Research Summary:
The generalized method of moments (GMM) provides a novel umbrella framework for the numerical solution of integral equations. GMM uses a partition of unity framework, thus allowing for the mixture of different classes of basis functions and meshes. This results in a scheme with a flexibility hitherto unavailable to the integral equation community.
Different classes of meshes, from point clouds to quadrilateral meshes, to standard triangulations can be trivially integrated to develop local surface approximations smooth to any desired order. Partitions of unity are then defined on these surfaces approximations to allow for the use of multiple classes of basis functions, ranging from local quasi-Helmholtz decompositions to high order functions and potentially singular basis functions; and mixtures of such functions. The GMM class of basis functions have been applied to a wide variety of scattering problems in acoustics and electromagnetics.
Examples:
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Locally smooth surafce approximations
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Non-conformal surafce discretizations
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Mixture of classes of basis functions
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h & p-adaptive basis functions
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Time Domain Low frequency stability
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Multiply connected objects
Recent Relevant Publications:
[1] Dault, Daniel, and B. Shanker. "A Mixed Potential MLFMA for Higher Order Moment Methods With Application to the Generalized Method of Moments."IEEE Transactions on Antennas and Propagation 64.2 (2016): 650-662.
[2] Crawford, Zane D., Daniel Dault, and B. Shanker. "Smooth Surface Blending for the 2-D Generalized Method of Moments." IEEE Antennas and Wireless Propagation Letters 15 (2016): 528-531.
[3] Dault, D., and B. Shanker. "An Interior Penalty Method for the Generalized Method of Moments." IEEE Transactions on Antennas and Propagation 63.8 (2015): 3561-3568.
[4] Li, Jie, et al. "Analysis of scattering from complex dielectric objects using the generalized method of moments." JOSA A 31.11 (2014): 2346-2355.