Elliptical Approximation Research Articles

3D reverse-time migration for pure P-wave in orthorhombic media

Compared with the transverse isotropic (TI) medium, the orthorhombic anisotropic medium has both horizontal and vertical symmetry axes and it can be approximated as a set of vertical fissures developed in a group of horizontal strata. Although the full-elastic orthorhombic anisotropic wave equation can accurately simulate seismic wave propagation in the underground media, a huge computational cost is required in seismic modeling, migration, and inversion. The conventional coupled pseudo-acoustic wave equations based on acoustic approximation can be used to significantly reduce the cost of calculation. However, these equations usually suffer from unwanted shear wave artifacts during wave propagation, and the presence of these artifacts can significantly degrade the imaging quality. To solve these problems, we derived a new pure P-wave equation for orthorhombic media that eliminates shear wave artifacts while compromising computational efficiency and accuracy. In addition, the derived equation involves pseudo-differential operators and it must be solved by 3D FFT algorithms. In order to reduce the number of 3D FFT, we utilized the finite difference and pseudo-spectral methods to conduct 3D forward modeling. Furthermore, we simplified the equation by using elliptic approximation and implemented 3D reverse-time migration (RTM). Forward modeling tests on several homogeneous and heterogeneous models confirm that the accuracy of the new equation is better than that of conventional methods. 3D RTM imaging tests on three-layer and SEG/EAGE 3D salt models confirm that the ORT media have better imaging quality.

Estimating body volumes and surface areas of animals from cross-sections.

Body mass and surface area are among the most important biological properties, but such information is lacking for some extant organisms and most extinct species. Numerous methods have been developed for body size estimation of animals for this reason. There are two main categories of mass-estimating approaches: extant-scaling approaches and volumetric-density approaches. Extant-scaling approaches determine the relationships between linear skeletal measurements and body mass using regression equations. Volumetric-density approaches, on the other hand, are all based on models. The models are of various types, including physical models, 2D images, and 3D virtual reconstructions. Once the models are constructed, their volumes are acquired using Archimedes' Principle, math formulae, or 3D software. Then densities are assigned to convert volumes to masses. The acquisition of surface area is similar to volume estimation by changing math formulae or software commands. This article presents a new 2D volumetric-density approach called the cross-sectional method (CSM). The CSM integrates biological cross-sections to estimate volume and surface area accurately. It requires a side view or dorsal/ventral view image, a series of cross-sectional silhouettes and some measurements to perform the calculation. To evaluate the performance of the CSM, two other 2D volumetric-density approaches (Graphic Double Integration (GDI) and Paleomass) are compared with it. The CSM produces very accurate results, with average error rates around 0.20% in volume and 1.21% in area respectively. It has higher accuracy than GDI or Paleomass in estimating the volumes and areas of irregular-shaped biological structures. Most previous 2D volumetric-density approaches assume an elliptical or superelliptical approximation of animal cross-sections. Such an approximation does not always have good performance. The CSM processes the true profiles directly rather than approximating and can deal with any shape. It can process objects that have gradually changing cross-sections. This study also suggests that more attention should be paid to the careful acquisition of cross-sections of animals in 2D volumetric-density approaches, otherwise serious errors may be introduced during the estimations. Combined with 2D modeling techniques, the CSM can be considered as an alternative to 3D modeling under certain conditions. It can reduce the complexity of making reconstructions while ensuring the reliability of the results.

Study on Properties of Glass-Fiber-Fabric-Reinforced Microwave-Absorbing Composites.

In this paper, the glass-fiber-fabric-reinforced resin-based absorbing composites were prepared, and their microwave-absorbing properties were studied via simulation and experiment. The simulation results show that the absorption bandwidth of the absorbing material can cover the X\C\S band, respectively, at different thicknesses. The minimum reflection loss (RL) of the composite with a thickness of 2.2 mm is -27.4 dB at 5.95 GHz. However, the experiment results are quite different from those of the simulation. The metallographic results indicate that it is the change of the mass fraction of the absorbents in the composites after curing that causes the difference. According to the metallographic results, three shape approximation methods were proposed to calculate the real mass ratio of the absorbents in the composites, namely, parallelogram approximation, bows approximation, and elliptical approximation. Meanwhile, the structural parameter Kf was introduced to optimize the calculation results. The electromagnetic parameters of the material based on the calculation results were measured, and the results show that the simulation results obtained via bow approximation have a better coincidence to the experiment results, and the mass ratio of the absorbent raises by around 9.95%, which lays a foundation for the subsequent design of microwave-absorbing composites.

Statistical analysis of parameters and adsorption isotherm models.

The present work intends to discuss parameter estimation and statistical analysis in adsorption. The Langmuir and Tóth isotherm models are compared for a set of carbon dioxide adsorption data on 13X zeolite from literature at different temperatures: 303, 323, 373, and 423 K. Statistical analyses were performed under frequentist and Bayesian perspectives. Under the frequentist statistical view, parameters were estimated using Maximum Likelihood estimation (MLE). Statistical analyses of parameters were performed by confidence regions in terms of elliptical approximation and likelihood region, while the evaluation of models was performed by chi-square statistics. The results showed that, for these nonlinear models, the elliptical region offers a poor approximation of the parameter estimates' confidence region, especially for the most correlated parameter pairs. Additionally, the four-parameter Tóth's equation yields less correlated parameters than the three-parameter Langmuir model. From a Bayesian perspective, the Markov chain Monte Carlo (MCMC) technique facilitated the reconstruction of the probability density functions of parameters as well as enabled the propagation of parametric uncertainties in the model responses. Finally, the accurate assessment of experimental uncertainty significantly influences the evaluation of models and their respective parameters.

Various-Order Low-Pass Filter with the Electronic Change of Its Approximation.

The design of a low-pass-frequency filter with the electronic change of the approximation characteristics of resulting responses is presented. The filter also offers the reconnection-less reconfiguration of the order (1st-, 2nd-, 3rd- and 4th-order functions are available). Furthermore, the filter offers the electronic control of the cut-off frequency of the output response. The feature of the electronic change in the approximation characteristics is investigated for the Butterworth, Bessel, Elliptic, Chebyshev and Inverse Chebyshev approximations. The design is verified by PSpice simulations and experimental measurements. The results are also supported by the transient domain response (response to the square waveform), comparison of the group delay, sensitivity analysis and implementation feasibility based on given approximation. The benefit of the proposed electronic change in the approximation characteristics feature (in general signal processing or for sensors in particular) is presented and discussed for an exemplary scenario.

Accuracy of Preoperative Cochlear Duct Length Estimation and Angular Insertion Depth Prediction.

In cochlear implantation with flexible lateral wall electrodes, a cochlear coverage of 70% to 80% is assumed to yield an optimal speech perception. Therefore, fitting the cochlear implant (CI) to the patient's individual anatomy has gained importance in recent years. For these reasons, the optimal angular insertion depth (AID) has to be calculated before cochlear implantation. One CI manufacturer offers a software that allows to visualize the AID of different electrode arrays. Here, it is hypothesized that these preoperative AID models overestimate the postoperatively measured insertion angle. This study aims to investigate the agreement between preoperatively estimated and postoperatively measured AID. Retrospective cross-sectional study. Single-center tertiary referral center. Patients undergoing cochlear implantation. Preoperative and postoperative high-resolution computed tomography (HRCT). The cochlear duct length was estimated by determining cochlear parameters ( A value and B value), and the AID for the chosen electrode was (i) estimated by elliptic circular approximation by the software and (ii) measured manually postoperatively by detecting the electrode contacts after insertion. A total of 80 HRCT imaging data sets from 69 patients were analyzed. The mean preoperative AID estimation was 662.0° (standard deviation [SD], 61.5°), and the mean postoperatively measured AID was 583.9° (SD, 73.6°). In all cases (100%), preoperative AID estimation significantly overestimated the postoperative determined insertion angle (mean difference, 38.1°). A correcting factor of 5% on preoperative AID estimation dissolves these differences. The use of an electrode visualization tool may lead to shorter electrode array choices because of an overestimation of the insertion angle. Applying a correction factor of 0.95 on preoperative AID estimation is recommended.

Accurate design of a TeO2 noncollinear acousto-optic tunable filter with refractive index correction.

The refractive index is a key factor in the design and analysis of noncollinear acousto-optic tunable filter (AOTF) devices. While previous studies have corrected and analyzed the effects of anisotropic birefringence and the rotatory property, they still rely on paraxial and elliptical approximations, which can introduce non-negligible errors (0.5° or more) into the geometric parameters of TeO2 noncollinear AOTF devices. In this paper, we address these approximations and their effects through refractive index correction. This fundamental theoretical research has significant implications for the design and application of noncollinear AOTF devices.

Semi-analytical solution for the elastic wave propagation due to a dislocation source in a transversely isotropic half-space

SUMMARY In this paper, the 3-D theoretical seismogram at the free surface of an elastic half-space with ‘vertical transverse isotropy (VTI)’ and an ‘elliptic anisotropy‘ is synthesized via semi-analytical formulation. To this end, the time-domain solution is obtained via the Hankel and Laplace integral transforms and the Cagniard contour integration scheme. The formulations include complete Green's functions expressed in compact and elegant formulations in terms of some elementary line integrals over the finite interval (0, π/2), due to the general point force and point double-couple source of arbitrary orientation, varying with time as Heaviside step function. This solution is further implemented to model seismic waves generated from moment tensor source with a ramp-type slip function. The solution clearly delineates compressional P waves, shear SV waves and SH waves, diffracted SPwaves and surface Rayleigh waves, with two critical distances in which the mode conversion happens. The first marks the conversion of the total reflection of SV wave into the diffracted SPwave travelling on the free surface with the velocity of compressional wave. The second, however, marks a location where the order of arrival times of SP and SH waves is reversed. The interesting phenomenon of shear wave splitting by the apparent time lag between the arrivals of SV and SHwaves is further demonstrated. Studied also is the effect of non-double-couple (non-DC) components of the moment tensor of the shear fault. Particularly, it is shown that non-DC components may lead to large amplitude of the P wave in the presence of anisotropy, resulting in the changed polarity of P wave at the onset of the motion. It is found that for the elliptic anisotropy, and for oblique-thrust faults, the isotropic moment (ISO) is always negative if the elastic constants satisfy C11 > C33 and C44 > C66, whereas it is always positive if C11 < C33 and C44 < C66 (for oblique-reverse faults, the situation is reversed). It is shown that for some sedimentary rocks, the decomposition of moment tensor in the general VTI and in its elliptic approximation are close to each other, and that the moment tensor decomposition in VTI media is very sensitive to the elastic coefficients, such that a small change in the elastic constants may lead to a remarkable change in the moment tensor decomposition. Particularly, the effect of elastic constant C13 is shown to be significant. While the time-domain solution obtained in this paper can be also degenerated to the solution in the corresponding isotropic half-space, the Mathematica code of the solution provided can be served as benchmark to other numerical solutions, applicable for the computation of the theoretical seismogram in the involved media.

Large-Degree Asymptotics of Rational Painlevé-IV Solutions by the Isomonodromy Method

The Painlevé-IV equation has two families of rational solutions generated, respectively, by the generalized Hermite polynomials and the generalized Okamoto polynomials. We apply the isomonodromy method to represent all of these rational solutions by means of two related Riemann–Hilbert problems, each of which involves two integer-valued parameters related to the two parameters in the Painlevé-IV equation. We then use the steepest-descent method to analyze the rational solutions in the limit that at least one of the parameters is large. Our analysis provides rigorous justification for formal asymptotic arguments that suggest that in general solutions of Painlevé-IV with large parameters behave either as an algebraic function or an elliptic function. Moreover, the results show that the elliptic approximation holds on the union of a curvilinear rectangle and, in the case of the generalized Okamoto rational solutions, four curvilinear triangles each of which shares an edge with the rectangle; the algebraic approximation is valid in the complementary unbounded domain. We compare the theoretical predictions for the locations of the poles and zeros with numerical plots of the actual poles and zeros obtained from the generating polynomials, and find excellent agreement.

Predictive capability of the logarithmic law for roughness-modeled large-eddy simulation of turbulent channel flows with rough walls

Direct numerical simulation (DNS) and large-eddy simulation (LES) resolving roughness elements are computationally expensive. LES employing the logarithmic law as the wall model, without the need to resolve the flow at the roughness element scale, provides an efficient alternative for simulating turbulent flows over rough walls. In this work, we evaluate the predictive capability of the roughness-modeled LES by comparing its predictions with those from the roughness-resolved DNS for turbulent channel flows with rough walls. A good agreement is observed for the mean streamwise velocity. The Reynolds stresses predicted by the roughness-modeled LES also reasonably agree with the roughness-resolved predictions. Differences, on the other hand, are observed for the dispersive Reynolds stresses, integral scales, and space-time correlations. The roughness-modeled LES fails to predict the dispersive stresses as one can expect. In the outer layer, the integral length scale predicted by the roughness-modeled LES is lower than the roughness-resolved prediction, which cannot be improved by refining the grid. As for the space-time correlations, discrepancies are shown for the streamwise velocity fluctuations, with a faster decay of the correlation in the outer layer observed in the roughness-modeled predictions. Examination of the space-time correlation using the elliptic approximation model shows that the roughness-modeled LES underpredicts the convection velocity in the near wall region while overpredicting the sweeping velocity in the outer layer with no improvements observed when refining the grid.

Active learning strategy: Computer aided numerical class project on pole-zero plot and transfer function of five low pass filter approximation functions

The conversion of the concept and the teaching of approximation function for analog low pass filter to computer aided numerical class project as an active learning strategy is presented. Governing equations for five approximation functions are presented and discussed. It is recommended that students are tasked to develop computer programs in any convenient computer programming language to make the pole-zero plot, obtain the transfer function, and plot the frequency response of low pass filters. Steps for such programs are suggested. Typical programs were developed and tested. The pole-zero plots and frequency response graphs generated are good illustrative and innovative teaching aids. Linear-phase response over the passband is a characteristic of Bessel approximation function but suffers less amplitude discrimination in the stopband. Chebyshev and Inverse Chebyshev approximation functions have the same order requirement which is usually greater than that of Elliptic approximation function but less than that of Butterworth approximation function. Elliptic approximation function is the best choice although it is more complex than the others. Through this computer aided numerical class project, the authors themselves gained more insight in the subject matter; certainly, students will learn actively.

Two-Dimensional Coupling-of-Modes Model for Surface Acoustic Wave Devices Considering Power Flow Angle

A new form of two-dimensional coupling-of-modes (2D-COM) equations considering power flow angle was proposed and verified experimentally. By adopting a wave vector relationship in elliptic approximation form, the parabolic 2D-COM equations were improved into an elliptic form which is applicable to the simulation of surface acoustic wave resonators considering power flow angle (PFA-SAWR). Combined the method with COMSOL PDE module, a complete simulation model for PFA-SAWR’s frequency response was established. Take the periodic grating structure on the AT-22° quartz substrate (PFA ≈5.4°) as an example, a case analysis was carried out based on this new simulation model. The accuracy of the model was confirmed by comparing the results of 3D FEM analysis. Then, in the experiment part, a PFA-SAWR on the AT-22° quartz substrate was fabricated and measured, the frequency responses obtained by simulation and experiments are in good agreement. Finally, a new resonator structure which can effectively suppress the transverse modes and improve the Q value of resonators was introduced. This new simulation model provides an effective technique for accurately analyzing and designing PFA-SAWRs.

Objective momentum barriers in wall turbulence

We use the recent frame-indifferent theory of diffusive momentum transport to identify internal barriers in wall-bounded turbulence. Formed by the invariant manifolds of the Laplacian of the velocity field, the barriers block the viscous part of the instantaneous momentum flux in the flow. We employ the level sets of single-trajectory Lagrangian diagnostic tools, the trajectory rotation average and trajectory stretching exponent, to approximate both vortical and internal wall-parallel momentum transport barrier (MTB) interfaces. These interfaces provide frame-indifferent alternatives to classic velocity-gradient-based vortices and high-shear boundaries between uniform momentum zones (UMZs). Indeed, we find that these elliptic manifold approximations and MTBs outperform standard vortices and UMZ interfaces in blocking diffusive momentum transport, which suggests our momentum barriers are physical features that may be the cause of coherence signatures in statistical and non-objective diagnostics. We also introduce normalized trajectory metrics that provide unprecedented visualizations of objective coherent structures by avoiding strong turbulence biases.

Object-Based Visual Camera Pose Estimation From Ellipsoidal Model and 3D-Aware Ellipse Prediction

In this paper, we propose a method for initial camera pose estimation from just a single image which is robust to viewing conditions and does not require a detailed model of the scene. This method meets the growing need of easy deployment of robotics or augmented reality applications in any environments, especially those for which no accurate 3D model nor huge amount of ground truth data are available. It exploits the ability of deep learning techniques to reliably detect objects regardless of viewing conditions. Previous works have also shown that abstracting the geometry of a scene of objects by an ellipsoid cloud allows to compute the camera pose accurately enough for various application needs. Though promising, these approaches use the ellipses fitted to the detection bounding boxes as an approximation of the imaged objects. In this paper, we go one step further and propose a learning-based method which detects improved elliptic approximations of objects which are coherent with the 3D ellipsoids in terms of perspective projection. Experiments prove that the accuracy of the computed pose significantly increases thanks to our method. This is achieved with very little effort in terms of training data acquisition - a few hundred calibrated images of which only three need manual object annotation. Code and models are released at https://gitlab.inria.fr/tangram/3d-aware-ellipses-for-visual-localization

Utilising angle-dependent critical current data in the electromagnetic modelling of HTS coils

A detailed methodology is presented for modelling the electromagnetic characteristics of HTS coils using angle-dependent critical current data obtained from experimental measurements of real wire samples. The results of such an analysis are contrasted with those obtained using more prevalent approaches such as a global minimum critical current method or an elliptical field magnitude-dependent functional approximation. Several interesting design consequences of significance to real-world devices that emerge only when the full anisotropy of real wires is taken into account are outlined and discussed. These include the beneficial impact on various performance metrics of the adoption of mixed conductor windings, the importance of coil orientation in optimising device performance, and the potential opportunity to derive a specific design benefit from the targeted use of conductors possessing inclined planarity.

ON OPTIMAL DESIGN OF REINFORCED CONCRETE COLUMNS SUBJECTED TO COMBINED BIAXIAL BENDING AND AXIAL LOAD

This paper presents a new procedure for designing rectangular columns subjected to combined biaxial bending and axial load. The new procedure is realized within a new comprehensive structural analysis software for building analysis and design. Compared to existing engineering packages, the new program does not require pre-setting the distribution of bars before designing and gives more detailed results: not only the required reinforcement amount, but also a distribution of bars of a certain diameter along the sides of the column section. The program starts by inserting four reinforcement bars into the corners of the column, and checks if this reinforcement bars provide required resistance to the applied loads. If the reinforced concrete column fails to withstand the applied loads, then the program increases the reinforcement by adding new bars and checks required resistance again. This process continues until the reinforced concrete column is able to withstand the applied loads. The proposed procedure is realized within following methods: (I) calculation of the so-called column capacity interaction volume using a large number of interaction curves (usually 24 curves); (II) the elliptical approximation of the column capacity interaction volume (the load contour equation). Within the second method, the program calculates two interaction curves only: the first curve is for an eccentric case in X direction and the second curve is for an eccentric case in Y direction. In both methods, the interaction curves are calculated according to American concrete institute standards. The use of the new procedure within the second method gives results very close to the first method (the most accurate), but it requires much less time for calculation. This is very important when designing large structures with multiple columns.

Boundary feedback stabilisation of a viscous incompressible fluid in a bounded domain

This paper presents a boundary feedback control design to globally practically exponentially stabilize a viscous incompressible fluid governed by Navier-Stokes equations in a bounded domain in three dimensional space. The control is implemented on a part of the rigid boundary and requires only boundary measurements. The Rothe method and an elliptic approximation are used to handle the time-dependent domain due to the boundary control in the proof of global existence of a weak solution of the closed-loop system. Due to consideration of less regular initial values of the fluid velocity, the forces induced by the fluid on the part of the control boundary are not possible to be bounded. Thus, the paper derives the bound on “fluid work” in stability and convergence analysis of the closed-loop system. The advantage of considering the weak solution is its global existence and less regularity of the initial data.

Homogenized characterization of cylindrical Li‐ion battery cells using elliptical approximation

Homogenization and finding the constitutive model of jellyroll in cylindrical lithium-ion batteries can be challenging because of their form factor. Taking samples out of the original jellyroll wounding or compressing cell assembly in its cylindrical coordinates are two possibilities for measuring the homogenized lateral strength of the cell. However, the former causes loss of accuracy due to changing constraints and electrolyte environment, and the latter requires complex fixtures that are not readily available or even practical to manufacture. Various approaches have been suggested by researchers to circumvent the above difficulties and allow the extraction of hardening curves. However, the precision of those approaches diminishes when the cells are under global compression vs local punch deformations. In this study, an updated homogenization method is established, using a lateral compression test on the jellyroll. The homogenization method is based on the assumption that the circular cross-section of the jellyroll under compression is deformed in an elliptical shape. Then the principle of virtual work is used to extract the hardening curve. To validate the above characterization model, isotropic and anisotropic finite element models were developed using crushable foam and modified honeycomb material models from the LS-DYNA library. Four sets of cell-level experiments were performed on cylindrical batteries using custom-designed fixtures, including flat lateral compression, rod indentation, hemispherical punch, and three-point bending. The voltage and surface temperature of the batteries were measured to capture the onset of short circuit during the tests. Comparison of the simulation results confirmed that the proposed homogenization method and the FE models can predict the behavior of cylindrical lithium-ion batteries with much higher accuracy compared to the currently available methods presented in the literature.

A Permeability Estimation Method Based on Elliptical Pore Approximation

Digital rock images may capture more detailed pore structure than the traditional laboratory methods. No explicit function can correlate permeability accurately for flow within the pore space. This has motivated researchers to predict permeability through the application of numerical techniques, e.g., using the finite difference method (FDM). However, in order to get better permeability calculation results, the grid refinement was needed for the traditional FDM and the accuracy of the traditional method decreased in pores with elongated cross sections. The goal of this study is to develop an improved FDM (IFDM) to calculate the permeabilities of digital rock images with complex pore space. An elliptical pore approximation method is invoked to describe the complex pore space. The permeabilities of four types of idealized porous media are calculated by IFDM. The calculated results are in sound agreement with the analytical solutions or semi-empirical solutions. What’s more, the permeabilities of the digital rock images after grid coarsening are calculated by IFDM in three orthogonal directions. These results are compared with the previously validated lattice-Boltzmann method (LBM), which indicates that the predicted permeabilities calculated by IFDM usually agree with permeabilities calculated by LBM. We conclude that the presented IFDM is suitable for complex pore space.

Determination of vertical well flow rate at elliptical approximation of external boundary

The external boundary typically has an arbitrary shape. The Dupuis formula obtained by circular approximation of a closed external boundary is usually used to determine the flow rate of a hydrodynamically perfect well. The paper proposes an elliptical approximation in which the wellhead may be located anywhere inside the bounding ellipse. The purpose of the study is to obtain a formula for calculating the well flow rate during elliptical approximation of the external boundary. Objects of study: a vertical well surrounded by an arbitrary external boundary. Research methods: differential and integral calculus. Results: formula for the flow rate of hydrodynamically perfect well is obtained based on known characteristics of formation, flowing fluid and parameters of an ellipse approximating the external boundary; an example is given showing the difference between the flow rates counted according to this formula and the Dupuis formula, which is its special case.