Wideband Electromagnetic Scattering And Radiation Problems Based On Integral Equation

As the highly integration and miniaturization of communication system, circuit structure becomes more complicated with multi-scale and multi-physical phenomena.Thus the modeling and analysing of multiscale chip-level circuit will become more and more important. Multiband, multifunction system has caused the mutual interference between the system and signal crosstalk problems. In addition, the electromagnetic inter-ference caused by the external electromagnetic radiation, even damage the system. Thus it is important to investigate the physical mechanism of electromagnetic wave in a broad frequency range, especially for the low frequency electromagnetic effect.In this dissertation, the problems are based on the time domain and frequency do-main integral equation respecitvely, including low frequency and middle freuquency solvers. There are some improvements and development for the fast solvers and novel preconditioning technique. They can provide powerful electromagnetic computing tools for broadband, multiband system and complex multi-scale circuit design.Firstly, the implementation of MOT (Marching on in Time) algorithm of time do-main integral equation has been introduced, and the criterion of MOT stability, disk the-ory for estimating the condition number of matrix, the requirements of the incident field are also discussed in this dissertaion.Secondly, accurate calculation of impedance matrix is an effective approach to im-prove the late time stability of the MOT algorithm, the inner source integral is transformed into calculating the temporal convolution between the retarded functions and the temporal basis function (or its derivation). And the outer integral is still using the gaussian integral method. Analytical convolution results for the polynomial temporal basis functions is given in details, However, as the order of temporal basis function increases or the expres-sion is very complex, analytic convolution solution is difficult to derive or even has no analytic results. A novel variable transformation based numerical method is presented to smooth and eliminate the singularity of the retarded potentials (scalar and vector poten-tial) and the curl of the vector potential. So the convolution between any regular temoral basis function and retarded potentials (or its curl) can be calculated quickly and accurately using the numerical integration method, the advantage is that it can be used in the MOT algorithm of the time-domain field integral equations, no matter which types the time ba-sis functions are. Compared to the analytical temporal convolution method, our porposed numerical integration method can accurately and quickly calculate the impedance matrix elements of MOT algorithm with any type of time basis function and different time steps.As several numerical results will demonstrate, it can largely improve the accuracy and the stability of the MOT algorithm.Later, the computation of the perturbation-based electric field integral equation of the formRn’1,n = 0,1,2, ... is accelerated by using fast Fourier transform technique.As an effective solution of the low frequency problem, the perturbation method employs the Taylor expansion of the scalar Green’s function in free space. However, multi-ple impedance matrices have to be solved at different frequency orders, and the com-putational cost becomes extremely high, especially for large-scale problems. Since the perturbed kernels still satisfy Toeplitz property on the uniform Cartesian grid, the fast Fourier transform based on Lagrange interpolation can be well incorporated to accelerate the multiple matrix vector products. Because of the non-singularity property of high-order kernels when n ≥1, we do not need to do any near field amendment. Later, the error of the FFT interpolation analysis is given for perturbation kernals with different or-der. Finally, the efficiency of the proposed method is validated in an iterative solver with numerical examples.Then, a constrained preconditioner for the time domain augmented electric field in-tegral equation (TD-AEFIE) is constructed to stabilize the marching-on-in-time (MOT)algorithm at low frequencies. Similar to its counterpart in the frequency domain, the TD-AEFIE system matrix also suffers from the saddle point matrix problem. It becomes ill conditioned when the frequency approaches zero and the time step △t→ ∞. By imple-menting the time normalization and applying a time-domain constrained preconditioner,the regularized system converges much faster than the original one. A theoretical anal-ysis shows that its condition number becomes independent with △t and the precision is also better than that without preconditioner. Finally, numerical examples demonstrate and validate the stability, accuracy, and efficiency of the proposed method.Finally, matrix pencil method (MPM), Prony decomposition, characteristic basis function Method and model order reduction are introduced and discussed, especially,mode order reduction method is first used in the MOT algorithm of time domain intergral equation. Numerical results demonstrate that this method can greatly improve compu-tational efficiency of MOT based TDIE. Meanwhile, compared with MPM, it is found that both methods satisfy the same laws of physics, and the selection of sampling points adhere to the same principle.