Complex Ray Research Articles

Hybrid Finite Element Method and Variational Theory of Complex Rays for Helmholtz Problems

In this paper, we consider the Weak Trefftz Discontinuous Galerkin (WTDG) method, which enables one to use at the same time the Finite Element Method (FEM) or Variational Theory of Complex Rays (VTCR) discretizations (polynoms and waves), for vibration problems. It has already been developed such that the FEM and the VTCR can be used in different adjacent subdomains in the same problem. Here, it is revisited and extended in order to allow one to use the two discretizations in the same subdomain, at the same time. Numerical examples illustrate the performances of such an approach.

Theory of optical-tweezers forces near a plane interface

Optical-tweezers experiments in molecular and cell biology often take place near the surface of the microscope slide that defines the bottom of the sample chamber. There, as elsewhere, force measurements require force-calibrated tweezers. In bulk, one can calculate the tweezers force from first principles, as recently demonstrated. Near the surface of the microscope slide, this absolute calibration method fails because it does not account for reverberations from the slide of the laser beam scattered by the trapped microsphere. Nor does it account for evanescent waves arising from total internal reflection of wide-angle components of the strongly focused beam. In the present work we account for both of these phenomena. We employ Weyl's angular spectrum representation of spherical waves in terms of real and complex rays and derive a fast-converging recursive series of multiple reflections that describes the reverberations, including also evanescent waves. Numerical simulations for typical setup parameters evaluate these effects on the optical force and trap stiffness, with emphasis on axial trapping. Results are in good agreement with available experimental data. Thus, absolute calibration now applies to all situations encountered in practice.

The use of seismic attributes to enhance imaging of salt structures in the Barents Sea

Seismic imaging and interpretation of salt bodies and surrounding minibasins have been always a challenge in areas with salt tectonics. Complex steeply dipping flanks along salt structures and adjacent stratigraphy remain problematic areas owing to the complex ray paths of the seismic waves travelling through the salt. This fact avoids a clear image of the salt-sediment interface, increasing the uncertainty of the interpretation of salt structures and surrounding minibasins. In this study, a 3D seismic cube located in the Nordkapp Basin (western Barents Sea), has been used to develop suitable interpretation attribute workflows which will allow the interpreter to: (1) improve interpretation and mapping techniques of salt bodies, salt-related structural elements, and adjacent minibasin stratigraphy; and (2) obtain a better understanding of salt kinematics, trapping geometry, and reservoir distribution adjacent to salt structures. The mapping and characterization of salt structures have been carried out using the attribute dip-illumination and variance. In contrast, chaos and anttrack highlight the presence of radial faults and have been used to better understand diapir kinematics. The surrounding minibasins have been interpreted using frequency filters and cosine of instantaneous phase, which have been essential to identify the different types of composite halokinetic sequences and the presence of sedimentary wedges which might act as potential reservoirs.

Extension of the variational theory of complex rays to orthotropic shallow shell structures

Nowadays, the interest of aerospace and automotive industries on virtual testing of medium-frequency vibrational behavior of shallow shell structures is growing. The development of software capable of predicting the vibrational response in such frequency range is still an open question because classical methods (i.e., FEM, SEA) are not fully suitable for the medium-frequency bandwidth. In this context the Variational Theory of Complex Rays (VTCR) is taking place as an ad-hoc technique to address medium-frequency problems. It is a Trefftz method based on a weak variational formulation. It allows great flexibility because any shape function that satisfies the governing equations can be used. This work further develops such theory. In particular, orthotropic materials are introduced in the VTCR formulation for shallow shell structures. A significant numerical example is proposed to show the strategy.

Application of the variational theory of complex rays to the determination of shock induced vibration of a RC structure

Security and safety are crucial aspects in the design of nuclear engineering structures. Civil engineering design and the qualification of materials to dynamic loads must consider the accelerations which they undergo. These accelerations could integrate not only seismic activity but also shaking movements consecutive to aircraft impacts with higher cutoff frequency. Current methodologies for handling such a shock in the calculation stage are based on transient analyzes using classical finite element methods associated with explicit numerical schemes or projection on modal basis. In both cases, to represent in a meaningful way a medium frequency content, a fine mesh is required, which is hardly compatible with the size of models of the civil engineering structures. In order to extend the current industrial methodologies and to allow a better representation of the behavior of the structure in the medium frequency range, an approach coupling a temporal and non-linear analysis of the impact area with a frequency approach for the treatment of the resulting shaking with the Variational Theory of Complex Rays (VTCR) has been developed [1]. The aim is to use the computational efficiency of the implemented strategy and to include the medium frequency range to calculate the nuclear structures response to an aircraft impact. 1 INTRODUCTION For nearly three years in the framework of pre-normative research in nuclear construction (RENON), the constructability research institute (IRC) focused some of its efforts on improving the characterization of floor response spectra in the case of aircraft impacts. The study of airplane crash in the design and verification of nuclear engineering structures has two important and distinct aspects: (i) the resistance of the structure subjected to an impact, loading, and (ii) the qualification of inner equipments to the vibrations induced. The calculation of the resistance of the structure and its design do not generally raise problem with current methods, however, the calculation of induced vibrations, although few harmful to inner equipments, requires special attention, especially with the lack of efficiency of current approaches. Indeed the calculation of floor response spectra (FRS) in this case generally exhibits a set of high magnitude accelerations within a frequency range that is generally much higher than the one observed when calculating the FRS due to an earthquake. The

Rigorous 3-D vectorial complex ray model applied to light scattering by an arbitrary spheroid

After a ray bundle passes a curved surface, the equal-phase wavefront associated with the refracted rays will be distorted. Consequently, the cross-section of a ray bundle with a curved wavefront during propagation in a homogeneous medium will vary with the ray-bundle propagation distance. Moreover, the phase of a ray bundle with convergent wavefront will undergo a phase shift of π/2 with each passage of a focal line. The contribution to the scattering amplitude by a ray bundle after passing a scatterer is determined by three elements: the cross-section variation of its wavefront, the total phase, and the refraction coefficients determined by Fresnel equations. In the geometric optics regime, the aforesaid three elements caused by a curved surface can be systematically quantified in terms of the vectorial complex ray-tracing technique. In this study, rigorous vectorial complex ray-tracing calculations are conducted for light scattering by a general spheroid and the results are validated in comparison with the benchmarks provided by the rigorous T-matrix method.

A quasi‐optimal coarse problem and an augmented Krylov solver for the variational theory of complex rays

SummaryThe variational theory of complex rays (VTCR) is an indirect Trefftz method designed to study systems governed by Helmholtz‐like equations. It uses wave functions to represent the solution inside elements, which reduces the dispersion error compared with classical polynomial approaches, but the resulting system is prone to be ill‐conditioned. This paper gives a simple and original presentation of the VTCR using the discontinuous Galerkin framework, and it traces back the ill‐conditioning to the accumulation of eigenvalues near zero for the formulation written in terms of wave amplitude. The core of this paper presents an efficient solving strategy that overcomes this issue. The key element is the construction of a search subspace where the condition number is controlled at the cost of a limited decrease of attainable precision. An augmented LSQR solver is then proposed to solve efficiently and accurately the complete system. The approach is successfully applied to different examples. Copyright © 2015 John Wiley & Sons, Ltd.

On some results about the variational theory of complex rays used close to the high frequency regime

This work presents some results about the variational theory of complex rays (VTCR) when the frequency increases. This numerical strategy has indeed be mainly only used in the literature on examples where there were about 10–20 wavelengths in the characteristic length of the problem. Few works concerned its use on examples with at least two or three times more wavelengths than that, then close to the high frequency regime, which is the case here. This work tries to show how to bypass the numerical difficulties one can suffer from when using the VTCR on examples with so many wavelengths. It also studies the evolution of the variational formulation of the VTCR when the frequency increases, which leads to some interesting physical interpretations and draws relations with some high frequency vibrations strategies.

Regular modes in a mixed-dynamics-based optical fiber.

A multimode optical fiber with a truncated transverse cross section acts as a powerful versatile support to investigate the wave features of complex ray dynamics. In this paper, we concentrate on the case of a geometry inducing mixed dynamics. We highlight that regular modes associated with stable periodic orbits present an enhanced spatial intensity localization. We report the statistics of the inverse participation ratio whose features are analogous to those of Anderson localized modes. Our study is supported by both numerical and experimental results on the spatial localization and spectral regularity of the regular modes.

''DOUBLE-HUMPED EFFECT'' IN THE TURBULENT COLLISION MAGNETIZED PLASMA

Statistical moments of the spatial power spectrum of multiple scattered ordinary and extraordinary waves in the turbulent collision magnetized plasma with aligned anisotropic electron density irregularities are investigated using modify smooth perturbation method taking into account diffraction effects. Correlation function and variances of the phase fluctuations are obtained for arbitrary correlation function of the electron density fluctuations. Double-humped Effect is investigated analytically and numerically using the anisotropic Gaussian spectral function of electron density irregularities for the polar ionospheric F-region applying the experimental data. Investigation of the anisotropy of the F layer irregularities is of great interest in ionospheric physics. It is well known that the irregularities are elongated in the direction of the geomagnetic field of lines, but especially at high latitudes they are also found to be anisotropic in the plane perpendicular to the external magnetic field vector. The anisotropy is usually investigated by correlation analysis of radio signals from satellites, observed by space receiver on the ground. The fluctuations in amplitude and phase (scintillations) of radio waves passing through the ionosphere are caused by spatial irregularities in the electron density. The spatial fluctuations in phase cause fluctuations of the angle of arrival at the observation points. Broadening of the spatial power spectrum (SPS) of scattered electromagnetic (EM) waves in the turbulent collision magnetized plasma for both power-law and anisotropic Gaussian correlation functions of electron density fluctuations was analyzed in (1) using the complex ray (-optics) approximation. Double-humped Effect, the features of the SPS, spectral width and shift of its maximum of multiple scattered EM waves in the turbulent collisionless magnetized plasma were considered in (2, 3). The influence of the collision frequency between plasma particles on the statistical characteristics of scattered EM waves in the turbulent magnetized plasma with both electron density and external magnetic field fluctuations was investigated in (4). The present paper reports the results of an analysis of the SPS, which is related to fluctuations of the radio refractive index in the F region. Analytical expressions of the correlation function of phase fluctuations of scattered both ordinary and extraordinary waves are obtained for the arbitrary correlation functions of electron density fluctuation using modify smooth perturbation method taking into account the diffraction effects. The Double-humped Effect in the collision magnetized plasma with elongated electron density irregularities is considered for the first time. Formation of a gap in the SPS of scattered radiation is analyzed numerically for the polar ionospheric F -region using the anisotropic Gaussian spectral function of electron density fluctuations. This spectrum contains angle of aligned plasma irregularities to the direction of an external magnetic field and anisotropy factor. The numerical calculation is computed based on experimental data. Finally, paper concludes the obtained results and their significance.

複素特性曲線法を用いた超音速自然層流翼の遷移解析

複素特性曲線法を用いた超音速自然層流翼の遷移解析

Variational theory of complex rays applied to shell structures: in-plane inertia, quasi-symmetric ray distribution, and orthotropic materials

Recently, interest of aerospace and automotive industries on medium-frequency vibrational behavior of composite shell structures has grown due to their high specific stiffness and fatigue resistance. Conventional methods such as the finite element method and the statistical energy analysis are not suitable for the medium-frequency bandwidth. Conversely, the variational theory of complex rays (VTCR) is taking place as an ad-hoc technique to tackle such frequency band. It is a Trefftz method based on a weak variational formulation. Equilibrium equations are met using exact solutions as shape functions. The variational problem imposes boundary conditions in weak form. The present paper extends VTCR to orthotropic shell structures. Moreover, several new enhancements are introduced. Now, we use a quasi-symmetric ray distribution which can greatly reduce computational costs, and addresses in-plane inertia which was neglected in previous works. Some relevant numerical examples are presented to show the strategy and results are compared with a FEM reference to study performances.

Nine toes; Mirror Foot Deformity.

Mirror foot is a very rare congenital anomaly, with only a few papers presenting definitive treatment for this entity. There are limited management recommendations. Most cases are treated before walking age. In our case, there were no associated developmental defects of the leg. The child underwent complex rays resection with medial foot reconstruction. After 7.5 years of followup, definitive surgical treatment was performed with satisfactory cosmetic and functional outcome.

The Variational Theory of Complex Rays applied to the shallow shell theory

In the last few decades the interest of aerospace and automotive industries towards the study of the medium-frequency response of complex shell structure frames has grown. Recently some dedicated “wave” computational approaches have been developed. Among them, a Trefftz technique called Variational Theory of Complex Rays (VTCR) is catching on as an ad hoc method to deal with such vibration problems. This work presents the development of the VTCR in the shallow shell theory to increase its effectiveness and flexibility. First, general theory is given and two key properties of the solution demonstrated. After that, two numerical examples are deeply analyzed.

Experimental validation of the vectorial complex ray model on the inter-caustics scattering of oblate droplets.

We report the first experimental validation of the Vectorial Complex Ray Model (VCRM) using the scattering patterns of large oblate droplets trapped in an acoustic field. The two principal radii and refractive index of the droplets are retrieved with a minimization method that involves VCRM predictions and experimental light scattering patterns. The latter are recorded in the droplet equatorial plane between the primary rainbow region and the associated hyperbolic-umbilic diffraction catastrophe. The results demonstrate that the VCRM can predict the fine and coarse stuctures of scattering patterns with good precision, opening up perspectives for the characterization of large non-spherical particles.

PGD-VTCR: A Reduced Order Model Technique to Solve Medium Frequency Broad Band Problems on Complex Acoustical Systems

The calculation of vibrational responses of complex systems on frequency bands appears to be more and more important in engineering simulation. This is particularly true in the medium frequency regime where the solution is very sensitive to the frequency. In this work, we propose a new path to determine the frequency response of a system at many frequencies. It is based on the Variational Theory of Complex Rays (VTCR)[1], a mid frequency dedicated numerical strategy, and the Proper Generalized Decomposition (PGD) [2], a model order reduction technique. The VTCR enables one to model the problem thanks to the use of waves, and the PGD expands the VTCR approximation over the frequency band through a separated variable representation. This strategy is illustrated on a 2-D acoustic car cavity example.

State-independent contextuality sets for a qutrit

We present a generalized set of complex rays for a qutrit in terms of parameter q=ei2π/k, a k-th root of unity. Remarkably, when k=2,3, the set reduces to two well-known state-independent contextuality (SIC) sets: the Yu–Oh set and the Bengtsson–Blanchfield–Cabello set. Based on the Ramanathan–Horodecki criterion and the violation of a noncontextuality inequality, we have proven that the sets with k=3m and k=4 are SIC sets, while the set with k=5 is not. Our generalized set of rays will theoretically enrich the study of SIC proofs, and stimulate the novel application to quantum information processing.

Modeling physical optics phenomena by complex ray tracing

Physical optics modeling requires propagating optical wave fields from a specific radiometric source through complex systems of apertures and reflective or refractive optical components, or even complete instruments or devices, usually to a focal plane or sensor. The model must accurately include the interference and diffraction effects allowed by the polarization and coherence characteristics of both the initial optical wave field and the components and media through which it passes. Like a spherical wave and a plane wave, a Gaussian spherical wave (or Gaussian beam) is also a solution to the paraxial wave equation and does not change its fundamental form during propagation. The propagation of a Gaussian beam is well understood and easily characterized by a few simple parameters. Furthermore, a paraxial Gaussian beam can be propagated through optical systems using geometrical ray-trace methods. The decomposition of arbitrary propagating wave fields into a superposition of Gaussian beamlets is, thus, an alternative to the classical methods of propagating optical wave fields. This decomposition into Gaussian beamlets has been exploited to significant advantage in the modeling of a wide range of physical optics phenomena.

Comparison of scattering diagrams of large non-spherical particles calculated by VCRM and MLFMA

The vectorial complex ray model (VCRM) developed in recent years improves considerably the precision and the applicability of the ray model in the light scattering by large non-spherical particles. But its results for particles of complex shape remain to be validated. The surface integral equation (SIE) with parallel multilevel fast multipole algorithm (MLFMA) is a rigorous numerical method so it permits us to obtain accurate results of particles of any shape. With new improvement we have made, our parallelized code of MLFMA allows us to calculate the scattering of non-spherical dielectric particles of size much larger than the incident wavelength. This paper is devoted to the validation of the VCRM by comparison of the scattering diagrams with the MLFMA results. The scattering diagrams of large ellipsoidal particles with different size parameters and aspect ratios are computed by using the two methods. Good agreements between them prove the ability of both MLFMA and VCRM for large non-spherical particles.

Pre-stack depth migration for improved imaging under seafloor canyons: 2D case study of Browse Basin, Australia

In the Browse Basin, as in many areas of the world, complex seafloor topography can cause problems with seismic imaging. This is related to complex ray paths, and sharp lateral changes in velocity. This paper compares ways in which 2D Kirchhoff imaging can be improved below seafloor canyons, using both time and depth domain processing.In the time domain, to improve on standard pre-stack time migration (PSTM) we apply removable seafloor static time shifts in order to reduce the push down effect under seafloor canyons before migration. This allows for better event continuity in the seismic imaging. However this approach does not fully solve the problem, still giving sub-optimal imaging, leaving amplitude shadows and structural distortion. Only depth domain processing with a migration algorithm that honours the paths of the seismic energy as well as a detailed velocity model can provide improved imaging under these seafloor canyons, and give confidence in the structural components of the exploration targets in this area. We therefore performed depth velocity model building followed by pre-stack depth migration (PSDM), the result of which provided a step change improvement in the imaging, and provided new insights into the area.