Geometric and energy-based methods for modeling and simulation of multi-physics systems in electrical engineering

L’objectif de ce travail de recherche est de proposer un nouveau concept pour la génération automatique de réseaux équivalents à partir de calculs de type éléments finis (TEF). La réduction d’ordre du modèle s’appuie sur des techniques informatiques de clustering qui segmentent la géométrie réelle en un nombre restreint d’ensembles à champs homogène. La distribution des champs dans la géométrie est calculée à partir de résultats d’une méthode TEF (plus précisément, il s’agit d’une Cell-Method) et à l’aide d’éléments de Whitney. Ces ensembles homogènes peuvent alors être regroupés (fusionner) pour générer les composants du réseau équivalent. Les techniques présentées dans ces thèses sont illustrées sur un exemple magnétostatique permettant, à partir du calcul du potentiel vecteur magnétique sur un maillage TEF, de générer un réseau de reluctance précis mais d’ordre très réduit par rapport au modèle initial.This PhD thesis deals with modeling and computational methods for the design of multi-physical devices in the context of Model-Based System Engineering (MBSE). The proposed methods are based on geometric and topological tools (differential and algebraic topology) as well as duality theory (energetic and differential forms) and advanced discretization and meshing techniques (Delaunay- Voroinoi mesh and computer graphics tools). Analogies so useful in multi-physics modeling are widely used in this work through Tonti's perspective and diagrams. Several standard approaches to engineering modelling are extended and complemented by machine learning and clustering algorithms to provide innovative methods for modeling, simulation and model order reduction. The main contribution of this work concerns a new concept of automated generation of lumped parameter (LP) models from field computations. The results of the field calculations are obtained with a cochain method inspired by the Discrete External Calculus and the Tonti's Cell-Method. This thesis proposes an original approach which generates a LP model and applies at the same time a model order reduction. The two objectives, the automatic generation of the LP model and the model reduction, are based on a topological analysis of the physical problem and Artificial Intelligence clustering algorithms. The final objective of this work is to propose a multi-scale model that can simplify the modeling and simulation of multi-physical devices. This multi-scale model should be suitable for both: solution analysis (computation of fields with the cochain model) and design, optimization, control synthesis (computation of physical quantities in a network of components with the LP model). Field computations are illustrated on a weakly coupled multi-physical case: Electrokinetic, then thermal, and finally mechanical field are computed. Model order reduction and automatic LP generation are illustrated on a magnetostatic case (based on an electromagnetic device): the magnetic H and B fields are calculated using the cochain method, then a reduced reluctance network is automatically generated from the field results. In this example, the number of degrees of freedom is almost reduced by a factor of one hundred