Spherical Expansion Research Articles

Application of Nonuniform FFT to Spherical Near-Field Antenna Measurements

This article investigates on the use of nonuniform fast Fourier transform (NUFFT) to solve spherical near-field transformation problems. Traditional postprocessing algorithms make use of conventional FFT for the transformation of measured grids with uniform sampling. More recently, the processing of irregular sampling grids has been introduced by means of matrix-based algorithms. The benefits of both type of algorithms are combined in this article with the implementation of the spherical wave expansion (SWE) summation and its adjoint with NUFFT. This allows the processing of fields with the efficiency of the FFT approaches and the versatility of the matrix operators. To this end, the NUFFT will be introduced and it will be shown how it can be implemented to accelerate the multidimensional summations required in the SWE calculations. Numerical investigations will be introduced to demonstrate the computational gains achieved with negligible approximation errors. Finally, the NUFFT will be tested on three different spherical near-field postprocessing problems: far-field transformation of near fields measured with spiral scanning, phaseless measurements on spherical surfaces with different sampling grids, and antenna diagnostics.

Stellar Speckle and Correlation Derived From Classical Wave Expansions for Spherical Antennas

Michelson phase and Hanbury Brown-Twiss intensity stellar interferometry require expressions for the first-and second-order correlation functions, respectively, of the fields radiated by stars in terms of their diameters and measured quasi-monochromatic wavelengths. Although our sun and most other stars are spherical in shape at optical wavelengths, previous determinations of speckle and correlation functions have modeled stars as circular disks rather than spheres because of the mathematical tools available for partially coherent fields on planar surfaces. However, with the incentive that most stars are indeed shaped like spheres and not disks, this article models a star as a spherical antenna composed of a random distribution of uncorrelated volume sources within a thin surface layer (photosphere). Working directly with the time-domain fields, a self-contained, straightforward, detailed derivation of speckle patterns and correlation functions is given based on a novel, angularly symmetric, spherical mode expansion with coefficients determined by the assumed Lambertian nature of the star's radiation and the uniform asymptotic behavior of the spherical Hankel functions. First-order spatially averaged and temporally averaged correlation functions are proven to be identical, and the normalized second-order correlation function is shown to equal one plus the square of the first-order correlation function. The direct time-domain approach reveals explicit expressions for the quasi-monochromatic wave-packet fields of stellar radiation as well as new criteria for the validity of the far-field approximation for the fields of incoherent sources that are much less restrictive than the Rayleigh-distance criterion for coherent sources.

An empirical approach for penetration of tandem warheads into concrete targets

We develop a semi-empirical or semi-analytical model based on the spherical cavity expansion theory in order to simulate the normal penetration of tandem warheads into concrete targets, which consist of two stages: penetration into a cavity region (or damaged region), and subsequent penetration into an undamaged region. To formulate the semi-empirical model, we consider the first impact of rigid projectiles on the damaged region with a pre-drilled cavity and the second on the undamaged region. For the penetration into the undamaged region corresponding to the second stage, an empirical constant used in the semi-empirical model is determined based on the penetration depth obtained by experiments in the literature. The semi-empirical model is then verified by comparing its results to experimental data and numerical results of the explicit finite element method (FEM) in terms of kinematic quantities of projectiles. For the penetration into damaged concrete targets in which a cavity is considered as a pre-drilled hole, which provides a basis of the formulation for the first stage, the results of the semi-empirical model are compared to experimental data, and an empirical constant is also estimated based on the numerical results of the explicit FEM. Finally, the concrete targets with a finite-length cavity, into which a tandem warhead penetrates, can be regarded as a combination of the damaged region with a finite-length cavity and the undamaged region. In the semi-empirical model for the penetration of tandem warheads, the empirical constants for the first and second stages are employed as those estimated for the damaged region with a cavity of finite length and for the undamaged region, respectively. The results from the semi-empirical model are compared with those from the explicit FEM due to the lack of experimental data. The good agreement of the results of the semi-empirical model and the explicit FEM demonstrates that the proposed semi-empirical model can predict the dynamic behavior of the tandem warheads in a simple and accurate manner.

Elasto-plastic solutions for expanding cavities in strain-hardening and/or softening soils

This paper presents an analytical finite strain solution for cavity expansion in strain-hardening and/or softening Mohr-Coulomb soil subjected to a non-associated flow rule. The evolutions of material parameters, i.e. deformation and strength parameters, are totally considered. The finite strain is applied to describe the plastic behavior with the logarithmic strain. By assuming constant material properties in a very small region, the plastic region is divided into a series of concentric annuli. The solution can be recursively obtained according to the closed-form solution of each annulus, and is suitable to soil with any complex material evolution types. For the cavity with the same cavity expansion a/a0, the required cavity pressure of strain-hardening and/or softening soils is larger and/or smaller than that of elasto-perfectly plastic soils for both spherical and cylindrical cavity expansions. With respect to the same cavity pressure, the displacement and stress around cavity of strain-hardening and/or softening models are larger and/or smaller than those of perfectly plastic model. With respect to the same cavity expansion displacement, the increases in the strength parameters including cohesive strength and friction enlarge the cavity pressure, and slightly decrease the plastic and residual radii.

Element phase centre and orientation compensation in conformal arrays during continuous deformations based on spherical mode expansion

Radiation field of conformal arrays can be obtained by simulation computation. There must be a trade-off between computation accuracy and complexity. With respect to applications where both high efficiency or even real-time applications and accuracy are required, such as adaptive beamformers together with time-varying array structures due to deformations or vibrations, more effective methodologies have to be employed for precise mainlobe direction maintenance and rigorous suppression of existing interferences. A method combining spherical mode expansion, rotation and translation of spherical waves, spatial displacement phase compensation, and approximations of constant element radiation patterns in local coordinate systems is proposed here, featuring one-time numerical effort for isolated element radiation patterns, fast coupling matrix calculation based on updated array configuration, and analytical radiation field expressions with limited eigenmode expansion coefficients storage requirements. Adaptive beamformers are utilised to validate the effects of remaining approximations including constant element patterns in local coordinate systems and constant mutual coupling matrix, proving the proposed method has the potential for aircraft-borne conformal antenna array radiation field analysis where deformations and vibrations occur continuously.

Effect of initial temperature on explosion characteristics of 2, 3, 3, 3–Tetrafluoropropene

The flammability of refrigerants is a major cause of refrigerant explosion incidents. Studying the explosion characteristics of refrigerants at different initial temperatures can provide significant benefits for solving the safety problems of refrigerants under actual working conditions. This paper studied the effects of the initial temperature and refrigerant concentration on the explosion characteristics of refrigerant 2, 3, 3, 3-tetrafluoropropene (R1234yf) at 0.1 MPa. The curves of explosion characteristics with different initial temperature revealed the same variation trend ranged from 25 °C to 115 °C. Specifically, as the refrigerant concentration was raised, the peak overpressure, the maximum rate of pressure rise, and laminar burning velocity increased initially and decreased afterwards, along with maximum values at the refrigerant concentration of 7.6%. When the refrigerant concentration was 7.6%, the peak overpressure declined exponentially with the initial temperature rise, while the maximum rate of pressure rise increased linearly. The laminar burning velocity calculated from the spherical expansion method indicated that the flame propagation was gradually accelerated by the increase of initial temperature, which coincided with the change of the maximum rate of pressure rise. Meanwhile, experiments and CHEMKIN simulation results demonstrated the effects of elevated temperature from 20 °C to 50 °C on the explosion limits of R1234yf. The lower explosion limit reduced and the upper explosion limit increased with rising initial temperature. In general, R1234yf exhibited moderate combustion and lower explosion risk, compared with traditional refrigerants.

Fast Spherical Near-Field to Far-Field Transformation for Offset-Mounted Antenna Measurements

In spherical near-field measurements, the number of measured samples grows with the radius of the minimum sphere enclosing the antenna under test (AUT). This leads to unnecessary long acquisition times for antennas mounted in offset positions with respect to the measurement sphere. In this letter, a general technique is proposed to reduce the sampling rate requirements of offset mounted antennas. The approach is based on centering the spherical wave expansion (SWE) over the AUT by a shift in the coordinate system to compensate for the offset. The purpose of this shift is to reduce the number of significant spherical waves required to represent the AUT field. In the subsequent steps of the algorithm, the coordinate system is brought to its original location to preserve the efficiency and probe-correction capabilities of the traditional spherical near-field to far-field transformation technique. This is performed by a proper translation of the SWE coefficients between both coordinate systems. Higher order probe correction is also supported intrinsically. The proposed algorithm is numerically tested to assess its performance. Electromagnetic simulation and anechoic chamber tests are used to validate it, showing reductions in measurement times with low transformation errors.

Acoustic radiation force on a compressible spheroid

The acoustic radiation force on a compressible spheroid is calculated using expansions of the scattered field in terms of both spherical and spheroidal wave functions. There is no restriction on the size or acoustic impedance of the spheroid, the structure of the incident field, or the orientation of the spheroid with respect to the incident field. The solution is the same as that developed previously for the radiation force on an elastic sphere, which is expressed as a summation of terms involving products of the coefficients in spherical wave expansions of the incident and scattered fields [Ilinskii etal., J. Acoust. Soc. Am. 144, 568–576 (2018)]. Spheroidal wave expansions are used to satisfy the boundary conditions on the compressible spheroid, and far-field asymptotes are used to relate the spheroidal and spherical wave scattering coefficients analytically. Explicit expressions for the scattering coefficients are available for spheroids with rigid or free surfaces. Results are compared with available analytical solutions for various limiting cases. The theoretical framework may be employed for objects of arbitrary shape provided the spherical wave expansion of the scattered field is known. [T.S.J. is supported by the Applied Research Laboratories Chester M. McKinney Graduate Fellowship in Acoustics.] a)Deceased.

Acoustic radiation force on a compressible spheroid.

The acoustic radiation force on a compressible spheroid is calculated using expansions of the scattered field in terms of both spherical and spheroidal wave functions that are matched analytically in the far field. There is no restriction on the size or impedance of the spheroid, the structure of the incident field, or the orientation of the spheroid with respect to the incident field. The form of the solution is the same as that developed previously for the radiation force on an elastic sphere, which is a summation of terms involving products of the coefficients in spherical wave expansions of the incident and scattered fields. Spheroidal wave expansions are employed to satisfy the boundary conditions and obtain the scattering coefficients. While the scattering coefficients must be obtained numerically for compressible spheroids, explicit expressions in terms of radial wave functions are available for spheroids with rigid or free surfaces. Results are compared with available analytical expressions for various limiting cases. The theoretical framework may be extended to objects of arbitrary shape.

Experimental and numerical study of the effects of oxygen-enriched air on the laminar burning characteristics of biomass-derived syngas

The laminar burning velocities (LBVs) of seven different composition stoichiometric mixtures (CO: 15%–20%, H2: 5%–20%, CH4: 5%–15%, CO2: 15%, N2: 50%) under different oxygen levels (O2: 21%–60%) were experimentally measured by spherical expansion flame method. Numerical calculations were conducted with the PREMIX code using seven detailed chemical reaction mechanisms (FFCM-1, GRI 3.0, USC-Ⅱ, San Diego + NOx, HP-mech, Li-2015, CRECK + NOx) and compared with experimental data. The CRECK mechanism provides an over-prediction of the LBV of CH4-free mixtures. San Diego is more prominent than CRECK with the addition of CH4, followed by GRI 3.0 and HP-mech. Three mechanisms—FFCM-1, Li-2015, USC-Ⅱ—that matched the experimental results well were chosen to perform the LBV calculations for a wider range of initial conditions (Φ: 0.6–1.6, Tu: 300 and 400 K, P0: 1 and 3 atm) for variable composition mixtures. The results show that for all cases, the LBV is considerably increased with increasing oxygen concentration. CH4-rich mixtures are more sensitive to oxygen concentrations and work to increase the LBVs mainly through thermal effects. An oxygen-rich environment makes the flammability limit wider even with high initial pressure. H2-rich mixtures accompanied by higher thermal diffusivity are more susceptible to the initial temperature, while chemical effects do not dominate in oxygen-rich environments. In oxygen-rich environments, diffusional-thermal instability is suppressed while the cellular structure of the flame is mainly affected by fuel composition and hydrodynamic instability. The formation of NO in flames of different composition mixtures in an oxygen-rich environment was analyzed.

The fundamental assumptions of cavity expansion models for penetration - revisited

The fundamental assumptions motivating the use of cylindrical (CCE) and spherical (SCE) cavity expansion solutions for the prediction of penetration mechanics are revisited. Predictions of an ovoid of Rankine model (OR), which has been validated relative to axisymmetric numerical simulations, are used as a reference solution for normal penetration into a metallic target. In previous papers it has been observed that the average axial resistance stress is independent of the penetration velocity until a critical value which denotes the onset of separation of the target material from the projectile’s surface. The main conclusion of this paper is that the velocity fields in the CCE and SCE models do not accurately model a realistic flow field for penetration of a projectile. The results in this paper quantify the errors caused by low estimates of the static resistance stress and compensation due to incorrect dependence of the resistance stress on the penetration velocity predicted by the CCE and SCE models.

Efficient estimation of EMI from stacked radiation sources by combining their spherical wave expansion coefficients

The objective of this study is the development of an efficient methodology, based on the spherical wave expansion theory, to evaluate the total unwanted electromagnetic radiation of multiple radiation sources starting from the knowledge of the radiation characteristic of every single source considered as isolated. The specific application domain deals with the estimation of the electromagnetic interference from stacked trays in server racks. After having computed the radiation of each isolated tray, by using the spherical expansion technique the proper expansion coefficients are calculated. They are combined in a set of new coefficients that are used to reconstruct the estimated total field from the server rack. This study discusses the proposed methodology step by step and offers practical considerations on its implementation when measured data are given as input. The adopted numerical examples show in a quantitative way, through different graphs and comparisons, the limits and the advantages of the technique. The proposed method has been shown to reliably predict the radiation from stacked trays with only a few dB of uncertainty. The examples are based on numerically computed data sets of electromagnetic emissions from trays and racks but the proposed procedure is valid also for measured data.

Study on Rock-Breaking Depth and Damage Area under Particle Jet Impact

Particle jet impact drilling technology is an efficient method which mainly uses high-velocity particles to break rock. As the important criterion for evaluating rock-breaking effect, rock-breaking depth and damage area were studied in this paper. Firstly, a particle jet impact rock-breaking test device was developed, and laboratory experiments have been carried out. Then, based on the spherical cavity expansion theory, the mathematical model of rock-breaking depth and damage area under particle jet impact was established. Afterward, the effect of water-jet impact velocity, impact angle, and particle diameter on rock-breaking depth and damage area was analyzed by comparing experimental results and mathematical calculation. The results show that rock-breaking depth and damage area would increase with increase of water-jet impact velocity and decrease slightly with increase of particle diameter. And the combination of 8° and 20° is recommended for nozzle layout. The experimental results and mathematical calculation are basically consistent, which could verify the correctness of the mathematical model. The study has significance for development and application of particle jet impact rock-breaking technology and perfection of theoretical research.

Solar Wind Turbulence from 1 to 45 au. III. Anisotropy of Magnetic Fluctuations in the Inertial Range Using Voyager and ACE Observations

Abstract We examine both Voyager and Advanced Composition Explorer magnetic field measurements at frequencies that characterize the inertial range and evaluate the anisotropy of the fluctuations as they relate to both the compressive component and underlying wavevector anisotropy of the turbulence. The magnetic fluctuation anisotropy as it relates to the compressive component is directly dependent upon both the plasma beta of the thermal proton component and the ratio of magnetic fluctuation magnitude to the strength of the mean magnetic field. This has been seen before at 1 au. The magnetic fluctuation anisotropy in the plane perpendicular to the mean magnetic field, which is a measure of the anisotropy of the underlying wavevector distribution, should depend on the angle between the mean magnetic field and the radial direction and should be confined to values between one and the index of the power spectrum, which is typically 5/3. Our results show that the average of this anisotropy exceeds the value of the spectral index and is out of bounds with the theory. Although the results are suggestive of past analyses, we find that spherical expansion of the turbulence may offer at least a partial explanation of the apparent amplification of this measured anisotropy.

The Non-Stationary Dynamic Analytical Solution of a Spherical/Cylindrical Cavity Expansion

Abstract A review of the pertinent literature related to the dynamic expansion of a spherical/cylindrical cavity shows that all the solutions with kinematic boundary conditions deal with a constant velocity at the cavity boundary. This paper develops a new general solution of the nonstationary dynamic problem of cavity expansion, which allows the application of time-dependent motion conditions at the cavity boundary. This solution can be used, for example, in the development of approximate approaches for projectiles penetrating with a non-constant velocity into different targets. Due to the complexity of the nonlinear nonstationary problem, an analytical solution of the problem may be developed if simplified constitutive relationships are used. In the present model, a simplified material model with a locked equation of state and a linear shear failure relationship is implemented. This solution may be applied to different materials such as concrete, soil, and rock. Special cases of the newly developed nonstationary solution are compared with different spherical and cylindrical cavity expansions solutions reported in the literature, and a good agreement is obtained. The capability of the present model is demonstrated in a following investigation of representative cases of cavity expansion with zero, constant, and variable acceleration of the cavity boundary. A significant difference in the stress variation for the different cases is shown. Along with the general solution which deals with an elastic–plastic region, a simplified solution which disregards the contribution of the elastic region is presented and the evaluation of the elastic region effect may be assessed.

Numerical and analytical investigations on projectile perforation on steel–concrete–steel sandwich panels

Steel–concrete–steel (SCS) sandwich panels have been achieving increasing attention due to its good impact resistant performance, but there still lacks related research concerning analytical model for SCS perforation. In this paper, numerical and analytical investigations were performed to study the impact resistance of SCS sandwich panels against rigid projectile penetration. FE simulations of concrete-steel perforation were conducted to validate the numerical model in terms of projectile residual velocity and damage mode. With same concrete core, five thickness combinations of front and rear steel plates were numerically studied to explore the sandwich structural effect on the perforation responses. Utilizing spherical cavity expansion analysis and plates petalling theory, a semi-empirical analytical model was developed to describe the perforation on SCS sandwich panels, which was characterized with 7 penetration stages. Agreement was reached between numerical simulation and theoretical model in terms of both projectile deceleration history and residual velocity. On the perspective of structurally absorbed energy, the thickness combinations of front and rear steel plate were further analyzed and discussed. For the same thickness, the rear steel plate was found to consume more energy than the front plate, which provides better protection against projectile impact loadings.

Magnetic-field modeling with surface currents. Part I. Physical and computational principles of bfieldtools

Surface currents provide a general way to model magnetic fields in source-free volumes. To facilitate the use of surface currents in magneto-quasistatic problems, we have implemented a set of computational tools in a Python package named bfieldtools. In this work, we describe the physical and computational principles of this toolset. To be able to work with surface currents of the arbitrary shape, we discretize the currents on triangle meshes using piecewise-linear stream functions. We apply analytical discretizations of integral equations to obtain the magnetic field and potentials associated with the discrete stream function. In addition, we describe the computation of the spherical multipole expansion and a novel surface-harmonic expansion for surface currents, both of which are useful for representing the magnetic field in source-free volumes with a small number of parameters. Lastly, we share examples related to magnetic shielding and the surface-coil design using the presented tools.

Under-Sampled Spherical Near-Field Antenna Measurements With Error Estimation

This article proposes an under-sampling scheme in φ for spherical near-field antenna measurements. Reducing the number of samples along one of the spherical coordinates allows to decrease the number of positioner steps. Therefore, the savings in the number of samples are almost directly translated into measurement time savings. Under-sampling usually leads to aliasing; however, it can be controlled by exploiting the properties of the spherical wave expansion. The axial symmetry of conventional antennas leads to narrow azimuthal spectra, where the influence of this aliasing is small. By designing an estimator of the error introduced by the aliasing, the required number of φ samples for a given accuracy can be determined, without any <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">a priori</i> knowledge about the antenna under test. This allows the application of this technique for any kind of antenna regardless of its level of axial symmetry, obtaining different values of measurement time savings depending on the level of symmetry. Several examples using the numerical and measured data are presented, showing significant reductions in the measured samples for different antennas. In addition, the algorithm is successfully tested against antennas with poor axial symmetry to demonstrate its robustness.

An elastoplastic solution for spherical cavity undrained expansion in overconsolidated soils

In this paper, a novel elastoplastic solution is developed for the undrained expansion of a spherical cavity in overconsolidated soils, with the aim of providing a mechanical model for the interpretation of cone penetration tests in overconsolidated soils. The well-established unified hardening (UH) model is adopted in the present solution to represent the unique mechanical behaviors of the overconsolidated soil during cavity expansion. The problem considered is formulated as a system of first order partial differential equations in terms of a Lagrangian description; these equations are solved as an initial value problem with the code developed based on the Runge-Kutta algorithm. The results, including the expansion-pressure curves, the distributions of the stress components, and the evolution of the potential strength during cavity expansion are presented to show the expansion responses in overconsolidated soils. The results reveal that the unique expansion responses in overconsolidated soil during cavity expansion, such as the development of a negative excess pore water pressure, the strain-hardening and softening behavior, the peak strength and the decay of the potential strength during cavity expansion could be well represented by the present solution.

Analysis of lateral response of existing single pile caused by penetration of adjacent pile in undrained clay

Quantifying the lateral displacement and bending moment of the existing pile caused by the penetration of the adjacent pile are of great engineering significance to ensure the safety and stability of the buildings. In this paper, a two-stage analysis approach for predicting the response of the existing pile due to the penetration of the adjacent pile is presented. First, a simplified model is proposed to determine the free-field lateral soil displacement induced by the pile penetration. This model assumes that the pile penetration process can be regarded as a series of displacement-controlled spherical cavity expansion in semi-infinite soil, and considers the large deformation effect of soil as well as the influence of the existence of the ground surface. The validity and accuracy of the proposed model are verified by comparing with the published results. Subsequently, the p-y curve method is adopted to analyze the nonlinear pile-soil interaction for solving the response of the existing pile induced by the lateral displacement of soil obtained in the first stage. Through a series of parametric investigations, it is indicated that the impact of the penetration of adjacent pile on the response of the existing pile rests with some factors. Finally, some design charts are presented by parametric analysis, which can be used to assess the response of the existing pile caused by the penetration of the adjacent pile in practical engineering.