Harmonic Basis Functions Research Articles

MolSym: A Python package for handling symmetry in molecular quantum chemistry.

A consideration of the point group symmetry of molecules is often advantageous from a computational efficiency standpoint and sometimes necessary for the correct treatment of chemical physics problems. Many modern electronic structure software packages include a treatment of symmetry, but these are sometimes incomplete or unusable outside of that program's environment. Therefore, we have developed the MolSym package for handling molecular symmetry and its associated functionalities to provide a platform for including symmetry in the implementation and development of other methods. Features include point group detection, molecule symmetrization, arbitrary generation of symmetry element sets and character tables, and symmetry adapted linear combinations of real spherical harmonic basis functions, Cartesian displacement coordinates, and internal coordinates. We present some of the advantages of using molecular symmetry as achieved by MolSym, particularly with respect to Hartree-Fock theory, and the reduction of finite difference displacements in gradient/Hessian computations. This package is designed to be easily integrated into other software development efforts and may be extended to further symmetry applications.

Fourier-Hankel-Abel Nyquist-limited tomography: A spherical harmonic basis function approach to tomographic velocity-map image reconstruction.

A new method for the fully generalized reconstruction of three-dimensional (3D) photoproduct distributions from velocity-map imaging (VMI) projection data is presented. This approach, dubbed Fourier-Hankel-Abel Nyquist-limited TOMography (FHANTOM), builds on recent previous work in tomographic image reconstruction [C. Sparling and D. Townsend, J. Chem. Phys. 157, 114201 (2022)] and takes advantage of the fact that the distributions produced in typical VMI experiments can be simply described as a sum over a small number of spherical harmonic functions. Knowing the solution is constrained in this way dramatically simplifies the reconstruction process and leads to a considerable reduction in the number of projections required for robust tomographic analysis. Our new method significantly extends basis set expansion approaches previously developed for the reconstruction of photoproduct distributions possessing an axis of cylindrical symmetry. FHANTOM, however, can be applied generally to any distribution-cylindrically symmetric or otherwise-that can be suitably described by an expansion in spherical harmonics. Using both simulated and real experimental data, this new approach is tested and benchmarked against other tomographic reconstruction strategies. In particular, the reconstruction of photoelectron angular distributions recorded in a strong-field ionization regime-marked by their extensive expansion in terms of spherical harmonics-serves as a key test of the FHANTOM methodology. With the increasing use of exotic optical polarization geometries in photoionization experiments, it is anticipated that FHANTOM and related reconstruction techniques will provide an easily accessible and relatively low-cost alternative to more advanced 3D-VMI spectrometers.

Robust methods for multiscale coarse approximations of diffusion models in perforated domains

For the Poisson equation posed in a domain containing a large number of polygonal perforations, we propose a low-dimensional coarse approximation space based on a coarse polygonal partitioning of the domain. Similarly to other multiscale numerical methods, this coarse space is spanned by locally discrete harmonic basis functions. Along the subdomain boundaries, the basis functions are piecewise polynomial. The main contribution of this article is an error estimate regarding the H1-projection over the coarse space; this error estimate depends only on the regularity of the solution over the edges of the coarse partitioning. For a specific edge refinement procedure, the error analysis establishes superconvergence of the method even if the true solution has a low general regularity. Additionally, this contribution numerically explores the combination of the coarse space with domain decomposition (DD) methods. This combination leads to an efficient two-level iterative linear solver which reaches the fine-scale finite element error in few iterations. It also bodes well as a preconditioner for Krylov methods and provides scalability with respect to the number of subdomains.

Nonlinear optimization for compact representation of orientation distributions based on generalized spherical harmonics

An orientation distribution is a necessary input in any crystal plasticity simulation. The computational time involved in crystal plasticity simulations scales linearly with the number of crystal orientations in the input distributions. Reducing the number of crystal orientations in representing the input orientation distributions quantitatively is a critical and necessary requirement for performing computationally efficient crystal plasticity simulations of deformation processes. A procedure for the compaction of orientation distribution functions (ODFs) relying on a spectral representation using series of generalized spherical harmonics (GSH) basis functions was recently developed. Linear fitting of the spectral representation of an ODF containing a compact set of weighted orientations towards a full-size ODF containing many crystal orientations was in the core of the procedure. This paper advances the compaction procedure by replacing the linear with a nonlinear optimization in Matlab for which a suitably defined error, gradient, and Hessian matrix are derived to allow for more efficient, accurate, and greater compactions. The utility of the new procedure is to allow for not only fitting the weights of a compacted set of orientations but to allow for optimizing weighted orientations or a combination of optimizing weights and orientations. The new compaction procedure has been successfully applied to compactions of large ODFs of cubic, hexagonal, and orthorhombic polycrystalline metals. In doing so, the evolution of texture, twinning, and stress-strain are predicted at large plastic strains with compact ODFs to agree with the corresponding full size ODFs using crystal plasticity models. In closing, guidance for effective texture compaction trading off the accuracy and computational gains are provided.

On the conjugate symmetry and sparsity of the harmonic decomposition of parametric surfaces with the randomised Kaczmarz method

The downside of increasing the resolution of surface scanning devices is that the amount of acquired data makes the morphological analysis of the scanned surfaces computationally challenging. This limitation is circumvented by using scalable and non–memory–intensive harmonic expansions. In this paper, the projection of parametric surfaces onto disk and spherical harmonics bases is investigated, and three novel computationally efficient algorithms are proposed based on the randomised Kaczmarz (RK). To boost the computational performance and convergence of the root mean square error (RMSE) of the reconstructed surfaces we exploited the conjugate symmetry property of the harmonic basis functions. Further, the sparsity of the signals is used for estimating the projection coefficients from an undersampled surface. The first algorithm only takes into consideration the conjugate symmetry property for enhancing the convergence of the RMSE. The second algorithm endows the sparse version of the RK algorithm with the conjugate symmetry property. The third algorithm combines the previous two to further accelerate the convergence of the RMSE. The performance of the developed algorithms is tested on three surfaces where we demonstrate that they outperform conventional reconstruction techniques in terms of processing time with comparable precision.

Dirichlet‐type absorbing boundary conditions for peridynamic scalar waves in two‐dimensional viscous media

AbstractConstruction of absorbing boundary conditions (ABCs) for nonlocal models is generally challenging, primarily due to the fact that nonlocal operators are commonly associated with volume constrained boundary conditions. Moreover, application of Fourier and Laplace transforms, which are essential for the majority of available methods for ABCs, to nonlocal models is complicated. In this paper, we propose a simple method to construct accurate ABCs for peridynamic scalar wave‐type problems in viscous media. The proposed ABCs are constructed in the time and space domains and are of Dirichlet type. Consequently, their implementation is relatively simple, since no derivatives of the wave field are required. The proposed ABCs are derived at the continuum level, from a semi‐analytical solution of the exterior domain using harmonic exponential basis functions in space and time (plane‐wave modes). The numerical implementation is done using a meshfree collocation approach employed within a boundary layer adjacent to the interior domain boundary. The modes satisfy the peridynamic numerical dispersion relation, resulting in a compatible solution of the interior region (near‐field) with that of the exterior region (far‐field). The accuracy and stability of the proposed ABCs are demonstrated with several numerical examples in two‐dimensional unbounded domains.

Wave packet enriched finite elements for thermo-electro-elastic wave propagation problems with discontinuous wavefronts

In this article, we study the numerical properties of the recently developed local (element-domain) harmonic basis function enriched finite element (FE) for wave propagation problems. The study includes the effect of the enrichment harmonics on the conditioning of system matrices and how the diagonal scaling mass lumping technique influences the possible ill-conditioning and the solution accuracy. Subsequently, a wave packet enriched thermoelectromechanical FE formulation is presented for axisymmetric and planar wave propagation problems in piezoelastic media. The conventional Lagrange interpolations for the displacement, temperature, and electric potential fields are enriched with the element-domain sinusoidal functions which satisfy the partition of unity condition. The extended Hamilton’s principle and constitutive relations of the Lord–Shulman and Green–Lindsay generalized piezothermoelasticity are employed to derive the coupled system of equations of motion which is solved using Newmark- direct time integration scheme. The performance of the proposed elements is assessed and wave characteristics are studied for the problems of thermoelastic shock waves in an elastic annular disk and thermoelectric shock waves in a piezoelectric hollow cylinder. The element shows significant improvement in the computational efficiency and accuracy over the conventional FE for these problems involving sharp discontinuities in the fields at the wavefronts.

Spherical-sector harmonics domain processing for wideband source localization using spherical-sector array of directional microphones

The spherical microphone array can be uneconomical for applications where the sources arrive only from a known section of the sphere. For this reason, the spherical-sector harmonics was developed for processing spherical sector array. The orthonormal spherical sector harmonics (SSH) basis functions which accounts for the discontinuity arising from sectioning the sphere have been developed and shown to work for the array of omnidirectional microphones. In this work, the SSH basis functions are applied to far-field wideband sound source localization using spherical-sector array of first-order directional microphones (cardioid microphones). The array manifold interpolation method is used to produce the steered covariance matrix and the MUSIC algorithm applied for the direction of arrival estimation. The root-mean-square error performance of this spherical-sector array of the first-order cardioid microphones is compared against that of the omnidirectional microphones for different directions and signal-to-noise ratio.

Local gravity field modeling using gravity difference observations along the line of sight of the GRACE-FO satellites and adjusted spherical cap harmonic basis functions

Local gravity field modeling using gravity difference observations along the line of sight of the GRACE-FO satellites and adjusted spherical cap harmonic basis functions

Correction of residual errors in configuration interaction electronic structure calculations.

Methods for correcting residual energy errors of configuration interaction (CI) calculations of molecules and other electronic systems are discussed based on the assumption that the energy defect can be mapped onto atomic regions. The methods do not consider the detailed nature of excitations but instead define a defect energy per electron that is unique to a specific atom. Defect energy contributions are determined from calculations on diatomic and hydride molecules and then applied to other systems. Calculated energies are compared with experimental thermodynamic and spectroscopic data for a set of 41 mainly organic molecules representing a wide range of bonding environments. The most stringent test is based on a severely truncated virtual space in which higher spherical harmonic basis functions are removed. The errors of the initial CI calculations are large, but in each case, including defect corrections brings calculated CI energies into agreement with experimental values. The method is also applied to a NIST compilation of coupled cluster calculations that employ a larger basis set and no truncation of the virtual space. The corrections show excellent consistency with total energies in very good agreement with experimental values. An extension of the method is applied to dmsn states of Sc, Ti, V, Mn, Cr, Fe, Co, Ni, and Cu, significantly improving the agreement of calculated transition energies with spectroscopic values.

Natural frequencies of a bubble near a solid sphere

This paper presents an analytical and computational method to describe natural frequencies of a spherical bubble residing near a solid sphere of an arbitrary size in an otherwise unbounded fluid. Under low capillary and Reynolds number limits, the relevant hydrodynamic fields are converted into time-invariant but frequency-dependent quantities by temporal Fourier transform. Then, the spatial variations in the velocity and the pressure can be expressed in terms of two sets of harmonic basis functions involving spherical coordinates centered around the particle and the bubble. A subsequent derivation of transformation coefficients between the aforementioned two sets allows a matrix equation relating the unknown amplitudes to the boundary conditions at all interfaces. Finally, natural frequencies corresponding to different modes of pulsation are obtained from the eigenvalues of the constructed matrix. The results show fast convergence of the computed frequencies with the increasing number of basis functions. These values change significantly with the distance of the bubble from the particle and even decay to zero for some modes when their surface-to-surface separation vanishes. Furthermore, bubble oscillation near a solid plate is also discussed when the radius of the solid sphere is increased to an infinitely large dimension. Thus, this article renders a comprehensive study of naturally pulsating submerged bubbles in the presence of a nearby solid surface of various kinds.

Benchmark forward gravity schemes: the gravity field of a realistic lithosphere model WINTERC-G

Abstract. Several alternative gravity forward modelling methodologies and associated numerical codes with their own advantages and limitations are available for the solid Earth community. With upcoming state-of-the-art lithosphere density models and accurate global gravity field data sets, it is vital to understand the opportunities and limitations of the various approaches. In this paper, we discuss the four widely used techniques: global spherical harmonics (GSH), tesseroid integration (TESS), triangle integration (TRI), and hexahedral integration (HEX). A constant density shell benchmark shows that all four codes can produce similar precise gravitational potential fields. Two additional shell tests were conducted with more complicated density structures: laterally varying density structures and a crust–mantle interface density. The differences between the four codes were all below 1.5 % of the modelled gravity signal suitable for reproducing satellite-acquired gravity data. TESS and GSH produced the most similar potential fields (<0.3 %). To examine the usability of the forward modelling codes for realistic geological structures, we use the global lithosphere model WINTERC-G that was constrained, among other data, by satellite gravity field data computed using a spectral forward modelling approach. This spectral code was benchmarked against the GSH, and it was confirmed that both approaches produce a similar gravity solution with negligible differences between them. In the comparison of the different WINTERC-G-based gravity solutions, again GSH and TESS performed best. Only short-wavelength noise is present between the spectral and tesseroid forward modelling approaches, likely related to the different way in which the spherical harmonic analysis of the varying boundaries of the mass layer is performed. The spherical harmonic basis functions produce small differences compared to the tesseroid elements, especially at sharp interfaces, which introduces mostly short-wavelength differences. Nevertheless, both approaches (GSH and TESS) result in accurate solutions of the potential field with reasonable computational resources. Differences below 0.5 % are obtained, resulting in residuals of 0.076 mGal standard deviation at 250 km height. The biggest issue for TRI is the characteristic pattern in the residuals that is related to the grid layout. Increasing the resolution and filtering allow for the removal of most of this erroneous pattern, but at the expense of higher computational loads with respect to the other codes. The other spatial forward modelling scheme, HEX, has more difficulty in reproducing similar gravity field solutions compared to GSH and TESS. These particular approaches need to go to higher resolutions, resulting in enormous computation efforts. The hexahedron-based code performs less than optimal in the forward modelling of the gravity signature, especially with a laterally varying density interface. Care must be taken with any forward modelling software as the approximation of the geometry of the WINTERC-G model may deteriorate the gravity field solution.

Variational vibrational states of HCOOH

Vibrational states of the formic acid molecule are converged using the GENIUSH–Smolyak approach and the potential energy surface taken from Tew and Mizukami (2016). The quantum nuclear motion is described by using the cis-trans torsional coordinate and eight curvilinear normal coordinates defined with respect to an instantaneous reference configuration changing as a function of the torsional degree of freedom. Harmonic oscillator basis functions are used for the curvilinear normal coordinates, a Fourier basis for the torsional coordinate, and a simple basis pruning condition is combined with a Smolyak integration grid. Trans, cis, and delocalized vibrational states are reported up to and slightly beyond the isomerization barrier.

SPHARM-PDM based image preprocessing pipeline for quantitative morphometric analysis (QMA) for in situ joint assessment in rabbit and rat models

The accessibility of quantitative measurements of joint morphometry depends on appropriate tibial alignment and volume of interest (VOI) selection of joint compartments; often a challenging and time-consuming manual task. In this work, we developed a novel automatic, efficient, and model-invariant image preprocessing pipeline that allows for highly reproducible 3D quantitative morphometric analysis (QMA) of the joint. The pipeline addresses the problem by deploying two modules: an alignment module and a subdivision module. Alignment is achieved by representing the tibia in its basic form using lower degree spherical harmonic basis functions and aligning using principal component analysis. The second module subdivides the joint into lateral and medial VOIs via a watershedding approach based on persistence homology. Multiple repeated micro-computed tomography scans of small (rat) and medium (rabbit) animal knees were processed using the pipeline to demonstrate model invariance. Existing QMA was performed to evaluate the pipeline’s ability to generate reproducible measurements. Intraclass correlation coefficient and mean-normalised root-mean-squared error of more than 0.75 and lower than 9.5%, respectively, were achieved for joint centre of mass, joint contact area under virtual loading, joint space width, and joint space volume. Processing time and technical requirements were reduced compared to manual processing in previous studies.

Using Spherical Harmonics for Navigating in Dynamic and Uncertain Environments

In this paper, we present a novel approach for local motion planning in an unknown environment populated by static and dynamic obstacles. A key component of our planner is the use of harmonic basis functions defined over the unit sphere, known as spherical harmonics (SH), to represent a collision-free space near an agent. In contrast with our previous approach, which only applies to unknown static environments, the approach proposed herein handles dynamic obstacles as well. It is done by efficiently calculating the coefficients of a spherical harmonics approximation of the local collision-free space for each step of a given planning horizon. Our method for approximating the collision-free space with spherical harmonics allows us to plan trajectories in challenging environments (e.g., planning through narrow passages) where other methods for generating search-spaces fail. To accurately approximate the collision-free space for future time steps along a planning horizon, we use a Kalman Filter (KF) to propagate the measured point cloud through time. Our use of the KF allows us to better approximate obstacles moving in space in comparison to inaccurate constant obstacle velocity assumptions used by other methods. Finally, we show the effectiveness of our proposed trajectory planner using two different non-trivial scenarios.

Learning developmental mode dynamics from single-cell trajectories.

Embryogenesis is a multiscale process during which developmental symmetry breaking transitions give rise to complex multicellular organisms. Recent advances in high-resolution live-cell microscopy provide unprecedented insights into the collective cell dynamics at various stages of embryonic development. This rapid experimental progress poses the theoretical challenge of translating high-dimensional imaging data into predictive low-dimensional models that capture the essential ordering principles governing developmental cell migration in complex geometries. Here, we combine mode decomposition ideas that have proved successful in condensed matter physics and turbulence theory with recent advances in sparse dynamical systems inference to realize a computational framework for learning quantitative continuum models from single-cell imaging data. Considering pan-embryo cell migration during early gastrulation in zebrafish as a widely studied example, we show how cell trajectory data on a curved surface can be coarse-grained and compressed with suitable harmonic basis functions. The resulting low-dimensional representation of the collective cell dynamics enables a compact characterization of developmental symmetry breaking and the direct inference of an interpretable hydrodynamic model, which reveals similarities between pan-embryo cell migration and active Brownian particle dynamics on curved surfaces. Due to its generic conceptual foundation, we expect that mode-based model learning can help advance the quantitative biophysical understanding of a wide range of developmental structure formation processes.

Spherical harmonic covariance and magnitude function encodings for beamformer design

Microphone and speaker array designs have increasingly diverged from simple topologies due to diversity of physical host geometries and use cases. Effective beamformer design must now account for variation in the array’s acoustic radiation pattern, spatial distribution of target and noise sources, and intended beampattern directivity. Relevant tasks such as representing complex pressure fields, specifying spatial priors, and composing beampatterns can be efficiently synthesized using spherical harmonic (SH) basis functions. This paper extends the expansion of common stationary covariance functions onto the SHs and proposes models for encoding magnitude functions on a sphere. Conventional beamformer designs are reformulated in terms of magnitude density functions and beampatterns along SH bases. Applications to speaker far-field response fitting, cross-talk cancelation design, and microphone beampattern fitting are presented.

Real-time locally injective volumetric deformation

We present a highly efficient method for interactive volumetric meshless shape deformation. Our method operates within a low dimensional sub-space of shape-aware C ∞ harmonic maps, and is the first method that is guaranteed to produce a smooth locally injective deformation in 3D. Unlike mesh-based methods in which local injectivity is enforced on tetrahedral elements, our method enforces injectivity on a sparse set of domain samples. The main difficulty is then to certify the map as locally injective throughout the entire domain. This is done by utilizing the Lipschitz continuity property of the harmonic basis functions. We show a surprising relation between the Lipschitz constant of the smallest singular value of the map Jacobian and the norm of the Hessian. We further carefully derive a Lipschitz constant for the Hessian, and develop a sufficient condition for the injectivity certification. This is done by utilizing the special structure of the harmonic basis functions combined with a novel regularization term that pushes the Lipschitz constants further down. As a result, the injectivity analysis can be performed on a relatively sparse set of samples. Combined with a parallel GPU-based implementation, our method can produce superior deformations with unique quality guarantees at real-time rates which were possible only in 2D so far.

Comparison of spherical cap and rectangular harmonic analysis of airborne vector gravity data for high-resolution (1.5 km) local geopotential field models over Tanzania

SUMMARYThis study examines local geopotential field modelling over a mountainous region in Tanzania using vector airborne gravity data. We use the adjusted spherical cap and rectangular harmonic analyses. Both methods are based on expansion of gravitational potential into a series of orthogonal harmonic basis functions of local support in such a way that the expansion coefficients are determined by gravity observations. All three components of gravity vector are simultaneously inverted to derive the geopotential coefficients. In order to evaluate the accuracy of the local models, independent checkpoints are selected within the study region and around its boundary and the computed gravity vectors are compared with the independent gravity observations. The results show an excellent agreement with root mean square error (RMSE) of < 1.6 mGal over the study area. On the contrary, the RMSEs of global geopotential models against the checkpoints data are 7 mGal for the models up to the maximum degree of 2190 (a resolution of ∼9.1 km) and 5 mGal to 5399 (∼3.7 km). Our local models are significantly more accurate than the state-of-the-art global models and fully exploit the airborne vector data with the measurement error of ∼1 mGal. We also present the regional models constrained only by radial (vertical) or by lateral (horizontal) gravity observations. Those models are considerably less accurate than the one from 3-D gravity data inversion. Lastly, the regional models are validated against topography data. It is found that the gravity–topography correlation is 0.8–0.9 at 100 km, 0.5 at 20 km and higher than the correlations of the global models at all frequencies. The gravity–topography admittances estimated from our regional models indicate ∼130 mGal km−1 and imply the effective density of 2500 kg m−3 for topographical mass.

Fully implicit spectral boundary integral computation of red blood cell flow

This paper is on an implicit time integration scheme for simulation of red blood cell (RBC) flow in an ambient fluid. The intra- and extracellular plasmas are modeled as Stokes flows and represented by boundary integral equations (BIE) written in a weakly singular form. The cell membrane is modeled as a thin elastic shell. Expressed in this way, the RBC flow model constitutes an implicit ordinary differential equation (IODE) in the cell shape. The cell shape and velocity field are discretized spatially by a spectral approach using spherical harmonic basis functions. It is then convenient to express the BIE in the Galerkin form with the spherical harmonics themselves as test functions. The key aspect in this paper is the recognition of the IODE structure of the RBC flow model and consequent application of a multi-step implicit solver for time integration. As with any implicit solver, a nonlinear equation in the cell shape is solved at each time step, for which Newton's method is applied. This requires the Jacobian of the IODE, or equivalently computation of Jacobian-vector products. An important contribution is the formulation of such Jacobian-vector products as evaluating a second BIE. The original weakly singular form is crucial in facilitating this formulation. The implicit solver employs variable order and adaptive time stepping controlled by truncation error and convergence of Newton iterations. Numerical examples show that larger time steps are possible and that the number of matrix-vector products is comparable to explicit methods. Source code is provided in the online supplementary material.