Reduced order modeling of high magnetic field magnets
Résumé
The Laboratoire National des Champs Magnétiques Intenses (LNCMI) is a French large scale facility enabling researchers to perform experiments in the highest possible magnetic field (up to 35 T static field). High magnetic fields are obtained by using water cooled resistive magnets connected with a 24 MW power supply. The design and optimization of these magnets requires from an engineering point of view the prediction of certain ''quantities of interest,'' or performance metrics, which we shall denote outputs --- namely magnetic field in the center, maximum stresses, maximum and average temperatures. These outputs are expressed as functionals of field variables associated with a set of coupled parametrized partial differential equations which describe the physical behavior of our magnets. The parameters, which we shall denote inputs, represent characterization variables --- such as physical properties heat transfer coefficients, water temperature and flowrate, and geometric variables in optimisation studies. To evaluate these implicit input--output relationships, solutions of a multi-physics model involving electro-thermal, magnetostatics, electro-thermal-mechanical and thermo-hydraulics are requested. It should be noted that this model is non-linear as the material properties depend on temperature. In practice these evaluations represents a huge of computational time but are mandatory to improve the magnet design as we cannot rely on common physical sense. To significantly reduce this computational time, we chose to use model order reduction strategies, and specifically to use the reduced basis method which is well adapted to the evaluation of input/output relationships (C. Prud'homme, D. V. Rovas, K. Veroy, L. Machiels, Y. Maday, A. T. Patera, G. Turinici, Reliable real-time solution of parametrized partial differential equations: Reduced-basis output bound methods, Journal of Fluids Engineering, 124 , 2002). We will present the reduced basis method applied to the non-linear electro-thermal coupled problem. To perform this work we rely on reduced basis framework set up by the feel++ library (http://www.feelpp.org/).