Solution Of Volume-surface Integral Equations Research Articles

Improving the Solution Accuracy of the Volume-Surface Integral Equation for the Open PEC Structures Embedded Within Dielectrics

In the solution of volume-surface integral equation (VSIE) for open perfect electric conductor (PEC) structures embedded within dielectric substrates, how to explicitly enforce the continuity condition (CC) of electric flux on the PEC–dielectric interfaces to improve the numerical accuracy is investigated. In the traditional VSIE formed by combining the surface electric field integral equation (EFIE) and volume integral equation (VIE), the fictitious surface charge, which does not exist physically, is accumulated on the PEC–dielectric interfaces. This will lead to the inaccuracy of numerical results, which can be eliminated by the CC. To correctly enforce the CC, the equivalent surface current distribution on both sides of open PEC should be distinguished. For this purpose, the combined field integral equation (CFIE) consisting of electric field and magnetic field integral equations is used to model the open PEC part. In the proposed approach, the number of surface unknowns needs to be doubled, which can be balanced by reducing the volumetric unknowns through the enforcement of CC. Furthermore, by fully considering the numerical symmetry between both sides of open PEC surface, the filling process of the matrix as well as the excitation vector in implementing the method of moments (MoMs) can be simplified. Numerical experiment demonstrates that the enforcement of CC can improve the VSIE accuracy obviously. For example, when calculating the frequency response of frequency selective surfaces (FSSs), the results obtained from the traditional VSIE often shift to higher frequencies, while the proposed approach gives reasonably good results.

On the Enforcement of Electric Field Boundary Condition in the Moment Method Solution of Volume-Surface Integral Equation

When the volume-surface integral equation (VSIE) is used to model the composite objects, the electric field boundary condition (BC) can be explicitly enforced on the dielectric-perfect electric conductor (PEC) interfaces to reduce the volumetric unknowns. In the proposed BC-based VSIE, using the least-square method, one of the four coefficients of SWG basis functions belonging to a same tetrahedron terminated by a PEC triangle is directly calculated by the other three, which is no longer a volumetric unknown. The matrix equation generated by the method of moments is iteratively solved in a straightforward manner, which can be accelerated by the multilevel fast multipole algorithm. Through reliable numerical results, it is illustrated that the enforcement of the BC can reduce the memory usage and the number of iterations without compromising accuracy, in addition to good versatility.

Accelerating Solution of Volume-Surface Integral Equations With Multiple Right-Hand Sides by Improved Skeletonization Techniques

Skeletonization is an effective method to accelerate the solution of linear systems with multiple right-hand sides (RHSs) by exploiting the rank-deficiency property of the RHS matrix. However, when the size of the RHS matrix is very large, the memory requirement would become a bottleneck of the skeletonization technique. To reduce the memory cost, this letter presents the theoretical basis of how to find out the skeleton basis functions (SBFs) to construct the skeleton RHS matrix more efficiently from a purely algebraic point of view and proposes a multilevel algorithm to figure out the SBFs for solving the integral equations with multiple RHSs. The numerical results show that the proposed method accelerates the solution of the volume–surface integral equation with multiple RHSs and reduces the memory cost significantly.

The Application of Improved Spherical Harmonics Expansion-Based Multilevel Fast Multipole Algorithm in the Solution of Volume-Surface Integral Equation

During the solution of volume-surface integral equation (VSIE), to reduce the core memory requirement of the radiation patterns (RPs) of the basis functions, an improved spherical harmonics expansion-based multilevel fast multipole algorithm (SE-MLFMA) using the mixed-potential representation and the triangle-/tetrahedron-based grouping scheme is applied. Numerical results show that accompanying with a faster speed, the memory requirement of the RPs in the improved SE-MLFMA is several times less than that in the conventional MLFMA without compromising accuracy. A result employing the OpenMP parallelization and vector arithmetic logic unit (VALU) hardware acceleration technique is also shown to illustrate the robustness and scalability of the improved SE-MLFMA method.

On the Use of Continuity Condition in the Fast Solution of Volume-Surface Integral Equation

The validity of the use of continuity condition (CC) is studied when it is combined with the volume-surface integral equation (VSIE) to solve the electromagnetic scattering/radiation of composite conductor-dielectric objects. It is found that the explicit enforcement of CC in VSIE may be inaccurate if the involved conductor is infinitesimally thin, but it works well for the composite structures that only contain conductors with closed surfaces. A new combined field integral equation-volume integral equation with enforced CC (CC-CFIE-VIE) is then proposed, and a convenient way to embed it into the context of the multilevel fast multipole algorithm is also provided. Numerical results show that CC-CFIE-VIE not only reduces the overall memory usage and the CPU time of a single iteration, but also decreases the number of iterations to reach desired accuracy.

Fast solution of volume-surface integral equation for scattering from composite conducting-dielectric targets using multilevel fast dipole method

This article presents a fast solution to the volume–surface integral equation for electromagnetic scattering from three-dimensional (3D) targets comprising both conductors and dielectric materials by using the multilevel fast dipole method (MLFDM). This scheme is based on the concept of equivalent dipole-moment method (EDM) that views the Rao–Wilton–Glisson and the Schaubert–Wilton–Glisson basis functions as dipole models with equivalent dipole moments. In the MLFDM, a simple Taylor's series expansion of the terms Rα (α = 1, −1, −2, −3) and R̂R̂ in the formulation of the EDM transforms the interaction between two equivalent dipoles into an aggregation–translation–disaggregation form naturally. Furthermore, benefiting from the multilevel grouping scheme, the matrix-vector product can be accelerated and the memory cost is reduced remarkably. Simulation results are presented to demonstrate the efficiency and accuracy of this method. © 2012 Wiley Periodicals, Inc. Int J RF and Microwave CAE, 2012.