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Approximation of maxwell equations by numerical methods on non-matching grids


Research objectives and content
The aim of the research project is the mathematical analysis and development of accurate numerical algorithms for the tridimensional modelling of non-stationary electromagnetic phenomena in geometries with moving boundaries. The system of time dependent Maxwell equations is approximate by combining time advancing schemes, mesh adaption procedures and the mortar element method, which is an optimal non-conforming finite element approach based on domain decomposition. The decomposition of the problem domain into subdomains is used to model complex geometries and different material properties. Moreover, different grid structures, different refinement levels and different degree of interpolation are considered to describe at best the problem characteristics in each subdomain. The spatial approximation is based upon finite element methods and, in each subdomain, it is carried out independently of the approximation within the adjacent subdomains Consequently, different (non-matching) sets of nodes are generated at the interfaces. The intertace matching constraints (trasmission of either boundary or continuity conditions) are imposed by using an auxilary mortar trace space defined over the union of all the intertaces and an appropriate variational operator. This method should provide a global error which is bounded by the sum of the local errors (optimality) and it allows to easily treat geometries with moving boundaries. Time advancing schemes based on finite differences are applied to solve the system of ordinary differential equations resulting from the spatial approximation Explicit and implicit schemes are analysed in relation to their ability in describing transient phenomena.
Reliable and robust error indicators are analysed to extend mesh adption techniques to non-stationary electromagnetics. Adaptive procedures are generally used to improve the quality of the computational mesh in order to compute the most accuratesolution at the smallest computational cost Finally, a numerical validation of the proposed approach will be made on a realistic electric engine problem, concluding the research project. Training content (objective, benefit and expected impact)
The proposed research work will provide a rigorous mathematical setup to approximate non-stationary electromagnetic phenomena in non-stationary geometries through a tridimensional approach much more flexible than the currently in use approaches Moreover, it is our belief that the contents of the project can be successfully used to enhance the performance of numericalcodes for electromagnetics. On the ground of the applicant experience, the research project would remarkably improve her knowledge on mathematical analysis and partial differential equations as well on applied mathematics and electromagnetics modelling.
Links with industry / industrial relevance (22)
An approach based on the combination of the mortar element method, time advancing schemes and mesh adaption has several interesting applications in the electrotechnic and electromagnetic fields. It can provide a very significant application fallout for electric apparata, by making a full scale simulation of their transient performances feasible.

Call for proposal

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Centre National de la Recherche Scientifique

91405 Orsay

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EU contribution
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Participants (1)