### Methods are combined.

Fast iterative algorithms are often used for solving Method of Moments (MoM) systems, having a large number of unknowns, to determine current distribution and other parameters. The most commonly used fast methods include the fast multipole method (FMM), the precorrected fast Fourier transform (PFFT), and low-rank QR compression methods. These methods reduce the *O*(*N*) memory and time requirements to *O*(*N log N*) by compressing the dense MoM system so as to exploit the physics of Green’s Function interactions. FFT-based techniques for solving such problems are efficient for space-filling and uniform structures, but their performance substantially degrades for non-uniformly distributed structures due to the inherent need to employ a uniform global grid. FMM or QR techniques are better suited than FFT techniques; however, neither the FMM nor the QR technique can be used at all frequencies.

This method has been developed to efficiently solve for a desired parameter of a system or device that can include both electrically large FMM elements, and electrically small QR elements. The system or device is set up as an oct-tree structure that can include regions of both the FMM type and the QR type. The system is enclosed with a cube at a 0-th level, splitting the cube at the 0-th level into eight child cubes. This forms cubes at a 1-st level, recursively repeating the splitting process for cubes at successive levels until a desired number of levels is created. For each cube that is thus formed, neighbor lists and interaction lists are maintained.

An iterative solver is then used to determine a first matrix vector product for any electrically large elements as well as a second matrix vector product for any electrically small elements that are included in the structure. These matrix vector products for the electrically large and small elements are combined, and a net delta for a combination of the matrix vector products is determined. The iteration continues until a net delta is obtained that is within the predefined limits. The matrix vector products that were last obtained are used to solve for the desired parameter. The solution for the desired parameter is then presented to a user in a tangible form; for example, on a display.

*This work was done by Vikram Jandhyala and Indranil Chowdhury of the University of Washington for Johnson Space Center. For further information, contact the Johnson Technology Transfer Office at (281) 483- 3809.*

*In accordance with Public Law 96-517, the contractor has elected to retain title to this invention. Inquiries concerning rights for its commercial use should be addressed to:*

*Janice C. WalshFederal Reporting Compliance Licensing SpecialistUW Tech Transfer*

*University of Washington*

*4322-11th Avenue, N.E., Suite 500*

*Seattle, WA 98105-4608*

*E-mail: This email address is being protected from spambots. You need JavaScript enabled to view it.*

*Refer to MSC-24439-1, volume and number of this NASA Tech Briefs issue, and the page number.*