Hierarchical Homogenization Including Artificial Neural Networks for the Non Linear Thermo-Mechanical Analysis of Superconducting Coils (plenary lecture)

The recently approved thermo-nuclear fusion reactor ITER (International Thermonuclear Experimental Reactor) requires for the operation high magnetic fields generated by means of superconducting (SC) coils which are fed by currents of some tens of kA. Such SC coils can be regarded as good examples of hierarchical structures where lower levels take part in the global behaviour. According to the current design, the SC alloy Nb3Sn (intermetallics) is formed into fine filaments which are embedded in a low-resistivity matrix of normal metal to make the elementary strand. More than one thousand strands are then twisted together according to a multi-level twisting scheme to form the final cable and wind the coil. Since the superconducting filaments are strain sensitive, it is extremely important to know the strain field after cool-down and under operating conditions. For this purpose we study the behaviour of the conductor by means of a hierarchical multi-scale procedure. The FE tools of theory of asymptotic homogenisation are here extended for the piecewise linear analysis of the SC fibrous composite with non-linear, temperature dependent components. We account also for local material yielding at the stage of microanalysis. To recover the strain inside each single component a suitable unsmearing technique is applied. The procedure requires the solution of many boundary value problems at the different scales. It may become cumbersome for longer loading histories because we have to repeat many times a FE solution 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 a sufficiently trained ANN which approximates the functional dependence of effective material properties on parameters describing the micro-structure. Examples will conclude the paper