Characteristic mode analysis with discontinuous Galerkin Surface integral equations

A discontinuous Galerkin (DG) surface integral equation method is proposed for electromagnetic scattering from targets with the impedance boundary condition (IBC). We present electric field integral equation (EFIE), magnetic field integral equation (MFIE), and self‐dual integral equation formulations for the problem and study their numerical performance. On the basis of the results of these experiments, DG is developed for EFIE and MFIE (DG‐EFIE and DG‐MFIE). The convergence of the iterative solutions and the solution accuracy of DG‐EFIE and DG‐MFIE are further investigated for a wide range of surface impedances. The numerical performance of these formulations is found to be nearly complementary with respect to the surface impedance. The capability of the proposed DG solution strategy for calculating scattering from large multiscale IBC targets is demonstrated.

Characteristic mode (CM) analysis and a waveport modeling method is combined for analyzing antenna radiation and scattering. Base on the mode matching method, a MoM waveports modeling method is presented for general metallic-dielectric structures. To analyze waveguide port fed antenna with characteristic modes analysis, a suitable weighted matrix is established based on energy relationship and the CMs can be obtained by solving the new eigenvalue problem. Once the CMs are found, modal weighted coefficients of CMs, equivalent electric and magnetic currents on the antenna can readily be obtained. Numerical results of a microstrip antenna with coaxial cable line feeding is given to validate the proposed method.

A single-layer broadband vertically polarized (VP) omnidirectional metasurface antenna is proposed in this communication. To guide the optimization and design of the antenna, characteristic mode (CM) analysis (CMA) is performed to investigate the intrinsic modal behaviors of the metasurface. A mode with VP omnidirectional radiation is first identified by analyzing an array of <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$4 \times 4$ </tex-math></inline-formula> squared patches. To excite this mode, a complex feeding network is required, which leads to gain loss and increased structural complexity. Furthermore, the mode currents change with the frequency. Consequently, this mode cannot be excited efficiently in a wide bandwidth using a fixed feeding structure. To solve these problems, the modal current is analyzed at different frequencies. The sizes of the four corner patches and the other four edge patches are adjusted to optimize the modal behaviors. After optimization, the mode becomes much more desirable and can be efficiently excited in a wide bandwidth using a single probe. Therefore, the proposed antenna can be designed on a single-layer substrate and excited by a simple feeding structure with broadband characteristics. To verify the design, the proposed metasurface antenna is simulated, fabricated, and evaluated. The results show that the proposed metasurface antenna can achieve a broad bandwidth of 64.2%.

The integration of electric field enhancement structures (EFESs) with Rydberg atomic sensors (RASs) has garnered considerable interest due to their potential to enhance detection sensitivity in quantum measurement systems. Despite this, there is a dearth of research on the directional response of EFES, and the analysis of the three-dimensional (3D) patterns of RAS remains a formidable challenge. RASs are employed in non-destructive measurement techniques, and are responsive to electric fields, primarily serving as reception devices. However, analyzing their reception patterns is a complex task that requires a sophisticated approach. To address this, we adopt characteristic mode (CM) analysis to illustrate the omnidirectional performance of RAS. According to the CM theory, the reception pattern can be calculated by a series of modal currents and their corresponding coefficients. The analytical representation of these coefficients negates the need for time-consuming full-wave (FW) numerical simulations, which are typically required to generate EFES patterns due to the necessity of scanning numerous angle parameters. This approach significantly reduces the complexity of solving EFES patterns, and provides insightful guidance for the design process. To validate the efficacy of our proposed method, we construct three prototypes. The results indicate that the final model resonates at 1.96 GHz, achieving an electric field gain of 25 dB and an out-of-roundness of 2.4 dB. These findings underscore the effectiveness of our method in analyzing EFES patterns, highlighting its potential for future applications in the field.

In this article, an approach combining the electric field integral equation (EFIE) and impedance boundary condition (IBC) is presented for the characteristic mode analysis (CMA) of composite metallic-dielectric structures. Different from published works, only the electric currents are used in this approach and the computed eigencurrents are in a similar form as those of purely metallic structures. To ensure completeness and orthogonality of the eigencurrents, symmetry of the IBC-EFIE operator has been carefully examined. For lossy structures, a new formulation is deduced for CMA. The presented IBC-EFIE formulation is able to handle a variety of problems, e.g., lossy metals, thin dielectrics, thin dielectric-coated conductors, and metasurfaces. Several numerical examples are provided for those problems, and full-wave CMA solutions are used to verify the IBC-EFIE formulation. It is shown that this method is fairly accurate while computationally efficient, which makes CMA a promising tool for large scale or complex composite structures.

Characteristic mode analysis has proven useful in the design of multi‐input multi‐output antennas. By identifying characteristic current modes associated with a conductive structure, the designer can place ports to replicate those current modes, which should generate orthogonal far fields with high isolation from one another. An algorithm that uses characteristic mode analysis is proposed to synthesise multi‐input multi‐output antennas from a conductive structure by determining port placement and excitations without visual inspection. Our algorithm uses a correlation metric to determine the proper excitation from a particular port, and uses a global optimisation algorithm to determine the optimal port placement. Given a particular structure and a domain of possible port locations, our algorithm places ports and generates excitations to synthesise the desired multi‐input multi‐output antenna. Our algorithm is used to synthesise and simulate a simple 3D cell phone model with a metal ground plane, metal rim, and dielectric. The result is an antenna with the desired six isolated channels at 2.5 GHz.

The complex impedance that would be measured by a four-point electrode on the intimal surface of type Va and Vb atherosclerotic lesions was simulated using two different electromagnetic software programs. One program used a nonconformal hexahedral mesh of 2.5 million elements to describe the lesion morphology, whereas the second program used a Conformal mesh of 320,000 elements. Even though the nonconformal mesh used more elements, the geometry of the lesion and electrodes was more accurately described by the Conformal mesh. The impedance of the lesions was calculated for the frequencies of 10/sup 5/,10/sup 6/,10/sup 7/, and 10/sup 8/ Hz. The simulated resistance and phase of the lesion for both the nonconformal and conformal mesh were very similar. The small difference in the results can be attributed to the more accurate modeling of the interface between the electrodes and the underlying layer.

This work presents a new formulation for conducting bodies of revolution (BOR) in layered media. It is derived using magnetic field integral equation (MFIE). The discrete complex image method is used to transform the Sommerfeld integrals into closed form solutions. For large closed objects, the use of the magnetic field formulation is shown to be more advantageous than the electric field integral equation (EFIE) concerning the convergence of the solution. The obtained formulation is applied to the contiguous half-spaces case. Results of both the magnetic and electric field integral equation formulations are presented.

Coupling local discontinuous and continuous galerkin methods for flow problems

Stéphane Lanteri and Francesca Rapetti Abstract—This paper is concerned with the design of a high order discontinuous Galerkin (DG) method for solving the 2D time-domain Maxwell equations on non-conforming triangu-lar meshes. The proposed DG method allows for using non-conforming meshes with arbitrary-level hanging nodes. This method combines a centered approximation for the evaluation of fluxes at the interface between neighboring elements of the mesh, with a leap-frog time integration scheme. Numerical experiments are presented which both validate the theoretical results and provide further insights regarding to the practical performance of the proposed DG method, particulary when non-conforming meshes are employed. Index Terms—Maxwell’s equations, discontinuous Galerkin method, non-conforming triangular meshes.

The aim of this work is to provide a solver for viscoelastic multi-phase flows within the Bounded Support Spectral Solver (BoSSS) currently under development at the Chair of Fluid Dynamics at the Technical University of Darmstadt. The discretisation in BoSSS consists of a high-order Discontinuous Galerkin (DG) method for single-phase flow and a high-order eXtended Discontinuous Galerkin (XDG) method for the multi-phase purpose. The solver shall be used to investigate numerically the behaviour of viscoelastic droplets. The macroscopic Oldroyd B model which is used in a wide range of applications is chosen as the constitutive model. A detailed derivation of the system of equations including the modeling principles for the Oldroyd B model is presented. A DG discretisation of the system of equations including the Local Discontinuous Galerkin (LDG) method is presented after introducing the field of the DG method. The derivation of appropriate flux functions for the constitutive equations and the extra stress tensor are one of the key derivations of this scientific work. Difficulties arising in the numerical solution of viscoelastic flow problems for higher Weissenberg numbers for different discretisation methods are due to the convection dominated, mixed hyperbolic-elliptic-parabolic nature of the system of equations. Several strategies are presented which overcome these problems and are known from the literature. A key achievement of this scientific work is the application of the LDG method, originally developed for a hyperbolic system of equations for a Newtonian fluid, on the viscoelastic system of equations which renders methods for preserving ellipticity unnecessary. Furthermore, various strategiesn to enhance and to support convergence of the solution of the DG discretised system are presented. These are the Newton method with different approaches determining the Jacobian of the system, a homotopy continuation method based on the Weissenberg number for a better initial guess for the Newton method, and a troubled cell indicator combined with an artificial diffusion approach or an adaptive mesh refinement strategy, respectively. For the completeness of this work the XDG method is presented using a sharp interface approach with a signed distance level-set function as it is implemented in BoSSS. The single-phase solver is combined with these methods and appropriate flux functions for the interface are implemented to enable multi-phase applications for viscoelastic fluid. Several numerical experiments are conducted to verify and to validate the viscoelastic singlephase solver and to show the capability of the viscoelastic multi-phase solver to simulate viscoelastic droplets. Advantages and disadvantages of the implementation and an outlook for future research can be found in the conclusion.

We further investigate the high order positivity-preserving discontinuous Galerkin (DG) methods for linear hyperbolic and radiative transfer equations developed in Yuan et al. (SIAM J Sci Comput 38:A2987–A3019, 2016). The DG methods in Yuan et al. (2016) can maintain positivity and high order accuracy, but they rely both on the scaling limiter in Zhang and Shu (J Comput Phys 229:8918–8934, 2010) and a rotational limiter, the latter may alter cell averages of the unmodulated DG scheme, thereby affecting conservation. Even though a Lax–Wendroff type theorem is proved in Yuan et al. (2016), guaranteeing convergence to weak solutions with correct shock speed when such rotational limiter is applied, it would still be desirable if a conservative DG method without changing the cell averages can be obtained which has both high order accuracy and positivity-preserving capability. In this paper, we develop and analyze such a DG method for both linear hyperbolic equations and radiative transfer equations. In the one-dimensional case, the method uses traditional DG space $$P^k$$ of piecewise polynomials of degree at most k. A key result is proved that the unmodulated DG solver in this case can maintain positivity of the cell average if the inflow boundary value and the source term are both positive, therefore the positivity-preserving framework in Zhang and Shu (2010) can be used to obtain a high order conservative positivity-preserving DG scheme. Unfortunately, in two-dimensions this is no longer the case. We show that the unmodulated DG solver based either on $$P^k$$ or $$Q^k$$ spaces (piecewise kth degree polynomials or piecewise tensor-product kth degree polynomials) could generate negative cell averages. We augment the DG space with additional functions so that the positivity of cell averages from the unmodulated DG solver can be restored, thereby leading to high order conservative positivity-preserving DG scheme based on these augmented DG spaces following the framework in Zhang and Shu (2010). Computational results are provided to demonstrate the good performance of our DG schemes.

This work analyses a low-profile, broadband monopolar patch antenna (which we call the Annular Coupled Patch) using Characteristic Mode Analysis. The full structure is divided into two resonators: a cylindrical cavity and shorted annular patch, which are electromagnetically coupled by the fringing fields about the annular ring of vias separating the two resonators. The analysis shows that Coupled Mode Theory accurately models the resonances of the full, coupled structure. The modal behavior is compared and contrasted with several other antennas that are also well-modeled by Coupled Mode Theory.

A new broadband circularly polarized (CP) metasurface (MS) antenna based on characteristic mode analysis (CMA) is proposed. The antenna consists of 4×4 truncated Γ-shaped unit cells and is fed by an improved slot coupling structure. The characteristic modes required to achieve CP are analyzed by CMA, and the CP performance is improved by changing the characteristic currents distribution. By introducing a stepped slot structure on the slot floor, the bandwidth of the antenna has been further broadened. By loading the microstrip line to feed the MS, left-handed circular polarization (LHCP) can be realized. In addition, the equivalent circuit method is used to explain how the designed MS antenna achieves CP. The overall size of the designed and fabricated MS antenna is 0.72 λ 0 ×0.72 λ 0 ×0.08 λ 0 at 6.2 GHz. The measured −10 dB impedance bandwidth of the antenna is 25.17% (5.21 GHz-6.71 GHz) and the measured 3 dB axial ratio bandwidth is 17.42% (5.87 GHz-6.99 GHz). Moreover, the highest gain in the working frequency band is 7.52 dBic. The proposed CP MS antenna has good application prospects in C-band satellite communications.

Characteristic mode analysis has traditionally been constrained to problems which utilize only perfect electric conductors (PEC). Through forced symmetry of a method of moments surface integral equation and newly proposed post-processing, characteristic modes can be solved for any material in a computationally efficient manner. As an example, the characteristic modes are solved for a mobile terminal consisting of both PEC and dielectric materials.