Surface Vector Field Research Articles

A decomposition theorem of surface vector fields and spectral structure of the Neumann-Poincaré operator in elasticity

We prove that the space of vector fields on the boundary of a bounded domain with the Lipschitz boundary in three dimensions is decomposed into three subspaces: elements of the first one extend to inside the domain as divergence-free and rotation-free vector fields, the second one to the outside as divergence-free and rotation-free vector fields, and the third one to both the inside and the outside as divergence-free harmonic vector fields. We then show that each subspace in the decomposition is infinite-dimensional. We also prove under a mild regularity assumption on the boundary that the decomposition is almost direct in the sense that any intersection of two subspaces is finite-dimensional. We actually prove that the dimension of intersection is bounded by the first Betti number of the boundary. In particular, if the boundary is simply connected, then the decomposition is direct. We apply this decomposition theorem to investigate spectral properties of the Neumann-Poincaré operator in elasticity, whose cubic polynomial is known to be compact. We prove that each linear factor of the cubic polynomial is compact on each subspace of decomposition separately and those subspaces characterize eigenspaces of the Neumann-Poincaré operator. We then prove all the results for three dimensions, decomposition of surface vector fields and spectral structure, are extended to higher dimensions. We also prove analogous but different results in two dimensions.

Erratum to “Navier–Stokes equations in a curved thin domain, Part III: thin-film limit”

There are two flaws in the third part [3] of our study on the Navier--Stokes equations in a curved thin domain. The first one is that the form of Ladyzhenskaya's inequality on a closed surface taken from the arXiv version [2] of the second part of our study is incorrect. The second one is that a wrong claim is used in the proofs of lemmas in [3, Section 6.3] which give estimates for the weighed potential part of a tangential surface vector field. This erratum corrects these flaws.

Retrieval of a Real‐Time Sea Surface Vector Field From SAR Doppler Centroid: 1. Ekman Current Retrieval

AbstractCurrent direction retrieval is a significant part of the Doppler Centroid Technique (DCT) for remote sensing of the sea surface vector field, and it remains an unsolved problem that has prevented its wide application for more than 15 years. In this paper, we review and update the core concept of the DCT, through which the Ekman scheme for current direction retrieval is obtained. In addition, the real sea surface vector field at approximately 24°N with weak background flow during Hurricane Sam is retrieved. We also prove that the measured velocity of the DCT is just the magnitude of the sea surface current vector, and thus, the DCT is the most promising technique for remote sensing of sea surface current dynamics.

Surface vector fields estimation using soft computing and remote sensing data at an open water marine renewable test site

Galway Bay has one of the world’s few open water marine renewable test sites and so is of national and international interest. A high frequency radar system has been deployed in Galway Bay area to monitor near real time surface currents and waves since July 2011. In this research, a soft computing approach was applied to estimate surface vector fields using the observed radar data. Results indicate that soft computing is a novel and promising method to estimate surface vector fields. It provides a potential way to obtain useful information of coastal water body for marine renewable energy development and assessment.

Second-kind boundary integral equations for electromagnetic scattering at composite objects

We consider electromagnetic scattering of time-harmonic fields in R3 at objects composed of several linear, homogeneous, and isotropic materials. Adapting earlier work on acoustic scattering (Claeys et al., 2015) we develop a novel second-kind direct boundary integral formulation for this scattering problem, extending the so-called Müller formulation for a homogeneous scatterer to composite objects. The new formulation is amenable to Galerkin boundary element discretization by means of discontinuous tangential surface vectorfields. A rigorous proof of its well-posedness is still missing. Yet numerical tests demonstrate excellent stability and competitive accuracy of the new approach compared with a widely used direct Galerkin boundary element method based on a first-kind boundary integral formulation. For piecewise constant approximation our experiments also confirm fast convergence of GMRES iterations independently of mesh resolution.

Image-Space Texture-Based Output-Coherent Surface Flow Visualization

Image-space line integral convolution (LIC) is a popular scheme for visualizing surface vector fields due to its simplicity and high efficiency. To avoid inconsistencies or color blur during the user interactions, existing approaches employ surface parameterization or 3D volume texture schemes. However, they often require expensive computation or memory cost, and cannot achieve consistent results in terms of both the granularity and color distribution on different scales. This paper introduces a novel image-space surface flow visualization approach that preserves the coherence during user interactions. To make the noise texture under different viewpoints coherent, we propose to precompute a sequence of mipmap noise textures in a coarse-to-fine manner for consistent transition, and map the textures onto each triangle with randomly assigned and constant texture coordinates. Further, a standard image-space LIC is performed to generate the flow texture. The proposed approach is simple and GPU-friendly, and can be easily combined with various texture-based flow visualization techniques. By leveraging viewpoint-dependent backward tracing and mipmap noise phase, our method can be incorporated with the image-based flow visualization (IBFV) technique for coherent visualization of unsteady flows. We demonstrate consistent and highly efficient flow visualization on a variety of data sets.

Tangent vectors to a 3-D surface normal: A geometric tool to find orthogonal vectors based on the Householder transformation

An important geometric and linear algebraic problem denoted as vector orthogonalization, fundamental to handle contact detection and contact force descriptions in engineering applications, is here considered. The problem is to find a set of linearly independent vectors that span the entire R3 Euclidean space given only one of the base vectors. This paper contains the explanation on how the Householder transformation, which is extensively used for matrix orthogonalization, provides an elegant analytical expression that solves the vector orthogonalization problem. Based on the QR matrix factorization method, the orthogonal vectors are produced using a Householder reflection that transforms the given vector into a multiple of the unit vector whose entries are all zero with the exception of the first. Based on efficiency, accuracy and numerical robustness criteria, the proposed technique is compared to other vector orthogonalization methods. The numerical results show that the Householder vector orthogonalization formula is the most efficient when it comes to outputting a set of orthonormal vectors, presenting speedups close to 1.017 times faster when compared to other efficient techniques. In addition, when dealing with Cn continuous implicit surfaces, with n>1, the Householder vector orthogonalization formula reveals to be particularly useful for vector calculus since it provides a set of differential operators to calculate, not only the normal, but also the tangent and binormal surface vector fields which can be used to calculate surface curvatures. The major contribution of this paper is to explicitize how the Householder transformation holds an analytical expression that calculates the tangent and binormal vectors from a given normal at a surface point vector, which is computationally efficient and numerically robust for real-time computational geometry and computer graphics applications, namely, for contact mechanics applications with implicit surfaces of engineering problems with multiple contacts. Such a vector orthogonalization technique also has direct applications in several CAD/CAM processes, ranging from the elaboration of rough solid models to the precise manufacturing of a product.

Nonlinear Force‐free Field Modeling of a Solar Active Region around the Time of a Major Flare and Coronal Mass Ejection

Solar flares and coronal mass ejections are associated with rapid changes in field connectivity and are powered by the partial dissipation of electrical currents in the solar atmosphere. A critical unanswered question is whether the currents involved are induced by the motion of preexisting atmospheric magnetic flux subject to surface plasma flows or whether these currents are associated with the emergence of flux from within the solar convective zone. We address this problem by applying state-of-the-art nonlinear force-free field (NLFFF) modeling to the highest resolution and quality vector-magnetographic data observed by the recently launched Hinode satellite on NOAA AR 10930 around the time of a powerful X3.4 flare. We compute 14 NLFFF models with four different codes and a variety of boundary conditions. We find that the model fields differ markedly in geometry, energy content, and force-freeness. We discuss the relative merits of these models in a general critique of present abilities to model the coronal magnetic field based on surface vector field measurements. For our application in particular, we find a fair agreement of the best-fit model field with the observed coronal configuration, and argue (1) that strong electrical currents emerge together with magnetic flux preceding the flare, (2) that these currents are carried in an ensemble of thin strands, (3) that the global pattern of these currents and of field lines are compatible with a large-scale twisted flux rope topology, and (4) that the ~1032 erg change in energy associated with the coronal electrical currents suffices to power the flare and its associated coronal mass ejection.

Southeast Florida Shelf circulation and volume exchange, observations of km-scale variability

Tides, wind and the Florida Current affect the dynamics of the South Florida continental shelf over a range of scales. Two field experiments in the winter (LARGO1) and summer (LARGO2) employed a high-frequency (HF) Ocean Surface Current Radar (OSCR), moored current meters and a boat-mounted Acoustic Doppler Current Profiler (ADCP) to study the circulation over the southeast Florida Shelf. The OSCR measured near-surface currents over an approximately 30 km by 30 km grid, with 1-km horizontal resolution, covering the shelf and extending offshore of the shelf break. By using observed 20-min surface vector field time series to backward track patches with defined volumes of shelf water, their recent origin was identified. The proportion of surface volume flux past various points on the shelf that originated offshore, along the shelf edge and inshore was then estimated. Onshore winds were linked to periods of transport of offshore water all the way to the inner shelf during both the summer and winter seasons. In LARGO2 the exchange of offshore water with water over the middle and inner shelf was closely linked to onshore and downcoast winds. Simulated particle trajectories revealed that Florida Current spin-off eddies caused off-shelf water to be advected onto the middle (LARGO1) or outer (LARGO2) shelf. However, because eddies had short residence times and small spatial scales their contribution to the overall exchange was much smaller than the wind.

Kinematic compatibility conditions of third order for a three-dimensional surface vector field

Kinematic compatibility conditions of third order for a three-dimensional surface vector field

Integrating the Maxwell equations, generalized functions, and nonlocal boundary conditions

The equivalence of the combined interior and exterior scattering problems with the Maxwell equations and boundary equations has been demonstrated [ J. Opt. Soc. Am. B68, 602– 610 ( 1978)]. The expressions for the fields inside and outside the medium were coupled through certain nonlocal boundary conditions that were shown to be generalizations of the Ewald–Oseen extinction theorem. I consider the Maxwell equations in the sense of generalized functions and obtain, without the use of potentials, expressions for the regional fields. These expressions involve four unknown surface-vector fields. Two nonlocal vector-boundary conditions are obtained directly from the regional field expressions. Two other local vector-boundary conditions emerge if we consider generalized functions acting on a certain space of test functions whose properties are described.

Some characteristics of the upper-air magnetic field and ionospheric currents

Characteristics of the upper-air magnetic field and ionospheric currents are determined through an analysis of published rocket data on the magnetic scalar intensity. For the region between the earth's surface and the E layer of the ionosphere, the observed values are compared with values obtained by extrapolating the surface vector field. The agreement between the two sets of values is very good for equatorial flights but only fair for a flight at White Sands, New Mexico. The equatorial, ionospheric current density, which has a maximum of about 21 amp/km2, varies considerably with time and location of the rocket flight. The equatorial electro jet has a current intensity of about 130 amp/km and flows practically along the magnetic equator. Some of the computed currents associated with the normal magnetic daily variation are at variance with those expected from considerations, such as harmonic analysis, of the surface transient field and sheet-current approximations. In the area around White Sands (geomagnetic latitude 41°N) a large negative magnetic anomaly exists that may contribute to the formation of the region of low intensity of radiation that lies between the two Van Allen radiation belts.