Study On Discontinuous Galerkin Methods And Model Order Reduction Techniques With Applications In Nanophotonics Simulation

Nanophotonics is an interdisciplinary subject between emerging nanotechnology and photonics,and it plays an important role in biomedicine,nanoantennas,metamaterials,etc.All of the above applications of nanophotonics require appropriate mathematical models to accurately describe the interaction of light waves with the medium.The semiclassical models are often used to simplify complexity.In the framework of classical mechanics,the coupling system of Maxwell equations describing electromagnetic propagation and partial differential equations（PDEs）describing the motion of electron clouds is mainly solved.In gerenal,the exact solutions to the resulting system of differential equations are not available.Thus,numerical treatment of these systems of PDEs is an important aspect in the scientific research and engineering design of nanophotonics.This paper focuses on the numerical simulations of classical electromagnetics and nanophotonics,and studies the high-order accuracy discontinuous Galerkin time-domain（DGTD）methods and high efficient model order reduction（MOR）techniques,which is to promote the development and application of computational electromagnetics and computational nanophotonics,and to bring new methods and theories to the field of scientific research and engineering computing.The main novelties and contents are as follows:1.The local DGTD formulation based on the completely centered flux,the Lagrange interpolation basis functions,and the second-order leap-frog（LF₂）time scheme is constructed for the differential form of the time-domain Maxwell equations.Then,by imposing the electromagnetic fields on the virtual elements and using the characteristics of the centered flux,the discrete boundary conditions satisfying the perfect electric conductor（PEC）and first-order Silver-Müller absorbing boundary conditions are obtained,and the global discontinuous Galerkin（DG）formulation is further established.Similarly,the local and global DGTD formulations are also constructed for the Maxwell–Drude model.In particular,the stability condition of the global DGTD scheme is analyzed by proving that the discrete energy is the positive definite quadratic form for the electric field,magnetic field and dipolar current vector.Numerical experiments for electromagnetic and nanophotonics problems verify the convergence of the DGTD method.2.The reduced-order models（ROM）based on the proper orthogonal decomposition（POD）method,the Galerkin projection technique,and the LF₂ time scheme are established for the global DG formulations of the time-domain Maxwell equations and Maxwell–Drude model（termed as the POD-DGTD method）.The stability conditions of the ROM are also analyzed.In particular,when the boundaries are the PEC conditions,the global energy of the discrete ROM respectively is conserved and bounded for the time-domain Maxwell equations and Maxwell-Drude model,which is consistent with the characteristics of global energy in the DGTD method.It is shown that the ROM based on Galerkin projection method can maintain the stability characteristics of the original high dimensional model（HDM）.Finally,the offline and online stages of the POD-DGTD method are summarized,and some 2-D and 3-D numerical experiments are presented to verify the accuracy,and to demonstrate the capability of the POD-DGTD method.3.In order to further improve the accuracy of the ROM in Part 2,the error bounds between the high fidelity model and the ROM are analyzed for the time-domain Maxwell equations,and then the adaptive snapshot selection algorithm exploiting the results of this analysis is proposed.A snapshot choosing rule aiming at keeping the error estimate close to a target selection error tolerance is proposed,which is similar to the standard rules found in adaptive time-stepping ordinary differential equations solvers.An incremental singular value decomposition（SVD）algorithm is used to update the SVD on-the-fly when a new snapshot is available.Numerical experiments show that the accuracy of the POD-DGTD method based on the adaptive snapshot selection algorithm is higher than that of the equispaced snapshot selection method under the same number of snapshots.4.For the parameterized electromagnetic scattering problems,the adaptive parameter selection algorithm based on the error estimation with the residual vector is proposed,and the reduced basis（RB）functions are obtained by combining with the POD method.Then,the parameterized ROM that can be scanned quickly is constructed by using the Galerkin projection method.In particular,an error scaling factor is designed to dynamically adjust the control error,which may result in fewer elements in the RB functions to satisfy the desired accuracy.Finally,the offline and online stages of the POD-DGTD method based on the adaptive parameter selection algorithm are summarized,and some numerical experiments including one-dimensional and multi-dimensional parameterized electromagnetic scattering problems are presented to demonstrate the behavior of the method.Numerical results show that the POD-DGTD method is more efficient at the non-training parameters than the DGTD method while ensuring high accuracy.5.Also for the parameterized electromagnetic scattering problems,the two-step POD method is employed to extract the RB functions,and the projection error of the snapshot vectors onto the subspace spaned by the RB functions is analyzed by using the Schmidt-Eckart-Young theorem.The time and parameter discrete modes of the reducedorder coefficient matrices are then obtained by using the SVD method.The non-intrusive ROM（termed as POD-CSI method）,which does not depend on the original HDM,is constructed by using the cubic spline interpolation（CSI）method to approximate the non-linear relationships between the time and the time discrete mode,and the parameter and the parameter discrete mode.The error bound of the POD-CSI method at the all training parameters is analyzed based on the projection error.In particular,the CSI-based continuous modes can be used directly for new time and parameter values to recover the reduced-order solutions during the online stage.Thus,the offline and online stages of the POD-CSI method are completely decoupled.Finally,the offline and online stages of the POD-CIS method are summarized.Numerical experiments for the scattering of a plane wave by a 2-D dielectric disk and a multilayer heterogeneous medium nicely illustrate the performance of the proposed POD-CSI method.