An adaptive multiresolution time-domain integral equation electromagnetic solver.

摘要

When the Method of Moments (MoM) is used to solve the time domain integral equations of transient scattering problems, the large cost results from the matrix filling and inversion. There are several ways to attack the problem. The first is to construct new algorithms to reduce the frequency scaling such as the Plane Wave Time Domain (PWTD) method. The second approach is using multiresolution analysis (MRA). MRA has found wide applications in recent years. The wavelet-based multiresolution analysis has proven to be an effective method in the analysis of electrically large structures. Wavelet-based MRA can capture the local as well as the global properties of the solutions efficiently. Most of the MRA research done in the time-domain has been applied to differential equations, such as Multiresolution Time-Domain (MRTD) method; while most of the work done with integral equations has been in frequency-domain. A third method that is discussed in this dissertation adopts adaptive basis functions to expand the input and the solution.; The essence of this method is to use only the most effective and necessary basis functions to represent the solution. It is based on the fact that, in transient applications, the induced current on the object varies greatly in amplitude with time and space in the computational domain. It is not necessary to explicitly compute the solution in the region where negligible responses is obvious. At the same time, there is no need to compute all the potentials (at every point or every patch) and integrate on the whole domain. Previous researchers have suggested using pulse-like basis functions and turning them "on" and "off" to save cost. However, it was noted that this could only be applied heuristically.; A new method is proposed to reduce the number of unknowns by using the multiresolution conception. This method applies multiresolution basis functions directly into the expansion of solution. One of the advantages of using multiresolution basis functions is that it is easy to add or discard local, higher order resolutions. The proper "turn on" of the high order coefficients provides accuracy; while the appropriate "turn off" guarantees the efficiency of the algorithm. We do not use matrix transformation between traditional and multiresolution basis functions as most other researchers do. Presented in this dissertation are marching-on-in-time schemes with adaptive basis functions at each time step. Both explicit and implicit schemes are discussed. They provide satisfactory results in the analysis of thin wire scatterers by using Haar wavelet basis functions. These schemes adaptively chose basis functions for the induced current in space and time. This method provides a systematic way to reduce the number of unknowns in transient application, such as in our pulsed power project, reducing computational cost, both in time and storage, with a controllable level of accuracy. It is thus efficient in transient/ultra wide band applications.