive matrix on the micro-structural data. Independent variables are here the mechanical properties of the components and some parameters are introduced to describe their geometrical repartition in the cell. The approximation, defined by the computed examples, substitutes the closed form expression. The ANN is used as a "functional formula" and replaces the solution of the BVP thus allowing a reduction in the time of computations for a multilevel composite. The Artificial Neural Network is constructed as follows:

• Input layer: to the first group of neurones of the input layer, the values of the constitutive parameters of the materials of the single cell are assigned. To the second group of neurones at the input the parameters describing the geometry of the cell are specified.
• Output layer: the neurones give the values of the effective constitutive parameters of the "homogenised" material.
• Hidden layers are constructed to assure the best approximation of the unknown relation between the material properties of components, their geometrical organisation and the effective material properties at the output.

For casual values of the materials data and for different geometries of the cell of periodicity the effective material characteristics are computed by a finite element (FE) solution of the BVP with periodic conditions and then suitably post-processed. The learning data for the training of the network are the pairs of sets: given random input and the computed corresponding output. The example is related to some practical application dealing with superconducting coils. In the present concept of ITER fusion reactor the toroidal field coils are made of Nb₃Sn based strands within cable-in-conduit-conductor (CICC) technology. The coils can be regarded as significant examples of a hierarchical composite, since the superconducting cable is made up of more than one thousand elementary strands, properly assembled using a multi level twisting system. The strand itself is a composite material, made of a high conductivity copper matrix where the Nb₃Sn superconducting filaments are embedded. There is a definite distinction between the micro scale of the strands and the meso scale of the cable, where the elementary wires can be regarded as homogeneous. First the ANN is trained to approximate well the effective material properties at micro and meso levels. For a given microstructure, the effective material parameters are computed. Then the global, thermo-mechanical problem of cooling down the coil is solved. The finite elements routine developed for the "unsmearing" process provide the real stress values over the single conductor, at the micro level. At the micro level, each of the homogeneous micro-components can change its mechanical properties, depending on the temperature or stress to which they are subjected. If this is a common feature for many micro-cells in a zone that can be considered as a meso domain, new effective properties must be calculated for this region. In practice, this is a region covered by a single element of the global or meso FE mesh. If yielding or other changes of properties of the homogeneous components occur, the micro cell is modified and the new effective parameters for the meso level are read from the ANN output, without repeating the costly BVP solution. The same procedure applies to the next structural level. In this case the new global effective data are obtained from the ANN, working in a recall mode with the new input data, obtained from the micro level. The global FE analysis can be repeated then with updated material data. The new unsmearing finishes the repetitive loop of the computational algorithm.

The approach becomes impracticable if we repeat the FE solution in order to obtain the effective constitutive data for each load step and for each element of the global or meso mesh. In contrast, the same chain of computations can be achieved within a reasonable time when the effective properties are read as an output signal from the sufficiently trained ANN. Because of this the application of the ANN presented, is crucial in our numerical practice.

We conclude the following: for a composite, the functional dependence of effective material properties on parameters describing the micro-structure cannot be obtained from asymptotic homogenisation but can be approximated by an ANN. This approximation is good enough to be recalled several times in order to compute, at any structural level, the effective characteristics for hierarchical composite, reducing thus the time of the computations in the homogenisation procedure.

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