Spherical Wave Decomposition Research Articles

The fast multipole method for solving integral equations ofthree-dimensional topography and basin problems

Summary The fast multipole method is developed for the solution of the boundary integral equations arising in wave scattering problems involving 3-D topography and 3-D basin problems. When coupled with an iterative solver for linear equations, the fast multipole method can significantly reduce memory requirements. The order of operations for the product of the matrix obtained from the discretization of the integral kernel and a vector is reduced from N2 to the order of pαN, where p is the order of the multipole expansion and α depends on the details of the implementation. In order to achieve efficient implementation the translation operators for the multipole expansion need to be used in the diagonal form based on a spherical wave decomposition. For problems with topography, the number of iterations required of the linear equation solver to achieve convergence is small and no preconditioning is necessary. However, for a basin problem, block-diagonal preconditioning is essential in the application of the iterative solver. Both the memory requirements and the CPU time are considerably reduced for topography problems. Although the memory requirement is reduced for a basin problem used in this numerical experiment, the CPU time would be still longer than that for the ordinary boundary element method if sufficient memory were available. These results indicate that the fast multipole method might be much more efficient than the ordinary method for 3-D elastic wave scattering problems with more than several tens of thousands of unknown variables.

Non-muffin-tin atomic scattering-matrices for semiconductor LEED-calculations

Non-muffin-tin potentials for LEED-calculations can be treated by the integration of the Lippmann-Schwinger equation as described by Nagano and Tong. We have solved this coupled system of equations numerically with a code for ordinary differential equations. The general scattering potential is generated by superimposing atomic Clementi-Roetti densities and Coulomb potentials for a cluster of crystal atoms. The exchange potential is modeled as a function of the superimposed electron density. Both, energy-dependent and energy-independent exchange potentials are investigated. The spherical wave decomposition of the potential entering the angular momentum representation of the Lippmann-Schwinger equation is performed with an efficient numerical integration algorithm. Spherical wave components of the superposition potential are shown to emphasize their influence on the LEED-calculation. Scattering amplitudes of this potential model are computed and compared with those of standard muffin-tin phase shifts of GaAs. Especially, surface configurations are considered within this procedure.

Spherical wave decomposition approach to ultrasonic field calculations

A simple, flexible, accurate, and comprehensive numerical method is presented for theoretically analyzing the diffraction field of a continuous wave transducer of arbitrary size, shape, and frequency. Using the extensively studied circular transducer for comparison, numerical results are shown for an unfocused transducer with uniform velocity excitation as well as for a focused transducer with Gaussian velocity excitation. Data concerning the execution time, program size, and convergence of the method are also presented for its implementation as a design tool on a minicomputer system.