Lebedev Grid Research Articles

A Fifty-Node Lebedev Grid And Its Applications To Ambisonics

This paper investigates the use of 50-node Lebedev grid for sound field capture and reproduction using Ambisonic approach. This grid is derived from rotationally-invariant quadrature rules, for which the underlying theory is recalled. Special attention is paid to spatial aliasing artifacts at capture and reproduction steps. While comparing a 50-node Lebedev grid with a Fliege and a t-design grid that both use almost the same number of nodes, it is shown that the Lebedev grid provides the best performance in terms of sound field capture and reproduction. Finally, a multi-band multi-order decoder is presented. These decoders take advantages of the inherent nested sub-grids when following the rotationally-invariant quadrature approach.

Tilted transverse isotropic reverse time migration with angle gathers: Implementation and efficiency

We have developed a 3D reverse time migration (RTM) implementation, with an extended imaging condition, in the presence of tilted transverse isotropy (TTI), using finite-difference time-domain solvers on the rotated staggered grid (RSG) and the Lebedev grid. We have evaluated dispersion analysis of both schemes for a sample TTI medium. Using synthetic and real seismic data in realistic execution configurations, we found surprising inconsistency with quantitative cost estimates in the literature. For fixed accuracy, the RSG scheme proved most efficient in our tests, and not two to three times less efficient, as had been posited. Of course, variability may arise with particular medium properties, computer architectures, or software implementations. Having analyzed wavefield modeling performance in detail, we then quantified the realistic costs associated with the imaging condition in RTM, especially extended imaging conditions. For large extensions of the imaging condition, the associated cost dominates the total execution time and adds very significantly to the total cost of the migration. Our implementation targets both shared and distributed memory parallelism, making it suitable for general CPU clusters, nonuniform memory access architectures, and Intel Xeon-Phi. We have also discussed other aspects of our implementation in detail. In particular, we have explored an alternative method to achieve concurrent communication and computation in RTM.

Fully anisotropic 3-D EM modelling on a Lebedev grid with a multigrid pre-conditioner

Fully anisotropic 3-D EM modelling on a Lebedev grid with a multigrid pre-conditioner

Collocated electrodynamic FDTD schemes using overlapping Yee grids and higher-order Hodge duals

The collocated Lebedev grid has previously been proposed as an alternative to the Yee grid for electromagnetic finite-difference time-domain (FDTD) simulations. While it performs better in anisotropic media, it performs poorly in isotropic media because it is equivalent to four overlapping, uncoupled Yee grids. We propose to couple the four Yee grids and fix the Lebedev method using discrete exterior calculus (DEC) with higher-order Hodge duals. We find that higher-order Hodge duals do improve the performance of the Lebedev grid, but they also improve the Yee grid by a similar amount. The effectiveness of coupling overlapping Yee grids with a higher-order Hodge dual is thus questionable. However, the theoretical foundations developed to derive these methods may be of interest in other problems.

An optimal scheme for numerical evaluation of Eshelby tensors and its implementation in a MATLAB package for simulating the motion of viscous ellipsoids in slow flows

To address the multiscale deformation and fabric development in Earth's ductile lithosphere, micromechanics-based self-consistent homogenization is commonly used to obtain macroscale rheological properties from properties of constituent elements. The homogenization is heavily based on the solution of an Eshelby viscous inclusion in a linear viscous medium and the extension of the solution to nonlinear viscous materials. The homogenization requires repeated numerical evaluation of Eshelby tensors for constituent elements and becomes ever more computationally challenging as the elements are deformed to more elongate or flattened shapes. In this paper, we develop an optimal scheme for evaluating Eshelby tensors, using a combination of a product Gaussian quadrature and the Lebedev quadrature. We first establish, through numerical experiments, an empirical relationship between the inclusion shape and the computational time it takes to evaluate its Eshelby tensors. We then use the relationship to develop an optimal scheme for selecting the most efficient quadrature to obtain the Eshelby tensors. The optimal scheme is applicable to general homogenizations. In this paper, it is implemented in a MATLAB package for investigating the evolution of solitary rigid or deformable inclusions and the development of shape preferred orientations in multi-inclusion systems during deformation. The MATLAB package, upgrading an earlier effort written in MathCad, can be downloaded online.

Three-dimensional finite volume simulation of the response of azimuth electromagnetic wave resistivity while drilling in inhomogeneous anisotropic formation

The azimuth electromagnetic wave resistivity while drilling is a new type of well logging technique. It can real-time detect the formation boundary, realize geosteering and borehole imaging in order to keep the tool always drilling in the some meaning reservoir. For effectively optimizing tool parameters, proper explanation and evaluation of the data obtained by azimuth electromagnetic wave resistivity while drilling, the efficient numerical simulation algorithm is required. In this paper, we use the finite volume algorithm in the cylindrical coordinate to establish the corresponding numerical method so that we can effectively simulate the response of the tool in various complex environments and investigate the influences of the change in formation and tool parameters on the tool response. Therefore, according to the typical coil architecture of the instrument of azimuth electromagnetic wave resistivity while drilling, we first introduce the electrical and magnetic dyadic Green's functions in inhomogeneous anisotropic formation by the electrical current source in the cylindrical coordinate. Through superposition principle, we derive the integral formula to compute the electric field intensity excited by tilted transmitter coils and the induction electrical potential on tilted receiving coils both mounded on the drill collar. Then, we use the coupled electrical potentials of the dyadic Green's functions to overcome the low induction number problem during modeling the electrical fields in inhomogeneous anisotropic formation. Furthermore, we use Lebedev grid in both and z directions to reduce the number of grid nodes, and the standard method to compute the equivalent conductivity in heterogeneous units for enhancing the discrete precision. On the basis, by the three-dimensional finite volume method, we discrete the equations about the coupled electrical potentials in the cylindrical coordinates and obtain the large sparse algebraic equation sets about the coupled electrical potentials field on the Lebedev grid. A combination of incomplete LU decomposition with the bi-conjugate gradient stabilization is used to solve the numerical solution. Finally, we validate the algorithm by comparing the numerical results obtained by two different methods, study the effects of the drill collar, anisotropy, the tilted angles of both coil, and borehole on the instrument response in inhomogeneous anisotropic formation. The numerical results show that the tool response obtained by the three-dimensional finite volume algorithm in the cylindrical coordinate system in anisotropic formation accord with that those obtained by other algorithms. The drill collar, inhomogeneous anisotropic n the formation will lead to both the smaller amplitude ratio and the smaller phase difference. In addition, when the coils of both transmitting and receiving coils are tilted, the amplitude ratio and phase difference of the tool are more sensitive to the change in formation parameter.

Electrostatic point charge fitting as an inverse problem: Revealing the underlying ill-conditioning.

Atom-centered point charge (PC) model of the molecular electrostatics-a major workhorse of the atomistic biomolecular simulations-is usually parameterized by least-squares (LS) fitting of the point charge values to a reference electrostatic potential, a procedure that suffers from numerical instabilities due to the ill-conditioned nature of the LS problem. To reveal the origins of this ill-conditioning, we start with a general treatment of the point charge fitting problem as an inverse problem and construct an analytical model with the point charges spherically arranged according to Lebedev quadrature which is naturally suited for the inverse electrostatic problem. This analytical model is contrasted to the atom-centered point-charge model that can be viewed as an irregular quadrature poorly suited for the problem. This analysis shows that the numerical problems of the point charge fitting are due to the decay of the curvatures corresponding to the eigenvectors of LS sum Hessian matrix. In part, this ill-conditioning is intrinsic to the problem and is related to decreasing electrostatic contribution of the higher multipole moments, that are, in the case of Lebedev grid model, directly associated with the Hessian eigenvectors. For the atom-centered model, this association breaks down beyond the first few eigenvectors related to the high-curvature monopole and dipole terms; this leads to even wider spread-out of the Hessian curvature values. Using these insights, it is possible to alleviate the ill-conditioning of the LS point-charge fitting without introducing external restraints and/or constraints. Also, as the analytical Lebedev grid PC model proposed here can reproduce multipole moments up to a given rank, it may provide a promising alternative to including explicit multipole terms in a force field.

Accurate small and wide angle x-ray scattering profiles from atomic models of proteins and nucleic acids.

A new method is introduced to compute X-ray solution scattering profiles from atomic models of macromolecules. The three-dimensional version of the Reference Interaction Site Model (RISM) from liquid-state statistical mechanics is employed to compute the solvent distribution around the solute, including both water and ions. X-ray scattering profiles are computed from this distribution together with the solute geometry. We describe an efficient procedure for performing this calculation employing a Lebedev grid for the angular averaging. The intensity profiles (which involve no adjustable parameters) match experiment and molecular dynamics simulations up to wide angle for two proteins (lysozyme and myoglobin) in water, as well as the small-angle profiles for a dozen biomolecules taken from the BioIsis.net database. The RISM model is especially well-suited for studies of nucleic acids in salt solution. Use of fiber-diffraction models for the structure of duplex DNA in solution yields close agreement with the observed scattering profiles in both the small and wide angle scattering (SAXS and WAXS) regimes. In addition, computed profiles of anomalous SAXS signals (for Rb(+) and Sr(2+)) emphasize the ionic contribution to scattering and are in reasonable agreement with experiment. In cases where an absolute calibration of the experimental data at q = 0 is available, one can extract a count of the excess number of waters and ions; computed values depend on the closure that is assumed in the solution of the Ornstein-Zernike equations, with results from the Kovalenko-Hirata closure being closest to experiment for the cases studied here.

Periodic boundary conditions for QM/MM calculations: Ewald summation for extended Gaussian basis sets

An implementation of Ewald summation for use in mixed quantum mechanics/molecular mechanics (QM/MM) calculations is presented, which builds upon previous work by others that was limited to semi-empirical electronic structure for the QM region. Unlike previous work, our implementation describes the wave function's periodic images using "ChElPG" atomic charges, which are determined by fitting to the QM electrostatic potential evaluated on a real-space grid. This implementation is stable even for large Gaussian basis sets with diffuse exponents, and is thus appropriate when the QM region is described by a correlated wave function. Derivatives of the ChElPG charges with respect to the QM density matrix are a potentially serious bottleneck in this approach, so we introduce a ChElPG algorithm based on atom-centered Lebedev grids. The ChElPG charges thus obtained exhibit good rotational invariance even for sparse grids, enabling significant cost savings. Detailed analysis of the optimal choice of user-selected Ewald parameters, as well as timing breakdowns, is presented.

Numerical implementation of constitutive model for arterial layers with distributed collagen fibre orientations

Several constitutive models have been proposed for the description of mechanical behaviour of soft tissues containing collagen fibres. Some of the commonly used approaches accounting for the dispersion of fibre orientations are based on the summation of (mechanical) contributions of differently oriented fibre families. This leads to the need of numerical integration on the sphere surface, and the related numerical consumption is the main disadvantage of this category of constitutive models. The paper is focused on the comparison of various numerical integration methods applied to a specific constitutive model applicable for arterial walls. Robustness and efficiency of several integration rules were tested with respect to application in finite element (FE) codes. Among all the analysed numerical integration rules, the best results were reached by Lebedev quadrature; the related parameters for the specific constitutive model are presented in the paper. The results were implemented into the commercial FE code ANSYS via user subroutines, and their applicability was demonstrated by an example of FE simulation with non-homogenous stress field.

FDTD Method on a Lebedev Grid for Anisotropic Materials

The finite-difference time-domain method is derived on a Lebedev grid for lossy anisotropic media. The Lebedev grid uses collocated field components and supports spurious solutions but an intuitive method for removing the extra solutions is presented. Update equations at material discontinuities and metal planes are derived and shown to take the same form as updates in bulk media. Additionally, a dispersion relation and stability criteria are presented. A numerical comparison with the Yee grid shows that the Lebedev grid suffers from greater numerical dispersion but better represents material discontinuities when compared using equal memory requirements. Furthermore, the small stencil on the Lebedev grid decreases the required number of calls to memory and simplifies the programming structure.

A fast ellipsoid model for asteroids inverted from lightcurves

Research about asteroids has recently attracted more and more attention, especially focusing on their physical structures, such as their spin axis, rotation period and shape. The long distance between observers on Earth and asteroids makes it impossible to directly calculate the shape and other parameters of asteroids, with the exception of Near Earth Asteroids and others that have passed by some spacecrafts. Photometric measurements are still generally the main way to obtain research data on asteroids, i.e. the lightcurves recording the brightness and positions of asteroids. Supposing that the shape of the asteroid is a triaxial ellipsoid with a stable spin, a new method is presented in this article to reconstruct the shape models of asteroids from the lightcurves, together with other physical parameters. By applying a special curvature function, the method calculates the brightness integration on a unit sphere and Lebedev quadrature is employed for the discretization. Finally, the method searches for the optimal solution by the Levenberg-Marquardt algorithm to minimize the residual of the brightness. By adopting this method, not only can related physical parameters of asteroids be obtained at a reasonable accuracy, but also a simple shape model of an ellipsoid can be generated for reconstructing a more sophisticated shape model.

Isotropic and nematic phases of the hard spheroid fluid: A virial approach

We use a high level virial expansion to investigate the properties of the isotropic and nematic phases of the hard spheroid fluid. We use the Monte Carlo techniques described previously to calculate the virial coefficients up to seventh order and we represent the dependence of these coefficients on particle orientations via a spherical harmonic expansion. The expansion coefficients are determined using Lebedev quadrature which carries out the angular integration required exactly. For fairly spherical spheroids (1/3<a/b<3, where a and b are the semiaxes of the spheroid) the virial expansion appears to be convergent in both the isotropic and nematic phases up to high density. For more aspherical particles, the predictions varied in a more erratic fashion upon increasing the number of virials. The use of a finite angular basis set to represent the orientationally dependent virial coefficients could be a contributory cause to this.

소리의 공간 제어를 위한 구형 다채널 스피커 어레이 설계

The objective of this paper is to design multichannel spherical loudspeaker array by considering various positioning methods such as Gaussian grid, Lebedev grid and packing method. For the spatial sound manipulation, which is to make desired sound field by controling multiple sound sources, the Kirchhoff- Helmholtz integral states that sound fields can be reproduced in terms of infinite control sources on the integral surface. But since we cannot control infinite number of sources for the implementation, we have to allocate finite number of sound sources which can approximately act as infinite number of sources. To manipulate sound field inside of a sphere (which is typical example of three dimensional array) by controlling sound sources on the surface, three methods of allocating sound sources, which are Gaussian grid, Lebedev grid and packing method, are reviewed. For each geometry, the performances of manipulation rendered by time-reversal operator and higher-order ambisonics are compared.

Optimal computation of brightness integrals parametrized on the unit sphere

We compare various approaches to find the most efficient method for the practical computation of the lightcurves (integrated brightnesses) of irregularly shaped bodies such as asteroids at arbitrary viewing and illumination geometries. For convex models, this reduces to the problem of the numerical computation of an integral over a simply defined part of the unit sphere. We introduce a fast method, based on Lebedev quadratures, which is optimal for both lightcurve simulation and inversion in the sense that it is the simplest and fastest widely applicable procedure for accuracy levels corresponding to typical data noise. The method requires no tessellation of the surface into a polyhedral approximation. At the accuracy level of 0.01 mag, it is up to an order of magnitude faster than polyhedral sums that are usually applied to this problem, and even faster at higher accuracies. This approach can also be used in other similar cases that can be modelled on the unit sphere. The method is easily implemented in lightcurve inversion by a simple alteration of the standard algorithm/software.

A comparison of the dispersion relations for anisotropic elastodynamic finite-difference grids

Several staggered grid schemes have been suggested for performing finite-difference calculations for the elastic wave equations. In this paper, the dispersion relationships and related computational requirements for the Lebedev and rotated staggered grids for anisotropic, elastic, finite-difference calculations in smooth models are analyzed and compared. These grids are related to a popular staggered grid for the isotropic problem, the Virieux grid. The Lebedev grid decomposes into Virieux grids, two in two dimensions and four in three dimensions, which decouple in isotropic media. Therefore the Lebedev scheme will have twice or four times the computational requirements, memory, and CPU as the Virieux grid but can be used with general anisotropy. In two dimensions, the rotated staggered grid is exactly equivalent to the Lebedev grid, but in three dimensions it is fundamentally different. The numerical dispersion in finite-difference grids depends on the direction of propagation and the grid type and parameters. A joint numerical dispersion relation for the two grids types in the isotropic case is derived. In order to compare the computational requirements for the two grid types, the dispersion, averaged over propagation direction and medium velocity are calculated. Setting the parameters so the average dispersion is equal for the two grids, the computational requirements of the two grid types are compared. In three dimensions, the rotated staggered grid requires at least 20% more memory for the field data and at least twice as many number of floating point operations and memory accesses, so the Lebedev grid is more efficient and is to be preferred.

Towards efficient ab initio calculations of electron scattering by polyatomic molecules: I. Efficient numerical quadrature of the UGT term

The UGU term was used as a model of the UGT term, and its evaluation by numerical quadrature was examined systematically with a training set of eight molecules. Minimum numbers of points have been determined for radial Gauss–Legendre and angular Lebedev quadratures that preserve the accuracy needed for practical applications. These quadratures are recommended for efficient calculation of electron scattering by polyatomic molecules.

Lebedev discrete variable representation

We describe the construction of a Lebedev discrete variable representation (DVR) (Dickinson and Certain 1968 J. Chem. Phys. 49 4209, Lill, Parker and Light 1982 Chem. Phys. Lett. 89 483, Light, Hamilton and Lill 1985 J. Chem. Phys. 82 1400). This DVR is based upon a new algorithm for constructing a generalized DVR in one or more dimensions. The novel part of this algorithm is a contraction of the underlying functional basis that is based upon the given points and weights. The contraction allows us to define a set of basis functions and a set of points, of equal number, for which the transformation between the ‘point basis’ and the ‘function basis’ is neither singular nor close to it. We present both a strictly orthogonal and a discretely orthogonal version of the Lebedev DVR, which are based on the octahedrally symmetric Lebedev quadratures for the sphere. The discretely orthogonal DVR is a classic generalized DVR (Light, Hamilton and Lill 1985 J. Chem. Phys. 82 1400) and the strictly orthogonal version is similar to a diagonalization DVR. These DVRs perform well in test calculations. The algorithm is general and should be applicable to nonseparable bases in many dimensions.

Efficient orientational averaging by the extension of Lebedev grids via regularized octahedral symmetry expansion

Gaussian spherical quadrature methods in the guise of the Lebedev sampling grids are highly efficient for some orientational (“powder”) averaging problems in solid state NMR. However, their applicability is currently restricted, as the sets of orientations are derived analytically and because they are not well adapted to simulate the broad peakshapes encountered, for example, in the NMR on static powders or on half-integer quadrupolar spins subject to second order quadrupolar interactions under magic-angle spinning conditions. We remedy these problems by (i) introducing the recursive procedure regularized octahedral symmetry expansion (ROSE), to which any existing Lebedev set may be subjected. Each recursive step gives a 9-fold enlarged set of orientations. (ii) We demonstrate that ROSE-expanded grids, in conjunction with spectral interpolation, is well suited for calculating broad peakshapes. These advances combine into the apparently most efficient general-purpose two-angle orientational averaging technique proposed to date for solid state NMR applications.

SG‐0: A small standard grid for DFT quadrature on large systems

We report the development of a new standard quadrature grid for DFT calculations. Standard Grid 0 (SG-0) is designed to be approximately half as large as, and to provide approximately half the accuracy of, the established SG-1 grid. It is based on MultiExp and Lebedev quadrature for radial and angular coordinates, respectively. We find that SG-0 is typically 50% faster than SG-1 for energy, gradient, and hessian calculations for the exchange-correlation energy. This leads to a 35-38% speedup in the total gradient and hessian computations, and we particularly recommend its use for preliminary calculations on moderately large biochemical systems. It has been implemented as the default grid for DFT calculations in the Q-Chem 3.0 package.