Lebedev Grid Research Articles

Efficient and accurate numerical-projection of electromagnetic multipoles for scattering objects

In this paper, we develop an efficient and accurate procedure of electromagnetic multipole decomposition by using the Lebedev and Gaussian quadrature methods to perform the numerical integration. Firstly, we briefly review the principles of multipole decomposition, highlighting two numerical projection methods including surface and volume integration. Secondly, we discuss the Lebedev and Gaussian quadrature methods, provide a detailed recipe to select the quadrature points and the corresponding weighting factor, and illustrate the integration accuracy and numerical efficiency (that is, with very few sampling points) using a unit sphere surface and regular tetrahedron. In the demonstrations of an isotropic dielectric nanosphere, a symmetric scatterer, and an anisotropic nanosphere, we perform multipole decomposition and validate our numerical projection procedure. The obtained results from our procedure are all consistent with those from Mie theory, symmetry constraints, and finite element simulations.Graphical

Filtered Lebedev quadrature method for robust and efficient beam shape coefficient estimation in acoustic tweezers calibration.

Acoustic tweezers offer a contactless, three-dimensional, and selective approach to trapping objects by harnessing the acoustic radiation force. Precise control of this technique requires accurate calibration of the force, which depends on the object's properties and the spherical harmonics expansion of the incident field through the beam shape coefficients. Previous studies showed that these coefficients can be determined using either the Lebedev quadrature or the angular spectrum methods. However, the former is highly susceptible to noise, while the latter demands extensive implementation time due to the number of required measurement points. A filtered method with a reduced number of points is introduced to address these limitations. Initially, we emphasize the implicit filtering in the angular spectrum method, allowing relative noise insensitivity. Subsequently, we present its unfiltered version, enabling force estimation of a standing field. Finally, we develop a filtered method based on the Lebedev quadrature, requiring fewer points, and apply it to focused vortex beams. Numerical evaluation of the radiation force demonstrates the method's resilience to noise and a reduced need for points compared to previous methods. The filtered Lebedev method paves the way for characterizing high-frequency acoustic tweezers, where measurement constraints necessitate rapid and robust beam shape coefficient estimation techniques.

2.5-Dimensional modeling of EM logging-while-drilling tool in anisotropic medium on a Lebedev grid

A 2.5D finite-difference (FD) algorithm for the modeling of the electromagnetic (EM) logging-while-drilling (LWD) tool in anisotropic media is presented. The FD algorithm is based on the Lebedev grid, which allows for the discretization of the frequency-domain Maxwell's equations in the anisotropic media in 2.5D scenarios without interpolation. This leads to a system of linear equations that is solved using the multifrontal direct solver which enables the simulation of multi-sources at nearly the cost of simulating a single source for each frequency. In addition, near-optimal quadrature derived from an optimized integration path in the complex plane is employed to implement the fast inverse Fourier Transform (IFT). The algorithm is then validated by both analytic and 3D solutions. Numerical results show that two Lebedev subgrid sets are sufficient for TI medium, which is common in geosteering environments. The number of quadrature points is greatly reduced by using the near-optimal quadrature method.

SO(3) quadratures in angular-momentum projection

While the angular-momentum projection is a common tool for theoretical nuclear structure studies, a large amount of computations are required particularly for triaxially deformed states. In the present work, we clarify the conditions of the exactness of quadratures in the projection method. For efficient computation, the Lebedev quadrature and spherical t-design are introduced to the angular-momentum projection. The accuracy of the quadratures is discussed in comparison with the conventional Gauss-Legendre and trapezoidal quadratures. We found that the Lebedev quadrature is the most efficient among them and the necessary number of sampling points for the quadrature, which is often proportional to the computation time, is reduced by a factor 2/3 in comparison with the conventional method.

Finite Difference Scheme Based on the Lebedev Grid for Seismic Wave Propagation in Fractured Media

Finite Difference Scheme Based on the Lebedev Grid for Seismic Wave Propagation in Fractured Media

Efficient Vector Interpolation Method Based on Lebedev Quadrature for MLFMA

In this letter, an efficient vector interpolation method based on Lebedev quadrature to accelerate the matrix-vector multiplications in far-field regions for the multilevel fast multipole algorithm (MLFMA) is presented. This approach avoids the singularity on sphere poles in the traditional interpolation method based on Gaussian quadrature and maintains the same accuracy without increasing computational costs. As a consequence, both memory consumption and processing time can be reduced considerably, providing a viable alternative to solve large-scale electromagnetic scattering problems using MLFMA.

High accuracy ab initio potential energy surface for the H2O-H van der Waals dimer.

A representation of the three-dimensional potential energy surface (PES) of the H2O-H van der Waals dimer is presented. The H2O molecule is treated as a rigid body held at its experimentally determined equilibrium geometry, with the OH bond length set to 1.809 650 34 a0 and the HOH bond angle set to 1.824 044 93 radians. Ab initio calculations are carried out at the coupled-cluster single, double, and perturbative triple level, with scalar relativistic effects included using the second-order Douglas-Kroll-Hess approximation. The ab initio calculations employ the aug-cc-pVnZ-DK series of basis sets (n = D, T, Q), which are recontracted versions of the aug-cc-pVnZ basis sets that are appropriate for relativistic calculations. The counterpoise method is used to reduce the basis set superposition error; in addition, results obtained using the aug-cc-pVTZ-DK and aug-cc-pVQZ-DK basis sets were extrapolated to the complete basis set (CBS) limit. The PES is based on calculations carried out at 1054 symmetry-unique H2O-H geometries for which the distance R between the H-atom and the H2O center of mass ranges from R = 2.5-9.0 Å. The reproduction of the PES along the orientational degrees of freedom was performed using Lebedev quadrature and an expansion in spherical harmonics. The mean absolute error of the reproduced PES is <0.02cm-1 for R ≥ 3.0 Å and <0.21cm-1 for R between 2.5 and 3.0 Å. The global minimum for the CBS PES is a coplanar H2O-H geometry, with R = 3.41 Å, in which the angle formed between the H2O C2 symmetry axis and the H-atom is 122.25°; the CBS binding energy for this geometry is 61.297cm-1. In addition, by utilizing the symmetry of the H2O molecule, the spherical harmonic expansion was simplified with no loss in accuracy and a speedup of ∼1.8 was achieved. The reproduced PES can be used in future molecular dynamics simulations.

Anisotropic elastic finite-difference modeling of sources and receivers on Lebedev grids

The Lebedev grid finite-difference (FD) method allows modeling of anisotropic elastic-wave propagation. On Lebedev grids, erroneous point-source excitations can create spurious (nonphysical) waves. The only known remedy for such artifacts in the literature is the Lisitsa-Vishnevsky method. This method uses a distributed array to create point sources and point receivers on the FD grid. However, the Lisitsa-Vishnevsky method does not fully eliminate spurious artifacts. A novel approach is found in the FD-consistent point source, which suppresses spurious artifacts entirely. The method requires no array recording to create point receivers. The advantage of this method over the Lisitsa-Vishnevsky method is determined with two anisotropic modeling examples.

Effects of Additive Noise in Binaural Rendering of Spherical Microphone Array Signals

Additive noise produced by the recording hardware will contribute to streamed signals from spherical microphone arrays under practical conditions. For the application of binaural reproduction and under the assumption that the noise is uncorrelated between the array channels, the spectral properties and the overall level of the rendered noise in the ear signals have been shown to be strongly influenced by the configuration of the array as well as of the processing pipeline. In a previous investigation, we determined the audibility thresholds for changes in the rendered noise due to listener head rotations as a function of the differences in noise level of individual array channels. In this article, we calibrate the instrumental metric of Composite Loudness Level to the perceptual data and predict audibility of changes in the additive noise due to head rotations for a broad set of array configurations and distributions of the noise levels across array channels. We demonstrate that some types of microphone layouts can produce audible variations even if the noise level is equal in all channels. This is particularly the case for sampling grids that exhibit negative quadrature weights such as the Lebedev and Fliege-Maier grids for some spherical harmonic orders. The analysis of configurations with unevenly distributed noise contributions show that the influence of the noise from individual array channels is determined by the proximity of their virtual location to the relative trajectory of the ears.

Anisotropic elastic finite-difference modeling of sources and receivers on Lebedev grids

Anisotropic elastic finite-difference modeling of sources and receivers on Lebedev grids

Computation of the radiation force exerted by the acoustic tweezers using pressure field measurements.

Acoustic tweezers allow for manipulation of small objects like elastic spheres with a force generated by the radiation pressure which arises from the nonlinear interaction between the incident and scattered waves by the object. The accurate control of the object by acoustic tweezers requires the study of the components of the three-dimensional (3D) force. If the physical properties of the elastic sphere are known, then the 3D components of the force can be calculated thanks to a decomposition of the incident acoustic field in the spherical functions basis. This study proposes evaluating the expansion coefficients. Three methods are used and compared. The first one consists of measuring the acoustic field on a spherical surface centered on the theoretical position of the object and to calculate the spherical functions decomposition by Lebedev quadratures. The second method is based on the measurement of the acoustic field at random points in a spherical volume and on the resolution of the inverse problem by a sparse method called the orthogonal matching pursuit. In the third method, the incident beam is measured on a transverse plane, decomposed into a sum of plane waves, and then the expansion coefficients are calculated. The results of the three methods will be presented and compared.

Implementation of General Dispersive Anisotropic Materials in Lebedev FDTD

A variant of the finite-difference time-domain (FDTD) algorithm—the Lebedev FDTD method—is augmented to allow for the simulation of electromagnetic waves in general dispersive anisotropic materials. Distinct dispersions may be applied to each of the principal axes of the anisotropic medium. The method is based on the auxiliary differential equation and is simple to implement. Because of the collocated nature of the algorithm, multiterm dispersions necessary in previous implementations are avoided. Lebedev grid has previously been studied in detail by different research groups for the simulation of anisotropy. In this paper, we specifically focus on extending frequency-dispersive materials to this method and provide multiple examples to demonstrate the accuracy, validity, and stability of the method.

Advancing the computational methodology of rigid rod and semiflexible polymer systems: A new solution to the wormlike chain model with rod‐coil copolymer calculations

ABSTRACTA wormlike chain model for rod type blocks in a rod‐coil diblock copolymer is implemented in the self‐consistent field theory (SCFT) formalism. A pseudo‐spectral method is used to solve for the single‐chain partition function of this copolymer system. Orientation degrees of freedom are discretized using Lebedev sphere rules such that orientation integrations are carried out through a Lebedev quadrature, an approach not used previously in tandem with the pseudo‐spectral method. Phase behavior in the rigid‐rod limit as a function of rod segment volume fraction, Flory–Huggins interaction parameter χ, degree of polymerization N, and rod contour length ratio β are examined in detail in one and two dimensions. Examples extending to three dimensions are included. Semiflexible behavior via the rod bending rigidity κ is explored. An approximation is used for rigid‐rods that do not need spherical harmonics leading to increased speed in finding equilibrium morphologies. The results show that standing vertical structures may be more easily produced with rigid‐rod blocks compared to coil‐coil lamellae, an important feature in nanolithographic applications. Suggestions are made for using the model in future molecular orientation studies where the model can be used with inverse search methods to measure the values of the model parameters for the real systems. © 2018 Wiley Periodicals, Inc. J. Polym. Sci., Part B: Polym. Phys. 2019, 57, 29–39

Hydrodynamic flows on curved surfaces: Spectral numerical methods for radial manifold shapes

We formulate hydrodynamic equations and spectrally accurate numerical methods for investigating the role of geometry in flows within two-dimensional fluid interfaces. To achieve numerical approximations having high precision and level of symmetry for radial manifold shapes, we develop spectral Galerkin methods based on hyperinterpolation with Lebedev quadratures for L2-projection to spherical harmonics. We demonstrate our methods by investigating hydrodynamic responses as the surface geometry is varied. Relative to the case of a sphere, we find significant changes can occur in the observed hydrodynamic flow responses as exhibited by quantitative and topological transitions in the structure of the flow. We present numerical results based on the Rayleigh-Dissipation principle to gain further insights into these flow responses. We investigate the roles played by the geometry especially concerning the positive and negative Gaussian curvature of the interface. We provide general approaches for taking geometric effects into account for investigations of hydrodynamic phenomena within curved fluid interfaces.

On Accuracy of Perfect Electric Conductor Implementation in Lebedev FDTD

The Lebedev grid is an alternative to Yee grid where, because the discretized fields lie at the same location in space, there is no need for further field interpolations when modeling anisotropic materials. In this letter, we focus on accurately implementing perfect electric conductors with planar interfaces, as well as 90° and 270° corners. Simulations are provided to demonstrate the overall improvements and also the second-order accuracy of the proposed methods.

Efficient Implementation of Variation after Projection Generalized Hartree-Fock.

Projected Hartree-Fock (PHF) theory can restore important symmetries to broken symmetry wave functions. Variation after projection (VAP) implementations make it possible to deliberately break and then restore a given symmetry by directly minimizing the projected energy expression. This technique can be applied to any symmetry that can be broken from relaxing constraints on single Slater determinant wave functions. For instance, generalized Hartree-Fock (GHF) wave functions are eigenfunctions of neither Ŝz nor S2. By relaxing these constraints, the wave function can explore a larger variational space and can reach lower energies than more constrained HF solutions. We have implemented spin-projected GHF (SGHF), which retains many of the advantages of breaking symmetry while also being a spin eigenfunction, with some notable improvements over previous implementations. Our new algorithm involves the formation of new intermediate matrices not previously discussed in the literature. Discretization of the necessary integration over the rotation group SO(3) is also accomplished much more efficiently using Lebedev grids. A novel scheme to incrementally build rotated Fock matrices is also introduced and compared with more standard approaches.

Lebedev acceleration and comparison of different photometric models in the inversion of lightcurves for asteroids

In the lightcurve inversion process where asteroid's physical parameters such as rotational period, pole orientation and overall shape are searched, the numerical calculations of the synthetic photometric brightness based on different shape models are frequently implemented. Lebedev quadrature is an efficient method to numerically calculate the surface integral on the unit sphere. By transforming the surface integral on the Cellinoid shape model to that on the unit sphere, the lightcurve inversion process based on the Cellinoid shape model can be remarkably accelerated. Furthermore, Matlab codes of the lightcurve inversion process based on the Cellinoid shape model are available on Github for free downloading. The photometric models, i.e., the scattering laws, also play an important role in the lightcurve inversion process, although the shape variations of asteroids dominate the morphologies of the lightcurves. Derived from the radiative transfer theory, the Hapke model can describe the light reflectance behaviors from the viewpoint of physics, while there are also many empirical models in numerical applications. Numerical simulations are implemented for the comparison of the Hapke model with the other three numerical models, including the Lommel-Seeliger, Minnaert, and Kaasalainen models. The results show that the numerical models with simple function expressions can fit well with the synthetic lightcurves generated based on the Hapke model; this good fit implies that they can be adopted in the lightcurve inversion process for asteroids to improve the numerical efficiency and derive similar results to those of the Hapke model.

Spectral Numerical Exterior Calculus Methods for Differential Equations on Radial Manifolds

We develop exterior calculus approaches for partial differential equations on radial manifolds. We introduce numerical methods that approximate with spectral accuracy the exterior derivative $\mathbf{d}$, Hodge star $\star$, and their compositions. To achieve discretizations with high precision and symmetry, we develop hyperinterpolation methods based on spherical harmonics and Lebedev quadrature. We perform convergence studies of our numerical exterior derivative operator $\overline{\mathbf{d}}$ and Hodge star operator $\overline{\star}$ showing each converge spectrally to $\mathbf{d}$ and $\star$. We show how the numerical operators can be naturally composed to formulate general numerical approximations for solving differential equations on manifolds. We present results for the Laplace-Beltrami equations demonstrating our approach.

Pseudo‐spectral method using rotated staggered grid for elastic wave propagation in 3D arbitrary anisotropic media

ABSTRACTStaggering grid is a very effective way to reduce the Nyquist errors and to suppress the non‐causal ringing artefacts in the pseudo‐spectral solution of first‐order elastic wave equations. However, the straightforward use of a staggered‐grid pseudo‐spectral method is problematic for simulating wave propagation when the anisotropy level is greater than orthorhombic or when the anisotropic symmetries are not aligned with the computational grids. Inspired by the idea of rotated staggered‐grid finite‐difference method, we propose a modified pseudo‐spectral method for wave propagation in arbitrary anisotropic media. Compared with an existing remedy of staggered‐grid pseudo‐spectral method based on stiffness matrix decomposition and a possible alternative using the Lebedev grids, the rotated staggered‐grid‐based pseudo‐spectral method possesses the best balance between the mitigation of artefacts and efficiency. A 2D example on a transversely isotropic model with tilted symmetry axis verifies its effectiveness to suppress the ringing artefacts. Two 3D examples of increasing anisotropy levels demonstrate that the rotated staggered‐grid‐based pseudo‐spectral method can successfully simulate complex wavefields in such anisotropic formations.

Coupling Yee Grid to Lebedev Grid in Two Dimensions

The standard Yee grid, though simple and robust, is not well suited to modeling anisotropic media. In contrast, the Lebedev grid is well suited, but is computationally more expensive. A methodology is presented to use the classical Yee grid in isotropic regions and take advantage of the Lebedev method as a subgridded region for anisotropic materials only. Two methods to couple the two regions are proposed and described in detail. The proposed methods show low reflection in a scenario where a normally incident plane wave hits a Yee–Lebedev interface. Also, the two techniques are shown to be both accurate and stable.