Xin Fu

Bio: Xin Fu is an academic researcher from University of Hong Kong. The author has contributed to research in topics: Integral equation & Equivalence (measure theory). The author has an hindex of 3, co-authored 12 publications receiving 35 citations. Previous affiliations of Xin Fu include Michigan State University.

Decoupled Potential Integral Equations for Electromagnetic Scattering From Dielectric Objects

Abstract: Recent work on developing novel integral equation formulations has involved using potentials as opposed to fields as unknown variables. This is a consequence of additional flexibility offered by potentials that enable development of well-conditioned systems. Until recently, most of the work in this area focused on formulations for analysis of scattering perfectly conducting objects. In this paper, we present well-conditioned decoupled potential integral equations (DPIEs) formulated for electromagnetic scattering from homogeneous dielectric objects. The formulation is based on decoupled boundary conditions derived for scalar and vector potentials. The resulting DPIE is a second kind integral equation, and does not suffer from either low frequency or dense mesh breakdown. Analytical properties of the DPIE are studied for spherical systems, and results provided demonstrate well-conditioned nature (and bounded spectrum) of the resulting linear system.

Decoupled Potential Integral Equations for Electromagnetic Scattering from Dielectric Objects

Abstract: Recent work on developing novel integral equation formulations has involved using potentials as opposed to fields. This is a consequence of the additional flexibility offered by using potentials to develop well conditioned systems. Most of the work in this arena has wrestled with developing this formulation for perfectly conducting objects (Vico et al., 2014 and Liu et al., 2015), with recent effort made to addressing similar problems for dielectrics (Li et al., 2017). In this paper, we present well-conditioned decoupled potential integral equation (DPIE) formulated for electromagnetic scattering from homogeneous dielectric objects, a fully developed version of that presented in the conference communication (Li et al., 2017). The formulation is based on boundary conditions derived for decoupled boundary conditions on the scalar and vector potentials. The resulting DPIE is the second kind integral equation, and does not suffer from either low frequency or dense mesh breakdown. Analytical properties of the DPIE are studied. Results on the sphere analysis are provided to demonstrate the conditioning and spectrum of the resulting linear system.

Generalized Debye Sources-Based EFIE Solver on Subdivision Surfaces

Abstract: The electric field integral equation is a well-known workhorse for obtaining fields scattered by a perfect electric conducting object. As a result, the nuances and challenges of solving this equation have been examined for a while. Two recent papers motivate the effort presented in this paper. Unlike traditional work that uses equivalent currents defined on surfaces, recent research proposes a technique that results in well-conditioned systems by employing generalized Debye sources (GDS) as unknowns. In a complementary effort, some of us developed a method that exploits the same representation for both the geometry (subdivision surface representations) and functions defined on the geometry, also known as isogeometric analysis (IGA). The challenge in generalizing GDS method to a discretized geometry is the complexity of the intermediate operators. However, thanks to our earlier work on subdivision surfaces, the additional smoothness of geometric representation permits discretizing these intermediate operations. In this paper, we employ both ideas to present a well-conditioned GDS-electric field integral equation. Here, the intermediate surface Laplacian is well discretized by using subdivision basis. Likewise, using subdivision basis to represent the sources results in an efficient and accurate IGA framework. Numerous results are presented to demonstrate the efficacy of the approach.

Generalized Debye Sources Based EFIE Solver on Subdivision Surfaces

Abstract: The electric field integral equation is a well known workhorse for obtaining fields scattered by a perfect electric conducting (PEC) object. As a result, the nuances and challenges of solving this equation have been examined for a while. Two recent papers motivate the effort presented in this paper. Unlike traditional work that uses equivalent currents defined on surfaces, recent research proposes a technique that results in well conditioned systems by employing generalized Debye sources (GDS) as unknowns. In a complementary effort, some of us developed a method that exploits the same representation for both the geometry (subdivision surface representations) and functions defined on the geometry, also known as isogeometric analysis (IGA). The challenge in generalizing GDS method to a discretized geometry is the complexity of the intermediate operators. However, thanks to our earlier work on subdivision surfaces, the additional smoothness of geometric representation permits discretizing these intermediate operations. In this paper, we employ both ideas to present a well conditioned GDS-EFIE. Here, the intermediate surface Laplacian is well discretized by using subdivision basis. Likewise, using subdivision basis to represent the sources, results in an efficient and accurate IGA framework. Numerous results are presented to demonstrate the efficacy of the approach.

Potential integral equations in electromagnetics

Abstract: In this work, a new integral equation (IE) based formulation is proposed using vector and scalar potentials for electromagnetic scattering. The new integral equations feature decoupled vector and scalar potentials that satisfy Lorentz gauge. The decoupling of the two potentials allows low-frequency stability. The formulation presented also results in Fredholm integral equations of second kind. The spectral properties of second kind integral operators leads to a well-conditioned system.

Computational Methods For Electromagnetics

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Field-Only Surface Integral Equations: Scattering from a Dielectric Body

Abstract: An efficient field-only nonsingular surface integral method to solve Maxwell’s equations for the components of the electric field on the surface of a dielectric scatterer is introduced. In this method, both the vector wave equation and the divergence-free constraint are satisfied inside and outside the scatterer. The divergence-free condition is replaced by an equivalent boundary condition that relates the normal derivatives of the electric field across the surface of the scatterer. Also, the continuity and jump conditions on the electric and magnetic fields are expressed in terms of the electric field across the surface of the scatterer. Together with these boundary conditions, the scalar Helmholtz equation for the components of the electric field inside and outside the scatterer is solved by a fully desingularized surface integral method. Compared with the most popular surface integral methods based on the Stratton–Chu formulation or the Poggio–Miller–Chew–Harrington–Wu–Tsai (PMCHWT) formulation, our method is conceptually simpler and numerically straightforward because there is no need to introduce intermediate quantities such as surface currents, and the use of complicated vector basis functions can be avoided altogether. Also, our method is not affected by numerical issues such as the zero-frequency catastrophe and does not contain integrals with (strong) singularities. To illustrate the robustness and versatility of our method, we show examples in the Rayleigh, Mie, and geometrical optics scattering regimes. Given the symmetry between the electric field and the magnetic field, our theoretical framework can also be used to solve for the magnetic field.

Equivalence Principle Algorithm With Body of Revolution Equivalence Surface for the Modeling of Large Multiscale Structures

Abstract: We propose a multiscale solver of equivalence principle algorithm with a body of revolution (BoR) equivalence surface (ES). First, the whole object is decomposed into subdomains; the ESs are defined as proper BoRs (e.g., spheres in this work) to enclose each subdomain. The Rao–Wilton–Glisson (RWG) and BoR basis functions are defined on each sphere, respectively. Second, the octree is constructed in each nonzero subdomain; the multilevel fast multipole algorithm (MLFMA) is employed to solve the equivalent currents on the ES individually, and then the equivalent currents are projected from RWG onto the BoR basis functions. The couplings between neighboring subdomains are evaluated directly using MLFMA, and the separated subdomains are substituted by the ES-to-ES couplings, and are evaluated efficiently by the BoR method of moments (BoR–MoM). To solve the equation system, a hybrid inner and outer iterative solver is employed, where the inner iteration is used to solve each local subdomains and the outer iteration is used to update the global solutions by collecting all the local solutions. Numerical results and discussions demonstrate the validity of the proposed work.

Physical characterization of air inclusions in river ice for the purpose of a backscatter model

Abstract: The physical characteristics of air inclusions embedded in the ice covers of the Saint Francois River (Quebec, Canada) and the Athabasca River (Alberta, Canada) are studied because of the importance of such inclusions to the analysis and interpretation of Synthetic Aperture Radar satellite images used to characterize river ice. Studies of ice cores sampled from these two rivers show that the concentration of air inclusions in the ice cover is highly dependent on both the ice type as well as the rate of freezing. When this rate is slow, the ice cover will have few air inclusions. However, when it is rapid and sustained in duration, the amount and cross sectional diameter of these inclusions increase. Air inclusions in the snow ice were found to be spherical and ranging in size from just a few millimetres up to about a centimetre. Generally, air inclusions formed in the sampled columnar ice were sparse and either spherical or tubular in shape. Air inclusions in the frazil ice were found to present a different structure from those formed in the other ice types. In most cases, their shape were irregular and their distribution was inhomogeneous. The study of these inclusions under a microscope showed them to have angular boundaries with a whitish appearance. These results will be valuable to the development of radar backscatter analysis algorithms for river ice characterization.

Decoupled Potential Integral Equations for Electromagnetic Scattering From Dielectric Objects

Abstract: Recent work on developing novel integral equation formulations has involved using potentials as opposed to fields as unknown variables. This is a consequence of additional flexibility offered by potentials that enable development of well-conditioned systems. Until recently, most of the work in this area focused on formulations for analysis of scattering perfectly conducting objects. In this paper, we present well-conditioned decoupled potential integral equations (DPIEs) formulated for electromagnetic scattering from homogeneous dielectric objects. The formulation is based on decoupled boundary conditions derived for scalar and vector potentials. The resulting DPIE is a second kind integral equation, and does not suffer from either low frequency or dense mesh breakdown. Analytical properties of the DPIE are studied for spherical systems, and results provided demonstrate well-conditioned nature (and bounded spectrum) of the resulting linear system.