Harmonic Expansion Coefficients Research Articles

A general and efficient harmonic balance method for nonlinear dynamic simulation

This paper presents a general and efficient harmonic balance (GE-HB) method to expeditiously evaluate the periodic responses of nonlinear systems. By employing the fast Fourier transform (FFT) technique, it rapidly computes harmonic expansion coefficients of the nonlinear terms and their corresponding constant tensor matrices. Notably, the nonlinear part of the Jacobian matrix is just the product of the coefficients and the constant tensor matrices, loop computations are no longer requisite for calculating the Jacobian matrix. This, facilitates accelerated determination of the nonlinear portion of the Jacobian matrix, thus obtaining the periodic responses of the system in an efficient manner. Furthermore, the constant tensor matrices are exclusively contingent upon the harmonic bases of the system's periodic response, devoid of reliance on the equation's specific formulation, facilitates obtaining the Jacobian matrix in a general manner. Simulation results on a 20-degree-of-freedom nonlinear oscillator chain system demonstrate that the efficiency of the GE-HB method is about tenfold that of the harmonic balance-alternating frequency/time domain (HB-AFT) method and twenty-fivefold that of the fourth-order Runge-Kutta (RK4) method. Further, the simulation of a 284-degree-of-freedom nonlinear rotor system exhibits the GE-HB method's computational efficiency surpassing HB-AFT method by over 500 times and Newmark method by over 3000 times. Moreover, in these two examples, the GE-HB method captures all solution branches, including unstable periodic solutions, providing comprehensive insights into periodic responses of the systems. The GE-HB method proposed here has the potential to explore dynamic characteristics for high-dimensional complex nonlinear systems in engineering.

Fast computation of the observer motion effects induced on monopole frequency spectra for tabulated functions

Context. Various methods have been studied to compute the boosting effects produced by the observer peculiar motion, which modifies and transfers the isotropic monopole frequency spectrum of the cosmic background radiation to higher multipoles. Explicit analytical solutions for the spherical harmonic expansion coefficients were already presented and applied to different types of background spectrum, strongly alleviating the computational effort needed for accurate theoretical predictions. The frequency spectra at higher multipoles are inherently led by higher-order derivatives of the monopole spectrum. Provided that it can be well described by analytic or semi-analytic functions, the computation of its transfer is not affected by numerical instabilities when evaluated at the required level of numerical accuracy. Instead, monopole frequency spectra described by tabulated functions are computed with a relatively poor frequency resolution in comparison with the Doppler shift, which necessitates interpolation of the tabular representation. The spectra are also affected by uncertainties related to intrinsic inaccuracies in the modelling or in the related observational data as well as to limited accuracy in their numerical computation. These uncertainties propagate and increase with the derivative order, possibly preventing the trustworthy computation of the transfer to higher multipoles and of the observed monopole. Aims. We study methods to filter the original function or its derivatives and the multipole spectra, to mitigate numerical instabilities, and to derive reliable predictions of the harmonic coefficients for different cosmic background models. Methods. From the analytical solutions, and assuming that the monopole spectrum can be expanded in Taylor’s series, we derive explicit expressions for the harmonic coefficients up to the multipole ℓmax = 6 in terms of monopole spectrum derivatives. We then consider different low-pass filters: prefiltering in Fourier space of the tabular representation; filtering in both real and Fourier space of the numerical derivatives; interpolation approaches; and a dedicated method based on amplification and deamplification of the boosted signal. We study the quality of these methods when applied to suitable analytical approximations of the tabulated functions, possibly polluted with simulated noise. These methods are then applied to the tabulations. Results. We consider two very different types of monopole spectra superimposed to the cosmic microwave background: the (smooth) extragalactic source microwave background signal from radio-loud active galactic nuclei and the (feature-rich) redshifted 21 cm line, and present our results in terms of spherical harmonic coefficients. The direct prediction of these coefficients can be noisy at ℓ > 1 or, depending on the uncertainty level, even at ℓ ≤ 1. Without assuming a functional form for the extragalactic background spectrum, the Gaussian prefiltering coupled to the sequential real-space filtering of derivatives allows us to derive accurate predictions up to ℓ ∼ 6, while a log–log polynomial representation, which is appropriate over several decades, gives accurate solutions at any ℓ. Instead, it is difficult to characterise the 21 cm line model variety, and so it is relevant to work without assumptions about the underlying function. Typically, the prefiltering provides accurate predictions up to ℓ ≃ 3 or 4, while the further sequential filtering of the derivatives or the boosting amplification and deamplification method improves the results up to ℓ = 4, while also allowing reasonable estimations of the spectrum at higher ℓ. Conclusions. The proposed methods can significantly extend the range of realistic cosmic background models manageable with a fast computation, beyond the cases characterised a priori by analytical or semi-analytical functions. These methods require only an affordable increase in computation time compared to the direct calculation via simple interpolation.

A modal expansion method for simulating reverberant sound fields generated by a directional source in a rectangular enclosure

The prediction of reverberant sound fields generated by a directional source is of great interest because practical sound sources are not omnidirectional, especially at high frequencies. For an arbitrary directional source described by cylindrical and spherical harmonics, this paper developed a modal expansion method for calculating the reverberant sound field generated by such a source in both two-dimensional and three-dimensional rectangular enclosures with finite impedance walls. The key is to express the modal source density using the cylindrical or spherical harmonic expansion coefficients of the directional source. A method based on the fast Fourier transform is proposed to enable the fast computation of the summation of enclosure modes when walls are lightly damped or rigid. This makes it possible to obtain accurate reverberant sound fields even in a large room and/or at high frequencies with a relatively low computational load. Numerical results with several typical directional sources are presented. The efficiency and the accuracy of the proposed method are validated by the comparison to the results obtained using the finite element method.

The l = 2 spherical harmonic expansion coefficients of the sky brightness distribution between 0.5 and 7 MHz

Context. The opacity of the ionosphere prevents comprehensive Earth-based surveys of low frequency ν ≲ 10 MHz astrophysical radio emissions. The limited available data in this frequency regime show a downturn in the mean sky brightness at ν ≲ 3 MHz in a divergence from the synchrotron emission power-law that is observed at higher frequencies. The turning over of the spectrum coincides with a shift in the region of maximum brightness from the Galactic plane to the poles. This implicates free-free absorption by interstellar ionized gas, whose concentration in the plane causes radiation that propagates in this region to suffer stronger absorption than radiation from the poles. Aims. Using observations from Parker Solar Probe (PSP), we evaluate the l = 0 and l = 2 spherical harmonic expansion coefficients of the radio brightness distribution at 56 frequencies between 0.5 and 7 MHz. These data quantify free-free absorption’s global effects on the brightness distribution, which provides new constraints on the distribution of free electrons in the Galaxy. Methods. The auto and cross spectra of the voltages induced on crossed short dipole antennas by radiation from a nonpolarized extended brightness distribution are linear combinations of the distribution’s l = 0 and l = 2 expansion coefficients. We extracted the least squares solution to these coefficients from PSP’s measurements of the radio background. Also, we generated hypothetical low frequency brightness maps that incorporated free-free absorption and tested their compatibility with the data. The maps primarily depended on models of the Galactic emissivity and distribution of free electrons. A comparison of the maps’ expansion coefficients with the empirical coefficients provided an indication of these input models’ accuracies. Results. An average reduced x2 ≈ 1.04 of the spherical harmonic analysis between 0.5 and 7 MHz indicates that PSP’s antennas act approximately as ideal short dipoles in this frequency band. The best-fit expansion coefficients show that, with decreasing frequency, the mean sky brightness decreases at ν < 3 MHz and the Galactic plane darkens relative to the poles. At ν > 0.6 MHz, these observations can be reproduced in synthetic brightness maps in which the Galactic emissivity maintains a power-law form and free-free absorption is modeled using free electron distributions derived from pulsar measurements. At lower frequencies, the empirical mean brightness falls below the mean in this model, possibly signifying a cutoff in the synchrotron power-law.

Continuum solutions to the two–center Coulomb problem in prolate spheroidal coordinates

We present a computational and theoretical framework for solving the Schrödinger equation (SE) for the two–center Coulomb problem in prolate spheroidal coordinates when the energy of the SE is positive. A general and robust computer code has been produced that calculates the separation constants, spheroidal harmonic expansion coefficients, regular quasi–radial two–center Coulomb wave functions, and two–center Coulomb phase shifts. These quantities can be calculated over a range of internuclear separations, angular momentum projections, and continuum electron momenta. A representative set of results are presented and compared with previous calculations, excellent agreement is found in many cases while significant disagreements are found in others. Program summaryProgram title:spheroidal-cwCPC Library link to program files:https://doi.org/10.17632/nf5gw7vjnh.1Code Ocean capsule:https://codeocean.com/capsule/7638744Licensing provisions: MITProgramming language: Fortran 90Nature of problem: A robust and accurate computer code for calculating Coulomb wave functions in prolate spheroidal coordinate is necessary for the calculation of cross sections for photonionization, and electron and positron scattering on charged diatomic targets.Solution method: The angular solution is expanded in a series of spherical harmonics leading to the diagonalization of a pentadiagonal matrix. The eigenvectors contain the expansion coefficients and the eigenvalues are the separation constants which, unlike the SE in spherical coordinates, are also functions of the energy. The radial two–center Coulomb wave functions are started using a power series solution and then propagated using a linear multistep method. An asymptotic expansion is used to calculate the two–center Coulomb phase shift, and to normalize the radial wave function.Additional comments including restrictions and unusual features: Two–center Coulomb radial wave functions and phase shifts can be accurately calculated for internuclear separations from 0.001 to 60.0 a0, angular momentum projection |m| from 0 to 44, momentum k from 0.01 to 100.0 a.u., and a combined nuclear charge Z+ from 1 to 12.

Acoustic radiation force and torque on a spheroid near an interface

Acoustic radiation force and torque on a spheroid of arbitrary size near a rigid or pressure release interface are calculated using expansions of the pressure field in terms of both spherical and spheroidal wave functions. The spheroidal expansion coefficients are obtained by satisfying the boundary conditions at the interface and at the surface of the spheroid. Conditions at the interface are satisfied using a virtual (image) spheroid, while satisfying conditions at the spheroid surface require the use of addition theorems for spheroidal wave functions. The radiation force and torque are expressed as a summation of terms involving products of the coefficients in spherical wave expansions of the incident and scattered fields [Ilinskii et al., J. Acoust. Soc. Am. 144, 568–576 (2018)]. Far-field asymptotes are used to relate the spheroidal and spherical harmonic expansion coefficients analytically. Results from the present analytical model are compared with those from a finite-element model. The effect of the interface on the radiation force and torque relative to that on a spheroid in a free field is discussed. [B.E.S. is supported by the Applied Research Laboratories Chester M. McKinney Graduate Fellowship in Acoustics.]

Simulation of Earth‐Ionosphere Cavity Resonances With Lightning Flashes Reported by OTD/LIS

AbstractLightning flashes cause a globally resonant electromagnetic wave field known as Earth‐ionosphere cavity resonances. This global wave field is simulated here by use of lightning flashes reported by the lightning imagers on board the Optical Transient Detector (OTD)/Lightning Imaging Sensor (LIS) space missions. The stochastic occurrence of global lightning flashes in time and space is described with a likelihood function composed of a series of delta functions. This likelihood function is generated by random variates calculated from inverse transform sampling of the global lightning distributions measured by OTD/LIS. The simulation uses spherical harmonic expansion coefficients, calculated from frequency and distance dependent magnetic field measurements, to describe the wave field around the Earth. The incoherent superposition of the waves at each location results in simulated time series and spectra. The simulations agree very well with actual measurements of magnetic fields with a radiometer at Arrival Heights in the Antarctic as part of the Stanford ELF/VLF Radio Noise Survey. The simulated relative radiant energy distributions are attributed to corresponding “Earth‐ionosphere cavity brightness temperatures” TEIC. The spherical harmonic expansion of these brightness temperatures exhibits a spectrum where the amplitudes of the spatial expansion coefficients decrease with increasing angular degree. The calculated Earth‐ionosphere cavity brightness temperatures can be used for comparison with atmospheric temperatures in future studies.

The analysis of Titan’s physical surface using multifractal geometry methods

Titan makes up 95% of the mass of all 82 satellites of Saturn. Titan’s diameter is 5152 km, which means that it is larger than the Moon by 50%, and it is also significantly larger than Mercury. On the satellite, a subsurface ocean is possible, the theory of the presence of which has already been advanced earlier by some scientists. It is located under a layer of ice and consists of 10% ammonia, which is a natural antifreeze for it and does not allow the ocean to freeze. On the one hand, the ocean contains a huge amount of salt, which makes the likelihood of life in it hardly possible. But on the other hand, since chemical processes constantly occur on Titan, forming molecules of complex hydrocarbon substances, this can lead to the emergence of the simplest forms of life. There are limitations on the probabilistic and statistical approaches, since not every process and not every result (form and structure of the system) is probabilistic in nature. In contrast to this, fractal analysis allows one to study the structure of complex objects, taking into account their qualitative specifics, for example, the relationship between the structure and the processes of its formation. When constructing a harmonic model of Titan, the method of decomposition of topographic information into spherical functions was used. As a result, based on the harmonic analysis of the Cassini mission data, a topographic model of Titan was created. In the final form, the model describing Titan’s surface includes the expansion of the height parameter depending on the spherical coordinates into a slowly converging regression series of spherical harmonics. For modeling surface details of the surface on a scale of 1 degree requires analysis of the (180 + 1)2 harmonic expansion coefficients. An over determined topographic information system was solved to meet the regression modelling conditions. In this case, a number of qualitative stochastic data, such as external measures, were used together with the standard postulation of the harmonic system of the Titan model. As a result of a sampling of self-similar regions (with close values of the self-similarity coefficients) on the surface of Titan, coinciding with the SRGB parameter (characterizes the color fractal dimension), the elements of the satellite’s surface were determined, which with a high degree of probability were evolutionarily formed under the action of the same selenochemical processes.

Spherical harmonic projections of the static structure factor of the dipolar hard sphere model: Theory vs simulations.

We investigate the static correlations of a dipolar fluid in terms of the irreducible coefficients of the spherical harmonic expansion of the static structure factor. To this end, we develop a theoretical framework based on a soft-core version of Wertheim's solution of the mean spherical approximation (MSA), which renders the analytical determination of such coefficients possible. The accuracy of this approximation is tested by a comparison against the results obtained with the assistance of extensive molecular dynamics simulations at different regimes of concentration and temperature. Crucial aspects for the comparison of the results provided by the two methods are carefully discussed, concerning the different reference frames used in theory and simulations to describe rotations and orientations, and leading to important differences in the behavior of correlation functions with the same combination of spherical harmonic indices. We find a remarkable agreement between the two approaches in the fluid regime, thus providing a first stringent comparison of the irreducible coefficients of the spherical harmonic expansion of the dipolar fluid's static structure factor, provided by the MSA theory and molecular dynamics simulations.

Documentation of TauSpinner approach for electroweak corrections in LHC Z rightarrow ll observables

The LHC Standard Model Z-boson couplings measurements approach the LEP legacy precision. The calculations of electroweak (EW) corrections available for the Monte Carlo generators become of relevance. Predictions of Z-boson production and decay require classes of QED/EW/QCD corrections and separatly from the production process QCD dynamics. At the LEP time electroweak form-factors and Improved Born Approximation were introduced for non QED genuine weak and line-shape corrections. This formalism was well suited for observables, so-called doubly-deconvoluted the Z-pole region where initial- and final-state QED real and virtual emissions were treated separately or were integrated over. The approach was convenient for implementation into Monte Carlo programs for LEP, Belle-BaBar and other future e^+e^- colliders and for invariant mass of outgoing lepton pair from a few GeV to well above WW and even t{bar{t}} threshold. We attempt now to profit from that, for the LHC pp and 70–150 GeV window for the outgoing lepton pair invariant mass. Our technical focus is on the EW corrections for LHC Z rightarrow ell ell observables. For this purpose the TauSpinner package, for the reweighting of previously generated events, is enriched with the genuine EW corrections (QED effects subtracted) of the Dizet electroweak library, taken from the LEP era KKMC Monte Carlo. Complete genuine EW O(alpha ) weak loop corrections and dominant higher-order terms are taken into account. For the efficiency and numerical stability look-up tables are used. For LHC observables: the Z-boson line-shape, the outgoing leptons forward-backward asymmetry, the effective leptonic weak mixing angles and finally for the spherical harmonic expansion coefficients of the lepton distributions, corrections are evaluated. Simplified calculations of Effective Born of modified EW couplings are compared with of Improved Born Approximation of complete set of EW form-factors. Approach uses LEP precision tests definitions and thus offers consistency checks. The package can be useful to evaluate of observables precision limits and to determine which corrections are then important for LHC and FCC projects phenomenology.

Improving chemical shift encoding-based water-fat separation based on a detailed consideration of magnetic field contributions.

PurposeTo improve the robustness of existing chemical shift encoding‐based water–fat separation methods by incorporating a priori information of the magnetic field distortions in complex‐based water–fat separation.MethodsFour major field contributions are considered: inhomogeneities of the scanner magnet, the shim field, an object‐based field map estimate, and a residual field. The former two are completely determined by spherical harmonic expansion coefficients directly available from the magnetic resonance (MR) scanner. The object‐based field map is forward simulated from air–tissue interfaces inside the field of view (FOV). The missing residual field originates from the object outside the FOV and is investigated by magnetic field simulations on a numerical whole body phantom. In vivo the spatially linear first‐order component of the residual field is estimated by measuring echo misalignments after demodulation of other field contributions resulting in a linear residual field. Gradient echo datasets of the cervical and the ankle region without and with shimming were acquired, where all four contributions were incorporated in the water–fat separation with two algorithms from the ISMRM water–fat toolbox and compared to water–fat separation with less incorporated field contributions.ResultsIncorporating all four field contributions as demodulation steps resulted in reduced temporal and spatial phase wraps leading to almost swap‐free water–fat separation results in all datasets.ConclusionDemodulating estimates of major field contributions reduces the phase evolution to be driven by only small differences in local tissue susceptibility, which supports the field smoothness assumption of existing water–fat separation techniques.

Generalized Stokes laws for active colloids and their applications

The force per unit area on the surface of a colloidal particle is a fundamental dynamical quantity in the mechanics and statistical mechanics of colloidal suspensions. Here we compute it in the limit of slow viscous flow for a suspension of N spherical active colloids in which activity is represented by surface slip. Our result is best expressed as a set of linear relations, the ‘generalized Stokes laws’, between the coefficients of a tensorial spherical harmonic expansion of the force per unit area and the surface slip. The generalized friction tensors in these laws are many-body functions of the colloidal configuration and can be obtained to any desired accuracy by solving a system of linear equations. Quantities derived from the force per unit area—forces, torques and stresslets on the colloids and flow, pressure and entropy production in the fluid—have succinct expressions in terms of the generalized Stokes laws. Most notably, the active forces and torques have a dissipative, long-ranged, many-body character that can cause phase separation, crystallization, synchronization and a variety of other effects observed in active suspensions. We use the results above to derive the Langevin and Smoluchowski equations for Brownian active suspensions, to compute active contributions to the suspension stress and fluid pressure, and to relate the synchrony in a lattice of harmonically trapped active colloids to entropy production. Our results provide the basis for a microscopic theory of active Brownian suspensions that consistently accounts for momentum conservation in the bulk fluid and at fluid-solid boundaries.

On numerical orientation averaging with spherical Fibonacci point sets and compressive scheme

For light scattering computations, we propose to perform orientation averaging with spherical Fibonacci point sets. Furthermore, a compressive scheme is introduced to recover light scattering quantities for all orientations based on the compressive sensing theory that exploits the priori information that these scattering quantities are sparse in the spherical harmonic domain. The scheme solves convex optimization problem by minimizing the l1-norm of spherical harmonic expansion coefficients of light scattering quantities. Combining with spherical Fibonacci point sets, this compressive scheme can achieve highly accurate recovery results with fewer orientations than conventional orientation averaging schemes. It is indicated that the proposed compressive orientation averaging scheme with Fibonacci point sets is straightforward to implement and shows good performance in error convergence.

Transformation between surface spherical harmonic expansion of arbitrary high degree and order and double Fourier series on sphere

In order to accelerate the spherical harmonic synthesis and/or analysis of arbitrary function on the unit sphere, we developed a pair of procedures to transform between a truncated spherical harmonic expansion and the corresponding two-dimensional Fourier series. First, we obtained an analytic expression of the sine/cosine series coefficient of the $$4 \pi $$ fully normalized associated Legendre function in terms of the rectangle values of the Wigner d function. Then, we elaborated the existing method to transform the coefficients of the surface spherical harmonic expansion to those of the double Fourier series so as to be capable with arbitrary high degree and order. Next, we created a new method to transform inversely a given double Fourier series to the corresponding surface spherical harmonic expansion. The key of the new method is a couple of new recurrence formulas to compute the inverse transformation coefficients: a decreasing-order, fixed-degree, and fixed-wavenumber three-term formula for general terms, and an increasing-degree-and-order and fixed-wavenumber two-term formula for diagonal terms. Meanwhile, the two seed values are analytically prepared. Both of the forward and inverse transformation procedures are confirmed to be sufficiently accurate and applicable to an extremely high degree/order/wavenumber as $$2^{30}\,{\approx }\,10^9$$ . The developed procedures will be useful not only in the synthesis and analysis of the spherical harmonic expansion of arbitrary high degree and order, but also in the evaluation of the derivatives and integrals of the spherical harmonic expansion.

Algorithm for Reconstruction of 3D Images of Nanorice Particles from Diffraction Patterns of Two Particles in Independent Random Orientations with an X-ray Laser

The method of angular correlations recovers quantities from diffraction patterns of randomly oriented particles, as expected to be measured with an X-ray free electron laser (XFEL), proportional to quadratic functions of the spherical harmonic expansion coefficients of the diffraction volume of a single particle. We have previously shown that it is possible to reconstruct a randomly oriented icosahedral or helical virus from the average over all measured diffraction patterns of such correlations. We point out in this paper that a structure of even simpler particles of 50 Å or so in diameter and consisting of heavier atomic elements (to enhance scattering) that has been used as a test case for reconstructions from XFEL diffraction patterns can also be solved by this technique. Even though there has been earlier work on similar objects (prolate spheroids), one advantage of the present technique is its potential to also work with diffraction patterns not only due to single particles as has been suggested on the basis on nonoverlapping delta functions of angular scattering. Accordingly, we calculated from the diffraction patterns the angular momentum expansions of the pair correlations and triple correlations for general particle images and reconstructed those images in the standard way. Although the images looked pretty much the same, it is not totally clear to us that the angular correlations are exactly the same as different numbers of particles due to the possibility of constructive or destructive interference between the scattered waves from different particles. It is of course known that, for a large number of particles contributing to a diffraction parttern, the correlations converge to that of a single particle. It could be that the lack of perfect agreement between the images reconstructed with one and two particles is due to uncancelling constructive and destructive conditions that are not found in the case of solution scattering.

Rectangular rotation of spherical harmonic expansion of arbitrary high degree and order

In order to move the polar singularity of arbitrary spherical harmonic expansion to a point on the equator, we rotate the expansion around the y-axis by $$90^{\circ }$$ such that the x-axis becomes a new pole. The expansion coefficients are transformed by multiplying a special value of Wigner D-matrix and a normalization factor. The transformation matrix is unchanged whether the coefficients are $$4 \pi $$ fully normalized or Schmidt quasi-normalized. The matrix is recursively computed by the so-called X-number formulation (Fukushima in J Geodesy 86: 271–285, 2012a). As an example, we obtained $$2190\times 2190$$ coefficients of the rectangular rotated spherical harmonic expansion of EGM2008. A proper combination of the original and the rotated expansions will be useful in (i) integrating the polar orbits of artificial satellites precisely and (ii) synthesizing/analyzing the gravitational/geomagnetic potentials and their derivatives accurately in the high latitude regions including the arctic and antarctic area.

Recording of extended sound fields using spherical microphone arrays and a-priori knowledge of the sound source positions

The acquisition of comprehensive sound field information is a central topic in spatial acoustics. In conventional systems, recording devices must be located at the listener’s viewpoint. This research introduces a sound field recording technique for spherical microphone arrays which makes use of a-priori information regarding the distribution of sound sources. The proposal generates sound field descriptions for viewpoints that are located away from the recording device, as long as there are no sound sources between the target viewpoint and the microphone array. Sound field descriptions, a set of spherical harmonic expansion coefficients, are generated from the array signals. A translation operator, calculated from approximate source positions known in advance, is defined so as to shift the plane wave decomposition of the partial fields associated with each expansion coefficient. The proposed method is compared with an existing technique based on a different kind of translation operator traditionally used in implementations of the boundary element method. These operators require no a-priori information, but their region of validity is limited by the spherical harmonic expansion order. The conventional method achieves greater accuracy in the proximity of the recording array; however, the proposed method can generate sound field descriptions covering a much larger region.

Evaluation of magnetic field homogeneity of a conduction-cooled REBCO magnet with a room temperature bore of 200 mm

Development of a high-temperature superconducting magnet wound with REBa 2 Cu 3 O 7-δ (REBCO)-coated conductor for ultrahigh-field magnetic resonance imaging (MRI) is in progress. Our final targets are 9.4-T MRI systems for whole-body and brain imaging. Since REBCO-coated conductors feature high mechanical strength under a tensile stress and high critical current density, superconducting magnets could be made smaller by using REBCO coils. Superconducting magnets for MRI require homogeneous stable magnetic fields. The homogeneity of the magnetic field is highly dependent on the size and current density of the coils. Furthermore, in REBCO magnets, the screening-current-induced magnetic field that changes the magnetic field distribution of the magnet is one of the critical issues. In order to evaluate the magnetic field homogeneity and the screening-current-induced magnetic field of REBCO magnets, a conduction-cooled REBCO magnet with a room-temperature bore of 200 mm was fabricated and tested. The REBCO coils were composed of 12 single pancakes, and the size of the homogeneous magnetic field region was 100-mm diameter spherical volume (DSV). The central magnetic field was as high as 1 T at 285 A. The magnetic field distribution on the z-axis was measured by using an NMR probe. The maximum error magnetic field was 470 parts per million (ppm) in the range from -50 to +50 mm, as well as in the coefficients of the spherical harmonic expansion for a 100-mm DSV. The error magnetic fields due to the screening-current-induced magnetic field were less than 5 ppm, because there was a sufficient distance between the coil and the homogeneous magnetic field region. The main reason for the error magnetic field was dimensional errors in the outer diameters and positions on the z-axis.

Thermodynamic stability in elastic systems: Hard spheres embedded in a finite spherical elastic solid

We determined the total system elastic Helmholtz free energy, under the constraints of constant temperature and volume, for systems comprised of one or more perfectly bonded hard spherical inclusions (i.e. "hard spheres") embedded in a finite spherical elastic solid. Dirichlet boundary conditions were applied both at the surface(s) of the hard spheres, and at the outer surface of the elastic solid. The boundary conditions at the surface of the spheres were used to describe the rigid displacements of the spheres, relative to their initial location(s) in the unstressed initial state. These displacements, together with the initial positions, provided the final shape of the strained elastic solid. The boundary conditions at the outer surface of the elastic medium were used to ensure constancy of the system volume. We determined the strain and stress tensors numerically, using a method that combines the Neuber-Papkovich spherical harmonic decomposition, the Schwartz alternating method, and Least-squares for determining the spherical harmonic expansion coefficients. The total system elastic Helmholtz free energy was determined by numerically integrating the elastic Helmholtz free energy density over the volume of the elastic solid, either by a quadrature, or a Monte Carlo method, or both. Depending on the initial position of the hard sphere(s) (or equivalently, the shape of the un-deformed stress-free elastic solid), and the displacements, either stationary or non-stationary Helmholtz free energy minima were found. The non-stationary minima, which involved the hard spheres nearly in contact with one another, corresponded to lower Helmholtz free energies, than did the stationary minima, for which the hard spheres were further away from one another.

Convergence of the hyperspherical harmonic expansion for crystallographic texture

Advances in instrumentation allow a material texture to be measured as a collection of spatially resolved crystallite orientations rather than as a collection of pole figures. However, the hyperspherical harmonic expansion of a collection of spatially resolved crystallite orientations is subject to significant truncation error, resulting in ringing artifacts (spurious oscillations around sharp transitions) and false peaks in the orientation distribution function. This article finds that the ringing artifacts and the accompanying regions of negative probability density may be mitigated or removed entirely by modifying the coefficients of the hyperspherical harmonic expansion by a simple multiplicative factor. An addition theorem for the hyperspherical harmonics is derived as an intermediate result.