A Study Of Fast Integration In Higher-order Methods Of Moments

As an accurate numerical method,Method of Moments(Mo M)has been widely used in electromagnetic scattering,antenna design and other fields for its excellent features,such as reducing the dimension of the problems to be solved and being easy to deal with the open domain problems.The main procedure of solving the problem with Mo M includes the mesh generation of the target surface,the interpolation of discrete elements,the calculation of matrix elements,the solution of matrix equation and so on.Among them,the computation of matrix elements and matrix solution occupy most of the time in the Mo M.Many fast algorithms for improving the speed of matrix solution have occurred.As for the calculation of matrix elements,especially singular elements,there is still room for improvement.According to different types of the mesh element and basis functions,Mo M could be divided into two categories.Lower-order Mo M uses planar elements for meshing,such as planar triangles and quadrilaterals.Lower-order basis functions are then defined on these elements.Higher-order Mo M combines curved elements and higher-order basis functions,such as higher-order interpolation basis functions or vector basis functions.The specific expressions of matrix elements are not the same for these two kinds of Mo M.Aiming at the fast calculation of singular integrals,near singular integrals and non-singular integrals,corresponding acceleration methods are proposed this paper.The main research contents are as follows:(1)Aiming at the singular problems in the lower-order Mo M,a complete analytical method for calculating the singular integrals is proposed.Firstly,the Taylor expansion of green’s function in the integral kernel is carried out.Afterwards,proper coordinate transformation is applied to derive the closed-form expression of each integral in the expansion.Through a series of linear combinations,the analytical formulations of singular integrals with Rao-Wilton-Glisson(RWG)and rectangular rooftop basis function are finally obtained.Compared with the traditional numerical methods,such as singularity subtraction and singularity cancellation method,the proposed method is more efficient.By changing the order of Taylor expansion,the numerical precision of the method could be controlled,thus improving the flexibility and adaptability of the algorithm.(2)As for the near-singular problems in the lower-order Mo M.The existing numerical method is optimized in this paper.The distribution of integration nodes is modified by changing the integration limits appropriately.Thus,massive number of invalid calculations are avoided and the efficiency could be improved.(3)When the bilinear quadrilateral elements are used for the mesh generation in higher-order Mo M,a semi-analytical method is proposed for singular terms.This method could reduce the computational complexity of quadruple integral.As to those non-singular terms,intermediate variables are precomputed and stored.Thus many repeated calculations could be reduced.The integral kernel is then rewritten to simplify the computation.And the goal of fast integration is finally achieved.(4)In the higher-order MOM based on Bezier patches and higher-order basis functions,this paper proposes a method to accelerate the calculation of matrix elements with GPU.On the basis of deriving the specific expression of matrix elements,variables in the integral are allocated to appropriate memory spaces according to the characteristics of different memory spaces on GPU.The algorithm flow is then optimized to improve the parallel efficiency.Finally,fast calculation of matrix elements is realized.Compared with the traditional numerical methods,the algorithms proposed in this paper could achieve ideal acceleration effect.