Higher-Order Moment Method Based On The Geometric Modeling Of NURBS Surfaces

In modern computational electromagnetics, the higher order moment method is a conventional method to analyze the objects of middle electrical size. The existing higher order moment methods often apply the bilinear patches to approach the structures, which is lack of accuracy and flexibility. Especially when the surface has large curvature, it has to be meshed into many small bilinear patches to achieve the accuracy in geometric modeling, thus make the higher order method degenerate to a low order method. Therefore, it is necessary to replace the geometric modeling of bilinear patches with the professional methods. The Non-Uniform Rational B-Spline (NURBS) technique was accepted as the only mathematical way to define shapes of products by the International Organization for Standardization (ISO) in 1991. And this technique is investigated deeply and is applied widely. Now it has already been adopted by more and more commercial CAD/CAM software.Combining with the projects of our team, this paper proposes a new higher order moment method based on the NURBS modeling. The method introduces the geometric modeling of NURBS surfaces to approach the structure of a practical problem, and obtains the system matrix by the higher order polynomial basis functions. Then the matrix can be solved directly or approximated by several techniques and then solved by iterative methods. It can be applied to solve problems of middle electrical size. The major work and contributions are as follows:As to the geometric modeling, this paper analyzes the interpolatory methods at first, which includes the bilinear patches and the higher order interpolatory patches. Then the NURBS patches are given and transformed to the Bezier patches, which is compared with the former method. At last, a technique is proposed to extract the data of the NURBS modeling based on the obj file, which is defined by Wavefront Company.This paper details the higher order basis functions defined on curvilinear quadrilaterals, which contains the higher order components defined on one patch and low order components defined on two patches. It is a very complicated problem to arrange these numerous basis functions properly. Then the paper introduces a simple and effective method to order them and judge the direction by several signs. To accelerate the iterating of the matrix solver, we can orthogonalize these basis functions. So the paper introduces the Legendre basis functions and the Maximiumally orthogonalized basis functions.There are seldom people who combine the complex geometric modeling with the higher order basis functions, owing to the very time-consuming process of filling the system matrix, especially the portion with singular quadrature. So this paper proposed a new formula to accelerate this quadrature, which is based on the Taylor's expansion, and it avoids most of the redundant operations. Numerical results show that this formula can accelerate the singular quadrature by more than 100 times. Furthermore, in order to accelerate the other quadrature, this paper proposes an approximate method to decompose these forth integrals into double integrals, thus dramatically decreases the time-expanditure, typically more than 50 times. By the above techniques, our new method can be as fast as the conventional higher order method, and it is applicable.But it is necessary to solve several special geometric structures to obtain a versatile solver, such as the generalized triangle, several patches connected on one common edge, and surfaces combined with line, etc. Numerical results show the effectiveness and flexibility of this method. When the structure has large curvature, the new method not only introduces much less error in geometric modeling, but also generates much fewer number of unknows, compared with the conventional higher order moment method. To reduce the number of unknows, this paper tries to use a new type of basis functions.This paper solves the matrix equation by the direct LU decomposition and the Krylov subspace methods, such as conjugate gradient method (CGNR), the Bi-conjugate gradient stabilized method (BICGSTAB), and the Generalized minimum residual method. Since the matrix generated by the higher order moment method combined with the electric field integral equation (EFIE) is often highly singular, it is hard to converge when solved by the iterative methods. So this paper proposes to apply the sparse approximate inverse preconditioner to reduce the condition number of the matrix. This preconditioner can speed up the convergence at the cost of only very small time and RAM.To reduce the requirement of RAM, this paper introduces the IE-FFT method, which interpolates the Green's function at the regular meshed points, and transforms it into a Toeplitz matrix. This Toeplitz matrix is used to reduce the memory requirement and accelerate the matrix vector multiplication. The effectiveness of the IE-FFT is shown by numerical examples. This method is often more efficient when used in the planar structures than in the 3-D problems.Finally, this paper introduces the ACA method to compress the system matrix. It makes use of the low-rank property of the matrix generated by the moment method,. This paper uses the direct methods and the iterative methods, combined with the out-of-core technique, to solve the compressed matrix equation, which can extend the higher order method to solve problems with larger electric size.