Spectral Differential Equation Approximation for Green's Functions
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Solution Overview
Problem
The evaluation of mixed potential Green's functions in electromagnetic analysis of uniaxially anisotropic multilayered media is computationally heavy due to the oscillatory, singular, and slowly-convergent behavior of Sommerfeld integrals, which existing numerical methods like the discrete complex image method lack robustness or are expensive.
Innovation Solution
The Spectral Differential Equation Approximation Method (SDEAM) is used to approximate Green's function spectra with rational functions, solving for electric and magnetic fields by transforming mixed potential Green's functions into pole-residue form and performing analytical inverse Fourier transforms, allowing for robust and error-controllable solutions using high-order basis functions and discretization refinement.
Engineering Contradictions & Design Principles
Engineering Contradiction Analysis
1Measurement precision
If Sommerfeld integrals are used to compute mixed potential Green's functions, then electromagnetic field analysis can be performed in multilayered media, but the computation becomes heavily expensive due to oscillatory, singular and slowly-convergent behavior of integrands
Solution Approach 1:
The patent transforms the Sommerford integrals into the spectral domain by changing the parameter representation from spatial domain to frequency domain. This parameter transformation allows the use of rational function approximations and pole-residue forms, which converge much faster than direct numerical integration of oscillatory Sommerfeld integrals, thereby resolving the contradiction between accuracy and computation speed
Solution Approach 2:
The patent replaces the direct numerical integration method (mechanical approach) with a spectral domain analytical method using rational function approximations. By substituting the mechanical integration process with algebraic operations in the spectral domain, the computation becomes significantly faster while maintaining accuracy, thus resolving the technical contradiction
2Productivity
If discrete complex image method is used to evaluate Sommerfeld integrals, then computation speed may be improved, but robustness is lost
Solution Approach 1:
The patent introduces an intermediary spectral domain representation as a bridge between the spatial domain problem and the final solution. By transforming the problem into the spectral domain where rational function approximations can be applied, the method achieves both the speed improvement desired by DCIM and the robustness of a more fundamental analytical approach, resolving the contradiction between speed and robustness
3Reliability
If robust extrapolation methods are used to evaluate Sommerfeld integrals, then reliability is improved, but computation becomes expensive
Solution Approach 1:
The patent segments the computation into two distinct stages: (1) spectral domain computation using rational function approximations and pole-residue forms, and (2) inverse transformation to obtain spatial domain results. This segmentation allows each stage to be optimized independently, achieving both robustness in the spectral domain and efficiency in the transformation process, thereby resolving the contradiction between reliability and productivity
Data Source
AI summary
The invention provides a method to compute the mixed potential contributions to electric and magnetic field in multilayered media, which can be applied in microwave engineering, integrated circuit analysis, and remote sensing. The Spectral Differential Equation Approximation Method (SDEAM) for solving Michalski-Zheng's mixed-potential Green's functions in fully shielded, partially open, and fully open multilayered media are described. The main advantage of SDEAM over other methods is that for a fixed location of the source elevation z′, it does not require fitting from scratch for every z, z′, and ρ combination. Because of this advantage, SDEAM can be superior in both performance and accuracy to other existing methods. The well-established boundary value problem numerical solvers also provide SDEAM with robustness for miscellaneous planar multilayered structures.