A Novel DG-FETD scheme based on parametric variational principle for nonlinear and multiscale electromagnetic simulations

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Conventional numerical techniques such as the finite-difference time-domain (FDTD) may face difficulties in solving nonlinear and multiscale electromagnetic problems. A major challenge is that the stable time step will be greatly decreased because of dense grids used for discretizing fine details in a multiscale structure, or when the material nonlinearity is strong, or both. In this study we will discuss a novel nonlinear discontinuous Galerkin finite-element time-domain (DG-FETD) scheme based on the parametric quadratic programming method for modeling transient multiscale electromagnetics problems with instantaneous nonlinear media. The proposed technique uses the highly parallel discontinuous Galerkin method for spatial discretization, and it facilitate non-conforming meshes between different subdomains to significantly improves the meshing flexibility in modeling complex and multiscale structures [1]. The element size in multiscale subdomains can be chosen flexibly to balance the computational cost. The domain decomposition strategy will separate the nonlinear and electrically fine structures from other linear subdomains, and special treatments will be applied to nonlinear subdomains to confine the additional computational costs. When handling the nonlinear part in time stepping, the proposed scheme is not based on iteration, but instead on the base exchanges in the solution of a standard quadratic programming problem. The nonlinear constructive law can be transformed into a set of linear complementary problems (LCPs) with parametric variables, which can be solved to a number of mature and efficient numerical tools. Compared with conventional numerical methods, the proposed scheme does not require updating system matrices at each time step; besides, it presents a very good convergence behavior even the problem to be solved has a strong nonlinearity. Several numerical examples will be given to demonstrate the effectiveness of this nonlinear DG-FETD scheme.