SEARCH · Engineering Papers

Results for “computational physics applications”

Search indexed NASA NTRS and DOE OSTI research on propulsion, heat transfer, battery materials and energy systems. Follow report and document links to the original sources.

SourceResearch connectionFrom yearThrough year

Quote a phrase for an exact phrase match. Source license links do not imply unrestricted reuse.

At least 955 records · Page 53

All sources NASA NTRS DOE OSTI

NASA NTRS1995

The Topographic Torque on a Bounding Surface of a Rotating Gravitating Fluid and the Excitation by Core Motions of Decadal Fluctuations in the Earth's Rotation

General expressions (with potential applications in several areas of geophysical fluid dynamics) are derived for all three components of the contribution made by the geostrophic part of the pressure field associated with flow in a rotating gravitating fluid to the topographic torque exerted by the fluid on a rigid impermeable bounding surface of any shape. When applied to the Earth's liquid metallic core, which is bounded by nearly spherical surfaces and can be divided into two main regions, the "torosphere" and "polosphere," the expressions reduce to formulae given previously by the author, thereby providing further support for his work and that of others on the role of topographic coupling at the core-mantle boundary in the excitation by core motions of Earth rotation fluctuations on decadal time scales. They also show that recent criticisms of that work are vitiated by mathematical and physical errors. Contrary to these criticisms, the author's scheme for exploiting Earth rotation and other geophysical data (either real or simulated in computer models) in quantitative studies of the topography of the core-mantle boundary (CMB) by intercomparing various models of (a) motions in the core based on geomagnetic secular variation data and (b) CMB topography based on seismological and gravity data has a sound theoretical basis. The practical scope of the scheme is of course limited by the accuracy of real data, but this is a matter for investigation, not a priori assessment.

Hide, Raymond↗

NASA NTRS2012

An Assessment of Comprehensive Code Prediction State-of-the-Art Using the HART II International Workshop Data

Despite significant advancements in computational fluid dynamics and their coupling with computational structural dynamics (= CSD, or comprehensive codes) for rotorcraft applications, CSD codes with their engineering level of modeling the rotor blade dynamics, the unsteady sectional aerodynamics and the vortical wake are still the workhorse for the majority of applications. This is especially true when a large number of parameter variations is to be performed and their impact on performance, structural loads, vibration and noise is to be judged in an approximate yet reliable and as accurate as possible manner. In this paper, the capabilities of such codes are evaluated using the HART II Inter- national Workshop data base, focusing on a typical descent operating condition which includes strong blade-vortex interactions. Three cases are of interest: the baseline case and two cases with 3/rev higher harmonic blade root pitch control (HHC) with different control phases employed. One setting is for minimum blade-vortex interaction noise radiation and the other one for minimum vibration generation. The challenge is to correctly predict the wake physics - especially for the cases with HHC - and all the dynamics, aerodynamics, modifications of the wake structure and the aero-acoustics coming with it. It is observed that the comprehensive codes used today have a surprisingly good predictive capability when they appropriately account for all of the physics involved. The minimum requirements to obtain these results are outlined.

vanderWall, Berend G.↗

NASA NTRS1994

A versatile nondestructive evaluation imaging workstation

Ultrasonic C-scan and eddy current imaging systems are of the pointwise type evaluation systems that rely on a mechanical scanner to physically maneuver a probe relative to the specimen point by point in order to acquire data and generate images. Since the ultrasonic C-scan and eddy current imaging systems are based on the same mechanical scanning mechanisms, the two systems can be combined using the same PC platform with a common mechanical manipulation subsystem and integrated data acquisition software. Based on this concept, we have developed an IBM PC-based combined ultrasonic C-scan and eddy current imaging system. The system is modularized and provides capacity for future hardware and software expansions. Advantages associated with the combined system are: (1) eliminated duplication of the computer and mechanical hardware, (2) unified data acquisition, processing and storage software, (3) reduced setup time for repetitious ultrasonic and eddy current scans, and (4) improved system efficiency. The concept can be adapted to many engineering systems by integrating related PC-based instruments into one multipurpose workstation such as dispensing, machining, packaging, sorting, and other industrial applications.

Chern, E. James↗

NASA NTRS2008

User's Manual for LEWICE Version 3.2

A research project is underway at NASA Glenn to produce a computer code which can accurately predict ice growth under a wide range of meteorological conditions for any aircraft surface. This report will present a description of the code inputs and outputs from version 3.2 of this software, which is called LEWICE. This version differs from release 2.0 due to the addition of advanced thermal analysis capabilities for de-icing and anti-icing applications using electrothermal heaters or bleed air applications, the addition of automated Navier-Stokes analysis, an empirical model for supercooled large droplets (SLD) and a pneumatic boot option. An extensive effort was also undertaken to compare the results against the database of electrothermal results which have been generated in the NASA Glenn Icing Research Tunnel (IRT) as was performed for the validation effort for version 2.0. This report will primarily describe the features of the software related to the use of the program. Appendix A has been included to list some of the inner workings of the software or the physical models used. This information is also available in the form of several unpublished documents internal to NASA. This report is intended as a replacement for all previous user manuals of LEWICE. In addition to describing the changes and improvements made for this version, information from previous manuals may be duplicated so that the user will not need to consult previous manuals to use this software.

Wright, William↗

NASA NTRS2016

Recommended Methods for Monitoring Skeletal Health in Astronauts to Distinguish Specific Effects of Prolonged Spaceflight

NASA uses areal bone mineral density (aBMD) by dual-energy X-ray absorptiometry (DXA) to monitor skeletal health in astronauts after typical 180-day spaceflights. The osteoporosis field and NASA, however, recognize the insufficiency of DXA aBMD as a sole surrogate for fracture risk. This is an even greater concern for NASA as it attempts to expand fracture risk assessment in astronauts, given the complicated nature of spaceflight-induced bone changes and the fact that multiple 1-year missions are planned. In the past decade, emerging analyses for additional surrogates have been tested in clinical trials; the potential use of these technologies to monitor the biomechanical integrity of the astronaut skeleton will be presented. OVERVIEW: An advisory panel of osteoporosis policy-makers provided NASA with an evidence-based assessment of astronaut biomedical and research data. The panel concluded that spaceflight and terrestrial bone loss have significant differences and certain factors may predispose astronauts to premature fractures. Based on these concerns, a proposed surveillance program is presented which a) uses Quantitative Computed Tomography (QCT) scans of the hip to monitor the recovery of spaceflight-induced deficits in trabecular BMD by 2 years after return, b) develops Finite Element Models [FEM] of QCT data to evaluate spaceflight effect on calculated hip bone strength and c) generates Trabecular Bone Score [TBS] from serial DXA scans of the lumbar spine to evaluate the effect of age, spaceflight and countermeasures on this novel index of bone microarchitecture. SIGNIFICANCE: DXA aBMD is a widely-applied, evidence-based predictor for fractures but not applicable as a fracture surrogate for premenopausal females and males <50 years. Its inability to detect structural parameters is a limitation for assessing changes in bone integrity with and without countermeasures. Collective use of aBMD, TBS, QCT, and FEM analysis for astronaut surveillance could accommodate NASA's aggressive schedule for risk definition and inform a NASA-developed model which assesses the probability of overloading bones during mechanically-loaded mission tasks and possibly for physical activities after return to Earth.

Vasadi, Lukas J.↗

NASA NTRSDate not supplied

Towards Accurate and Efficient Predictions of Martensitic Transition Temperatures for Shape Memory Alloys from First Principles

Recent rapid progresses in physics theory and computational power have made it possible to predict the martensitic transition temperatures (MTTs) in shape memory alloys (SMAs) from first principles [1-3]. In particular, rigorous while time-consuming thermodynamic integration has been employed to compute the anharmonic phonon free energies, which play a crucial role in determining martensitic phase transitions in SMAs. However, this approach has only been applied to simple binaries, and its accuracy is unsatisfying for certain SMAs such as the most commonly used NiTi. In this work, we report on several new developments to our method that bring first-principles theory and experiment much closer into agreement including the MTT of NiTi, and that improve the computational efficiency significantly. We have applied our refined approach to investigate the Ni0.5Ti0.5-xHfx and PdxNi0.5-xTi0.5 ternaries, and the predicted MTT for each composition is within 100K compared with experiment. We will address various techniques to overcome the difficulty encountered in studying ternaries. Our theoretical approach is expected to be a broadly applicable and predictive theory for designing complex SMAs with desirable properties. [1] J.B. Haskins, A.E. Thompson,and J.W. Lawson, Phys. Rev B 94, 214110 (2016). [2] J.B. HaskinsandJ.W. Lawson, J. App. Phys. 121, 205103 (2017). [3] J.B. Haskins, H. Malmir, S. J. Honrao, L. A. Sandoval, and J.W. Lawson, Acta Materialia 212, 116872 (2017).

Zhigang Wu↗

NASA NTRSDate not supplied

Towards Accurate and Efficient Predictions of Martensitic Transition Temperatures for Shape Memory Alloys from First Principles

Zhigang Wu↗

NASA NTRS1986

An analysis of finite-difference and finite-volume formulations of conservation laws

Finite-difference and finite-volume formulations are analyzed in order to clear up the confusion concerning their application to the numerical solution of conservation laws. A new coordinate-free formulation of systems of conservation laws is developed, which clearly distinguishes the role of physical vectors from that of algebraic vectors which characterize the system. The analysis considers general types of equations--potential, Euler, and Navier-Stokes. Three-dimensional unsteady flows with time-varying grids are described using a single, consistent nomeclature for both formulations. Grid motion due to a non-inertial reference frame as well as flow adaptation is covered. In comparing the two formulations, it is found useful to distinguish between differences in numerical methods and differences in grid definition. The former plays a role for non-Cartesian grids, and results in only cosmetic differences in the manner in which geometric terms are handled. The differences in grid definition for the two formulations is found to be more important, since it affects the manner in which boundary conditions, zonal procedures, and grid singularities are handled at computational boundaries. The proper interpretation of strong and weak conservation-law forms for quasi-one-dimensional and axisymmetric flows is brought out.

Vinokur, Marcel↗

NASA NTRS1989

An analysis of finite-difference and finite-volume formulations of conservation laws

Finite-difference and finite-volume formulations are analyzed in order to clear up the confusion concerning their application to the numerical solution of conservation laws. A new coordinate-free formulation of systems of conservation laws is developed, which clearly distinguishes the role of physical vectors from that of algebraic vectors which characterize the system. The analysis considers general types of equations: potential, Euler, and Navier-Stokes. Three-dimensional unsteady flows with time-varying grids are described using a single, consistent nomenclature for both formulations. Grid motion due to a non-inertial reference frame as well as flow adaptation is covered. In comparing the two formulations, it is found useful to distinguish between differences in numerical methods and differences in grid definition. The former plays a role for non-Cartesian grids, and results in only cosmetic differences in the manner in which geometric terms are handled. The differences in grid definition for the two formulations is found to be more important, since it affects the manner in which boundary conditions, zonal procedures, and grid singularities are handled at computational boundaries. The proper interpretation of strong and weak conservation-law forms for quasi-one-dimensional and axisymmetric flows is brought out.

Vinokur, Marcel↗

NASA NTRS1997

Three-Dimensional Viscous Alternating Direction Implicit Algorithm and Strategies for Shape Optimization

A gradient-based shape optimization based on quasi-analytical sensitivities has been extended for practical three-dimensional aerodynamic applications. The flow analysis has been rendered by a fully implicit, finite-volume formulation of the Euler and Thin-Layer Navier-Stokes (TLNS) equations. Initially, the viscous laminar flow analysis for a wing has been compared with an independent computational fluid dynamics (CFD) code which has been extensively validated. The new procedure has been demonstrated in the design of a cranked arrow wing at Mach 2.4 with coarse- and fine-grid based computations performed with Euler and TLNS equations. The influence of the initial constraints on the geometry and aerodynamics of the optimized shape has been explored. Various final shapes generated for an identical initial problem formulation but with different optimization path options (coarse or fine grid, Euler or TLNS), have been aerodynamically evaluated via a common fine-grid TLNS-based analysis. The initial constraint conditions show significant bearing on the optimization results. Also, the results demonstrate that to produce an aerodynamically efficient design, it is imperative to include the viscous physics in the optimization procedure with the proper resolution. Based upon the present results, to better utilize the scarce computational resources, it is recommended that, a number of viscous coarse grid cases using either a preconditioned bi-conjugate gradient (PbCG) or an alternating-direction-implicit (ADI) method, should initially be employed to improve the optimization problem definition, the design space and initial shape. Optimized shapes should subsequently be analyzed using a high fidelity (viscous with fine-grid resolution) flow analysis to evaluate their true performance potential. Finally, a viscous fine-grid-based shape optimization should be conducted, using an ADI method, to accurately obtain the final optimized shape.

Pandya, Mohagna J.↗

NASA NTRS2010

NASA Tech Briefs, June 2010

Topics covered include: Situational Awareness from a Low-Cost Camera System; Data Acquisition System for Multi-Frequency Radar Flight Operations Preparation; Mercury Toolset for Spatiotemporal Metadata; Social Tagging of Mission Data; Integrating Radar Image Data with Google Maps; Demonstration of a Submillimeter-Wave HEMT Oscillator Module at 330 GHz; Flexible Peripheral Component Interconnect Input/Output Card; Interface Supports Lightweight Subsystem Routing for Flight Applications; MMIC Amplifiers and Wafer Probes for 350 to 500 GHz; Public Risk Assessment Program; Particle Swarm Optimization Toolbox; Telescience Support Center Data System Software; Update on PISCES; Ground and Space Radar Volume Matching and Comparison Software; Web-Based Interface for Command and Control of Network Sensors; Orbit Determination Toolbox; Distributed Observer Network; Computer-Automated Evolution of Spacecraft X-Band Antennas; Practical Loop-Shaping Design of Feedback Control Systems; Fully Printed High-Frequency Phased-Array Antenna on Flexible Substrate; Formula for the Removal and Remediation of Polychlorinated Biphenyls in Painted Structures; Integrated Solar Concentrator and Shielded Radiator; Water Membrane Evaporator; Modeling of Failure for Analysis of Triaxial Braided Carbon Fiber Composites; Catalyst for Carbon Monoxide Oxidation; Titanium Hydroxide - a Volatile Species at High Temperature; Selective Functionalization of Carbon Nanotubes: Part II; Steerable Hopping Six-Legged Robot; Launchable and Retrievable Tetherobot; Hybrid Heat Exchangers; Orbital Winch for High-Strength, Space-Survivable Tethers; Parameterized Linear Longitudinal Airship Model; and Physics of Life: A Model for Non-Newtonian Properties of Living Systems.

Source record↗

NASA NTRS2022

IRIS: High-fidelity Perception Sensor Modeling for Closed-Loop Planetary Simulations

Perception plays a key role in autonomous and semi-autonomous planetary exploration vehicles. For instance, landers can use computer vision techniques for identifying safe landing locations, aerial vehicles use cameras as navigation sensors, and planetary rovers use them for localization and hazard detection. Engineering simulations of such systems requires the accurate modeling of perception and vision sensors for simulating autonomy scenarios. In addition, the modeling of sensors for landers, aerial and ground vehicles requires the ability to handle large and high-resolution terrains, the accurate modeling of illumination, hi-fidelity rendering via ray/path tracing and the inclusion of sensor characteristics. Vision sensor models strive to simulate sensor reality by using physics principles to model the interaction of light and objects. Furthermore, high frame rate performance is highly desirable for in-the-loop simulations involving vehicle dynamics and control software. In this paper we describe a new sensor modeling capability called Inter-planetary Rendering for Imaging and Sensors (IRIS) that meets these requirements for the real-time and high-fidelity simulation of vision sensors for planetary aerospace and robotics applications.

Elmquist, Asher↗

NASA NTRS2003

Glenn-ht/bem Conjugate Heat Transfer Solver for Large-scale Turbomachinery Models

A coupled Boundary Element/Finite Volume Method temperature-forward/flux-hack algorithm is developed for conjugate heat transfer (CHT) applications. A loosely coupled strategy is adopted with each field solution providing boundary conditions for the other in an iteration seeking continuity of temperature and heat flux at the fluid-solid interface. The NASA Glenn Navier-Stokes code Glenn-HT is coupled to a 3-D BEM steady state heat conduction code developed at the University of Central Florida. Results from CHT simulation of a 3-D film-cooled blade section are presented and compared with those computed by a two-temperature approach. Also presented are current developments of an iterative domain decomposition strategy accommodating large numbers of unknowns in the BEM. The blade is artificially sub-sectioned in the span-wise direction, 3-D BEM solutions are obtained in the subdomains, and interface temperatures are averaged symmetrically when the flux is updated while the fluxes are averaged anti-symmetrically to maintain continuity of heat flux when the temperatures are updated. An initial guess for interface temperatures uses a physically-based 1-D conduction argument to provide an effective starting point and significantly reduce iteration. 2-D and 3-D results show the process converges efficiently and offers substantial computational and storage savings. Future developments include a parallel multi-grid implementation of the approach under MPI for computation on PC clusters.

Divo, E.↗

NASA NTRS1994

Accurate interlaminar stress recovery from finite element analysis

The accuracy and robustness of a two-dimensional smoothing methodology is examined for the problem of recovering accurate interlaminar shear stress distributions in laminated composite and sandwich plates. The smoothing methodology is based on a variational formulation which combines discrete least-squares and penalty-constraint functionals in a single variational form. The smoothing analysis utilizes optimal strains computed at discrete locations in a finite element analysis. These discrete strain data are smoothed with a smoothing element discretization, producing superior accuracy strains and their first gradients. The approach enables the resulting smooth strain field to be practically C1-continuous throughout the domain of smoothing, exhibiting superconvergent properties of the smoothed quantity. The continuous strain gradients are also obtained directly from the solution. The recovered strain gradients are subsequently employed in the integration o equilibrium equations to obtain accurate interlaminar shear stresses. The problem is a simply-supported rectangular plate under a doubly sinusoidal load. The problem has an exact analytic solution which serves as a measure of goodness of the recovered interlaminar shear stresses. The method has the versatility of being applicable to the analysis of rather general and complex structures built of distinct components and materials, such as found in aircraft design. For these types of structures, the smoothing is achieved with 'patches', each patch covering the domain in which the smoothed quantity is physically continuous.

Tessler, Alexander↗

NASA NTRS2020

Probability of Decompression Sickness and Venous Gas Emboli from 49 NASA Hypobaric Chamber Tests with Reference to Exploration Atmosphere

Introduction: Decompression sickness (DCS) is a complex biophysical event; it combines human perception of pain, for instance, and the presence of a gas phase in the tissues. Living tissues are complex and dynamic. Micronuclei and later bubbles may or may not form given what appears to be the same conditions. Even when bubbles grow, symptoms may or may not develop under what appears to be the same conditions. Therefore, at this time it is appropriate to consider DCS as a probabilistic rather than a deterministic event. Methods: Probabilistic models about hypobaric DCS and venous gas emboli (VGE) require a large amount of quality research data, a definition of decompression dose using physical and physiologic variables, and a flexible analytical approach that can quantify the association between each outcome and all covariates of interest (assuming independence between DCS and VGE) and then ultimately be extended to acknowledge dependencies between DCS and VGE. Our DCS and VGE data are from 1,031 hypobaric decompressions from 1983 to 2016. A total of 577 humans participated in 49 hypobaric chamber tests to evaluate denitrogenation procedures used by astronauts in the Space Shuttle and International Space Station programs. We defined decompression dose as the ratio of computed nitrogen tension in a theoretical 360-minute half-time compartment to ambient pressure, which accounts for denitrogenation and exposure pressure as well as explanatory variables such as age, sex, body mass index, and the presence or absence of ambulation as part of exercise at the exposure pressure. A parametric survival model, using a log-logistic distribution, was used to quantify the time to development of DCS, VGE, and Grade IV VGE. Results: Our survival estimates are applicable to simple hypobaric decompressions, such as depressurizations in 5 to 30 minutes to exposure pressures between 4 and 10 pounds per square inch absolute (psia) and after minutes to hours of denitrogenation, either under resting or exercise conditions to accelerated denitrogenation. The regressions are applicable to exposures between 2 to 6 hours and under conditions of ambulation or no ambulation as part of exercise at the test pressure. We estimate that an exposure to 4.3 psia with simulated extravehicular activity (EVA) that includes ambulation after equilibration to the exploration atmosphere at 8.2 psia with a 34% oxygen atmosphere will result in 3.1% DCS (1.8% to 5.2 95% confidence interval), 23.2% VGE (16.7 to 31.2%), and 8.5% Grade IV VGE (4.7 to 14.7%) in equal samples of men and women exposed for 6 hours. Discussion: Probabilistic models for DCS, VGE, and Grade IV VGE can be used to inform those that plan future EVAs. Their applications are useful to quantify the risk of DCS and VGE in astronauts that perform EVAs in low-pressure space suits while in space or while exploring the surfaces of the moon or Mars.

Johnny Conkin↗

NASA NTRS2016

Full Field X-Ray Fluorescence Imaging Using Micro Pore Optics for Planetary Surface Exploration

Many planetary surface processes leave evidence as small features in the sub-millimetre scale. Current planetary X-ray fluorescence spectrometers lack the spatial resolution to analyse such small features as they only provide global analyses of areas greater than 100 mm(exp 2). A micro-XRF spectrometer will be deployed on the NASA Mars 2020 rover to analyse spots as small as 120m. When using its line-scanning capacity combined to perpendicular scanning by the rover arm, elemental maps can be generated. We present a new instrument that provides full-field XRF imaging, alleviating the need for precise positioning and scanning mechanisms. The Mapping X-ray Fluorescence Spectrometer - "Map-X" - will allow elemental imaging with approximately 100μm spatial resolution and simultaneously provide elemental chemistry at the scale where many relict physical, chemical and biological features can be imaged in ancient rocks. The arm-mounted Map-X instrument is placed directly on the surface of an object and held in a fixed position during measurements. A 25x25 mm(exp 2) surface area is uniformly illuminated with X-rays or alpha-particles and gamma-rays. A novel Micro Pore Optic focusses a fraction of the emitted X-ray fluorescence onto a CCD operated at a few frames per second. On board processing allows measuring the energy and coordinates of each X-ray photon collected. Large sets of frames are reduced into 2d histograms used to compute higher level data products such as elemental maps and XRF spectra from selected regions of interest. XRF spectra are processed on the ground to further determine quantitative elemental compositions. The instrument development will be presented with an emphasis on the characterization and modelling of the X-ray focussing Micro Pore Optic. An outlook on possible alternative XRF imaging applications will be discussed.

micro-XRF spectrometer↗

NASA NTRS1991

Requirements for a network storage service

Sandia National Laboratories provides a high performance classified computer network as a core capability in support of its mission of nuclear weapons design and engineering, physical sciences research, and energy research and development. The network, locally known as the Internal Secure Network (ISN), comprises multiple distributed local area networks (LAN's) residing in New Mexico and California. The TCP/IP protocol suite is used for inter-node communications. Scientific workstations and mid-range computers, running UNIX-based operating systems, compose most LAN's. One LAN, operated by the Sandia Corporate Computing Computing Directorate, is a general purpose resource providing a supercomputer and a file server to the entire ISN. The current file server on the supercomputer LAN is an implementation of the Common File Server (CFS). Subsequent to the design of the ISN, Sandia reviewed its mass storage requirements and chose to enter into a competitive procurement to replace the existing file server with one more adaptable to a UNIX/TCP/IP environment. The requirements study for the network was the starting point for the requirements study for the new file server. The file server is called the Network Storage Service (NSS) and its requirements are described. An application or functional description of the NSS is given. The final section adds performance, capacity, and access constraints to the requirements.

Kelly, Suzanne M.↗

NASA NTRS1994

Errors induced by the neglect of polarization in radiance calculations for Rayleigh-scattering atmospheres

Although neglecting polarization and replacing the rigorous vector radiative transfer equation by its approximate scalar counterpart has no physical background, it is a widely used simplification when the incident light is unpolarized and only the intensity of the reflected light is to be computed. We employ accurate vector and scalar multiple-scattering calculations to perform a systematic study of the errors induced by the neglect of polarization in radiance calculations for a homogeneous, plane-parallel Rayleigh-scattering atmosphere (with and without depolarization) above a Lambertian surface. Specifically, we calculate percent errors in the reflected intensity for various directions of light incidence and reflection, optical thicknesses of the atmosphere, single-scattering albedos, depolarization factors, and surface albedos. The numerical data displayed can be used to decide whether or not the scalar approximation may be employed depending on the parameters of the problem. We show that the errors decrease with increasing depolarization factor and/or increasing surface albedo. For conservative or nearly conservative scattering and small surface albedos, the errors are maximum at optical thicknesses of about 1. The calculated errors may be too large for some practical applications, and, therefore, rigorous vector calculations should be employed whenever possible. However, if approximate scalar calculations are used, we recommend to avoid geometries involving phase angles equal or close to 0 deg and 90 deg, where the errors are especially significant. We propose a theoretical explanation of the large vector/scalar differences in the case of Rayleigh scattering. According to this explanation, the differences are caused by the particular structure of the Rayleigh scattering matrix and come from lower-order (except first-order) light scattering paths involving right scattering angles and right-angle rotations of the scattering plane.

Mishchenko, M. I.↗