Nodal Discontinuous Galerkin Finite Element Method And Its Application In Computational Electromagnetics

Computational Electromagnetics (CEM) has been made a great leap and been extensively applied to electromagnetic (EM) industry and academic research since the subject emerged in 1960s. CEM community has developed a few popular and effective numerical methods to solve all kinds of EM problems in the past four decades. These methods mainly include Finite-Difference Time-Domain (FDTD) method, Finite Element Method (FEM), Finite Volume Method (FVM), and Method of Moment (MoM), etc. All these methods are extensively employed to solve almost all kinds of different EM problems. However, they have different disadavantages in these methods. For example, FDTD method is not suitable for handling structures with complex geotry; FEM and MOM methods are difficult to be parallelized because of the coupling relationship between on element to all other elements; FVM can not use high-order basis functions. To overcome the shortcomings of these traditional numerical methods, we studied a class of relatively new and novel numerical method– Discontinuous Galerkin Finite Element Method (DG-FEM) and implemented nodal based DG-FEM. We studied how to resolve EM resonator problems, EM wave propogation problems, EM scattering problems and stochastic EM wave scattering problems for object with uncertain shape, based on nodal DG-FEM.EM resonator problem is one of the fundemantal problems in electromagnetics. We solved EM resonator problems from one dimension to three dimensions using nodal DG-FEM.EM wave propagation is another of essential problems in electromagnetics. First of all, we solved the EM wave propagation in free space using nodal DG-FEM. Then we utilized nodal DG-FEM to deal with EM wave propagation in a lossy dielectric medium.Next, we focused on how to deal with EM wave scattering problem using nodal DG-FEM. EM wave scattering problem is a very hot research topic without any doubt in CEM community. In particularly, how to quantify the scattering characteristic of an object with uncertain shape is very important for the stealth and anti-stealth of military target since RADAR was invented in World War II. First, we solved the 2 dimensional (2-D) metallic cylinder scattering problem using nodal DG-FEM. Second, we solved a 2-D square cylinder scattering problem. Finally, we solved object with uncertain shape stochastic scattering problem by combining with the Sparse Grid integration method as well as the Stochastic Collocation method. Sparse Grid method is a very effective method to reduce computation cost for high dimensional integration problems. And stochastic collocation method is an easy-programing method as Monte-Carlo method for stochastic problem, but it has higher convergence rate than the Monte-Carlo method does. To speed up the computation, we proposed an adaptive sparse grid method based on the Richardson extrapolation method. The key idea is to predict the result of the next level through those of the current level and the previous level. The program will be terminated when the error between the predicted result and that of the current level is below a given tolerance; otherwise, the program will continue to compute the next level. The adaptive sparse grid will save more CPU time when the problem to be solved has higher dimensionality.Scientific computing and simulation is one of three methods in scientific research. Simulations are indispensable for theoretical and experimental research. There are gradually increasing interests to simulate enormous problem in EM research and industry societies as the science and technology evolve. One single personal computer (PC) or workstation is not powerful enough to satisfy the computational resource requirements. Therefore parallel computing is the only way for conquering huge EM problem. It is very easy to implement a parallel version of DG-FEM because DG-FEM exchanges information between elements through numerical flux and one element only relates with its neighbours. We implemented a 3-D parallel nodal DG-FEM EM solver under supercomputer environment. Our implementation bases on the Open Message-Passing Interface (OpenMPI) library and the ParMETIS parallel graph partition library for load balancing.DG-FEM usually employs high-order polynomials in space discretization. To match the high-order accuracy in space discretization in DG-FEM, it also adopts high-order time discretization schemes, for instance, second order frog leap scheme, second to fourth order Runge-Kutta method, predicator-corrector method in multi-step methods, etc. However, these methods are derived via analytical or semi-analytical methods and their accuracies are fixed. It is very difficult or even impossiable to derive more accuracy similar methods. A new highly accuracy time discretization scheme is implemented in this work. The new scheme is fully numerically constructured. And the form of the new schem is same as the traditional predictor-corretor method. But the new method approximates the solution of ordinary differential equations (ODEs) using exponential functions, not polynomials functions in traditional-corretor method.All of our contributions have laid a good foundation for further development using DG-FEM for EM analysis and application.