Abstract. Description of complex materials, in particular complex fluids, involves numerous computational challenges. Today, atomistic descriptions are too expensive from the computational point of view, motivating that this kind of analysis is restricted to extremely small systems. The next scale introduces some simplificative hypotheses, leading to coarse grained descriptions. At this level, Brownian dynamics simulations are usually employed. However, this level of description requires intensive computation resources with significant unfavorable impact on the simulation performances (CPU time). For these reasons sometimes kinetic theory descriptions are preferred. In those descriptions, the molecular conformation is described from a probability distribution function (PDF) whose evolution is governed by the Fokker-Planck equation. This approach, despite its mathematical simplicity introduces some computational challenges. First, the distribution function is defined in a multidimensional space, and then the associated partial differential equations must be solved in a multidimensional domain (some times involving thousands dimensions). Secondly, the analysis of transient models needs intensive computation, in particular when the system response under small amplitude oscillations (of high and very high frequency) is concerned. In some of our former works , 144, 98-121, 2007] we proposed a technique based on the separated representation of the unknown field able to circumvent the curse of dimensionality. In this paper, we are addressing the second challenging point, the one related to the transient behavior. For this purpose we propose a separated representation of transient models leading to a non-incremental strategy, allowing impressive CPU time savings.

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Purpose The purpose of this paper is to apply the method of separation of variables to obtain the current distribution in the slot of an electrical machine, taking into account the skin effect.
Design/methodology/approach A slot in an electrical machine, filled with a solid conductor, and fed with an externally imposed density current, presents a current distribution that depends on the skin effect. The magnetic potential vector is formulated and solved using a separate representation as a finite sum of unidimensional (space and time) functions, taking into account the boundary conditions. The proposed method obtains the transient and permanent distribution of the current in the interior of the slot, both in transient and steady regime, and the results at the end of the transient are compared with the analytic ones in permanent regime.
Findings The magnetic potential vector in the interior of a slot filled with a solid conductor can be expressed as a finite sum of just 16 modes, which capture the evolution of the field during the transient and permanent regime. These modes are expressed as the product of space and time functions, which have been obtained automatically by the separation of variables algorithm. Instead of solving multiple field problems, one for each time instant, the proposed method just solves two single variable differential equations, one in the time domain and other in the spatial one.
Research limitations/implications The application of the proposed method to non sinusoidal currents, such as those generated by variable speed drives, would allow to compute the field taking into account both the very small time scale of the pulse width modulation pulses, in the range of kiloHz, and the wide time scale of the currents envelope, in the range of 0 100 Hz. Extension to 2D and 3D spatial configurations is also under consideration.
Originality/value Using the method of separation of variables to solve electromagnetic problems provides a new method which can simplify and speed up the computation of transient fields in multidimensional time and space domains

This paper proposes a general framework for expressing parametrically quantities of interest related to the solution of complex structural mechanics models, in particular the ones involved in crash analyses where strongly coupled nonlinear and dynamic behaviors coexist with space-time localized mechanisms. Advanced nonlinear regressions able to proceed in the low-data limit, enabling to accommodate heterogeneous parameters, will be proposed and their performances evaluated in the case of crash simulations. As soon as these parametric expressions will be determined, they can be used for generating large amounts of realizations of the quantity of interest for different choices of the parameters, for supporting data-analytics. On the other hand, such parametric representations allow the use advanced optimization techniques, evaluate sensitivities and propagate uncertainty all them under the stringent real-time constraint.

Solving mechanical problems in large structures with rich localized behaviors remains a challenging issue despite the enormous advances in numerical procedures and computational performance. In particular, these localized behaviors need for extremely fine descriptions, and this has an associated impact in the number of degrees of freedom from one side, and the decrease of the time step employed in usual explicit time integrations, whose stability scales with the size of the smallest element involved in the mesh. In the present work we propose a data-driven technique for learning the rich behavior of a local patch and integrate it into a standard coarser description at the structure level. Thus, localized behaviors impact the global structural response without needing an explicit description of that fine scale behaviors.