Spherical Multipole Expansion Research Articles

Decoupling method of self-gravity compensation based on spherical multipole expansion for space gravitational wave detectors

In space gravitational wave detection missions, the gravity and its gradients produced by the spacecraft on the two Test Masses (TMs) are commonly referred to as the Self-Gravity(SG). It is an important source of TM disturbances in gravitational wave detection and other drag-free space missions and will affect the TM acceleration noise in many ways. The SG can be reduced by adding Balance Masses (BMs). But for typical space gravitational wave detectors, in which the sensitive axes of the two TMs are at an angle of 60°, the couplings of different SG components of two TMs make the gravity compensation process complicated in practice, which is normally an iterative process. This paper analyses the correspondence between the SG components of the two TMs and the spherical harmonics of different orders, and proposes a compensation method based on spherical multipole expansion. This method allows independent design of the BMs for most of the main SG components, without couplings and iterations. To verify this method, a self-gravity compensation simulation is carried out by using a demonstrating spacecraft structural model for TianQin gravitational wave detection mission. Three sets of BMs are designed on the outer surface of the inertial sensor vacuum chamber, to compensate for the two linear accelerations and one linear gradient that exceed the requirements. The results show that the SG components after compensation are two orders of magnitude lower than the initial level, and all the components meet the preliminary requirements of TianQin mission. This study could provide reference for the engineering design and development of the spacecraft and inertial sensor payload for space gravitational wave detection missions.

Estimating the Net Magnetic Moment of Geological Samples From Planar Field Maps Using Multipoles

AbstractRecent advances in magnetic microscopy have enabled studies of geological samples whose weak and spatially nonuniform magnetizations were previously inaccessible to standard magnetometry techniques. A quantity of central importance is the net magnetic moment, which reflects the mean direction and the intensity of the magnetization states of numerous ferromagnetic crystals within a certain volume. The planar arrangement of typical magnetic microscopy measurements, which originates from measuring the field immediately above the polished surface of a sample to maximize sensitivity and spatial resolution, makes estimating net moments considerably more challenging than with spherically distributed data. In particular, spatially extended and nonuniform magnetization distributions often cannot be adequately approximated by a single magnetic dipole. To address this limitation, we developed a multipole fitting technique that can accurately estimate net moment using spherical harmonic multipole expansions computed from planar data. Given that the optimal location for the origin of such expansions is unknown beforehand and generally unconstrained, regularization of this inverse problem is critical for obtaining accurate moment estimates from noisy experimental magnetic data. We characterized the performance of the technique using synthetic sources under different conditions (noiseless data, data corrupted with simulated white noise, and data corrupted with measured instrument noise). We then validated and demonstrated the technique using superconducting quantum interference device microscopy measurements of impact melt spherules from Lonar crater, India and dusty olivine chondrules from the CO chondrite meteorite Dominion Range 08006.

Low-frequency directional characteristics of a gamelan gong

The structural modes of gamelan gongs often have clear impacts on their far-field directivity patterns with the number of directional lobes corresponding to the associated structural mode shapes. Many of the lowest modes produce dipole-like radiation with the dipole moment determined by the positions of the nodal and antinodal regions. Spherical harmonic and multipole expansions facilitate further understanding of the gongs' low-frequency directional characteristics. The expansions also yield practical simplifications to model their radiation.

Magnetic-field modeling with surface currents. Part I. Physical and computational principles of bfieldtools

Surface currents provide a general way to model magnetic fields in source-free volumes. To facilitate the use of surface currents in magneto-quasistatic problems, we have implemented a set of computational tools in a Python package named bfieldtools. In this work, we describe the physical and computational principles of this toolset. To be able to work with surface currents of the arbitrary shape, we discretize the currents on triangle meshes using piecewise-linear stream functions. We apply analytical discretizations of integral equations to obtain the magnetic field and potentials associated with the discrete stream function. In addition, we describe the computation of the spherical multipole expansion and a novel surface-harmonic expansion for surface currents, both of which are useful for representing the magnetic field in source-free volumes with a small number of parameters. Lastly, we share examples related to magnetic shielding and the surface-coil design using the presented tools.

Quadratic quantities in acoustics: Scattering cross-section and radiation force

Time-harmonic acoustic waves interact with a scatterer: this interaction is calculated using linear, first-order theory. However, there are quadratic, second-order quantities that are of interest. These include the scattering cross-section and the steady radiation force; these quantities can be expressed as integrals of products of first-order quantities over a sphere. These integrals are evaluated exactly. The results are infinite series of products of the coefficients in the spherical multipole expansions of the incident and scattered fields; they do not depend on the radius of the spherical integration surface. For a specific scattering problem, the coefficients can be connected by an appropriate T-matrix. In most previous work, the spherical surface is moved to infinity so that far-field quantities can be introduced: it is shown that this process is not straightforward and it may introduce spurious difficulties.

Model-Based Calibration for Magnetic Manipulation

Model-based calibration of a magnetic workspace not only provides a smooth representation of the field and its gradient matrix, but also uses physical constraints to smooth the calibration measurements. This paper presents the first model-based technique to calibrate a magnetic manipulation system by using nonlinear least squares to solve for a scalar potential for each source. The performance of the method is verified by comparison to numerical finite element simulation and a case study calibration of a real system, where it is able to achieve an $R^{2}$ value of 0.9997. Furthermore, the analytical representations for the first three spatial derivatives of a spherical multipole expansion are provided for convenience, which correspond to the torque, force, and force-spatial-rate-of-change on a magnetic dipole in the workspace.

Atom-partitioned multipole expansions for electrostatic potential boundary conditions

Applications such as grid-based real-space density functional theory (DFT) use the Poisson equation to compute electrostatics. However, the expected long tail of the electrostatic potential requires either the use of a large and costly outer domain or Dirichlet boundary conditions estimated via multipole expansion. We find that the oft-used single-center spherical multipole expansion is only appropriate for isotropic mesh domains such as spheres and cubes. In this work, we introduce a method suitable for high aspect ratio meshes whereby the charge density is partitioned into atomic domains and multipoles are computed for each domain. While this approach is moderately more expensive than a single-center expansion, it is numerically stable and still a small fraction of the overall cost of a DFT calculation. The net result is that when high aspect ratio systems are being studied, form-fitted meshes can now be used in lieu of cubic meshes to gain computational speedup.

Derivation and analysis of the analytical velocity and vortex stretching expressions for an O(N log N)-FMM

In the current paper, a method for deriving the analytical expressions for the velocity and vortex stretching terms as a function of the spherical multipole expansion approximation of the vector potential is presented. These terms are essential in the context of 3D Lagrangian vortex particle methods combined with fast summation techniques. The convergence and computational efficiency of this approach is assessed in the framework of an O(N log N)-type Fast Multipole Method (FMM), by using vorticity particles to simulate a system of coaxial vortex rings for which also the exact results are known. It is found that the current implementation converges rapidly to the exact solution with increasing expansion order and acceptance factor. An investigation into the computational efficiency demonstrated that the O(N log N)-type FMM is already viable for a particle size of only several thousands and that this speedup increases significantly with the number of particles. Finally, it is shown that the implementation of the FMM with the current analytical expressions is at least twice as fast as when opting for using even the simplest implementation of finite differences instead.

Computation of Galerkin Double Surface Integrals in the 3-D Boundary Element Method

Many boundary element integral equation kernels are based on the Green's functions of the Laplace and Helmholtz equations in three dimensions. These include, for example, the Laplace, Helmholtz, elasticity, Stokes, and Maxwell's equations. Integral equation formulations lead to more compact, but dense linear systems. These dense systems are often solved iteratively via Krylov subspace methods, which may be accelerated via the fast multipole method. There are advantages to Galerkin formulations for such integral equations, as they treat problems associated with kernel singularity, and lead to symmetric and better conditioned matrices. However, the Galerkin method requires each entry in the system matrix to be created via the computation of a double surface integral over one or more pairs of triangles. There are a number of semi-analytical methods to treat these integrals, which all have some issues, and are discussed in this paper. We present novel methods to compute all the integrals that arise in Galerkin formulations involving kernels based on the Laplace and Helmholtz Green's functions to any specified accuracy. Integrals involving completely geometrically separated triangles are non-singular and are computed using a technique based on spherical harmonics and multipole expansions and translations, which results in the integration of polynomial functions over the triangles. Integrals involving cases where the triangles have common vertices, edges, or are coincident are treated via scaling and symmetry arguments, combined with automatic recursive geometric decomposition of the integrals. Example results are presented, and the developed software is available as open source.

Electric field in media with power-law spatial dispersion

In this paper, we consider electric fields in media with power-law spatial dispersion (PLSD). Spatial dispersion means that the absolute permittivity of the media depends on the wave vector. Power-law type of this dispersion is described by derivatives and integrals of non-integer orders. We consider electric fields of point charge and dipole in media with PLSD, infinite charged wire, uniformly charged disk, capacitance of spherical capacitor and multipole expansion for PLSD-media.

Multipolar Ewald methods, 1: theory, accuracy, and performance.

The Ewald, Particle Mesh Ewald (PME), and Fast Fourier–Poisson (FFP) methods are developed for systems composed of spherical multipole moment expansions. A unified set of equations is derived that takes advantage of a spherical tensor gradient operator formalism in both real space and reciprocal space to allow extension to arbitrary multipole order. The implementation of these methods into a novel linear-scaling modified “divide-and-conquer” (mDC) quantum mechanical force field is discussed. The evaluation times and relative force errors are compared between the three methods, as a function of multipole expansion order. Timings and errors are also compared within the context of the quantum mechanical force field, which encounters primary errors related to the quality of reproducing electrostatic forces for a given density matrix and secondary errors resulting from the propagation of the approximate electrostatics into the self-consistent field procedure, which yields a converged, variational, but nonetheless approximate density matrix. Condensed-phase simulations of an mDC water model are performed with the multipolar PME method and compared to an electrostatic cutoff method, which is shown to artificially increase the density of water and heat of vaporization relative to full electrostatic treatment.

Characterization of the radiation pattern of antennas using time-domain spherical-multipole and moment expansion

This paper presents results of a time-domain spherical-multipole near-to-far-field transformation together with a moment expansion applied to antenna radiation patterns. The equivalence principle is applied to the tangential fields all over the FDTD/WP-PML spatial grid and the use of the spherical-multipole expansion leads to the near-to-far-field transformation, and energy and power norms of temporal signals are used to obtain the antenna radiation patterns for transient and steady-state responses. Such approach is employed to obtain UWB antenna radiated fields and radiation patterns directly in time-domain, it being more convenient to perform an unified characterization in time and frequency domains.

Electromagnetic diffraction and scattering of a complex-source beam by a semi-infinite circular cone

Abstract. A complex-source beam (CSB) is used to investigate the electromagnetic scattering and diffraction by the tip of a perfectly conducting semi-infinite circular cone. The boundary value problem is defined by assigning a complex-valued source coordinate in the spherical-multipole expansion of the field due to a Hertzian dipole in the presence of the PEC circular cone. Since the incident CSB field can be interpreted as a localized plane wave illuminating the tip, the classical exact tip scattering problem can be analysed by an eigenfunction expansion without having the convergence problems in case of a full plane wave incident field. The numerical evaluation includes corresponding near- and far-fields.

RADIATION BOUNDARY CONDITIONS FOR THE HELMHOLTZ EQUATION FOR ELLIPSOIDAL, PROLATE SPHEROIDAL, OBLATE SPHEROIDAL AND SPHERICAL DOMAIN BOUNDARIES

One of the most popular radiation boundary conditions for the Helmholtz equation in exterior 3-D regions has been the sequence of operators developed by Bayliss et al.1 for computational domains with spherical exterior boundaries. The present paper extends those spherical operators to triaxial ellipsoidal boundaries by utilizing two mathematical constructs originally developed for ellipsoidal acoustic infinite elements.2 The two constructs are: (i) a radial/angular coordinate system for ellipsoidal geometry, and (ii) a convergent ellipsoidal radial expansion for exterior fields, analogous to the classical spherical multipole expansion. The ellipsoidal radial and angular coordinates are smooth generalizations of the traditional radial and angular coordinates used in spherical, prolate spheroidal and oblate spheroidal systems. As a result, all four coordinate systems and their corresponding radiation boundary conditions are included within this single ellipsoidal system, varying smoothly from one to the other. The geometric flexibility of this system enables the exterior boundary of the computational domain to closely circumscribe objects with a wide range of aspect ratios, thereby reducing the size and cost of 3-D computational models.

Improved measurement surface for MEG using magnetic-dipole sources and a spherical-multipole expansion

Abstract. The multipole representation of Magnetoencephalography (MEG) signals is known as a useful tool for distinguishing between magnetic fields arising from the brain and external disturbances. In this contribution we extend this concept and show that a closed double-layer surface with magnetometer probes is better suited to determine the corresponding multipole amplitudes αlm than a conventional single-layer surface with gradiometers and magnetometer probes. For two different source configurations we show that the αlm rapidly converge to the exact values. This proof of concept motivates to further optimize the geometry of the double-layer surface and the sensors' positions.

Scattering of a transversely confined Neumann beam by a spherical particle

Various properties of an electromagnetic wave whose spherical multipole expansion contains only Riccati-Neumann functions are examined. In particular, the novel behavior of the beam phase during diffractive spreading is discussed. When a Neumann beam is scattered by a spherical particle, the diffraction and external reflection portions of the scattering amplitude constructively interfere for large partial waves. As a result, a set of rapidly decreasing beam shape coefficients is required to cut off the partial wave sum in the scattering amplitudes. Because of its strong singularity at the origin, a Neumann beam can be produced by a point source of radiation at the center of a spherical cavity in a high conductivity metal, and Neumann beam scattering by a spherical particle can occur for certain partial waves if the sphere is placed at the center of the cavity as well.

Time domain near-field to near-field transformation using a spherical-multipole approach

[1] The time domain electromagnetic field of an arbitrary localized radiating structure can be efficiently obtained by means of a time domain spherical-multipole expansion valid outside a minimum sphere enclosing all radiating elements. The method is based on the Fourier transform of the frequency-domain spherical-multipole expansion and on a finite expansion of the spherical Hankel function of the 2nd kind leading to a triple sum of multipoles instead of the well-known double sum in case of the frequency-domain multipole expansion. It is shown that those time domain multipole amplitudes which are relevant only in the near-field can be recursively deduced from the time domain amplitudes dominant in the far field. The latter can be obtained by a recently proposed spherical-multipole based time domain near-field to far-field algorithm which has been shown to be particularly suited for the Finite-Difference Time Domain (FDTD) method.

Convergence of the Electrostatic Interaction Based on Topological Atoms

An atom−atom partitioning of the electrostatic energy between unperturbed molecules is proposed on the basis of the topology of the electron density. Atom−atom contributions to the electrostatic energy are computed exactly, i.e., via a novel six-dimensional integration over two atomic basins, and by means of the spherical tensor multipole expansion, up to total interaction rank L = lA + lB + 1 = 6. The convergence behavior of the topological multipole expansion is compared with that using distributed multipole analysis (DMA) multipole moments for a set of van der Waals complexes at the B3LYP/6-311+G(2d,p) level. Within the context of the Buckingham−Fowler model it is shown that the topological and DMA multipole moments converge to a very similar interaction energy and geometry (average absolute discrepancy of 1.3 kJ/mol and 1.3°, respectively) and are both in good to excellent agreement with supermolecule calculations.

Atom–atom partitioning of intramolecular and intermolecular Coulomb energy

An atom–atom partitioning of the (super)molecular Coulomb energy is proposed on the basis of the topological partitioning of the electron density. Atom–atom contributions to the molecular intra- and intermolecular Coulomb energy are computed exactly, i.e., via a double integration over atomic basins, and by means of the spherical tensor multipole expansion, up to rank L=lA+lB+1=5. The convergence of the multipole expansion is able to reproduce the exact interaction energy with an accuracy of 0.1–2.3 kJ/mol at L=5 for atom pairs, each atom belonging to a different molecule constituting a van der Waals complex, and for nonbonded atom–atom interactions in single molecules. The atom–atom contributions do not show a significant basis set dependence (3%) provided electron correlation and polarization basis functions are included. The proposed atom–atom Coulomb interaction energy can be used both with post-Hartree–Fock wave functions and experimental charge densities in principle. The Coulomb interaction energy between two molecules in a van der Waals complex can be computed by summing the additive atom–atom contributions between the molecules. Our method is able to extract from the supermolecule wave function an estimate of the molecular interaction energy in a complex, without invoking the reference state of free noninteracting molecules. We provide computational details of this method and apply it to (C2H2)2; (HF)2; (H2O)2; butane; 1,3,5-hexatriene; acrolein and urocanic acid, thereby covering a cross section of hydrogen bonds, and covalent bonds with and without charge transfer.

The dielectric response of a lattice of dielectric spheres: convergence of multipole expansion methods

The response to an external applied static electric field of a large spherical sample of a lattice of dielectric spheres embedded in a dielectric medium is analysed. The lattice cells are simple cubic and contain one dielectric sphere per cell. The response gives a dielectric constant for the lattice array. The electric field problem is written as an integral equation for an effective charge density on the surface of the dielectric spheres. The interactions of the responses of the spheres with one another are represented via lattice sums of external spherical harmonic potentials. The numerical implementation of the problem is carried out by truncating a spherical harmonic multipole expansion for this charge density at a set of increasing angular momentum numbers. The convergence of the solution as the truncation number increases is examined numerically. Some implications for the theoretical basis of simulations of the dielectric constant and viscosity of suspensions are discussed.