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Journal articles on the topic 'Spherical Harmonic method'

Author: Grafiati
Published: 18 May 2024

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1

Snape-Jenkinson, C. J., S. Crozier, and L. K. Forbes. "NMR shim coil design utilising a rapid spherical harmonic calculation method." ANZIAM Journal 43, no. 3 (January 2002): 375–86. http://dx.doi.org/10.1017/s1446181100012578.

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AbstractA rapid spherical harmonic calculation method is used for the design of Nuclear Magnetic Resonance shim coils. The aim is to design each shim such that it generates a field described purely by a single spherical harmonic. By applying simulated annealing techniques, coil arrangements are produced through the optimal positioning of current-carrying circular arc conductors of rectangular cross-section. This involves minimizing the undesirable harmonics in relation to a target harmonic. The design method is flexible enough to be applied for the production of coil arrangements that generate fields consisting significantly of either zonal or tesseral harmonics. Results are presented for several coil designs which generate tesseral harmonics of degree one.
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2

Doicu, Adrian, and Dmitry S. Efremenko. "Linearizations of the Spherical Harmonic Discrete Ordinate Method (SHDOM)." Atmosphere 10, no. 6 (May 28, 2019): 292. http://dx.doi.org/10.3390/atmos10060292.

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Linearizations of the spherical harmonic discrete ordinate method (SHDOM) by means of a forward and a forward-adjoint approach are presented. Essentially, SHDOM is specialized for derivative calculations and radiative transfer problems involving the delta-M approximation, the TMS correction, and the adaptive grid splitting, while practical formulas for computing the derivatives in the spherical harmonics space are derived. The accuracies and efficiencies of the proposed methods are analyzed for several test problems.
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Kudlicki, Andrzej, Małgorzata Rowicka, Mirosław Gilski, and Zbyszek Otwinowski. "An efficient routine for computing symmetric real spherical harmonics for high orders of expansion." Journal of Applied Crystallography 38, no. 3 (May 13, 2005): 501–4. http://dx.doi.org/10.1107/s0021889805007685.

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A numerically efficient method of constructing symmetric real spherical harmonics is presented. Symmetric spherical harmonics are real spherical harmonics with built-in invariance with respect to rotations or inversions. Such symmetry-invariant spherical harmonics are linear combinations of non-symmetric ones. They are obtained as eigenvectors of an appropriate operator, depending on symmetry. This approach allows for fast and stable computation up to very high order symmetric harmonic bases, which can be used in e.g. averaging of non-crystallographic symmetry in protein crystallography or refinement of large viruses in electron microscopy.
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4

Crowley, John W., and Jianliang Huang. "A least-squares method for estimating the correlated error of GRACE models." Geophysical Journal International 221, no. 3 (March 9, 2020): 1736–49. http://dx.doi.org/10.1093/gji/ggaa104.

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SUMMARY A new least-squares method is developed for estimating and removing the correlated errors (stripes) from the Gravity Recovery and Climate Experiment (GRACE) and GRACE Follow-On (GRACE-FO) mission data. This method is based on a joint parametric model of the correlated errors and temporal trends in the spherical harmonic coefficients of GRACE models. Three sets of simulation data are created from the Global Land Data Assimilation System (GLDAS), the Regional Atmospheric Climate Model 2.3 (RACMO2.3) and GRACE models and used to test it. The results show that the new method improves the decorrelation method by Swenson & Wahr significantly. Its application to the release 5 (RL05) and new release 6 (RL06) spherical harmonic solutions from the Center for Space Research (CSR) at The University of Texas at Austin demonstrates its effectiveness and provides a relative assessment of the two releases. A comparison to the Swenson & Wahr and Kusche et al. methods highlights the deficiencies in past destriping methods and shows how the inclusion and decoupling of temporal trends helps to overcome them. A comparison to the CSR mascon and JPL mascon solutions demonstrates that the new method yields global trends that have greater amplitude than those produced by the CSR RL05 mascon solution and are of comparable quality to the JPL RL06 mascon solution. Furthermore, these results are obtained without the need for a priori information, scale factors or complex regularization methods and the solutions remain in the standard form of spherical harmonics rather than discrete mascons. The latter could introduce additional discretization error when converting to the spherical harmonic model, upon which many post-processing methods and applications are built.
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Sun, Huiyuan, Thushara D. Abhayapala, and Prasanga N. Samarasinghe. "Time Domain Spherical Harmonic Processing with Open Spherical Microphones Recording." Applied Sciences 11, no. 3 (January 25, 2021): 1074. http://dx.doi.org/10.3390/app11031074.

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Spherical harmonic analysis has been a widely used approach for spatial audio processing in recent years. Among all applications that benefit from spatial processing, spatial Active Noise Control (ANC) remains unique with its requirement for open spherical microphone arrays to record the residual sound field throughout the continuous region. Ideally, a low delay spherical harmonic recording algorithm for open spherical microphone arrays is desired for real-time spatial ANC systems. Currently, frequency domain algorithms for spherical harmonic decomposition of microphone array recordings are applied in a spatial ANC system. However, a Short Time Fourier Transform is required, which introduces undesirable system delay for ANC systems. In this paper, we develop a time domain spherical harmonic decomposition algorithm for the application of spatial audio recording mainly with benefit to ANC with an open spherical microphone array. Microphone signals are processed by a series of pre-designed finite impulse response (FIR) filters to obtain a set of time domain spherical harmonic coefficients. The time domain coefficients contain the continuous spatial information of the residual sound field. We corroborate the time domain algorithm with a numerical simulation of a fourth order system, and show the proposed method to have lower delay than existing approaches.
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6

Dwivedi, Priyadarshini, Gyanajyoti Routray, and Rajesh M. Hegde. "Spherical harmonics domain-based approach for source localization in presence of directional interference." JASA Express Letters 2, no. 11 (November 2022): 114802. http://dx.doi.org/10.1121/10.0015243.

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This paper presents a learning-based method for source localization in the presence of directional interference under reverberant and noisy conditions. The proposed method operates on the spherical harmonic decomposition of the spherical microphone array recordings to yield spherical harmonics coefficients as the features. An attention mechanism is incorporated through a binary mask that filters out the dominant undesired source components from the features before training. A convolutional neural network is trained to map the phase and magnitude of the filtered coefficients with the location class. Hence, the objective is to develop the binary mask followed by source localization.
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7

SHOJAEI, M. R., A. A. RAJABI, and H. HASANABADI. "HYPER-SPHERICAL HARMONICS AND ANHARMONICS IN m-DIMENSIONAL SPACE." International Journal of Modern Physics E 17, no. 06 (June 2008): 1125–30. http://dx.doi.org/10.1142/s0218301308010398.

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In quantum mechanics the hyper-spherical method is one of the most well-established and successful computational tools. The general theory of harmonic polynomials and hyper-spherical harmonics is of central importance in this paper. The interaction potential V is assumed to depend on the hyper-radius ρ only where ρ is the function of the Jacobi relative coordinate x1, x2,…, xn which are functions of the particles' relative positions.
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Yanagawa, Kazunori, Ayane Fujihira, Hideki Yamaguchi, and Nozomu Yoshizawa. "Describing the characteristics of light field in architectural spaces using spherical harmonic function." IOP Conference Series: Earth and Environmental Science 1099, no. 1 (November 1, 2022): 012014. http://dx.doi.org/10.1088/1755-1315/1099/1/012014.

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Abstract The direction, density, and diffusivity of light are important indicators of spatial characteristics in describing a three-dimensional light environment. Mury presented a method for describing, measuring, and visualizing the structure of light fields using spherical harmonics in terms of changes in the density and direction of light in three-dimensional space. We extended this study by using higher-order spherical harmonics, which would represent more diverse characteristics of the light environment. We also quantitatively described the light environment as numerical values and investigated the correspondence between these numerical values and human perceptual quantities. As a result, we confirmed that there is a certain degree of correspondence between the “complexity” quantified by the spherical harmonic and the “complexity” perceived by people when observing real space.
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Wang, Jian Qiang, Hao Yuan Chen, and Yin Fu Chen. "The Analysis of the Associated Legendre Functions with Non-Integral Degree." Applied Mechanics and Materials 130-134 (October 2011): 3001–5. http://dx.doi.org/10.4028/www.scientific.net/amm.130-134.3001.

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Spherical cap harmonic (SCH) theory has been widely used to format regional model of fields that can be expressed as the gradient of a scalar potential. The functions of this method consist of trigonometric functions and associated Legendre functions with integral-order but non-integral degree. Evidently, the constructing and computing of Legendre functions are the core content of the spherical cap functions. In this paper,the approximated calculation method of the normalized association Legendre functions with non-integral degree is introduced and an analysis of the entire order of associated non-Legendre function calculation is presented. Besides, we use the Muller method to search out for all intrinsic values. The results showed that the highest order of spherical harmonic function for constructing regional model of fields is limited, thus high-resolution spherical harmonic structure of local gravity field need to be improved.
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Xiao-yong, Zhang, and Guo Ben-yu. "Spherical Harmonic–Generalized Laguerre Spectral Method for Exterior Problems." Journal of Scientific Computing 27, no. 1-3 (January 19, 2006): 523–37. http://dx.doi.org/10.1007/s10915-005-9056-6.

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11

Gao, Jiabao, Fubin Tu, Chengbao Hu, Daosheng Ling, and Zhijiao Zeng. "Rockfall simulation via spherical harmonic based discrete element method." Computers & Geosciences 186 (April 2024): 105573. http://dx.doi.org/10.1016/j.cageo.2024.105573.

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12

Kolyanova, A. S. "Restoration of the orientation distribution function for materials with low lattice and sample symmetry using the harmonic method." Industrial laboratory. Diagnostics of materials 89, no. 9 (September 24, 2023): 34–40. http://dx.doi.org/10.26896/1028-6861-2023-89-9-34-40.

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A lot of the properties polycrystalline materials depend on their crystallographic texture. The most complete information about the texture can be obtained from the orientation distribution function (ODF). We present the results of recovering ODF using series expansion technique for materials with low crystal and sample symmetry. The technique of ODF restoration is based on its Fourier series expansion with symmetrical spherical harmonic functions. Real spherical harmonics which are linear combinations of general spherical harmonics were used. The model single-component texture as well as the real texture of magnesium alloy sample subjected to equal-channel angular pressing have been studied. Textures are characterized by hexagonal crystal symmetry and triclinic sample symmetry. In both cases RP-factors and ODF calculation errors that were used as reliability criteria of ODF reconstruction showed good agreement between the calculated and experimental data. It was also revealed that the ODF of a magnesium alloy sample subjected to equal-channel angular pressing contains two texture components (1216)[1211] and (1216)[1211] with maximum intensity values of 13.81 and 2.23, respectively. The results obtained can be used for texture studies of ceramics, rocks and other non-metallic materials characterized by a lower symmetry.
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13

Chen, Wenjin, and Robert Tenzer. "Reformulation of Parker–Oldenburg's method for Earth's spherical approximation." Geophysical Journal International 222, no. 2 (April 24, 2020): 1046–73. http://dx.doi.org/10.1093/gji/ggaa200.

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SUMMARY Parker–Oldenburg's method is perhaps the most commonly used technique to estimate the depth of density interface from gravity data. To account for large density variations reported, for instance, at the Moho interface, between the ocean seawater density and marine sediments, or between sediments and the underlying bedrock, some authors extended this method for variable density models. Parker–Oldenburg's method is suitable for local studies, given that a functional relationship between gravity data and interface geometry is derived for Earth's planar approximation. The application of this method in (large-scale) regional, continental or global studies is, however, practically restricted by errors due to disregarding Earth's sphericity. Parker–Oldenburg's method was, therefore, reformulated also for Earth's spherical approximation, but assuming only a uniform density. The importance of taking into consideration density heterogeneities at the interface becomes even more relevant in the context of (large-scale) regional or global studies. To address this issue, we generalize Parker–Oldenburg's method (defined for a spherical coordinate system) for the depth of heterogeneous density interface. Furthermore, we extend our definitions for gravity gradient data of which use in geoscience applications increased considerably, especially after launching the Gravity field and steady-state Ocean Circulation Explorer (GOCE) gravity-gradiometry satellite mission. For completeness, we also provide expressions for potential. The study provides the most complete review of Parker–Oldenburg's method in planar and spherical cases defined for potential, gravity and gravity gradient, while incorporating either uniform or heterogeneous density model at the interface. To improve a numerical efficiency of gravimetric forward modelling and inversion, described in terms of spherical harmonics of Earth's gravity field and interface geometry, we use the fast Fourier transform technique for spherical harmonic analysis and synthesis. The (newly derived) functional models are tested numerically. Our results over a (large-scale) regional study area confirm that the consideration of a global integration and Earth's sphericty improves results of a gravimetric forward modelling and inversion.
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14

Jia, Maoshen, Wenbei Wang, and Ziyu Yang. "2.5D Sound Field Reproduction Using Higher Order Loudspeakers." Cybernetics and Information Technologies 15, no. 6 (December 1, 2015): 5–15. http://dx.doi.org/10.1515/cait-2015-0063.

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Abstract Using 3-Dimensional (3D) sound sources as secondary sources to 2-Dimensional (2D) sound field reproduction, it is termed 2.5-Dimensional (2.5D) sound field reproduction which is currently drawing broad interest in acoustic signal processing. In this paper we propose a method to reproduce a 2D sound field, using a circular array of 3D High Order (HO) loudspeakers, which provides a mode matching solution based on 3D wave field translation. Using the spherical addition theorem, we first obtain a spherical harmonics representation of a 2D sound field reproduced by an array of HO loudspeakers. Then, the corresponding relationship between the reproduced sound field and the desired sound field is established by spherical/cylindrical harmonic expansions. Finally, the modal weights of HO loudspeakers are designed by using a least squares method. Simulation results show that the proposed method extends the reproduction region and significantly reduces the required minimum number of loudspeakers over the other referenced methods.
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15

Han, Chao, Tianshan Dong, Ming Cheng, Yu Wang, and Shenggang Liu. "A New Normalization Form Based on a General Unified Normalization Computing for the Gravitational Potential Tensor of Arbitrary Orders." Journal of Physics: Conference Series 2235, no. 1 (May 1, 2022): 012074. http://dx.doi.org/10.1088/1742-6596/2235/1/012074.

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Abstract Taking a general unified recursive algorithm for computing the normalized gravitational potential based on Cunningham's formula into account, the numerical stability of both the recursion for spherical harmonic series and the pre-processing for the spherical harmonic coefficients are discussed. Then a new method of normalizing the associated Legendre polynomial is proposed.
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16

Pail, R., and G. Plank. "Comparison of numerical solution strategies for gravity field recovery from GOCE SGG observations implemented on a parallel platform." Advances in Geosciences 1 (June 17, 2003): 39–45. http://dx.doi.org/10.5194/adgeo-1-39-2003.

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Abstract. The recovery of a full set of gravity field parameters from satellite gravity gradiometry (SGG) is a huge numerical and computational task. In practice, parallel computing has to be applied to estimate the more than 90 000 harmonic coefficients parameterizing the Earth’s gravity field up to a maximum spherical harmonic degree of 300. Three independent solution strategies, i.e. two iterative methods (preconditioned conjugate gradient method, semi-analytic approach) and a strict solver (Distributed Non-approximative Adjustment), which are operational on a parallel platform (‘Graz Beowulf Cluster’), are assessed and compared both theoretically and on the basis of a realistic-as-possible numerical simulation, regarding the accuracy of the results, as well as the computational effort. Special concern is given to the correct treatment of the coloured noise characteristics of the gradiometer. The numerical simulations show that there are no significant discrepancies among the solutions of the three methods. The newly proposed Distributed Nonapproximative Adjustment approach, which is the only one of the three methods that solves the inverse problem in a strict sense, also turns out to be a feasible method for practical applications.Key words. Spherical harmonics – satellite gravity gradiometry – GOCE – parallel computing – Beowulf cluster
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Medjber, Salim, Hacene Bekkar, Salah Menouar, and Jeong Ryeol Choi. "Quantization of a 3D Nonstationary Harmonic plus an Inverse Harmonic Potential System." Advances in Mathematical Physics 2016 (2016): 1–6. http://dx.doi.org/10.1155/2016/3693572.

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The Schrödinger solutions for a three-dimensional central potential system whose Hamiltonian is composed of a time-dependent harmonic plus an inverse harmonic potential are investigated. Because of the time-dependence of parameters, we cannot solve the Schrödinger solutions relying only on the conventional method of separation of variables. To overcome this difficulty, special mathematical methods, which are the invariant operator method, the unitary transformation method, and the Nikiforov-Uvarov method, are used when we derive solutions of the Schrödinger equation for the system. In particular, the Nikiforov-Uvarov method with an appropriate coordinate transformation enabled us to reduce the eigenvalue equation of the invariant operator, which is a second-order differential equation, to a hypergeometric-type equation that is convenient to treat. Through this procedure, we derived exact Schrödinger solutions (wave functions) of the system. It is confirmed that the wave functions are represented in terms of time-dependent radial functions, spherical harmonics, and general time-varying global phases. Such wave functions are useful for studying various quantum properties of the system. As an example, the uncertainty relations for position and momentum are derived by taking advantage of the wave functions.
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Chaitanya, S. K., Siddharth Sriraman, Srinath Srinivasan, and Srinivasan K. "Equivalent source method based Near-field acoustic holography using machine learning." INTER-NOISE and NOISE-CON Congress and Conference Proceedings 265, no. 6 (February 1, 2023): 1545–53. http://dx.doi.org/10.3397/in_2022_0213.

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The equivalent source method has been one of the most commonly used methods for sound source localization. It involves equivalent sources spread over the source plane (or region). The pressure fields from these equivalent sources are usually spherical harmonics. But, the spherical harmonic fields are derived for the Sommerfeld boundary condition with no reflection or reverberation. Data-driven methods help perform sound source localization in a reverberant environment when no prior information about the surroundings is available. The methods studied are linear regression (LR) with Adam, linear regression with L-BFGS, multi-layer perceptron (MLP) with one and two hidden layers. The simulations are conducted for two monopoles in rooms with different reverberation times and compared with one norm convex optimization (L1CVX). It is observed that overall, LR with L-BFGS gave the best results. Also, for low reverberation time, LR with L-BFGS was able to localize the sources better than L1CVX.
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Josef, J. A., and J. E. Morel. "Simplified spherical harmonic method for coupled electron-photon transport calculations." Physical Review E 57, no. 5 (May 1, 1998): 6161–71. http://dx.doi.org/10.1103/physreve.57.6161.

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Guo, Ben-Yu, and Wei Huang. "Mixed Jacobi-spherical harmonic spectral method for Navier–Stokes equations." Applied Numerical Mathematics 57, no. 8 (August 2007): 939–61. http://dx.doi.org/10.1016/j.apnum.2006.09.003.

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Gimbutas, Z., and L. Greengard. "A fast and stable method for rotating spherical harmonic expansions." Journal of Computational Physics 228, no. 16 (September 2009): 5621–27. http://dx.doi.org/10.1016/j.jcp.2009.05.014.

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22

Awrejcewicz, J., V. A. Krysko, and I. V. Kravtsova. "Dynamics and statics of flexible axially symmetric shallow shells." Mathematical Problems in Engineering 2006 (2006): 1–25. http://dx.doi.org/10.1155/mpe/2006/35672.

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In this work, we propose the method for the investigation of stochastic vibrations of deterministic mechanical systems represented by axially symmetric spherical shells. These structure members are widely used as sensitive elements of pressure measuring devices in various branches of measuring and control industry, machine design, and so forth. The proposed method can be easily extended for the investigation of shallow spherical shells, goffer-type membranes, and so on. The so-called charts of control parameters for a shell subjected to a transversal uniformly distributed and local harmonic loading force and resistance moment are constructed. The scenarios of the transition of vibration of shallow-type system into chaotic state are investigated with the use of the theory of differential equations and the theory of nonlinear dynamics. The method of the control of chaotic vibrations of flexible spherical shells subjected to a transversal harmonic load through a synchronized action of either harmonic resistance moment or force is proposed, illustrated, and discussed.
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Liu, Zhenghong, Haocheng Zhou, Xiyu Song, Mei Wang, and Liuqing Weng. "Multi-Objective NSGA-II Optimization for Broadband Beamforming with Spherical Harmonic Domain Assistance." Sensors 23, no. 20 (October 12, 2023): 8403. http://dx.doi.org/10.3390/s23208403.

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Sidelobe suppression is a major challenge in wideband beamforming for acoustic research, especially in high noise and reverberation environments. In this paper, we propose a multi-objective NSGA-II wideband beamforming method based on a spherical harmonic domain for spherical microphone arrays topology. The method takes white noise gain, directional index and maximum sidelobe level as the optimization objectives of broadband beamforming, adopts the NSGA-II optimization strategy with constraints to estimate the Pareto optimal solution, and provides three-dimensional broadband beamforming capability. Our method provides superior sidelobe suppression across different spherical harmonic orders compared to commonly used multi-constrained single-objective optimal beamforming methods. We also validate the effectiveness of our proposed method in a conference room setting. The proposed method achieves a white noise gain of 8.28 dB and a maximum sidelobe level of −23.42 dB at low frequency, while at high frequency it yields comparable directivity index results to both DolphChebyshev and SOCP methods, but outperforms them in terms of white noise gain and maximum sidelobe level, measuring 16.14 dB and −25.18 dB, respectively.
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Norris, A. N., and A. L. Shuvalov. "Elastodynamics of radially inhomogeneous spherically anisotropic elastic materials in the Stroh formalism." Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences 468, no. 2138 (October 12, 2011): 467–84. http://dx.doi.org/10.1098/rspa.2011.0463.

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A method for solving elastodynamic problems in radially inhomogeneous elastic materials with spherical anisotropy is presented, i.e. materials having c ijkl = c ijkl ( r ) in a spherical coordinate system { r , θ , ϕ }. The time-harmonic displacement field u ( r , θ , ϕ ) is expanded in a separation of variables form with dependence on θ , ϕ described by vector spherical harmonics with r -dependent amplitudes. It is proved that such separation of variables solution is generally possible only if the spherical anisotropy is restricted to transverse isotropy (TI) with the principal axis in the radial direction, in which case the amplitudes are determined by a first-order ordinary differential system. Restricted forms of the displacement field, such as u ( r , θ ), admit this type of separation of variables solution for certain lower material symmetries. These results extend the Stroh formalism of elastodynamics in rectangular and cylindrical systems to spherical coordinates.
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Song, Hyo-Jong, In-Hyuk Kwon, and Junghan Kim. "Characteristics of a Spectral Inverse of the Laplacian Using Spherical Harmonic Functions on a Cubed-Sphere Grid for Background Error Covariance Modeling." Monthly Weather Review 145, no. 1 (January 1, 2017): 307–22. http://dx.doi.org/10.1175/mwr-d-16-0134.1.

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Abstract In this study, a spectral inverse method using spherical harmonic functions (SHFs) represented on a cubed-sphere grid (SHF inverse) is proposed. The purpose of the spectral inverse method studied is to help with data assimilation. The grid studied is the one that results from a spectral finite element decomposition of the six faces of the cubed sphere on Gauss–Legendre–Lobatto (GLL) points with equiangular gnomonic projection. For a given discretization of the cube in this form, as the total wavenumber of the test functions increases, there comes a point at which the cube’s eigenstructure fails to be able to replicate the spherical harmonic functions. The authors call this point a limit wavenumber in using the SHF inverse. In common with the authors’ previous research, the allowable total wavenumber of the SHF inverse increases more effectively with an enhanced polynomial order. The use of the eigenvectors and eigenvalues of the Laplacian, discretized on the grid spacing used in this study, to the Poisson equation is compared with the benchmark set by using the spherical harmonics solution to the problem. In terms of accuracy, the SHF inverse is superior to a direct inverse of the Laplacian using eigendecomposition. The feasibility of SHF inverse in operational implementation is examined under a massive computational environment.
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Suzuki, Yukika, Izumi Tsunokuni, and Yusuke Ikeda. "2.5 dimensional sound field reproduction based on mode matching and equivalent sources considering primary reflections." INTER-NOISE and NOISE-CON Congress and Conference Proceedings 268, no. 1 (November 30, 2023): 7203–9. http://dx.doi.org/10.3397/in_2023_1079.

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Many sound field reproduction methods using many loudspeakers (secondary sources) have been studied to reproduce a physically-accurate sound field. In our previous research, we proposed a pressure matching method using the modeled transfer functions of the secondary sources based on the sparse equivalent source method (ESM). By modeling the sound field emitted from the loudspeaker, the number of measurement points required for pressure matching can be significantly reduced. Moreover, another well-known method of sound field reproduction is Mode Matching (MM). In the MM method, the driving functions are derived such that the weight coefficients of the spherical harmonics are matched between the desired and reproduced sound fields. In this study, we proposed an implementation of the MM method based on the ESM considering the sound reflections and loudspeakers' directivity. We estimate the spherical harmonic coefficients of the loudspeakers and sound reflections based on the superposition of equivalent sources. In the simulation experiments, we evaluate the reproduction accuracy achieved with the first-order sound reflections.
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Burnham, Christian J., and Niall J. English. "A New Relatively Simple Approach to Multipole Interactions in Either Spherical Harmonics or Cartesians, Suitable for Implementation into Ewald Sums." International Journal of Molecular Sciences 21, no. 1 (December 31, 2019): 277. http://dx.doi.org/10.3390/ijms21010277.

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We present a novel derivation of the multipole interaction (energies, forces and fields) in spherical harmonics, which results in an expression that is able to exactly reproduce the results of earlier Cartesian formulations. Our method follows the derivations of Smith (W. Smith, CCP5 Newsletter 1998, 46, 18.) and Lin (D. Lin, J. Chem. Phys. 2015, 143, 114115), who evaluate the Ewald sum for multipoles in Cartesian form, and then shows how the resulting expressions can be converted into spherical harmonics, where the conversion is performed by establishing a relation between an inner product on the space of symmetric traceless Cartesian tensors, and an inner product on the space of harmonic polynomials on the unit sphere. We also introduce a diagrammatic method for keeping track of the terms in the multipole interaction expression, such that the total electrostatic energy can be viewed as a ‘sum over diagrams’, and where the conversion to spherical harmonics is represented by ‘braiding’ subsets of Cartesian components together. For multipoles of maximum rank n, our algorithm is found to have scaling of n 3.7 vs. n 4.5 for our most optimised Cartesian implementation.
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Vadon, A., J. J. Heizmann, and C. Laruelle. "A Quantitative Texture Analysis of Pluri-Crystals by Texture Goniometry." Advances in X-ray Analysis 30 (1986): 429–37. http://dx.doi.org/10.1154/s0376030800021583.

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To understand how a material evolves - its crystal growth, topotaxy, twinning, phase transformation, plastic deformation, microstress, etc. - it is important to know the crystal orientations, either between them or in respect to the sample.The crystal texture of the material is quantified by the Orientation Distribution Function (O.D.F.). This function represents the part of the material volume having a given orientation. To compute this O.D.F, we must first measure one or several complete or incomplete pole figures and then analyse them either with Roe-Bunge's harmonic method or with Vadon, Ruer, Baro's vector method. In the case of a very sharp texture, the results obtained with the harmonic method are not good because developing a DIRAC function in spherical harmonics requires a high rank, hence a large number of pole figures. On the contrary, with the vector method, the results are good since discretizing amounts to developing on a step-function basis.
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Wang, Zhong-Qing, Rong Zhang, and Ben-Yu Guo. "Spherical harmonic-generalized Laguerre pseudospectral method for three-dimensional exterior problems." International Journal of Computer Mathematics 87, no. 9 (July 2010): 2123–42. http://dx.doi.org/10.1080/00207160802617020.

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Huang, Wei, and Ben-yu Guo. "Fully discrete Jacobi-spherical harmonic spectral method for Navier-Stokes equations." Applied Mathematics and Mechanics 29, no. 4 (April 2008): 453–76. http://dx.doi.org/10.1007/s10483-008-0404-1.

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31

Mikucki, Michael, and Yongcheng Zhou. "Fast Simulation of Lipid Vesicle Deformation Using Spherical Harmonic Approximation." Communications in Computational Physics 21, no. 1 (December 5, 2016): 40–64. http://dx.doi.org/10.4208/cicp.oa-2015-0029.

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AbstractLipid vesicles appear ubiquitously in biological systems. Understanding how the mechanical and intermolecular interactions deform vesicle membranes is a fundamental question in biophysics. In this article we develop a fast algorithm to compute the surface configurations of lipid vesicles by introducing surface harmonic functions to approximate themembrane surface. This parameterization allows an analytical computation of the membrane curvature energy and its gradient for the efficient minimization of the curvature energy using a nonlinear conjugate gradient method. Our approach drastically reduces the degrees of freedom for approximating the membrane surfaces compared to the previously developed finite element and finite difference methods. Vesicle deformations with a reduced volume larger than 0.65 can be well approximated by using as small as 49 surface harmonic functions. The method thus has a great potential to reduce the computational expense of tracking multiple vesicles which deform for their interaction with external fields.
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32

Wittmann, Ronald C., Carl F. Stubenrauch, and Michael H. Francis. "Using Truncated Data Sets in Spherical-Scanning Antenna Measurements." International Journal of Antennas and Propagation 2012 (2012): 1–6. http://dx.doi.org/10.1155/2012/979846.

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We discuss the mitigation of truncation errors in spherical-scanning measurements by use of a constrained least-squares estimation method. The main emphasis is the spherical harmonic representation of probe transmitting and receiving functions; however, our method is applicable to near-field measurement of electrically small antennas for which full-sphere data are either unreliable or unavailable.
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Yin, Weidong, Leizheng Shu, Yang Yu, and Yu Shi. "Free-Vertex Tetrahedral Finite-Element Representation and Its Use for Estimating Density Distribution of Irregularly-Shaped Asteroids." Aerospace 8, no. 12 (November 30, 2021): 371. http://dx.doi.org/10.3390/aerospace8120371.

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In this article, we present a free-vertex tetrahedral finite-element representation of irregularly shaped small bodies, which provides an alternative solution for estimating asteroid density distribution. We derived the transformations between gravitational potentials expressed by the free-vertex tetrahedral finite elements and the spherical harmonic functions. Inversely, the density of each free-vertex tetrahedral finite element can be estimated via the least-squares method, assuming a spherical harmonic gravitational function is present. The proposed solution is illustrated by modeling gravitational potential and estimating the density distribution of the simulated asteroid 216 Kleopatra.
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Valdez, Marc Andrew, Alex J. Yuffa, and Michael B. Wakin. "On-grid compressive sampling for spherical field measurements in acoustics." Journal of the Acoustical Society of America 152, no. 4 (October 2022): 2240–56. http://dx.doi.org/10.1121/10.0014628.

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We derive a compressive sampling method for acoustic field reconstruction using field measurements on a pre-defined spherical grid that has theoretically guaranteed relations between signal sparsity, measurement number, and reconstruction accuracy. This method can be used to reconstruct band limited spherical harmonic or Wigner D-function series (spherical harmonic series are a special case) with sparse coefficients. Contrasting typical compressive sampling methods for Wigner D-function series that use arbitrary random measurements, the new method samples randomly on an equiangular grid, a practical and commonly used sampling pattern. Using its periodic extension, we transform the reconstruction of a Wigner D-function series into a multi-dimensional Fourier domain reconstruction problem. We establish that this transformation has a bounded effect on sparsity level and provide numerical studies of this effect. We also compare the reconstruction performance of the new approach to classical Nyquist sampling and existing compressive sampling methods. In our tests, the new compressive sampling approach performs comparably to other guaranteed compressive sampling approaches and needs a fraction of the measurements dictated by the Nyquist sampling theorem. Moreover, using one-third of the measurements or less, the new compressive sampling method can provide over 20 dB better de-noising capability than oversampling with classical Fourier theory.
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35

Getman, Andriy, and Oleksandr Konstantinov. "Finding an analytical solution for the cylinder’s fluxmetric demagnetizing factor using spherical harmonics." Eastern-European Journal of Enterprise Technologies 2, no. 5 (128) (April 30, 2024): 33–41. http://dx.doi.org/10.15587/1729-4061.2024.301008.

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The work examines an analytical solution for calculating the fluxmetric demagnetizing factor of cylindrical magnets at large values of magnetic susceptibility and an arbitrary value of elongation. The application of the analytical solution for calculating the demagnetizing factor significantly simplifies the modeling and calculation of magnetic characteristics of cylindrical technical objects. A simplified analytical model of the scalar potential of the magnetic field of a cylinder with infinite magnetic favorability, inductively magnetized in a uniform magnetic field, was constructed using an approximate representation of the distribution of fictitious magnetic charges on its surface. The method of spherical harmonic analysis for the magnetic field was used, which made it possible to obtain an analytical representation of the demagnetization field in the central cross section of the cylinder. Limitation of the harmonic series of this representation by seven first harmonics is proposed, and an additional amplitude factor is applied to correct the contribution of the first harmonic to the demagnetization field. This made it possible to compensate for the distortion of the magnetic field near the ends of the cylinder and bring the simplified analytical model closer to the target mathematical model with a uniform demagnetization magnetic field. The reliability of the results of calculating the fluxmetric demagnetizing factor according to the derived formula was evaluated by comparing them with the known results obtained using the numerical method of calculation and according to empirical formulas. It is shown that the proposed approach makes it possible to obtain reliable results of calculating the fluxmetric demagnetizing factor with a deviation of up to 5 % at infinite favorability in the range of cylinder elongation values from 0.01 to 500
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36

Famoriji, Oluwole John, and Thokozani Shongwe. "Spherical Atomic Norm-Inspired Approach for Direction-of-Arrival Estimation of EM Waves Impinging on Spherical Antenna Array with Undefined Mutual Coupling." Applied Sciences 13, no. 5 (February 27, 2023): 3067. http://dx.doi.org/10.3390/app13053067.

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A spherical antenna array (SAA) is an array-designed arrangement capable of scanning in almost all the radiation sphere with constant directivity. It finds recent applications in aerospace, spacecraft, vehicular and satellite communications. Therefore, estimation of the direction-of-arrival (DoA) of electromagnetic (EM) waves that impinge on an SAA with unknown mutual coupling called for research attention. This paper proposed a spherical harmonic atomic norm minimization (SHANM) for DoA estimation using an SAA configuration. The gridless sparse signal recovery problem is considered in the spherical harmonic (SH) domain in conjunction with the atomic norm minimization (ANM). Because of the unavailability of the Vandermonde structure in the SH domain, the theorem of Vandermonde decomposition that is the mathematical basis of the traditional ANM methods finds no application in SH. Addressing this challenge, a low-dimensional semidefinite programming (SDP) approach implementing the SHANM method is developed. This approach is independent of Vandermonde decomposition, and directly recovers the atomic decomposition in SH. The numerical experimental results show the superior performance of the proposed method against the previous methods. In addition, accounting for the impacts of mutual coupling, an experimental measured data, which is the generally accepted ground of testing any method, is employed to illustrate the efficacy and robustness of the proposed methods. Finally, for achieving DoA estimation with sufficient localization accuracy using a SAA, the proposed SHANM-based method is a better option.
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Shinobu, Naoki, Toma Yoshimatsu, Hiroaki Itou, Shihori Kozuka, Noriyoshi Kamado, and Yoichi Haneda. "Time domain virtual sensing method based on a rigid-sphere transfer function for active noise control headrests." Journal of the Acoustical Society of America 154, no. 4_supplement (October 1, 2023): A123. http://dx.doi.org/10.1121/10.0022993.

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Active noise control (ANC) is expected to be used in public transportation such as airplanes and trains. While using these applications, passengers prefer not to wear earphones or ear microphones, if possible, to reduce ear strain. However, conventional ANC systems require a physical microphone, which acts as an error sensor, at the ear position. To address this issue, a virtual sensing technique has been studied that uses a microphone mounted on a seat’s headrest to estimate the sound pressure at the ear position. In this study, we propose a virtual sensing technique that interpolates the sound pressure at the ear position using a spherical microphone array surrounding the head. This method improves robustness to variations in noise arrival direction using spherical harmonic coefficients without directional dependence. In general, the interpolation formula using spherical harmonic expansion is obtained in the frequency domain. By contrast, the proposed method designs the interpolation process as an IIR filter by considering the minimum phase and estimates the noise signal directly in the time domain, achieving the real-time processing required for ANC. The effectiveness of the proposed method in reducing noise was investigated for various noise arrival directions.
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Lakeev, Ivan Y. "STUDY OF SPATIAL DISTRIBUTION OF EARTH GRAVITATIONAL FIELD BY SPECTRAL WINDOW LIMITING METHOD." Vestnik SSUGT (Siberian State University of Geosystems and Technologies) 25, no. 4 (2020): 37–44. http://dx.doi.org/10.33764/2411-1759-2020-25-4-37-44.

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The article shows how windows are formed by successively combining harmonics и into a separate group, which act as a lowpass or high harmonic bandpass filter. Manipulation of spectral window width allows to get information about the nature of gravity field spatial distribution in certain frequency ranges. The degree range of the harmonic window was selected so that the result was oriented towards the actual source of the gravity disturbance. Calculations of spectrozonal models of quasi-geoid field height of Western Siberia, Fennoscandia, and Central Russia territories with degree series limiting by values N1-2 from N1 = 2 to N2 = 200, from N1 = 9 to N2 = 22 and N1 = 30 to N2 = 200, were carried out in GeoUnd 1.0 software. This software is used to calculate quasi-geoid height by expansion of gravity field coefficients in spherical functions row. These spectrozonal models are presented in graphical version for illustration and analysis. Results show that using of global quasi-geoid spectrozonal models obtained from Earth's gravity potential combined models is a modern and productive method for detecting current and future vertical Earth crust movements in local and regional areas.
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39

Baykiev, Eldar, Dilixiati Yixiati, and Jörg Ebbing. "Global High-Resolution Magnetic Field Inversion Using Spherical Harmonic Representation of Tesseroids as Individual Sources." Geosciences 10, no. 4 (April 16, 2020): 147. http://dx.doi.org/10.3390/geosciences10040147.

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In this study, we present a novel approach combining the advantages of tesseroids in representing geophysical structures though their voxel-like discretization features with a spherical harmonic representation of the magnetic field. Modelling of the Earth lithospheric magnetic field is challenging since part of the spectra is hidden by the core field and the forward modeled field of a lithospheric magnetization is always biased by the spectral range used. In our approach, a spherical harmonic representation of the magnetic field of spherical prisms (tesseroids) is used for high-resolution magnetic inversion of lithospheric field models. The use of filtered spherical harmonic models of the magnetic field of each tesseroid ensures that the resulting field matches the spectral range of the input data. For the inversion, we use the projected gradient method. The projected gradient method easily allows us to assign an initial guess (i.e., a-priori assumption) for the inversion and avoids negative values of susceptibilities. The latter is providing more plausible models since induced magnetization is assumed to be dominant over the continents and, for the oceans, a remanence model can be subtracted. We show an application of the technique to a synthetic dataset and a satellite-derived lithospheric field model where the model geometry is based on seismic information. We also demonstrate a proof-of-concept for high-resolution tile-wise inversion for the Bangui anomaly in Africa.
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BERMÚDEZ, ALFREDO, LUIS HERVELLA-NIETO, ANDRÉS PRIETO, and RODOLFO RODRÍGUEZ. "VALIDATION OF ACOUSTIC MODELS FOR TIME-HARMONIC DISSIPATIVE SCATTERING PROBLEMS." Journal of Computational Acoustics 15, no. 01 (March 2007): 95–121. http://dx.doi.org/10.1142/s0218396x07003238.

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The aim of this paper is to study the time-harmonic scattering problem in a coupled fluid-porous medium system. We consider two different models for the treatment of porous materials: the Allard–Champoux equations and an approximate model based on a wall impedance condition. Both models are compared by computing analytically their respective solutions for unbounded planar obstacles, considering successively plane and spherical waves. A numerical method combining an optimal bounded PML and finite elements is also introduced to compute the solutions of both problems for more general axisymmetric geometries. This method is used to compare the solutions for a spherical absorber.
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41

Zhong, Jiaxin, Haishan Zou, and Jing Lu. "A modal expansion method for simulating reverberant sound fields generated by a directional source in a rectangular enclosure." Journal of the Acoustical Society of America 154, no. 1 (July 1, 2023): 203–16. http://dx.doi.org/10.1121/10.0020070.

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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.
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42

Morris, P. R., and R. E. Hook. "Comparison of Incomplete Pole-Figure Methods for Surfaces Perpendicular to Rolling, Transverse and Normal Directions." Textures and Microstructures 19, no. 1-2 (January 1, 1992): 75–80. http://dx.doi.org/10.1155/tsm.19.75.

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Coefficients for a generalized-spherical-harmonic expansion of the crystallite orientation distribution function (ODF) through L=16 were obtained by an incomplete pole-figure method from a deep-drawing aluminum-killed sheet steel sample with surface perpendicular to the sheet-normal direction (ND). These coefficients were subsequently transformed from the RD, TD, ND reference frame to –ND, TD, RD and ND, RD, TD reference frames. Spherical-surface-harmonic expansions of incomplete {110}, {100}, and {112} pole-figures were calculated for each reference frame and used as input data to calculate ODF coefficients for each frame. The thus-calculated coefficients were transformed to the RD, TD, ND frame in each case. Series expansions of pole-figures and ODF for each frame are compared with the initial data.
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43

Ilham, M. Syirojudin, R. Margiono, A. Marsono, and N. Ardiana. "Indonesian Earth’s Lithospheric Magnetic Field modelling using Spherical Cap Harmonic Analysis Method." IOP Conference Series: Earth and Environmental Science 873, no. 1 (October 1, 2021): 012030. http://dx.doi.org/10.1088/1755-1315/873/1/012030.

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Abstract The earth’s lithospheric magnetic field is part of the main earth’s magnetic field. The lithospheric field has a very small value compared to the Earth’s main magnetic field, approximately less than 1%, and this field is generated at the earth’s crust and upper mantle. Modelling of lithospheric field is useful mainly for predicting the distribution of the value of lithospheric fields and to determine the magnetic anomaly. In this research, modelling the Earth’s lithospheric magnetic field uses Spherical Cap Harmonic Analysis (SCHA) method and this method can do modelling using regional magnetic data. The data used for the modelling are magnetic repeat station data in Indonesia region (BMKG’s Epoch) and SWARM satellite data. The results of the modelling using integrated SWARM satellite and repeat station data produce RMSE values of 64.0834 nT and the expansion of index K is 70. In addition, the results of the modelling resolution is 1.50. The value’s range of modelling’s result are -987.192 – 998.239 nT for X component, -968.189 – 949.438 nT for Y component, -981.266 – 608.676 nT for Z component, and -904.151 – 997.389 nT for total intensity are.
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44

Gazazyan, E. D., and M. I. Ivanyan. "Toward a theory of antenna characteristic measurement by the spherical harmonic method." Radiophysics and Quantum Electronics 30, no. 10 (October 1987): 897–901. http://dx.doi.org/10.1007/bf01034852.

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45

Inanc, Feyzi. "Verification of the three-dimensional modular nodal method for spherical harmonic equations." Annals of Nuclear Energy 23, no. 7 (May 1996): 613–16. http://dx.doi.org/10.1016/0306-4549(95)00058-5.

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46

Mathews, Jonathan, and Jonas Braasch. "A method for real-time, multiple-source localization using spherical harmonic decomposition." Journal of the Acoustical Society of America 140, no. 4 (October 2016): 3450. http://dx.doi.org/10.1121/1.4971144.

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47

Morena, Carlos De La, Y. A. Nefedyev, A. O. Andreev, E. N. Ahmedshina, A. A. Arkhipova, E. V. Kronrod, and N. Y. Demina. "The analysis of Titan’s physical surface using multifractal geometry methods." Journal of Physics: Conference Series 2103, no. 1 (November 1, 2021): 012017. http://dx.doi.org/10.1088/1742-6596/2103/1/012017.

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Abstract 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.
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48

SULAIMAN, SAIFUL AMAN HJ, Nur Sofia Erina Ariff, Nurzaitie Aflah Abdullah, Adolfientje Kasenda Olesen, and Muhammad Daud Mahdzur. "Accuracy Assessment of High-Degree Geopotential Models in Peninsular Malaysia." ASM Science Journal 18 (December 19, 2023): 1–13. http://dx.doi.org/10.32802/asmscj.2023.1018.

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The selection of the accurate geoid model is essential for geoid determination in specific regions. Many improvements to the basic theory of more reliable data are available for numerical modelling studies. All of the innovations have led to the development of a sequence of global geopotential models of increasing spherical harmonic degree and order and the resolution of the geopotential models. There are hundreds of geopotential models that can be downloaded in ICGEM. All geopotential ICGEM models have accuracy and resolution based on the degree and order of spherical harmonic coefficients. This study has two high-degree geopotential models: the latest one, XGM2019e_2159, and another geopotential model, EGM2008. The accuracy evaluation of two geopotential models needs to be evaluated based on terrestrial gravity data and existing GPS points data to choose which geopotential model is the best-fit geopotential models with different degrees and order in terms of spherical harmonic coefficients using the Root Mean Square method. The statistical analysis of the geopotential model derived based on the existing GPS points shows that the degree and order 720 for XGM2019e_2019 are the best-fit geopotential models in Peninsular Malaysia. The result also indicates that the gravity anomaly derived from EGM2008 with the maximum degree and order of 2190 regarding the spherical harmonic coefficient is the most accurate geoid model that can be used as a reference over Peninsular Malaysia. Overall, it can be concluded that XGM2019e_2019 and EGM2008 are the best-fit geopotential models for Peninsular Malaysia.
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Michael, J. Robert, and Anatoliy Volkov. "Density- and wavefunction-normalized Cartesian spherical harmonics forl≤ 20." Acta Crystallographica Section A Foundations and Advances 71, no. 2 (January 23, 2015): 245–49. http://dx.doi.org/10.1107/s2053273314024838.

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The widely used pseudoatom formalism [Stewart (1976).Acta Cryst. A32, 565–574; Hansen & Coppens (1978).Acta Cryst.A34, 909–921] in experimental X-ray charge-density studies makes use of real spherical harmonics when describing the angular component of aspherical deformations of the atomic electron density in molecules and crystals. The analytical form of the density-normalized Cartesian spherical harmonic functions for up tol≤ 7 and the corresponding normalization coefficients were reported previously by Paturle & Coppens [Acta Cryst.(1988), A44, 6–7]. It was shown that the analytical form for normalization coefficients is available primarily forl≤ 4 [Hansen & Coppens, 1978; Paturle & Coppens, 1988; Coppens (1992).International Tables for Crystallography, Vol. B,Reciprocal space, 1st ed., edited by U. Shmueli, ch. 1.2. Dordrecht: Kluwer Academic Publishers; Coppens (1997).X-ray Charge Densities and Chemical Bonding. New York: Oxford University Press]. Only in very special cases it is possible to derive an analytical representation of the normalization coefficients for 4 <l≤ 7 (Paturle & Coppens, 1988). In most cases forl> 4 the density normalization coefficients were calculated numerically to within seven significant figures. In this study we review the literature on the density-normalized spherical harmonics, clarify the existing notations, use the Paturle–Coppens (Paturle & Coppens, 1988) method in the WolframMathematicasoftware to derive the Cartesian spherical harmonics forl≤ 20 and determine the density normalization coefficients to 35 significant figures, and computer-generate a Fortran90 code. The article primarily targets researchers who work in the field of experimental X-ray electron density, but may be of some use to all who are interested in Cartesian spherical harmonics.
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50

Gupta, Raj K., S. K. Patra, and W. Greiner. "Structure of 294,302120 Nuclei Using the Relativistic Mean-Field Method." Modern Physics Letters A 12, no. 23 (July 30, 1997): 1727–36. http://dx.doi.org/10.1142/s021773239700176x.

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We have studied the structure of superheavy nuclei 294,302120 in the framework of a relativistic mean-field formalism, using three different parameter sets (NL1, NL–SH and TM1) in an axially deformed harmonic oscillator basis. The calculated shapes are found to be parameter-dependent, e.g. NL1 parameter set predicts 302120 as a spherical and 294120 a very weakly oblate deformed nucleus whereas NL–SH and TM1 parameter sets predict both the nuclei with strongly prolate/oblate deformed configurations, in their respective ground states. This result, coupled with the calculated single-particle energy spectrum for NL1 parameter set, supports for Z=120 nuclei the spherical magic shell more at N=184 than at N=172. Even for the spherical 302120 nucleus, many new closed shells are predicted and some of the known magic numbers are found absent. Also, the binding energies of the various isotopes of Z=104–111 nuclei are calculated whose comparisons with experimental data favor the NL1 parameter set.
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