Canonical Grid Research Articles

Fast Coupling Iterative Algorithm for Media Rough Surface with Multiple Targets Below

This paper proposes a fast mutual coupling iteration algorithm for analyzing the composite electromagnetic scattering characteristics of a Gaussian rough surface with multiple targets below it. Firstly, an electromagnetic scattering model of Gaussian rough surface with multiple targets underneath is established. To improve the efficiency of electromagnetic scattering calculations, the (banded matrix iterative approach canonical grid - conjugate gradient method (BMIA/CAG-CMG) algorithm is used to calculate the surface scattering of the rough surface, while for the scattering of the targets, the traditional numerical algorithm moment of methods (MoM) is used for the calculation. Its main acceleration principle is to decompose the matrix into banded and Toeplitz matrices during the solution process, and then use fast Fourier transform (FFT) for fast solution based on their characteristics. By comparing the calculation results of this algorithm with those of the MoM algorithm, the correctness of this algorithm is verified. Several examples are studied, and the impact of different parameters on the surface current and composite scattering characteristics is discussed, which have important significance for deepening the understanding of scattering characteristics in complex electromagnetic environments.

Electromagnetic Scattering and Emission From Large Rough Surfaces With Multiple Elevations Using the MLSD-SMCG Method

Electromagnetic scattering and emission from 1-D rough surfaces with multiple elevations are studied using full-wave simulations. Both the root-mean-square (rms) heights and the surface length are large compared to the wavelength. A novel multilevel steepest decent-sparse matrix canonical grid (MLSD-SMCG) method is proposed to address limitations in the original SMCG. The uniform Nystrom method and neighborhood impedance boundary condition (NIBC) are also incorporated in solving the dual surface integral equations (SIEs) of the method of moments (MoM). Simulation results are illustrated at L-band for soil and ocean surfaces. The surface rms heights and lengths are up to 1.43 and 243.8 m corresponding to 6 and 1024 wavelengths at 1.26 GHz, respectively. For ocean surfaces, the wind speeds up to 20 m/s are considered, and the entire spectrum is included to capture all relevant surface length scales. Numerical results indicate the proposed approach is computationally efficient and accurate. Energy conservation checks in simulations are at $10^{-4}$ for ocean scattering and emission. Also, the effects of wind-driven roughness on ocean emissivity are further investigated using the proposed approach in terms of wind speed and observation angle for both polarizations.

Accurate Calculations of Emissivities of Polar Ocean Surfaces Between 0.5 and 2 GHz Using an NIBC/Nystrom/SMCG Method

We use full-wave simulations of rough surfaces to calculate the emissivities of a polar ocean between 0.5 GHz and 2 GHz for applications in ocean salinity. High accuracy is required because the emissivity variations between flat and rough surfaces are at the order of magnitude of $10^{-3}$ . In addition, the changes of emissivity due to salinity change are only about $4.2\times 10^{-4}$ /psu to $2.6\times 10^{-3}$ /psu for 0.5–2 GHz over the polar region. An enhanced method is proposed by combining neighborhood impedance boundary condition (NIBC), Nystrom, and sparse matrix canonical grid (SMCG) to solve the dual surface integral equations (SIEs) using the method of moments (MoM). To simulate large-scale ocean waves using a complete ocean spectrum, the surface root-mean-square (RMS) heights are up to 22.5 cm corresponding to 1.5 times wavelength of 2 GHz. Thus, the surface lengths used are up to 333 wavelengths to account for the large RMS height. Simulation results are illustrated for 0.5–2 GHz. We show that the energy conservations are obeyed to $10^{-4}$ . The effects of roughness and salinity on broadband polar ocean emissivity are studied with the proposed approach. Numerical simulations show that the sensitivity to salinity is still preserved in spite of roughness.

The discrete Gaussian free field on a compact manifold

In this article we define the discrete Gaussian free field (DGFF) on a compact manifold. Since there is no canonical grid approximation of a manifold, we construct a random graph that suitably replaces the square lattice Zd in Euclidean space, and prove that the scaling limit of the DGFF is given by the manifold continuum Gaussian free field (GFF). Furthermore using Voronoi tessellations we can interpret the DGFF as element of a Sobolev space and show convergence to the GFF in law with respect to the strong Sobolev topology.

Sea Surface Radar Scattering at L-Band Based on Numerical Solution of Maxwell’s Equations in 3-D (NMM3D)

Radar scattering from ocean surfaces is investigated by 3-D numerical solution of Maxwell’s equations [numerical Maxwell’s model in 3-D (NMM3D)] using the ocean surface profiles stochastically generated from a 3-D Durden–Vesecky ocean spectrum. The surface integral equations (SIEs) are formulated for dielectric surfaces using Green’s functions of the air and the ocean permittivities with the surface tangential electric and magnetic fields as the unknowns. In solving the SIEs using the method of moment, a fast matrix solver of the sparse matrix canonical grid is used in conjunction with Rao–Wilton–Glisson basis functions. The computation has been implemented on a high-performance parallel computing cluster for problems with up to six million surface unknowns. Unlike the two-scale model (TSM) approximation, NMM3D does not require division of the surface spectrum into large- and small-scale ocean waves. The results of backscattering simulations are compared to Aquarius satellite radar measurements for wind speeds of 5, 8, and 10 m/s and for incidence angles of 29°, 39°, and 46°. The results show that NMM3D ocean backscattering solutions at L-band are in good agreement with Aquarius satellite radar data for co-polarized VV, HH, and cross-polarized VH returns as well as for the VV/HH ratio. The azimuthal dependence of L-band backscatter is also assessed. Finally, NMM3D results are compared to TSM solutions and are shown to lie close to Aquarius data in observed VV/HH ratio, and their azimuthal dependencies.

Rough Surface and Volume Scattering of Soil Surfaces, Ocean Surfaces, Snow, and Vegetation Based on Numerical Maxwell Model of 3-D Simulations

In this paper, we give an overview and an update on the recent progress of our research group in numerical model of Maxwell equations in three dimensions (NMM3D) on random rough surfaces and discrete random media and their applications in active and passive microwave remote sensing. The random rough surface models were applied to soil surfaces and ocean surfaces. The discrete random media models were applied to snow and vegetation. For rough surface scattering, we use the surface integral equations of Poggio–Miller–Chang–Harrington–Wu–Tsai that are solved by the method of moments using the Rao–Wilton–Glisson basis functions. The sparse matrix canonical grid method is used to accelerate the matrix column multiplications. In modeling the rough surfaces, we use the exponential correlation functions for soil surfaces and the Durden–Vesecky ocean spectrum for ocean surfaces. In scattering by terrestrial snow and snow on sea ice, we use the volume integral equations formulated with the dyadic half-space Green's function. The microstructure of snow is modeled by the bicontinuous media. In scattering by vegetation, we use the discrete scatterers of cylinder. The NMM3D formulation is based on the Foldy–Lax multiple scattering equations in conjunction with the body of revolution for a single scatterer. For rough surface scattering, simulations results are compared with advanced integral equation model, small slope approximation, small perturbation method, and two scale model. For volume scattering by snow, results are compared with the bicontinuous dense media radiative transfer. For scattering by vegetation, results are compared with distorted Born approximation and radiative transfer equation. Comparisons are also made with experiments.

On the arc index of cable links and Whitehead doubles

In this paper, we construct an algorithm to produce canonical grid diagrams of cable links and Whitehead doubles, which give sharper upper bounds of the arc index of them. Moreover, we determine the arc index of [Formula: see text]-cable links and Whitehead doubles of all prime knots with up to eight crossings.

Velocity derivative skewness in fractal-generated, non-equilibrium grid turbulence

The evolution of the velocity derivative skewness, S(∂u/∂x), is investigated along two streamwise axes and four transverse positions in the wake of a square-fractal-element grid. In the near-field, the produced turbulence exhibits non-equilibrium characteristics including Cϵ∼ReMα/ReLβ. In the far-field, the turbulence agrees with canonical grid turbulence results and Cϵ is approximately constant. It is found that in the non-equilibrium region, the value of −S(∂u/∂x) is dependent on both streamwise and transverse positions, but after a sufficient decay period, it takes on a near constant value in the far-field. It is demonstrated that the evolution Cϵ approximately corresponds to that of −S(∂u/∂x), which is suggestive that some of the non-equilibrium properties are likely a result of residual strain from the turbulence generating conditions.

Porting bioinformatics applications from grid to cloud: A macromolecular surface analysis application case study

In this paper we describe our experience in exploiting different cloud-based environments for an actual use case taken from the bioinformatics domain – the molecular surfaces analysis - that identifies similarities and possible complementarities in the protein surfaces. The analysis of macromolecular surfaces is important since protein surface conformations drive many biological reactions. We developed a workflow that performs the macromolecular surfaces analysis and provides interesting results from a scientific point of view. An important issue is represented by the fact that it is highly compute-intensive, therefore it cannot be run on a single CPU system for meaningful use cases and a parallel infrastructure is required to obtain reasonable execution time. For a decade grid infrastructures have represented suitable solutions to achieve cost effective computational power for Bioinformatics applications. However, these solutions do not offer an adequate customisation of the computational environment (e.g. installing databases and configuring virtual network) due to the rigid organisation of the storage and computational sites. Running applications on customised machines obtained by user-defined images simplifies the computing model, decreases the failure rates and therefore reduces waiting times for production analysis with respect to the canonical grid computations. For these reasons a cloud-based approach is more suitable than a pure grid paradigm. We experimented using two cloud-based approaches, based on the Worker Node On Demand Service and on OpenStack, to run the molecular surfaces analysis use case and we compared the results in terms of performance, efficiency and efforts to build the computing model with respect to grid computing.

Erratum: Navigating comics: an empirical and theoretical approach to strategies of reading comic page layouts

This General Commentary provides a corrected version of the caption for Figure 2. The text should read: (A) Canonical grid layout stereotypically read in a “Z-path.” (B) Layout where a vertical panel “blocks” the creation of a row of panels. (C) Layout where panels are separated by a wide space. (D) Layout where panels overlap each other. (E) Layout where panels are staggered to no longer retain a contiguous gutter.In addition, the link to the supplementary analysis of the Steranko page has been changed to: http://www.visuallanguagelab.com/P/ECS_Supplement.pdf

QEx

The most popular and actively researched class of quad remeshing techniques is the family of parametrization based quad meshing methods . They all strive to generate an integer-grid map , i.e. a parametrization of the input surface into R 2 such that the canonical grid of integer iso-lines forms a quad mesh when mapped back onto the surface in R 3 . An essential, albeit broadly neglected aspect of these methods is the quad extraction step, i.e. the materialization of an actual quad mesh from the mere "quad texture". Quad (mesh) extraction is often believed to be a trivial matter but quite the opposite is true: numerous special cases, ambiguities induced by numerical inaccuracies and limited solver precision, as well as imperfections in the maps produced by most methods (unless costly countermeasures are taken) pose significant challenges to the quad extractor. We present a method to sanitize a provided parametrization such that it becomes numerically consistent even in a limited precision floating point representation. Based on this we are able to provide a comprehensive and sound description of how to perform quad extraction robustly and without the need for any complex tolerance thresholds or disambiguation rules. On top of that we develop a novel strategy to cope with common local fold-overs in the parametrization. This allows our method, dubbed QEx , to generate all-quadrilateral meshes where otherwise holes, non-quad polygons or no output at all would have been produced. We thus enable the practical use of an entire class of maps that was previously considered defective. Since state of the art quad meshing methods spend a significant share of their run time solely to prevent local fold-overs, using our method it is now possible to obtain quad meshes significantly quicker than before. We also provide libQEx, an open source C++ reference implementation of our method and thus significantly lower the bar to enter the field of quad meshing.

Six strategies for defeating the Runge Phenomenon in Gaussian radial basis functions on a finite interval

Radial basis function (RBF) interpolation is a “meshless” strategy with great promise for adaptive approximation. Because it is meshless, there is no canonical grid to act as a starting point for function-adaptive grid modification. Uniform grids are therefore common with RBFs. Like polynomial interpolation, which is the limit of RBF interpolation when the RBF functions are very wide (the “flat limit”), RBF interpolants are vulnerable to the Runge Phenomenon, whether the grid is uniform or irregular: the N-point interpolant may diverge as N→∞ on the interval spanned by the interpolation points even for a function f(x) that is analytic everywhere on and near this interval. In this work, we discuss six strategies for defeating the Runge Phenomenon, specializing to a uniform grid on a finite interval and one particular species of RBFs in one space dimension: ϕ(x)=exp(−[α2/h2]x2) where h is the grid spacing. Three strategies fail, but three are successful. One good strategy is to vary α with the number of interpolation points N as N−1/4. Unfortunately this gives both a subgeometric rate of convergence for the approximation (error falling as exp(−qN) for some constant q) and a matrix condition number that grows as exp(pN) for some p>0. (In order to explain why fixed α, independent of N, is not a convergent strategy, we digress to discuss error saturation, and show experimentally that the saturation error on the finite interval is about 0.6exp(−0.47/α2)‖f‖∞; if a user-chosen error tolerance δ is acceptable, then the optimum choice is αoptimum(δ)=1/−2log(δ/0.06).) The second good strategy is RBF Extension, which uses RBF functions with centers on a interval slightly larger than the target interval to approximate f(x). A third strategy is to split the interval into two “boundary layers” and a large middle interval and apply separate RBF approximations on each. The slow-decrease-of-α strategy is much cheaper, but RBF Extension is much more resistant to ill-conditioning and therefore can achieve much lower errors. The three-interval method is the least accurate, but is robust and inexpensive.

Backscattering Coefficients, Coherent Reflectivities, and Emissivities of Randomly Rough Soil Surfaces at L-Band for SMAP Applications Based on Numerical Solutions of Maxwell Equations in Three-Dimensional Simulations

We used Numerical Maxwell Model in 3-D Simulations (NMM3D) to study the backscattering coefficients, coherent reflectivities, and emissivities of soil surfaces using Gaussian random rough surfaces with exponential correlation functions. The surface area used is 8 by 8 square wavelengths. A total of close to 200 cases are computed by varying rms height, correlation length, and soil permittivity. We consider a 40° incidence angle. For each case, 15 realizations of rough surface profiles are generated, and 30 solutions of Maxwell equations are computed because of two polarizations. The method for solving the Maxwell equations is based on the Method of Moments (MoM) with Rao-Wilton-Glisson (RWG) basis functions. The solutions are accelerated by the sparse matrix canonical grid method implemented on parallel computing. The rms height varies up to 0.126 wavelength. The results are compared with the Dubois formulation, Small Perturbation Method (SPM), Kirchhoff Approximation (KA), and Advanced Integral Equation Model (AIEM). The NMM3D results are also compared with VV and HH backscatter data of soil surfaces where ground truth rms heights and correlation lengths were both measured. Good agreement is found between the NMM3D results and experimental measurement data. Based on the computed cases, interpolation tables are derived that can be directly applied to L-band active and passive microwave remote sensing of soil moisture, such as for the upcoming Soil Moisture Active and Passive (SMAP) mission.

BISTATIC SCATTERING FROM TWO-DIMENSIONAL DIELECTRIC OCEAN ROUGH SURFACE WITH A PEC OBJECT PARTIALLY EMBEDDED BY USING THE G-SMCG METHOD

An e-cient approach called general sparse matrix canonical grid (G-SMCG) method is proposed to analyze the electromagnetic scattering from 2-D dielectric rough surface with a conducting object partially buried. In this paper, the scattering of 3-D arbitrarily shaped object is computed by using the traditional method of moments (MoM) with RWG basis function, and the scattering of rough surface is analyzed by using the SMCG method. The coupling interactions between an object and rough surface are calculated by iterative method. Combing the ocean rough surface with Pierson Moskowitz (PM) spectrum, the bistatic scattering coe-cients of typical objects buried in the ocean surface have been computed by using the proposed method. Then the accuracy and e-ciency of this method are discussed. Finally, the bistatic scattering coe-cients of a ship located on ocean surface are calculated, and the in∞uence of sea state and wind direction on the scattering coe-cients is indicated.

Implementation and validation of the curvilinear coordinate method for the scattering of electromagnetic waves from two-dimensional dielectric random rough surfaces

The curvilinear coordinate method is applied for analysing 2-D dielectric random rough surfaces. The theory is based on Maxwell's equations written in a non-orthogonal coordinate system. For each medium, this method leads to an eigenvalue system. The scattered fields within two media are expanded as linear combinations of eigensolutions satisfying the outgoing wave condition. The boundary conditions allow the scattering amplitudes to be determined. The coherent and incoherent intensities are estimated by averaging the scattering amplitudes over several realizations. The theory is verified by comparison with results obtained by other exact method. A discussion on the C-method and the Sparse-Matrix CAnonical Grid method is proposed in terms of accuracy and computation time.

MICROWAVE EMISSION FROM TWO-DIMENSIONAL INHOMOGENEOUS DIELECTRIC ROUGH SURFACES BASED ON PHYSICS-BASED TWO-GRID METHOD

Numerical simulations of emission for two-dimensional randomly rough surfaces with an inhomogeneous layered medium are presented. The inhomogeneous layered medium is modeled by a generalized n-layered stratified media. The numerical method was adopted from the physics-based two-grid method (PBTG). To ensure the strict accuracy requirement while to relief the memory and CPU resources, the PBTG in conjunction with the sparse-matrix canonical grid method (SMCG) was used in this paper. In so doing the reflected terms of the dyadic Green's function that accounts for layered media were added into the integral equations governing the surface tangential fields. Since the reflected part of the dyadic Green's function does not

Application of stochastic second-degree method to electromagnetic scattering from randomly rough surfaces

Application of a stochastic second-degree method in combination with the banded matrix canonical grid (BMIA/CAG) method for two-dimensional electromagnetic scattering from PEC randomly rough surfaces is presented. This method can improve convergence while preserving the computational attractiveness of the BMIA/CAG method. Numerical examples illustrate the effectiveness of the proposed method.

Application of the Stochastic Second-degree Iterative Method to EM Scattering from Randomly Rough Surfaces

Application of a stochastic second-degree method in combination with the banded matrix canonical grid (BMIA/CAG) method for two-dimensional electromagnetic scattering from PEC randomly rough surfaces is presented. This method can improve convergence over while preserving the computational attractiveness of the BMIA/CAG method. Numerical examples illustrate the effectiveness of the proposed method.

Study of BGP Peering Session Attacks and Their Impacts on Routing Performance

We present a detailed study of the potential impact of border gateway protocol peering session attacks and the resulting exploitation of route flap damping (RFD) that cause network-wide routing disruptions. We consider canonical grid as well as down-sampled realistic autonomous system (AS) topologies and address the impact of various typical service provider routing policies. Our modeling focuses on three dimensions of routing performance sensitivity: 1) protocol aware attacks (e.g., tuned to RFD); 2) route selection policy; and 3) attack-region topology. Analytical results provide insights into the nature of the problem and potential impact of the attacks. Detailed packet-level simulation results complement the analytical models and provide many additional insights into specific protocol interactions and timing issues. Finally, we quantify the potential effect of the BGP graceful restart mechanism as a partial mitigation of the BGP vulnerability to peering session attacks.

Scattering by rough surface using a hybrid technique combining the multilevel UV method with the sparse matrix canonical grid method

A procedure is developed for combining efficiently the multilevel UV method with the sparse matrix canonical grid (SMCG) method in the computation of three‐dimensional (3‐D) wave scattering from random rough surface. It handles near‐ and intermediate‐field interactions by the multilevel UV method and far‐field interactions by the SMCG method. This hybrid UV/SMCG method removes the large memory requirement of both the UV method in the far field and the SMCG method in the near field. The computational complexity for the hybrid method is O(NlogN). The tradeoffs in computer memory requirement and CPU time between the UV part and the SMCG part are examined for Gaussian random rough surfaces with Gaussian correlation function and with exponential correlation function. For a surface area of 44 × 44 square wavelengths with RMS of 1 wavelength and 123,904 surface unknowns, the UV/SMCG method requires CPU of 52.2 s per iteration with total CPU of 46.6 min for 22 iterations on a single processor of CPU speed of 2.66 GHz.