Research On Model Pricing In Computational Electromagnetics

This thesis mainly discusses the application of model reduction method in computational electromagnetics,and enriches the high-performance numerical method for solving time harmonic Maxwell equations.The numerical computation of various electromagnetics problems is based on the proper boundary conditions for solving the partial differential equations(PDE)in the region of interest,which can be discretized by the finite element,finite difference,discontinuous Galerkin method or finite volume method,then we can calculation the numerical solution of the obtained discrete system.With the classic continuous finite element,one can have different orders of approximation in each element,thereby enabling local changes in both size and order,known as hp-adaptivity.However,the schemes result in the introduction of computationally expensive matrices(the globally defined mass matrix and stiffness matrix),and because the semidiscrete scheme is implicit,the mass matrix must be inverted,this is a clear disadvantage compared to finite difference and finite volume methods.The discontinuous Galerkin finite element method(DG-FEM)is an intelligent combination of finite element method and finite volume method.It provides a lot of required properties by imitating the foundation space and test function of the finite volume method.Therefore,the DG method has the following characteristics: the unstructured mesh generation area is the same as the finite element method;like the finite volume method,the unit uses the numerical flux(numerical flux)to exchange information,ensures the stability of the format,and realizes the weak addition of the boundary conditions;all operations are based on the unit local.In the DG finite element method,the solution of the boundary between the units can be discontinuous,which is continuous in the inner element.In this thesis,we aim at combining the model reduction methods with the high order discontinuous Galerkin method for the solution of 2D time-harmonic Maxwell’s equations,then reduces the large original system to a very low dimensional one.There are one strategies considered:the reduced basis method.The reduced basis method offers a way to construct approximations to such input-output relationships,which can be evaluated very fast.The key is an online-offline decomposition.In a so-calledoffline phase the reduced model is built self-adaptively.In an actual application(online phase)only the reduced model is solved.Rigorous error estimation techniques allow to control and quantify the accuracy of the approximative reduced model,such that reduced basis solutions are reliable.