Discontinuous Galerkin Integral Equation Methods For Solving Multiscale Electromagnetic Problems

Based on the discontinuous Galerkin method,a systematic and in-depth research has been conducted on the discontinuous Galkerin surface integral equation for multiscale electromagnetic problems.An intuitive form of discontinuous Galerkin surface integral equation and a discontinuous Galerkin augmented electric field integral equation for solving low frequency breakdown problems are proposed in this thesis.Aiming at solving complex multiscale targets in practical engineering,the integral equation based domain decomposition method,which combines the fast algorithms and the parallel techniques,is studied and realized.The basic concepts of electromagnetic theory are introduced at first of the thesis.According to the equivalent principle and Green function in frequency domain,the surface integral equations are constructed for perfectly electric conduting and dielectric body.Then,the process of solving the integral equations by the moment of method is reviewed.Finally,the Loop-Flower basis functions are applied in overcoming the lowfrequency breakdown problems.In addition,the spectral behavior of the Gram matrix of Loop-Flower basis function is studied in detail,and the condition number of the Gram matrix can be theoretically predicted.In order to solve complex multiscale targets efficiently,an intuitive form of discontinuous Galerkin surface integral equation method is proposed in the thesis.The infinitely large line-line integral introduced by the discontinuous basis function,can be transformed into bounded term by deleting the ? neighborhood area of the singular point.In addition,the line-line integral under arbitrary spatial position is derived.Finally the discontinuous Galerkin electric field integral equation,the discontinuous Galerkin magnetci field integral equation and the discontinuous Galerkin combined field integral equation are obtained.The discontinuous Galerkin integral equation method can analyze complex multiscale targets based on non-conformal meshing accurately,which greatly simplifies the modeling and mesh preprocessing.The discontinuous Galerkin electric field integral equation encounters low-frequency breakdown at low frequencies.In order to overcome this issue,a discontinuous Galerkin augmented electric field integral equation is proposed.In the proposed method,line charge basis functions are introduced to discribe the discontinuity of current,and the current continuity equation is enforced,in additon,a preconditioner is constructed to improve the conditioning and the final linear equation is solved by perturbation method.The electromagnetic problem can be solved by the discontinuous Galerkin augmented electric field integral equation correctly with fast convergence at low frequencies.The proposed method provides a powerful tool for calcluating the low-frequency multiscale problem.Aiming at the complex multiscale electromagnetic scattering problem,a domain decomposition method based on integral equation is proposed.The half RWG basis functions are defined on the boundary and the RWG basis functions are defined inside the subdomain.The discontinuous Galerkin technique is applied to ensure that the current continuity at the boundary.Then,the diagonal block preconditioner and the basis function rearrangement techniques are implemented to improve the conditioning of the impedance matrix,and the linear equations can be solved at very fast convergence rate.Finally,adaptive cross-approximation method is adopted to compresses the coupling matrix between subunits,and impedance matrix filling process is accelerated by the OpenMP technique.The method proposed in this chapter can accurately and fastly solve the complex multiscale target electromagnetic problems.In this thesis,discontinuous Galerkin integral equations and their applications in practical engineering are studied in detail.They enriche the theoretical study of the integral equation methods,and also provide a powerful tool for multiscale complex electromagnetic numerical simulation.