Cartesian Derivatives Research Articles

Fast evaluation of spherical harmonics with sphericart.

Spherical harmonics provide a smooth, orthogonal, and symmetry-adapted basis to expand functions on a sphere, and they are used routinely in physical and theoretical chemistry as well as in different fields of science and technology, from geology and atmospheric sciences to signal processing and computer graphics. More recently, they have become a key component of rotationally equivariant models in geometric machine learning, including applications to atomic-scale modeling of molecules and materials. We present an elegant and efficient algorithm for the evaluation of the real-valued spherical harmonics. Our construction features many of the desirable properties of existing schemes and allows us to compute Cartesian derivatives in a numerically stable and computationally efficient manner. To facilitate usage, we implement this algorithm in sphericart, a fast C++ library that also provides C bindings, a Python API, and a PyTorch implementation that includes a GPU kernel.

Degenerate kernels of polyharmonic and poly-Helmholtz operators in polar and spherical coordinates

This paper presents the degenerate kernels for the polyharmonic and poly-Helmholtz partial differential operators in the polar and spherical coordinates. These degenerate kernels are essential if the analytical/semi-analytical solutions and the mathematical degeneracies involving these operators are conducted by the boundary integral equations. In addition, a two-dimensional creeping flow problem is considered to illustrate the use of the degenerate kernels for problems governed by coupled partial differential equations. The coupled governing equations of the creeping flow problems are converted into a biharmonic equation by using the Hörmander linear partial differential operator theory and the Cartesian partial derivatives on the harmonic and biharmonic degenerate kernels are used to obtain the degenerate kernels of two-dimensional creeping flow problems for BIE analyses. Finally, numerical experiments are conducted to study the convergence of the polyharmonic and poly-Helmholtz degenerate kernels.

Cartesian spatial derivatives of boundary element solutions on the exact free surface of fully nonlinear numerical wave tanks

The convergence and stability of the fully nonlinear simulation in a three-dimensional potential numerical wave tank is vulnerable along the time marching algorithm of the free surface, as integrating the boundary element's solution into the high-order time integration of the free surface can trigger the numerical instability in the material node approach. Although this matter was identified in some of the prior studies on three-dimensional wave tanks, the approximation of the velocity components plays a crucial role in the sustainability of a simulation. Therefore, a methodology for deriving the Cartesian components of the fluid particles’ velocity over the exact free surface is proposed in this paper. Various circumstances come across while determining the velocity components of fluid particles, which are the function of the location of the free surface nodal points. Since plenty of prior techniques have been based on the tangential derivatives of the free surface boundary value from the previous iteration, the disruption in the convergence of the model occurs most often. Hence, achieving a generic algorithm that merely depends on the boundary element solutions, is a highly important achievement for retaining the stability and simultaneously the convergence of the solution.

PDE-Based Multidimensional Extrapolation of Scalar Fields over Interfaces with Kinks and High Curvatures

We present a PDE-based approach for the multidimensional extrapolation of smooth scalar quantities across interfaces with kinks and regions of high curvature. Unlike the commonly used method of [T. Aslam, J. Comput. Phys., 193 (2004), pp. 349--355], in which normal derivatives are extrapolated, the proposed approach is based on the extrapolation and weighting of Cartesian derivatives. As a result, second- and third-order accurate extensions in the $L^\infty$ norm are obtained with linear and quadratic extrapolations, respectively, even in the presence of sharp geometric features. The accuracy of the method is demonstrated on a number of examples in two and three spatial dimensions and compared to the approach of [T. Aslam, J. Comput. Phys., 193 (2004), pp. 349--355]. The importance of accurate extrapolation near sharp geometric features is highlighted on an example of solving the diffusion equation on evolving domains.

Implementation and benchmarking of a crosstalk-free method for wavefront Zernike coefficients reconstruction using Shack-Hartmann sensor data

In wavefront characterization, often the combination of a Shack-Hartmann sensor and a reconstruction method utilizing the Cartesian derivatives of Zernike circle polynomials (the least-squares method, to be called here Method A) is used, which is known to introduce crosstalk. In \citep{janssen2014zernike} a crosstalk-free analytic expression of the LMS estimator of the wavefront Zernike coefficients is given in terms of wavefront partial derivatives (leading to what we call Method B). Here, we show an implementation of this analytic result where the derivative data are obtained using the Shack-Hartmann sensor and compare it with the conventional least-squares method.

Efficient and robust recurrence relations for the Zernike circle polynomials and their derivatives in Cartesian coordinates.

For some time it has been known and recommended that the calculation of Zernike polynomials to radial orders higher than 8 to 10 should be performed using recurrence relations rather than explicit expressions due increasingly large cancellation errors. This paper presents a set of simple recurrence relations that can be used for the unit-normalized Zernike polynomials in polar coordinates and easily adapted to Cartesian coordinates as well. The recurrence relations are also well suited for the calculation of the Cartesian derivatives of the Zernike polynomials. The recurrence relations are easily extended to arbitrarily high orders. Assessments of the precision achievable with standard 64-bit floating point arithmetic show that Zernike polynomials up to radial order 30 can be calculated over the unit disc with errors not exceeding 5E-14, and up to radial order 50 with errors not exceeding 1.2E-13. Comparison with the Zernike capability in OpticStudio (Zemax) shows that the recurrence relations are superior in performance (both speed and precision) over the existing algorithm implemented in the software. General pseudo-code for the calculation of Zernike polynomials and their derivatives is also presented.

Spheroidal models of the exterior gravitational field of Asteroids Bennu and Castalia

Gravitational field of small bodies can be modeled e.g. with mascons, a polyhedral model or in terms of harmonic functions. If the shape of a body is close to the spheroid, it is advantageous to employ the spheroidal basis functions for expressing the gravitational field. Spheroidal harmonic models, similarly to the spherical ones, may be used in navigation and geophysical tasks. We focus on modeling the exterior gravitational field of oblate-like Asteroid (101955) Bennu and prolate-like Asteroid (4769) Castalia with spheroidal harmonics. Using the Gauss–Legendre quadrature and the spheroidal basis functions, we converted the gravitational potential of a particular polyhedral model of a constant density into the spheroidal harmonics. The results consist of (i) spheroidal harmonic coefficients of the exterior gravitational field for the Asteroids Bennu and Castalia, (ii) spherical harmonic coefficients for Bennu, and (iii) the first and second-order Cartesian derivatives in the local spheroidal South-East-Up frame for both bodies. The spheroidal harmonics offer biaxial flexibility (compared with spherical harmonics) and low computational costs that allow high-degree expansions (compared with ellipsoidal harmonics). The obtained spheroidal models for Bennu and Castalia represent the exterior gravitational field valid on and outside the Brillouin spheroid but they can be used even under this surface. For Bennu, 5 m above the surface the agreement with point-wise integration was 1% or less, while it was about 10% for Castalia due to its more irregular shape. As the shape models may produce very high frequencies, it was crucial to use higher maximum degree to reduce the aliasing. We have used the maximum degree 360 to achieve 9–10 common digits (in RMS) when reconstructing the input (the gravitational potential) from the spheroidal coefficients. The physically meaningful maximum degree may be lower (≪360) but its particular value depends on the distance and/or on the application (navigation, exploration, etc.).

The anharmonic quartic force field infrared spectra of three polycyclic aromatic hydrocarbons: Naphthalene, anthracene, and tetracene.

Current efforts to characterize and study interstellar polycyclic aromatic hydrocarbons (PAHs) rely heavily on theoretically predicted infrared (IR) spectra. Generally, such studies use the scaled harmonic frequencies for band positions and double harmonic approximation for intensities of species, and then compare these calculated spectra with experimental spectra obtained under matrix isolation conditions. High-resolution gas-phase experimental spectroscopic studies have recently revealed that the double harmonic approximation is not sufficient for reliable spectra prediction. In this paper, we present the anharmonic theoretical spectra of three PAHs: naphthalene, anthracene, and tetracene, computed with a locally modified version of the SPECTRO program using Cartesian derivatives transformed from Gaussian 09 normal coordinate force constants. Proper treatments of Fermi resonances lead to an impressive improvement on the agreement between the observed and theoretical spectra, especially in the C-H stretching region. All major IR absorption features in the full-scale matrix-isolated spectra, the high-temperature gas-phase spectra, and the most recent high-resolution gas-phase spectra obtained under supersonically cooled molecular beam conditions in the CH-stretching region are assigned.

A low frequency elastodynamic fast multipole boundary element method in three dimensions

This paper presents a fast multipole boundary element method (FMBEM) for the 3-D elastodynamic boundary integral equation in the `low frequency' regime. New compact recursion relations for the second-order Cartesian partial derivatives of the spherical basis functions are derived for the expansion of the elastodynamic fundamental solutions. Numerical solution is achieved via a novel combination of a nested outer---inner generalized minimum residual (GMRES) solver and a sparse approximate inverse preconditioner. Additionally translation stencils are newly applied to the elastodynamic FMBEM and an implementation of the 8, 4 and 2-box stencils is presented, which is shown to reduce the number of translations per octree level by up to $$60\,\%$$60%. This combination of strategies converges 2---2.5 times faster than the standard GMRES solution of the FMBEM. Numerical examples demonstrate the algorithmic and memory complexities of the model, which are shown to be in good agreement with the theoretical predictions.

Numerical differentiation method to calculate molecular properties at ground and excited states – Application to Julolidinemalononitrile

A general method to evaluate the Cartesian derivatives of the dipole moment and polarizability of electronic excited states is presented. This method, based on successive numerical differentiations of the dipole moment, is implemented in a locally developed package of programs called VibKit. An application to describe the vibrational signatures of the first excited state of julolidinemalononitrile in comparison to those of its ground state has been performed using time-dependent density functional theory (TDDFT). The modifications of the IR and Raman signatures are shown to be related to the delocalization of the charge density and geometry modifications upon excitation, as evidenced by the use of group coupling matrices (GCMs).

Enhanced Sampling of DNA Step Parameters: Impact of Methylation on DNA Shape and Flexibility

We present a novel approach for the selection of DNA step parameters as reaction coordinates in umbrella sampling simulations. Simplified representation of DNA that uses only three atoms per base, allowed for highly efficient calculations of the step parameters and their Cartesian derivatives in the molecular dynamics simulations. Good correlation between the actual and calculated twist, roll, tilt, rise, slide and shift was obtained. The method is illustrated through its application to the unmethylated and methylated DNA systems. Impact of the methylation on the shape and flexibility of DNA depending on the location of the methyl group is discussed.

Smoothed finite element method implemented in a resultant eight-node solid-shell element for geometrical linear analysis

A smoothed finite element method formulation for the resultant eight-node solid-shell element is presented in this paper for geometrical linear analysis. The smoothing process is successfully performed on the element mid-surface to deal with the membrane and bending effects of the stiffness matrix. The strain smoothing process allows replacing the Cartesian derivatives of shape functions by the product of shape functions with normal vectors to the element mid-surface boundaries. The present formulation remains competitive when compared to the classical finite element formulations since no inverse of the Jacobian matrix is calculated. The three dimensional resultant shell theory allows the element kinematics to be defined only with the displacement degrees of freedom. The assumed natural strain method is used not only to eliminate the transverse shear locking problem encountered in thin-walled structures, but also to reduce trapezoidal effects. The efficiency of the present element is presented and compared with that of standard solid-shell elements through various benchmark problems including some with highly distorted meshes.

Enhanced sampling simulations of DNA step parameters.

A novel approach for the selection of step parameters as reaction coordinates in enhanced sampling simulations of DNA is presented. The method uses three atoms per base and does not require coordinate overlays or idealized base pairs. This allowed for a highly efficient implementation of the calculation of all step parameters and their Cartesian derivatives in molecular dynamics simulations. Good correlation between the calculated and actual twist, roll, tilt, shift, and slide parameters is obtained, while the correlation with rise is modest. The method is illustrated by its application to the methylated and unmethylated 5'-CATGTGACGTCACATG-3' double stranded DNA sequence. One-dimensional umbrella simulations indicate that the flexibility of the central CG step is only marginally affected by methylation.

Recurrence relations for the Cartesian derivatives of the Zernike polynomials

A recurrence relation for the first-order Cartesian derivatives of the Zernike polynomials is derived. This relation is used with the Clenshaw method to determine an efficient method for calculating the derivatives of any linear series of Zernike polynomials.

Derivatives of elegant Laguerre–Gaussian beams: vortex structure and orbital angular momentum

The commutation between the paraxial wave equation and the derivative operator allows us to generate novel beam solutions. In this work, we analyze the solutions generated by the derivatives with respect to Cartesian coordinates of elegant Laguerre–Gaussian beams. We present compact expressions for the derivatives of arbitrary integer order and study the resulting orbital angular momentum (OAM) and phase structure. We found that the derivative operator preserves OAM but the topological structure is modified. The resulting topological charge depends on the initial seed beam and the order of the derivative. We also introduce a two-parameter differential operator resulting from the linear combination of Cartesian derivatives ∂x and ∂y. In analogy with the Poincaré sphere for polarized beams, this operator can be mapped on the surface of a unit sphere. The results can find applications in the generation and control of optical vortex structures.

Fast and robust computation of Cartesian derivatives of Zernike polynomials

This work deals with a problem of an effective and robust evaluation of Cartesian partial derivatives of Zernike polynomials. Calculations based on explicit relations suffer from high computational costs and are numerically instable for higher orders. Thus, fast and numerically stable methods for calculation of derivatives of Zernike polynomials are needed. Such methods employing recursive techniques based on the relation of Zernike radial polynomials to Jacobi polynomials are proposed. The performance of proposed algorithms is compared to the standard calculation using explicit relations.

Three-Dimensional Wave Gradiometry for Polarized Seismic Waves

Abstract We expand the theoretical development of seismic‐wave gradiometry to 3D using the polarized wavefield. First, we develop a map that relates the Cartesian spatial derivatives to the spatial derivatives in spherical coordinates. We then develop a set of partial differential equations that relate the spherical derivatives of the observed wavefield to the body‐wave radiation patterns. Because the spherical derivatives of the wavefield depend on the direction of wave propagation, we also develop relationships to find the propagation direction using simple ratios of the Cartesian‐based wave derivatives. Finally, we test the robustness of estimating the wave‐propagation direction, and its sensitivity to noise.

Arbitrary order El'yashevich–Wilson B tensor formulas for the most frequently used internal coordinates in molecular vibrational analyses

In recent years, internal coordinates have become the preferred means of expressing potential energy surfaces. The ability to transform quantities from chemically significant internal coordinates to primitive Cartesian coordinates and spectroscopically relevant normal coordinates is thus critical to the further development of computational chemistry. In the present work, general nth order formulas are presented for the Cartesian derivatives of the five most commonly used internal coordinates--bond stretching, bond angle, torsion, out-of-plane angle, and linear bending. To compose such formulas in a reasonably understandable fashion, a new notation is developed that is a generalization of that which has been used previously for similar purposes. The notation developed leads to easily programmable and reasonably understandable arbitrary order formulas, yet it is powerful enough to express the arbitrary order B tensor of a general, N-point internal coordinate, as is done herein. The techniques employed in the derivation of such formulas are relatively straightforward, and could presumably be applied to a number of other internal coordinates as needed.

Atomic charges from IR intensity parameters: theory, implementation and application

A computational method to extract atomic charges (IR charges) from DFT/ab initio-computed atomic polar tensors is presented and compared with commonly available population schemes. The procedure adopted and its implementation in a (freely available) code are also reported and commented. Thanks to the procedure developed, infrared charges can be now readily gathered also by non-experts, provided that Cartesian dipole derivatives from a quantum chemical calculation of the IR spectrum are available. The method has been applied for the calculation of IR charges of about 50 molecules: It performs well in describing peculiar intramolecular interactions, providing a picture of the molecular charge distribution coherent with the chemical intuition. A nice agreement is also found with charges obtained from Hirshfeld algorithm and from the fitting of the electrostatic potential (CHELPG and MK schemes). On this basis, we propose IR charges as a reliable, physically sound and easily accessible alternative to other charge parameters currently adopted.

Solutions to Maxwell's equations using spheroidal coordinates

Analytical solutions to the wave equation in spheroidal coordinates in the short wavelength limit are considered. The asymptotic solutions for the radial function are significantly simplified, allowing scalar spheroidal wave functions to be defined in a form which is directly reminiscent of the Laguerre–Gaussian solutions to the paraxial wave equation in optics. Expressions for the Cartesian derivatives of the scalar spheroidal wave functions are derived, leading to a new set of vector solutions to Maxwell's equations. The results are an ideal starting point for calculations of corrections to the paraxial approximation.