Two-dimensional Rough Surfaces Research Articles

Powerful numerical method for analysis of electromagnetic scattering from multiple 3D coated targets buried under rough surface

ABSTRACT In this work, an efficient numerical method is proposed for three-dimensional (3D) electromagnetic scattering from multiple coated targets buried under a two-dimensional dielectric rough surface. In the proposed method, the interior region of each coated target is regarded as an independent subregion. The field within each coated target is modeled by 3D vector finite element method (FEM). The multiple scattering interactions between the buried targets and between targets and rough surface are considered by the electric field integral equation and the Poggio–Miller–Chang–Harrington–Wu integral equations, respectively. The fields inside and outside the targets are coupled together via boundary conditions on the boundary of each target. To improve the calculation efficiency and reduce the memory requirement, the fast multipole technique is used in our method. The presented method is numerically validated by comparing it with method of moments and conventional FEM-boundary integral method (BIM). The results indicate that, compared with traditional FEM-BIM, our method has significant advantages in computational efficiency and memory requirements.

Roughness implementation and convective heat transfer coefficient computation toward ice accretion simulation

In the simulation of icing, the calculation of the convective heat transfer coefficient for rough surfaces has significant importance. Our study investigates the extension of the γ-Reθt transition model for rough icing surfaces and aims to provide an alternative to the integral boundary layer method for facilitating the calculation of the heat transfer coefficient in three-dimensional icing simulations. Initially, the performance of the γ-Reθt transition model in predicting the transition characteristics and heat transfer for smooth surfaces is examined. Subsequently, to compensate for the deficiency of the γ-Reθt transition model in predicting the premature transition caused by roughness, an additional model equation Ar is introduced to simulate the influence of the surface roughness. Three test cases, including a clean cylinder, an iced cylinder, and a swept wing, are used to validate the applicability of the γ-Reθt-Ar transition model for heat transfer computations on rough surfaces. The results demonstrate that the γ-Reθt-Ar model is superior to other turbulence models in computing convective heat transfer for two-dimensional and three-dimensional icing rough surfaces and shows its potential for ice accretion simulation.

Analyzing Vortex Light Beam Scattering Characteristics from a Random Rough Surface

The propagation and scattering of vortex light beams in complex media have significant implications in the fields of laser imaging, optical manipulation, and communication. This paper investigates the scattering characteristics of vortex light beams from a random rough surface. Firstly, a two-dimensional Gaussian rough surface is generated using the Monte Carlo method combined with the linear filtering method. Subsequently, the vortex beams are decomposed into the superposition of infinite plane waves, and the scattering of each plane wave from the rough surface is calculated using the Kirchhoff Approximation method. Numerical results of the angle distribution and spatial distribution of OAM scattering Laser Radar Cross Section (LRCS) are presented, varying with different surface roughness parameters for a rough aluminum surface and the beam’s parameters. The results demonstrate that the scattering of vortex beams is influenced by the beam’s parameters, such as Orbital Angular Momentum (OAM) mode number and elevation angle, which may bring new insights into vortex wave-matter interactions and their applications in high resolution imaging.

Analysis of Doppler spectrum of a spaceborne Doppler scatterometer using an echoed signal simulation model

ABSTRACT The Doppler spectrum of a microwave scatterometer signal is determined by the relative motion between the scatterometer and the ocean surface. It is also determined by the modulation from the ocean surface backscattering coefficient. The relative motion includes contributions mainly from the motion of the platform, the phase velocity of the resonant Bragg waves, the orbital motions of ocean waves, and the ocean surface current. Estimating the Doppler centroid from the complex Doppler spectrum of the received signal is an important issue for the application of a spaceborne Doppler scatterometer in ocean surface current retrieval. In this study, we use an echoed signal simulation model based on a two-dimensional rough dynamic ocean surface to estimate the Doppler spectrum of a spaceborne Doppler scatterometer. We analyze the Doppler spectrum with different observation geometry and ocean state, and the effects of modulation on the Doppler spectrum with the pulse-pair method. Our results show that the high platform velocity and the modulation of the backscattering coefficient from the ocean surface are the main influencing factors that affect the shape of the Doppler spectrum of the received signal, which affects the accuracy of estimation of the Doppler centroid. For a Ka-band pencil-beam rotating Doppler scatterometer using the pulse-pair method, the modulation can lead to a measurement error of 1.2 m·s−1.

A Simplified Lattice Boltzmann Boundary Conditions for Gas Transport in Self-Affine Microchannels with an Inherent Roughness of in a Tight Reservoir

A simplified method of determining lattice Boltzmann boundary conditions based on self-affine microchannels with an inherent roughness in a tight reservoir is presented in this paper to address nonlinear efficiency problems in fluid simulation. This approach effectively combines the influence of rough surfaces in the simulation of the flow field, the description of L-fractal theory applied to rough surfaces, and a generalized lattice Boltzmann method with equivalent composite slip boundary conditions for inherent roughness. The numerical simulations of gas slippage in a two-dimensional plate model and rough surfaces to induce gas vortex reflux flow are also successfully carried out, and the results are in good agreement with the simulation results, which establishes the reliability and flexibility of the proposed simplified method of rough surfaces. The effects of relative average height and fractal dimensions of the rough surfaces under exact boundary conditions and equivalent coarsened ones are investigated from three perspectives, namely those of the average lattice velocity, the lattice velocity at average height position at the outlet, and the coefficient of variation for lattice velocity at average height position. It was found that the roughness effect on gas flow behavior was more obvious when it was associated with the enhanced rarefaction effect. In addition, the area of gas seepage was reduced, and the gas flow resistance was increased. When the fractal dimension of the wall was about 1.20, it has the greatest impact on the fluid flow law. In addition, excessive roughness of the wall surface tends to lead to vortex backflow of the gas in the region adjacent to the wall, which greatly reduces its flow velocity. For gas flow in the nanoscale seepage space, wall roughness hindered gas migration rate by 84.7%. For pores larger than 200 nm, the effects of wall roughness on gas flow are generally negligible.

Simulation of a Radio Signal Scattered from a Rough Surface by the Method of Stationary Points

Introduction. Many radar and radio navigation problems require adequate (i.e., corresponding to the physics of radio wave scattering under a certain set of conditions) simulation of a radio signal scattered from a rough surface. At present, four categories of mathematical models are used for these purposes, with each corresponding to a particular group of methods for solving the scattering problem. These methods include those based on a rigorous solution of the electrodynamic equations (methods of moments, finite elements, finite differences); approximate methods for solving electrodynamic problems (methods of small perturbations, tangent plane (Kirchhoff method), two-scale model); those of statistical equivalents; and phenomenological models. In the former two categories, the electromagnetic field is computed on a surface grid, whose step must be much smaller than the signal wavelength. This makes the simulation of a scattered radio signal effectively unrealizable for two-dimensional rough surfaces. The method of statistical equivalents is based on the calculation of the probabilistic characteristics of a scattered signal. Difficulties arising in this case narrow the possibilities of simulation, largely due to the impossibility of obtaining closed expressions for the required characteristics without significant simplifications. In addition, carrying out analytical calculations, the solution is restricted to the class of surfaces with the Gaussian distribution of roughness heights. Phenomenological models, although providing for a qualitative analysis of signal scattering from a rough surface, fail to take the physics of the scattering process into account.Aim. To create a simple and adequate model of a signal scattered from a two-dimensional rough surface using the tangent plane approximation of the boundary conditions (the Kirchhoff method), which can be realized on modern computers.Materials and methods. The proposed model of the scattered signal is based on its representation as a sum of signals reflected from stationary points on the surface, i.e., those points where the law of specular reflection of the incident electromagnetic wave is locally fulfilled.Results. The implementation of the model is reduced to determination of the positions of stationary points on a twodimensional rough surface. This problem is solved on modern computers quite simply without computations with complex numbers. It turns out that the number of stationary points, even with a large surface roughness, rarely exceeds two. This fact, along with the simplicity of finding the coordinates of stationary points, makes this model suitable for long series of statistical runs.Conclusion. The proposed model is easy to implement and can be used to study the accuracy of low-altitude target estimation, altimeters, Doppler and correlation speed and drift angle estimators.

Accuracy of tracking radio altimeters working over a 2D rough surface

Low altitude tracking radio altimeters use periodic frequency modulated continuous signals and the altitude measurement is based on estimation of the mean frequency of the beat signal. To realize the estimation, a closed automatic control loop is organized in radio altimeter. The loop maintains the frequency of the beat signal at a fixed value by adjusting some parameters of the transmitted signal. There is one more way to build up a tracking contour for the altitude estimate, the operation of which is based on the use of a measuring phase locked loop. The circuit adjusts the reference signal ‑ a copy of the beat signal ‑ to maximize the phase structure of the beat signal and the reference signal. When measuring the height over a rough surface, in addition to errors caused by the internal noise in the receiver, the range noise begins to affect the accuracy of the estimate. The range noise is due to the formation of a signal scattered by a rough surface as sum of reflections from stationary points of the surface where the local normal is directed towards the phase center of the radio altimeter antenna. The article aim is a comparative analysis of the accuracy of three types of tracking radio altimeters working over a rough two-dimensional surface: radio altimeter with a phase-locked loop and tracking altimeters with adjustment of the frequency deviation or the modulation period of the transmitted signal. The signal reflected from a two-dimensional rough surface was simulated by the method of stationary points, which is a particular case of the Kirchhoff method. As a result of the simulation, the biases and the root-mean-square errors of height estimates were obtained for the three types of tracking altimeters and different values of the surface roughness parameter. For all altimeters, the estimate bias is small and does not depend on the roughness parameter. The root-mean-square error of height estimation for all considered radio altimeters increases with increase of the roughness parameter. The best results were shown by the radio altimeter with the measuring phase locked loop. Its error is approximately 3 times less than that of the radio altimeters with adjustment of the transmitted signal parameters.

Experimental and numerical studies of turbulent flows over two-dimensional and three-dimensional rough surfaces under an adverse pressure gradient

The effects of surface roughness and other surface topographical feature on turbulent boundary layer (TBL) development have been widely explored, especially for flow cases with a constant streamwise pressure gradient distribution. In the case where the TBL is exposed to the action of an adverse pressure gradient (APG), flow separation is expected to occur and the classical mechanisms responsible for energy transport between the inner and outer part of the TBL are invalid. The combined effect of roughness and APG on the TBL is not easy to predict, especially due to the lack of databases containing relevant research data. The main aims of the present study were to conduct experimental investigations and to perform large eddy simulations to determine the effects of different surface topographies (two-dimensional (2D) and three-dimensional (3D)) on turbulent flow development over a smooth wall under a strong APG downstream of the corrugation. The investigation was performed with a fixed value of the Reynolds number Reθ = 4900. In particular, the differences in the production and dynamics of near-wall vortical structures under different corrugation geometries were studied. It was found that the higher amplitude of 2D wavy surface the greater production of turbulence and the earlier separation of the TBL. The response of the TBL differed according to the 3D topography, where it was characterized by a greater streamwise velocity deficit, lower growth of all components of the velocity fluctuations, and a smaller increase in the normal velocity gradient relative to the 2D case with comparable amplitude. The results also demonstrated that the 3D case generated smaller and less energetic structures on the flat plate downstream of the corrugation compared with the 2D case.

Investigation on dynamics performance of multibody system with rough surface

This investigation aims to validate the effect of the rough contact surface on the dynamic responses of the multibody system based on the fractal theory. Firstly, a two-dimensional rough surface is described using the Weierstrass-Mandelbrot (W-M) function based on the fractal theory. The contact properties of a single asperity are estimated based on the Morag and Etsion (ME) model and Hertz law. The novel contact stiffness with roughness is derived from the size distribution function of the Majumdar and Bhushan (M-B) model. Secondly, a new hysteresis damping factor is derived to represent the energy dissipation during contact. Subsequently, a more general contact force model is proposed based on a nonlinear spring-damper vibration system, in which the novel contact stiffness is associated with the new hysteresis damping factor. The simulation results show that the newly introduced contact force model is a more generalized contact model and the existing contact force models are a special case under the condition of an ideal contact surface. Finally, the new contact force model is utilized to predict the dynamic responses of the one-dimension granular chain and the slider-crank mechanism with a clearance joint, which is validated by the experimental data.

Study of Two-Frequency Mutual Coherence Function From Two-Dimensional Rough Surfaces

Study of pulse scattering from two-dimensional rough surfaces is of great theoretical and engineering significance. In this paper the analytical expression of two-frequency mutual coherence function of the scattered wave from two-dimensional rough surfaces is derived with the integral equation method, which includes both the single scattering and the multiple scattering and therefore is applicable to surfaces with small or large-scale roughness. Then the numerical simulation of the two-frequency mutual coherence function from two-dimensional surfaces with Gaussian distribution is performed, and the results are compared with those with Kirchhoff approximation.

Study of Pulse Scattering from Two-Dimensional Non-Gaussian Rough Ocean Surfaces

In this paper the analytical expression of two-frequency mutual coherence function of the scattered wave from two-dimensional non-Gaussian rough ocean surfaces is derived first by the Kirchhoff approximation. The Gram-Charlier distribution instead of the conventional Gaussian distribution is utilized in carrying out the ensemble average operation in order to represent the asymmetric characteristics of ocean surfaces. Then the scattered pulse for arbitrary incident pulse is derived utilizing the two-frequency mutual coherence function for wide-sense stationary uncorrelated scattering channel. Finally the two-frequency mutual coherence function and the scattered pulse with a Gaussian modulated or rectangular incident pulse from two-dimensional non-Gaussian rough ocean surfaces for different wind speed are numerically simulated, and the pulse broadening phenomenon is discussed.

The effects of size distribution functions on contact between fractal rough surfaces

In the fractal rough surface characterized by Weierstrass-Mandelbrot function, the amount of asperities with frequency index n is γ times the amount of asperities with frequency index n-1. According to the property, the revised size distribution functions and truncated size distribution functions of asperities with frequency index n are developed for the two-dimensional and three-dimensional fractal rough surfaces, respectively. The relations between real contact area and total contact load are obtained. The comparison between predictions of present model, the revised MB model and experimental data are given. The results show these asperities whose frequency indexes range from nmin to nmin+5 are major contribution to real contact area and total contact load. For given initial frequency index and total contact load, the real contact area of present model is smaller than that of the revised MB model. With an increase in initial frequency index, the results of present model gradually approach to that of the revised MB model. When initial frequency index nmin is less than the elastic critical frequency index nec, the trends of present model are in better agreement with experiment than that of the revised MB model.

Reliable approach for bistatic scattering of three-dimensional targets from underlying rough surface based on parabolic equation**Project supported by the National Natural Science Foundation of China (Grant No. 61771407).

A parabolic equation (PE) based method for analyzing composite scattering under an electromagnetic wave incidence at low grazing angle, which composes of three-dimensional (3-D) electrically large targets and rough surface, is presented and discussed. A superior high-order PE version is used to improve the accuracy at wider paraxial angles, and along with the alternating direction implicit (ADI) differential technique, the computational efficiency is further improved. The formula of bistatic normalized radar cross section is derived by definition and near-far field transformation. Numerical examples are given to show the validity and accuracy of the proposed approach, in which the results are compared with those of Kirchhoff approximation (KA) and moment of method (MoM). Furthermore, the bistatic scattering properties of composite model in which the 3-D PEC targets on or above the two-dimensional Gaussian rough surfaces under the tapered wave incidence are analyzed.

Synthesis and Performance of Iron Oxide-Based Ceramsite in a Biotrickling Filter for Nitrogen Oxides Removal

A novel medium containing iron oxide-based porous ceramsite (modified ceramsite) and commercial ceramsite was used in two bench-scale biotrickling filter for nitrogen oxides removal to evaluate the performance of the modified ceramaite. In this study, we adopted the method of surface coating by choosing FeCl3·6H2O as modifier. Under the calcination time was 4 h at the temperature of 500°C, the modified filler presented excellent adsorption for microbial growth and the mass dry weight of biofilm reached a peak of 1.28 mg/g. Results showed that the density increased by 17% and porosity increased by about 15%, and the isoelectric point PI increased more than 4 times, while the surface pH value reduced to 3.46. The surface of pH is much lower than the isoelectric point, to ensure that its surface is electropositive, besides, the modified ceramsite had a more rough surface compared with ceramsite and changed the two-dimensional rough surface into three-dimensional surface. It only took 8 days for the biotrickling filter to start up with the modified ceramsite while the commercial ceramsite was about 22 days, which greatly shortened the start-up period. Packing with the modified filler, the NO removal performance stayed quite stable and efficient, the removal efficiency of NO did not apparently decrease under high inlet concentration of NO and basically maintained an average value of 92.8% during the whole operation period of treatment. While the commercial ceramsite presented an obvious removal decrease comparing with the modified, especially with the high NO inlet concentration of above 1600 mg/m3 and the removal efficiency was less than 80%. The iron oxide-based ceramsite proved to have the capability for improving the performance of the biofilter for NO removal. Furthermore, the fall off of the surface coating is not obvious under the experimental condition and the property of the coating is relatively stable in a long period of operation. Thus, our findings support the modified filler of iron oxide-based ceramsite as a material for use in filter media in in a biotrickling filter for pollutant gas removal.

Polarimetric Scattering from Two-Dimensional Dielectric Rough Sea Surface with a Ship-Induced Kelvin Wake

Based on the polarimetric scattering model of second-order small-slope approximation (SSA-II) with tapered wave incidence for reducing the edge effect caused by limited surface size, monostatic and bistatic polarimetric scattering signatures of two-dimensional dielectric rough sea surface with a ship-induced Kelvin wake is investigated in detail by comparison with those of sea surface without ship wake. The emphasis of this paper is on an investigation of depolarized scattering and enhanced backscattering of sea surface with a ship wake that changes the sea surface geometric structure especially for low wind conditions. Numerical simulations show that in the plane of incidence rough sea surface scattering is dominated by copolarized scattering rather than cross-polarized scattering and that under low wind conditions a larger ship speed gives rise to stronger enhanced backscattering and enhanced depolarized scattering. For both monostatic and bistatic configuration, simulation results indicate that electromagnetic scattering signatures in the presence of a ship wake dramatically differ from those without ship wake, which may serve as a basis for the detection of ships in marine environment.

A Hybrid 3DMLUV-FIA Method for Scattering from a 3D Dielectric Object above a 2D Dielectric Rough Surface

The electromagnetic scattering from the composite model of a three-dimensional (3D) dielectric object located above a two-dimensional (2D) dielectric rough surface is analyzed in this work. Poggio, Miller, Chang, Harrington, Wu, and Tsai (PMCHWT) integral equations are discretized by the method of moments (MoM) into a matrix which is solved by Biconjugate Gradients Stabilized (BICGSTAB) method. Method of 3DMLUV was used for PEC object located above rough surface. Comparing to the case when object and rough surface are both PEC, the memory requirement and computational complexity for dielectric models are increased due to doubled unknown number. Moreover, compared to dielectric object in free space, the coupling between dielectric object and dielectric rough surface will result in complicated numerical simulation. To solve this problem, the updated rank based 3D Multilevel UV (3DMLUV) method is employed to reduce memory consumption and CPU time overhead. The 3DMLUV has been successfully applied in the scattering of PEC targets; however, when the object or rough surface becomes dielectric, the coupling between dielectric object and dielectric rough surface will lead to slow constriction. Therefore, the Fast Iterative Approach (FIA) is applied to further speed up the constricted speed of the matrix required in 3DMLUV. The efficiency, stability, and accuracy of the proposed method are demonstrated in a variety of scattering problems.

A COMPLEX MIX-SHIFTED PARALLEL QR ALGORITHM FOR THE C-METHOD

The C-method is an exact method for analyzing gratings and rough surfaces. This method leads to large-size dense complex non-Hermitian eigenvalue. In this paper, we introduce a parallel QR algorithm that is specifically designed for the C-method. We define the early shift for the matrix according to the observed properties. We propose a combination of the early shift , Wilkinson's shift and exceptional shift together to accelerate convergence. First, we use the early shift in order to have quick deflation of some eigenvalues. The multi-window bulge chain chasing and parallel aggressive early deflation are used. This approach ensures that most computations are performed in level 3 BLAS operations. The aggressive early deflation approach can detect deflation much quicker and accelerate convergence. Mixed MPI-Open MP techniques are used for performing the codes to hybrid shared and distributed memory platforms. We validate our approach by comparison with experimental data for scattering patterns of two-dimensional rough surfaces.

The validity of Kirchhoff theory for scattering of elastic waves from rough surfaces

The Kirchhoff approximation (KA) for elastic wave scattering from two-dimensional (2D) and three-dimensional (3D) rough surfaces is critically examined using finite-element (FE) simulations capable of extracting highly accurate data while retaining a fine-scale rough surface. The FE approach efficiently couples a time domain FE solver with a boundary integration method to compute the scattered signals from specific realizations of rough surfaces. Multiple random rough surfaces whose profiles have Gaussian statistics are studied by both Kirchhoff and FE models and the results are compared; Monte Carlo simulations are used to assess the comparison statistically. The comparison focuses on the averaged peak amplitude of the scattered signals, as it is an important characteristic measured in experiments. Comparisons, in both two dimensions and three dimensions, determine the accuracy of Kirchhoff theory in terms of an empirically estimated parameter σ 2 /λ 0 ( σ is the RMS value, and λ 0 is the correlation length, of the roughness), being considered accurate when this is less than some upper bound c , ( σ 2 /λ 0 < c ). The incidence and scattering angles also play important roles in the validity of the Kirchhoff theory and it is found that for modest incidence angles of less than 30°, the accuracy of the KA is improved even when σ 2 /λ 0 > c . In addition, the evaluation results are compared using 3D isotropic rough surfaces and 2D surfaces with the same surface parameters.

CIRCULARLY POLARIZED WAVE SCATTERING FROM TWO-DIMENSIONAL DIELECTRIC ROUGH SEA SURFACE

Based on the polarimetric scattering model of second-order small-slope approximation (SSA-II) with tapered wave incidence under linear and circular polarization, monostatic and bistatic scattering from two-dimensional dielectric rough sea surface is investigated. The emphasis of the present study is on the polarization signature of the scattered wave under circularly polarized wave incidence, which is related to the Brewster angle. Numerical simulations show that for bistatic configuration under circularly polarized wave incidence, the polarization state of scattering wave strongly depends on incident angle, scattering angle, as well as their relation to the Brewster angle associated with medium permittivity.

Scattering Studies for Two-Dimensional Exponential Correlation Textured Rough Surfaces Using Small-Slope Approximation Method

Two-dimensional exponential correlation rough surfaces characterized by textures are combined with the small-slope approximation (SSA) method to comparatively study electromagnetic (EM) scattering features of textured surfaces. The normalized copolarized radar cross section from 2-D exponential correlation rough surfaces characterized by stripe texture and block texture, respectively, is analyzed. Several numerical results show the effects of incident angle, texture angle, correlation length, and root-mean-square height on the copolarimetric scattering from the textured rough surface. The validity of the SSA method is verified by comparisons of theoretical value and measured data. Moreover, normalized amplitude distributions of backscattering fields from cells in a scene are studied through its statistical distribution and space correlation function, which are particularly useful for analysis and simulation of remote sensing images. Finally, based on the statistical distribution and space correlation function, the zero memory nonlinear transformation method is utilized to simulate EM scattering from very large scenes. The simulated scene coincides with the original one quite well.