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A novel Model Order Reduction (MOR) technique is developed to compute high-dimensional parametric solutions for electromagnetic fields in synchronous machines. Specifically, the intrusive version of the Proper Generalized Decomposition (PGD) is employed to simulate a Permanent-Magnet Synchronous Motor (PMSM). The result is a virtual chart allowing real-time evaluation of the magnetic vector potential as a function of the operation point of the motor, or even as a function of constructive parameters, such as the remanent flux in permanent magnets. Currently, these solutions are highly demanded by the industry, especially with the recent developments in the Electric Vehicle (EV). In this framework, standard discretization techniques require highly time-consuming simulations when analyzing, for instance, the noise and vibration in electric motors. The proposed approach is able to construct a virtual chart within a few minutes of off-line simulation, thanks to the use of a fully separated representation in which the solution is written from a series of functions of the space and parameters coordinates, with full space separation made possible by the use of an adapted geometrical mapping. Finally, excellent performances are reported when comparing the reduced-order model with the more standard and computationally costly Finite Element solutions. © 2021 by the authors. Licensee MDPI, Basel, Switzerland.

[EN] AC lines of industrial busbar systems are usually built using conductors with rectangular cross sections, where each phase can have several parallel conductors to carry high currents. The current density in a rectangular conductor, under sinusoidal conditions, is not uniform. It depends on the frequency, on the conductor shape, and on the distance between conductors, due to the skin effect and to proximity effects. Contrary to circular conductors, there are not closed analytical formulas for obtaining the frequency-dependent impedance of conductors with rectangular cross-section. It is necessary to resort to numerical simulations to obtain the resistance and the inductance of the phases, one for each desired frequency and also for each distance between the phases' conductors. On the contrary, the use of the parametric proper generalized decomposition (PGD) allows to obtain the frequency-dependent impedance of an AC line for a wide range of frequencies and distances between the phases' conductors by solving a single simulation in a 4D domain (spatial coordinates x and y, the frequency and the separation between conductors). In this way, a general "virtual chart" solution is obtained, which contains the solution for any frequency and for any separation of the conductors, and stores it in a compact separated representations form, which can be easily embedded on a more general software for the design of electrical installations. The approach presented in this work for rectangular conductors can be easily extended to conductors with an arbitrary shape., This work was supported by the Spanish "Ministerio de Economia y Competitividad" in the framework of the "Programa Estatal de Investigacion, Desarrollo e Innovacion Orientada a los Retos de la Sociedad" (project reference DPI2014-60881-R).

International audience; This article deals with the efficient construction of approximations of fields and quantities of interest used in geometric optimisation of complex shapes that can be encountered in engineering structures. The strategy, which is developed herein, is based on the construction of virtual charts that allow, once computed offline, to optimise the structure for a negligible online CPU cost. These virtual charts can be used as a powerful numerical decision support tool during the design of industrial structures. They are built using the proper generalized decomposition (PGD) that offers a very convenient framework to solve parametrised problems. In this paper, particular attention has been paid to the integration of the procedure into a genuine engineering design process. In particular, a dedicated methodology is proposed to interface the PGD approach with commercial software.

International audience; The paper presents the concept of "Virtual Charts" applied to computational structural mechanics. Based on the Proper Generalized Decomposition (PGD), a model reduction method, the idea is to create a numerical tool to help engineers to perform shape optimization, where quantities of interest, computed for all the geometrical configurations, will be stored. Two different approaches have been developed in order to introduce geometry variations as parameters in the PGD. Promising results were obtained for academic examples.