Spherical Harmonic Expansion Method Research Articles

Rockfall simulation via spherical harmonic based discrete element method

The trajectory of falling rock is significantly affected by its shape. In order to capture the falling rock shape, spherical harmonic function expansion method is used. The reasonable order of spherical harmonic function expansion items is determined according to the error analysis. Spherical harmonic based discrete element method is utilized to simulate the rockfall process of cylindrical and irregular shape rocks. The influences of the rock and terrain shapes on the rockfall trajectory are analyzed. The results show that the shape of falling rock can be described precisely by spherical harmonic function expansion method, and only a little data should be stored. The predicted trajectory of cylindrical rockfall is close to the experimental result. Spherical and block falling rocks usually show more regular horizontal displacement, lateral offset and jump height during the motion process. Those motion quantities for long strip and flaky falling rocks are more sensitive to their initial rotations. Terrain has a strong influence on rockfall trajectory, falling rocks are prone to move along or rest at mountain valleys.

Direct discontinuous Galerkin method for potential magnetic field solutions

In this paper, we employ the direct discontinuous Galerkin (DDG) method for the first time to extrapolate the coronal potential magnetic field (PF) with the source surface (SS) and call the developed numerical model as the DDG-PFSS solver. In this solver, the Laplace’s equation is solved by means of the time-dependent method, i.e., introducing a pseudo-time term into the Laplace’s equation and changing the boundary value problem into the initial-boundary value problem. The steady-state solution of the initial-boundary value problem is the solution of the Laplace’s equation to be solved. This formulation facilitates the implementation of the DDG discretization. In order to validate the DDG-PFSS solver, we test a problem with the exact solution, which demonstrates the effectiveness and third-order accuracy of the solver. Then we apply it to the extrapolation for the coronal potential magnetic field. We use the integral GONG synoptic magnetogram of Carrington rotation (CR) 2060 as the boundary condition and achieve the global potential magnetic field solution by the DDG-PFSS solver. The numerical results such as the coronal holes and streamer belts derived from the DDG-PFSS solver are in good agreement with those obtained from the spherical harmonic expansion method. Also, based on the numerical magnetic field and Wang-Sheeley-Arge model, the obtained solar wind speed is found to basically capture the structures of the high- and low-speed streams observed at 1 AU. These results suggest that the DDG-PFSS solver can be seen as a contribution to the numerical methods for obtaining the global potential magnetic field solutions of the solar corona.

Damage profile evolution model based on the Boltzmann transport equation for silicon micromachining with the focused helium ion beam

In this work, a damage profile evolution model based on the Boltzmann transport equation was developed for silicon micromachining using the focused helium ion beam. The energy grouping method, the spherical harmonic function expansion method, and the discrete ordinate method were used to solve the model numerically to obtain the implanted ion distribution and the amorphous damage profile. The influence of the ion beam energy and the ion dose on the evolution of the amorphous damage profile and the helium ion distribution in the silicon substrate were investigated. The implantation depth of the ions was positively correlated with the beam energy, and the influence of the ion dose on the implantation depth of the ions was very slight. The amorphous damage profiles predicted by the model were in good agreement with the experimental transmission electron microscope images. The results showed that the amorphous damage profile evolved into an approximately cylindrical shape for low beam energy (<10 keV), and an approximate “vase” shape for high beam energy (>10 keV). For the specified beam energy, the maximum amorphous width increased linearly and the amorphous depth first increased rapidly and then slowly increased with the increasing ion dose, and there was a limit on the amorphous depth. The computational efficiency of the model was not limited by the number of particles, which made up for a common shortcoming of the Monte Carlo and molecular dynamics methods.

Three-dimensional acoustic energy flow from rotating sound sources

A detailed study of the three-dimensional sound fields around a monopole point source in subsonic and supersonic rotation is investigated. We calculate the sound pressure, the acoustic velocity, as well as the instantaneous and time-averaged acoustic intensity fields with the advanced time approach. The advanced time approach is applied for sound sources at subsonic and supersonic rotation and yield comparable results as those obtained from the spherical harmonic series expansion method or the solution of the retarded time equation. Further, the direction of the time-averaged acoustic energy flow is derived from the acoustic intensity vectors. It is shown, that the direction of the energy flow path differs from the radial direction and depends on the rotational direction and rotational velocity. Those findings are useful, for example, to improve the acoustic absorption performance of sound absorbers and acoustic liners around turbomachines.

Acoustic ray method derived with the concept of analogue gravity for the calculation of the sound field due to rotating sound sources

Acoustics in a moving fluid is a simple model of an analogue space–time. The analysis shows that the null geodesics of the metric gμν in that space–time correspond to the rays in the eikonal approximation of geometrical acoustics. The physical connection to general relativity is the propagation of sound along the null geodesics that corresponds to the propagation of a scalar field in a background gravitational field. In this framework, we derive a three-dimensional acoustic ray method in a rotating frame of reference and calculate an acoustic metric, acoustic rays, travel times, and amplitudes for the fluctuating part of the pressure. Further the application to rotating beamforming, i.e., localization and estimation of the strength of rotating sound sources with a microphone array, is presented. Analytical calculations are compared with the spherical harmonic series expansion method for the sound propagation in the rotating frame of reference. Two benchmark cases on rotating sound sources with generic time-data and measured time-data of an axial fan demonstrate its applicability to rotating beamforming.

A Generalized Solution for Parallelized Computation of the Three-dimensional Gravitational Potential on a Multipatch Grid in Spherical Geometry

Abstract We present a generalized algorithm based on a spherical harmonics expansion method for efficient computation of the three-dimensional gravitational potential on a multipatch grid in spherical geometry. Instead of solving for the gravitational potential by superposition of separate contributions from the mass density distribution on individual grid patches, our new algorithm directly computes the gravitational potential due to contributions from all grid patches in one computation step, thereby reducing the computational cost of the gravity solver. This is possible by considering a set of angular weights that are derived from rotations of spherical harmonics functions defined in a global coordinate system that is common for all grid patches. Additionally, our algorithm minimizes data communication between parallel computing tasks by eliminating its proportionality to the number of subdomains in the grid configuration, making it suitable for parallelized computation on a multipatch grid configuration with any number of subdomains. Test calculations of the gravitational potential of a triaxial ellipsoidal body with constant mass density on the Yin–Yang two-patch overset grid demonstrate that our method delivers the same level of accuracy as a previous method developed for the Yin–Yang grid while offering improved computation efficiency and parallel scaling behavior.

Analysis of Small-Angle Neutron Scattering Spectra from Deformed Polymers with the Spherical Harmonic Expansion Method and a Network Model

We combine the recently proposed spherical harmonic expansion (SHE) method [Phys. Rev. X 2017, 7, 031003] with a modified network model to analyze the small-angle neutron scattering (SANS) spectra ...

On eigenvalues of the linearization of a free boundary problem modeling two-phase tumor growth

In this paper we study eigenvalues of the linearization of a free boundary problem modeling the growth of a tumor containing two species of cells: proliferating cells and quiescent cells. Such eigenvalues are potential bifurcation points from which nonradial solutions of the free boundary problem might bifurcate from the radial solution. A special feature of this problem is that it contains a singular ordinary differential equation which causes the main difficulty of this problem. By using the spherical harmonic expansion method combined with some techniques for solving singular differential integral equations developed in some previous literature, eigenvalues of the linearized problem are completely determined. Invertibility of some linear operators related to the linearized problem in suitable function spaces is also studied which might be useful in the analysis of the original free boundary problem.

Effects of surface rates for the series reaction A → B → C on successive separated spherical sites

A steady - state series reaction A →B → C, consisting of a first reaction A →B on sphere 1 of radius a1 with first order kinetic rate constant kA and reactant diffusivity DA, followed by a second reaction B → C on sphere 2 of radius a2 with a first order kinetic rate constant kB and a B chemical species diffusivity DB, occurs on two discrete chemically active spheres a center - to – center distance d apart. The twin spherical harmonic expansion method with a Neumann iterative solution of the coefficient equations is used to generate a rigorous solution for the rate of series reaction in terms of an expansion in the inverse dimensionless center - to - center sphere separation d (=d⁄((a_1+a_2 ))) up to the inverse thirteenth power. The expansion coefficients depend on three dimensionless parameters (γ,λ_A,λ_B), where the dimensionless inverse surface kinetic constants are λ_A=D_A⁄a_1 kA on sphere 1, λ_B=D_B⁄(a_2 k_B ) on the sphere 2, while the geometry enters as the radius ratio γ=a_1⁄a_2 . Using these results an equivalent site kinetic change is imposed, first to λ_Aon site 1, then to λ_Bon site 2, to determine which site is more influential for the overall series reaction rate. Effects of γ geometry modification at several fixed kinetic surface site rates are examined. When either one of reactive spheres is diffusion controlled, various levels of λ kinetic controls at the other site or γ site geometries are generated to study the effects of site kinetics or geometry control on the series reaction rate. As the “other site” reaction rate also increases toward diffusion control, inflection points are observed to develop as precursors to a local maximum on the series reaction rate versus sphere separation curves. An application to the isomerization of n-pentane is presented.

Solvation Structure of Surface-Supported Amine Fragments: A Molecular Dynamics Study

Amine-grafted silica gel is an efficient heterogeneous catalyst for the Knoevenagel condensation and draws much attention in green chemistry for applications like heavy metal adsorption and CO2 fixation. Despite its successful usage in diverse areas, fundamental questions remain on how the silica substrate affects the local chemical environment of the tethered amines. In this work, we use all-atom molecular dynamics simulation to investigate the solvation structures of two primary amines tethered onto a silica surface at different pHs of aqueous solutions. The atomic density profiles in the solvation shell are analyzed with a spherical harmonics expansion method for both isolated and silica-supported amines in different aqueous environments. The simulation results provide direct evidence for the strong influence of the silica surface on the hydration structure that is often ignored in the theoretical analysis of surface reactions. The surface effect becomes less prominent on the tethered amine as the alky...

Imaging of Broadband Noise from Rotating Sources in Uniform Axial Flow

A method is presented for imaging rotating broadband sources with a microphone array. The microphone array rotates virtually with the same angular frequency as the source such that the pressure data are mathematically transformed into a rotating reference frame and the influence of the moving source (Doppler shift) is compensated. In contrast with other works, this method works completely in the frequency domain by using the spherical harmonic series expansion method to calculate a modified free-space Green function in the rotating system. An analytical solution for the sound radiation from the rotating sound source, which considers motion compensation and a uniform subsonic axial flow, is deduced. Simulations on a rotating point source considering axial flow are performed. Measurements with a fan are examined for different frequencies with standard delay and sum beam-forming and high-resolution algorithms.

Series reactions A → B → C on successive spheres

The twin spherical harmonic expansion method with iterative solution of the coefficient equations is used to generate a rigorous analytical solution for the rate of series reaction, A→B→C, occurring, respectively, on two successive spheres of radius a1 and a2, a center-to-center distance d apart. To investigate the influences of the intersphere diffusion and geometry, diffusion-controlled reactions are considered. Results are presented as a series expansion of the dimensionless reaction rate R in terms of the dimensionless center-to-center sphere separation d¯(=d/(a1+a2)) reciprocals, and for various sphere radius ratios γ(=a1/a2). When the sphere radius ratio γ is less than unity, a maximum in the series reaction rate is found for a d¯ of about 1.05. Also an exact value of the series dimensionless reaction rate of ln2 is obtained in the limit γ→0 (very large a2 or very small a1) for spheres in contact. Results suggest that the plots of reaction rates for contacting spheres can be extrapolated versus γ to the ln2 limit at γ→0, and that the rate maximum effect is large in the γ→0 limit.

Analytical Acoustic Power Spectrum Formulations for Rotating Monopole and Dipole Point Sources

Analytical acoustic power spectrum formulations for the rotating monopole and dipole point sources are proposed by employing the spherical harmonic series expansion method. Both the analytical acoustic power spectra and the overall acoustic power show a good agreement with the results obtained from other methods. A nondimensional acoustic power ratio (APR) is employed to investigate the effects of the rotational Mach number, the direction of the dipole source, and the number of sources on the acoustic power output.

Differential speed rolling of twin-roll-cast 6xxx al alloy strips and its influence on the sheet formability

We demonstrate the feasibility of a technique that combines twin-roll strip casting, asymmetric rolling, and subsequent heat treatment, in obtaining Al alloy sheets with high-strength/high-formability. The precipitation- hardening Al alloy sheet thus obtained exhibited an excellent formability (\(\bar r\)= 1.20, Δr = 0.17) and mechanical properties (σTS = 265 MPa, e = 35%), which cannot be readily obtained via the conventional route based on direct-chill casting and heavy rolling operation. In this study, we examined the effects of the various process conditions used at different stages of the process that contribute to the development of specific textures. Simulation studies based on the generalized spherical harmonic series expansion method and the viscoplastic self-consistent (VPSC) model were conducted to arrive at a comprehensive understanding of the factors associated with the high formability realized in Al alloy sheet. It was found that specific textures evolved via twin-roll strip casting, asymmetric rolling, and heat treatment canceled out the anisotropic characteristics of the individual textures, resulting in the high sheet formability.

Accelerated Method for Predicting Acoustic Far Field and Acoustic Power of Rotating Source

Visualization of acoustic pressure and acoustic velocity field and computation of acoustic power are meaningful to investigate noise radiation and propagation mechanism of turbomachines. However, the conventional surface integral method is very time-consuming because the acoustic response for numerous source–observer pairs should be numerically computed one by one. This paper presents an accelerated method for predicting acoustic pressure and acoustic velocity in the acoustic far field radiated by rotating sources. The accelerated method is derived based on a spherical harmonic series expansion method, which explicitly describes the variation of the acoustic pressure and acoustic velocity with the radial coordinate and azimuthal angle of the observer in acoustic far field. In the proposed accelerated method, only the acoustic pressure of a few reference observer points with different polar angles is directly computed by numerical integration, whereas the acoustic pressure and acoustic velocity of the other observers are fast computed using the interpolation method. Moreover, an accelerated method for computing the acoustic power is also developed, which avoids both the numerical integral of the observer surface and computation of the acoustic velocity. Numerical illustrations indicate that the accelerated method has a good accuracy and is more efficient than the conventional numerical integral method.

Spherical-harmonics expansion method for density-matrix simulations of quantum electron dynamics in continuum states

Spherical-harmonics expansion method is proposed to solve the quantum time-evolution equations for density matrices numerically in momentum space. This method facilitates efficient real-time simulations of quantum electron dynamics in continuum states through extension of our previous formalism developed for discrete states. Numerical accuracy and efficiency are demonstrated through two examples: (i) multiphoton ionization of a one-electron atom in intense laser field, and (ii) real-time dynamics of plasma oscillations in electron liquids. In case (i), coupled dynamics of density matrices for bound and continuum states reveals an enhancement of multiphoton ionization over the Keldysh approximation through resonant intermediate states. © 2015 Wiley Periodicals, Inc.

Comparison of analytic distribution function models for hot-carrier degradation modeling in nLDMOSFETs

We analyze the applicability of different analytic models for the carrier distribution function (DF), namely the heated Maxwellian, the Cassi model, the Hasnat approach, the Reggiani model, and our own concept, to describe hot-carrier degradation (HCD) in nLDMOS devices. As a reference, we also obtain the carrier distribution function as a direct solution of the Boltzmann transport equation using the spherical harmonics expansion method. The DFs evaluated with these models are used to simulate the interface state generation rates, the interface state density profiles, and changes of the linear and saturation drain currents as well as the threshold voltage shift. We show that the heated Maxwellian approach leads to an underestimated HCD at long stress times. This trend is also typical for the Cassi and Hasnat models but in these models HCD is underestimated in the entire stress time window. While the Reggiani model gives good results in the channel and drift regions, it cannot properly represent the high-energy tails of the DF near the drain, and thus leads to a weaker curvature of the degradation traces. We show finally that our model is capable of capturing DFs with very good accuracy and, as a result, the change of the device characteristics with stress time.

Modeling of Hot-Carrier Degradation in nLDMOS Devices: Different Approaches to the Solution of the Boltzmann Transport Equation

We propose two different approaches to describe carrier transport in n-laterally diffused MOS (nLDMOS) transistor and use the calculated carrier energy distribution as an input for our physical hot-carrier degradation (HCD) model. The first version relies on the solution of the Boltzmann transport equation using the spherical harmonics expansion method, while the second uses the simpler drift-diffusion (DD) scheme. We compare these two versions of our model and show that both approaches can capture HCD. We, therefore, conclude that in the case of nLDMOS devices, the DD-based variant of the model provides good accuracy and at the same time is computationally less expensive. This makes the DD-based version attractive for predictive HCD simulations of LDMOS transistors.

Analytical solution for sound radiated from the rotating point source in uniform subsonic axial flow

The paper presents an analytical solution for sound radiated from the rotating monopole and dipole point sources in uniform subsonic axial flow by employing the spherical harmonic series expansion method. The analytical results obtained from the present series expansion formulations have a good agreement with the numerical results obtained from the time-domain and frequency-domain numerical methods. The results also indicate that the effect of the freestream Mach number on the directivity pattern is various for different types of the rotating sources.

Modeling of deep-submicron silicon-based MISFETs with calcium fluoride dielectric

We model the main characteristics of metal-insulator-silicon field-effect transistors (MISFETs) with different gate insulators using the carrier energy distribution function calculated with a Spherical Harmonics Expansion method. In addition to standard devices with Silicon Dioxide or Oxynitride we study a hypothetical MISFET with a rather new crystalline dielectric-Calcium Fluoride. The real physical parameters of the $$\hbox {CaF}_{2}$$ CaF 2 /Si tunnel barrier are used in our simulations. The obtained characteristics of the transistors with $$\hbox {CaF}_{2}$$ CaF 2 are, in some details, better than those of the devices with traditional oxides. Being a step forward in the context of the industrial implementation of fluorite, this work opens the possibility of simulating the characteristics of different silicon-based devices with crystalline insulators.