Auxiliary Differential Equation Method Research Articles

Rotationally symmetric transverse magnetic vector wave propagation for nonlinear optics

Rotationally symmetric transverse magnetic vector wave propagation for nonlinear optics

A high-order explicit time-domain FEM using 15-node tetrahedral elements for room acoustics modeling: Basic performance

Time-domain room acoustic modeling using mass-lumped higher-order tetrahedral finite elements with an explicit time-marching scheme is highly attractive because of its excellent geometrical flexibility with unstructured meshing and applicability into massively parallel computing. However, the standard mass-lumped tetrahedral elements yield an unstable scheme, and only implicit time-marching schemes are available. This study proposes a novel wave-based room acoustics solver based on a high-order explicit time-domain FEM. The present solver uses mass-lumped 15-node tetrahedral elements for spatial discretization and a dissipation-free two-stage partitioned Runge-Kutta time integration for time discretization. The 15-node tetrahedral elements had been developed to apply mass-lumping methods to 10-node tetrahedral elements, which are the most common finite element in engineering applications, but its applicability in room acoustics problems is unknown. Higher accuracy and efficiency of the proposed method over the standard method using 10-node tetrahedral elements is presented through an eigenvalue analysis of a long duct model and acoustic simulations in a small room. Additionally, frequency-dependent sound absorbing boundary is implemented by revising an auxiliary differential equation method.

Auxiliary differential equation (ADE) method based complying-divergence implicit FDTD method for simulating the general dispersive anisotropic material.

The preceding works introduced the leapfrog complying divergence implicit finite-difference time-domain (CDI-FDTD) method, which exhibits high accuracy and unconditional stability. In this study, the method is reformulated to simulate general electrically anisotropic and dispersive media. The auxiliary differential equation (ADE) method is employed to solve the equivalent polarization currents, which are then integrated into the CDI-FDTD method. The iterative formulae are presented, and the calculation method is similar to that of the traditional CDI-FDTD method. Additionally, the Von Neumann method is utilized to analyze the unconditional stability of the proposed method. To evaluate the performance of the proposed method, three numerical cases are conducted. These include calculating the transmission and reflection coefficients of a monolayer graphene sheet and a monolayer magnetized plasma, as well as the scattering properties of a cubic block plasma. The numerical results obtained by the proposed method demonstrate its accuracy and efficiency in simulating general anisotropic dispersive media, compared to both the analytical method and the traditional FDTD method.

Relativistic Bistatic Scattering of a High-Speed Moving Plasma Coated Object

Accurate modeling of relativistic electromagnetic scattering characteristics from high-speed motion of plasma coated objects is crucial for the development of hypersonic aircraft and their applications in the identification and surveillance of moving stealth targets. Nevertheless, a solution for bistatic polarized radar cross sections (RCSs) from a 3-D object with plasma coated layer in motion has yet to be obtained. This manuscript proposes a solution to this problem by employing a combination of the auxiliary differential equation (ADE) method with Lorentz finite-difference time-domain (FDTD) method. Utilizing the Lorentz transformation, this paper presents the transformation of parameters of the incident plane wave and dimensions of the object between the laboratory system that remains static and the rest system that remains stationary relative to the object in high-speed motion. The near-zone electromagnetic fields near the object are computed using the ADE method in the rest system, after which the near-field to far-field (NF-FF) transformation is employed to obtain the far-zone polarized scattered field. By applying Lorentz transformation to the coordinates, this paper presents a solution for the polarized scattering from moving plasma coated objects. Especially, radial components of the polarized scatterings are analyzed. The proposed method is validated through several numerical experiments, demonstrating its efficiency and accuracy.

A Dispersive Formulation of Time-Domain Meshless Method With PML Absorbing Boundary Condition for Analysis of Left-Handed Materials

We have proposed a dispersive formulation of scalar-based meshless method for time-domain analysis of electromagnetic wave propagation through left-handed materials (LHMs). Moreover, we have incorporated Berenger’s perfectly matched layer (PML) absorbing boundary condition (ABC) into the dispersive formulation to truncate open-domain structures. In general, the dispersive formulations of conventional numerical methods are proper tools for analysis of left-handed (LH) media, whenever the LH media can be characterized by considering spatial or frequency dispersion effect for the constitutive parameters. In comparison to the grid-based numerical methods, it is proven that meshless methods not only are strong tools for accurate approximation of derivatives in Maxwell’s equations but also can provide more flexibility in modeling the spatial domain of problems. However, we have not seen any reports on using dispersive forms of meshless methods for simulation of wave propagation in metamaterials and applying any PML ABCs to dispersive formulation of meshless method. The proposed formulation enables us to take advantage of meshless methods in analysis of LH media with frequency-dependent parameters. For modeling the frequency behavior of the medium, we have used auxiliary differential equation (ADE) method based on the relations between electromagnetic fields intensities and current densities. Effectiveness of the proposed formulation is verified by analysis of 2-D and 3-D numerical examples; also, some basic factors which affect the accuracy, stability, and computational cost of the simulations are studied.

Numerical Accuracy of Finite-Difference Time-Domain Formulations for Magnetized Plasma

The finite-difference time-domain (FDTD) has been widely used to analyze electromagnetic (EM) wave propagation in complex dispersive media. Over the past three decades, a variety of FDTD approaches for the EM wave propagation in magnetized plasma has been presented. In this work, we perform a comprehensive study on the numerical accuracy of four FDTD formulations for magnetized plasma including the JE convolution (JEC) method, the exponential time differencing (ETD) method, the H-J collocated auxiliary differential equation (ADE) method, and the E-J collocated ADE method. Toward this purpose, the numerical permittivity tensor of magnetized plasma in the four FDTD formulations are derived and then we analyze them to determine which approach can provide the best accuracy. It is found that the E-J collocated ADE method can lead to the best accuracy. Numerical examples awere performed to validate our investigations.

Dynamical plane wave solutions for the Heisenberg model of ferromagnetic spin chains with beta derivative evolution and obliqueness

The oblique plane waves with their dynamical behaviors for a (2+1)-dimensional nonlinear Schrödinger equation (NLSE) having beta derivative spatial-temporal evolution are investigated. In order to study such phenomena, NLSE is converted to a nonlinear ordinary differential equation with a planar dynamical system by considering the variable wave transform with obliqueness and the properties of the beta derivative. Some more new general forms of analytical solutions, like bright, dark, singular, and pure periodic solutions of NLSE are constructed by employing the auxiliary ordinary differential equation method and the extended simplest equation method. The effect of obliqueness and beta derivative parameter on several types of wave structures along with the phase portrait diagrams are reported by considering some special values of parameters for the existence of attained solutions. It is found that the planar dynamical system is not supported by any type of orbit for Θ=450. It is also confirmed from the obtained solutions that no plane waves are generated for Θ=450. The presented studies on bifurcation analysis and analytical solutions for (2+1)-dimensional NLSE would be very useful to understand the physical scenarios of nonlinear spin dynamics in magnetic materials for Heisenberg models of ferromagnetic spin chains.

Numerical methods for design of metamaterial photonic crystals and random metamaterials

Two-dimensional metamaterial photonic crystals (2DMPCs) composed of dispersive metamaterials in a positive-refractive-index medium were investigated by incorporating finite-difference time-domain calculations into the auxiliary differential equation method. A distinct band gap was formed and the effects of positional and size disorder when the dispersive metamaterials are aligned in air were elucidated. In addition, using the self-consistent finite-difference frequency-domain method, an eigenmode analysis of 2DMPCs with positional disorder was performed. Finally, a numerical method for the inverse design of binary random metamaterial multilayers was proposed.

Novel higher order convolutional perfectly matched layer based on complex envelope ADI‐FDTD algorithm for truncating the unmagnetized plasma

An improved higher order convolutional perfectly matched layer (HO-CPML) algorithm is proposed, which combines the complex envelope (CE) method and the alternating direction implicit (ADI) method. This algorithm is solved by the auxiliary differential equation method in the unmagnetized plasma, so as to simulate the unconditionally stable finite-difference-time-domain (FDTD). The novel HO-CPML with fractional decomposition method different from the previous one is adopted to truncate the unmagnetized plasma, making full use of the characteristics of the HO-CPML. The numerical example in plasma simulation shows the characteristics of the algorithm directly. According to the simulation results, the algorithm is effective, and the absorption effect is better than the conventional ADI (CL-ADI) algorithm and the first order CPML (FO-CPML) algorithm.

Multiscale modeling and simulation methods for electromagnetic and multiphysics problems

Multiscale modeling and simulation methods for electromagnetic and multiphysics problems

Implicit Approximate Crank–Nicolson Theory for Anisotropic Ferrite Structure Simulation with Enhanced Absorption

AbstractTo simulate ferrite structures with anisotropic characteristics efficiently, an unconditionally stable implementation is proposed by incorporating the approximate Crank–Nicolson (CN) theory and a complex‐frequency‐shifted perfectly matched layer (CFS‐PML) formulation in the finite‐difference time‐domain lattice. More specifically, the proposed implementation combines the CN approximate factorization splitting procedure with bilinear transformation in the z‐domain within the CFS‐PML regions. To simulate the unique anisotropic characteristic of the ferrite structure, the auxiliary differential equation (ADE) method, which is modified at the integer time step, is introduced during the formulation. Numerical examples are presented to demonstrate the effectiveness and efficiency theoretically and experimentally. According to the results, it can be concluded that the proposed scheme takes full advantage of the approximate CN theory, CFS‐PML formulation, and modified ADE method in terms of the considerable computational accuracy, outstanding absorbing performance, and remarkable efficiency. Moreover, it can be demonstrated that the proposed algorithm can maintain its stability with the dissatisfaction of the Courant–Friedrichs–Lewy condition.

An explicit time-domain FEM for acoustic simulation in rooms with frequency-dependent impedance boundary: Comparison of performance in 2D simulation with frequency-domain FEM

Accurate boundary modelings that address the frequency-dependent sound absorption characteristics of various sound absorbers are crucial for wave-based room acoustic simulation. In time-domain simulations, however, a computationally demanding convolution appears in frequency-dependent impedance boundary conditions. The present paper proposes a room acoustic solver with a fourth-order accurate explicit TD-FEM, incorporating a frequency-dependent absorbing boundary condition efficiently using a recursive convolution method, namely the auxiliary differential equation (ADE) method. Its performance against the fourth-order accurate frequency-domain FEM is examined via 2D real-scale room acoustic problems, solving a sound propagation in an office room up to 4.5 kHz. Firstly, we describe briefly the formulation of the proposed room acoustics solver based on the explicit TD-FEM. Then, the discretization error property of the proposed method is evaluated via an impedance tube problem, including a frequency-dependent impedance boundary of porous sound absorber. Finally, the accuracy and efficiency of the proposed method are demonstrated with the comparison of frequency-domain FEM solver, which uses a sparse direct solver for the solution of the linear system at each frequency. Results showed the proposed method can perform an acoustic simulation with significantly low computational costs compared to the frequency-domain solver while keeping an acceptable level of accuracy.

Plane Wave Solutions With Dynamical Behaviors for Heisenberg Model of Ferromagnetic Spin Chain With Beta Derivative Evolution and Obliqueness

The plane waves with their dynamical behaviors of (2+1)-dimensional nonlinear Schrodinger equation (NLSE) having beta derivative spatial-temporal evolution are investigated. In order to study such phenomena, NLSE is converted to a nonlinear ordinary differential equation with a planar dynamical system (PDS) by considering the variable wave transform and the properties of the beta derivative. By employing two distinct solution methods, namely, the auxiliary ordinary differential equation method and the extended simplest equation method, some more new general form of analytical solutions of NLSE are constructed. The effect of beta derivative parameter and obliqueness on several types of wave structures along with the phase portrait diagrams are reported. It is found that the PDS is not supported by any type of orbits for Θ = 45°. It is also confirmed from the obtained solutions that no plane waves are generated for Θ = 45°. The presented studies on bifurcation analysis and analytical solutions for (2+1)-dimensional NLSE would be very useful to understand the physical scenarios for Heisenberg models of ferromagnetic spin chain with magnetic interactions.

3D Time-domain electromagnetic full waveform inversion in Debye dispersive medium accelerated by multi-GPU paralleling

Electromagnetic full waveform inversion in Debye dispersive medium (EFWI-D) is a promising technique to reconstruct the inner structure and electrical properties of the medium such as soil, rock and biological tissues. Same as conventional full waveform inversion, EFWI-D requires high computational cost, especially in the 3D case. To reduce the long computation time, we design and implement the EFWI-D algorithm in time domain using multiple GPU cards. The inversion method is based on the L-BFGS optimization algorithm, which can increase the convergence of the misfit function, while the auxiliary differential equation (ADE) method is employed for modeling the Debye dispersive medium by using exponential time differencing (ETD) finite-difference time-domain (FDTD) approach. Moreover, a multi-stream strategy is performed in the workflow to improve the computation performance. Numerical results illustrate the improvement of the computational performance and the preliminarily feasibility of the proposed inversion algorithm. Program summaryProgram title: 3D Electromagnetic Full Waveform Inversion in Debye Dispersive Medium Based on Multi-GPU ParallelingCPC Library link to program files:https://doi.org/10.17632/mjd9pp5dcm.1Licensing provisions: LGPLProgramming language: C/C++, CUDANature of problem: Electromagnetic FWI derives high-resolution distribution of electrical parameters by using additional information provided by the amplitude and phase of the received signals. It goes beyond ray-tracing tomography techniques, which use only the travel time kinematics of the signals. Mathematically, electromagnetic FWI is a kind of conditional extremum problem which is to minimize the difference between observed and modeled waveform of received signals under the condition of Maxwell's equation.Solution method: The basis of inversion is the solution of the forward problem. In this program, the forward simulation is based on Auxiliary Differential Equation (ADE) method. Since the inverse problem is nonlinear, it is solved by using iterative solution which is an optimization method. As it is well known that 3D FWI requires tremendous computation, multi-GPU paralleling is used in this program to accelerate the computing process.Additional comments including Restrictions and Unusual features: The program is mainly designed to solve the inverse problem in Debye dispersive medium, but it also can be used for solving the forward and inverse problem in nondispersive medium by setting the input parameter: poles_of_debye_dispersive_medium(non_dispersive=0,dispersive=1-5)=0.

Time-domain simulations of sound propagation in a flow duct with extended-reacting liners

A general methodology to simulate acoustic propagation in ducts with extended-reacting liners in the time domain is presented, including a generic perforated sheet on the air-material interface. The Linearized Euler Equations (LEE) with a mean flow profile are solved in the duct and the linearized equations on an equivalent fluid are solved in the liner material. The auxiliary differential equation method (ADE) is used to prevent the computation of convolution integrals, and leads to a formulation compatible with high-order numerical schemes. The methodology is illustrated for the case of liners consisting of rigid-frame porous materials and a prototype of locally-resonant acoustic metamaterials. A one-dimensional (1D) test case is first used to validate the algorithm and assess the numerical error. The numerical order of the algorithm is the expected one, independently of the interface, as long as the number of poles retained in the partial fraction expansions involved in the formulation is high enough. The algorithm is then applied to a realistic two-dimensional (2D) configuration in a duct with flow, and is used to illustrate the restrained validity of the locally-reacting approximation. Finally, the impact of the flow Mach number on the acoustic performance of porous and metamaterial extended-reacting liners is briefly assessed.

Discontinuous Galerkin Time Domain Method with Dispersive Modified Debye Model and its Application to the Analysis of Optical Frequency Selective Surfaces

We develop a discontinuous Galerkin time domain (DGTD) algorithm with an experimentally validated modified Debye model (MDM) to take metal dispersion into consideration. The MDM equation is coupled with Maxwell’s equations and solved together through the auxiliary differential equation (ADE) method. A Runge-Kutta time-stepping scheme is proposed to update the semi-discrete transformed Maxwell’s equations and ADEs with high order accuracy. Then we employ the proposed algorithm to analyze an infinite doubly periodic frequency selective surface (FSS) operating in the optical regime that exhibits transmission enhancement due to the surface plasmatic effect. The accuracy and the efficiency enhancements are validated through a comparison with commercial simulation software. This work represents the first integration of MDM with DGTD, which enables the DGTD algorithm to efficiently analyze metallic structures in the optical regime.

Hybrid Implicit-Explicit Procedure With Improved Absorption for Anisotropic Magnetized Plasma in Bandpass Problem

Although several algorithms have been developed for the simulation of anisotropic magnetized plasma, they still show decrement of efficiency especially with fine structures along a single direction. By incorporating the complex envelope method, higher order concept, one-step leapfrog hybrid implicit- explicit (HIE) procedure and perfectly matched layer (PML) formulation, such algorithm is proposed to take advantages of them in bandpass simulation of fine structures. In addition, in order to simulate the magnetized plasma with unique anisotropic property, auxiliary differential equation method is modified according to the proposed unconditionally stable scheme. From the aspect of both theoretically and numerically, corresponding numerical examples and experiment are carried out for the further demonstration of effectiveness. The results indicate that the proposed scheme which can break the Courant-Friedrichs-Levy (CFL) condition shows considerable entire performance in the bandpass anisotropic magnetized plasma simulation with fine structure along a single direction.

Exponential time differencing implementation of the stretched coordinate perfectly matched layer

AbstractThe exponential time differencing (ETD) formulation for the stretched coordinate perfectly matched layer (PML) is proposed through the Taylor expansion. This general D and B formulation can be used to truncate arbitrary media in the FDTD simulations. The numerical experiments validate the effectiveness of the ETD schemes. The computational efficiency of the ETD algorithm is identical to that of simple auxiliary differential equation method. The ETD is superior to any other implementation of the stretched coordinate and uniaxial PMLs. The optimum value of the effective relative permittivity for the inhomogeneous media can reduce the peak relative reflection error of the PML by about 15 dB relative to the average value. The performance of the PML is improved remarkably.

ADE-TLM Algorithm for Modeling Wave Propagation in Biological Tissues with Debye Dispersion

In this paper we present a Transmission Line Matrix (TLM) algorithm for the simulation of electromagnetic wave interaction with a Debye dispersive medium. This new formulation is based on the use of the polarization currents in the medium. The auxiliary differential equation (ADE) method is considered to deal with dispersion after the classical discretization. The accuracy and efficiency of this approach were tested on 1D Debye medium by calculating the reflection coefficient on an air-dielectric interface. The potential of the developed algorithm to model the existence of tumors in a human breast is also demonstrated. The obtained results compared with the analytic model show a good agreement. The number of operations needed for each iteration has been reduced, hence the computational time in comparison with time convolution techniques, while maintaining a comparable numerical accuracy.

Traveling wave with beta derivative spatial-temporal evolution for describing the nonlinear directional couplers with metamaterials via two distinct methods

This work is reported the analytical solutions for describing the nonlinear directional couplers with metamaterials by including spatial–temporal fractional beta derivative evolution. The auxiliary ordinary differential equation method and the generalized Riccati method with the properties of beta derivative are implemented to secure such solutions. The solutions are obtained in the new forms by involving of some useful mathematical functions. The constraint conditions among the traveling wave parameters are evaluated. Some of the obtained solutions are presented graphically to illustrate the effectiveness of beta derivative parameter and mathematical techniques. It is investigated that the amplitudes of soliton are increased with the increase of fractional beta derivative parameter. The obtained results would be very useful to examine and understand the physical issues in nonlinear optics, especially in twin-core couplers with optical metamaterials.