Search Results (253)

The present paper proposes a three-dimensional (3D) spherical shell model for the magneto-electro-elastic (MEE) free vibration analysis of simply supported multilayered smart shells. A mixed curvilinear orthogonal reference system is used to write the unified 3D governing equations for cylinders, cylindrical panels and spherical shells. The closed-form solution of the problem is performed considering Navier harmonic forms in the in-plane directions and the exponential matrix method in the thickness direction. A layerwise approach is possible, considering the interlaminar continuity conditions for displacements, electric and magnetic potentials, transverse shear/normal stresses, transverse normal magnetic induction and transverse normal electric displacement. Some preliminary cases are proposed to validate the present 3D MEE free vibration model for several curvatures, materials, thickness values and vibration modes. Then, new benchmarks are proposed in order to discuss possible effects in multilayered MEE curved smart structures. In the new benchmarks, first, three circular frequencies for several half-wave number couples and for different thickness ratios are proposed. Thickness vibration modes are shown in terms of displacements, stresses, electric displacement and magnetic induction along the thickness direction. These new benchmarks are useful to understand the free vibration behavior of MEE curved smart structures, and they can be used as reference for researchers interested in the development of of 2D/3D MEE models. Full article

We introduce the concept of the magnetic metapole—a theoretical extension of classical multipole theory involving a fractional j pole count (related to the harmonic degree n as j = 2ⁿ). Defined by a scalar potential with colatitudinal dependence and no radial variation, the metapole yields a magnetic field that decays as 1/r and is oriented along spherical surfaces. Unlike classical multipoles, the metapole cannot be described as a point source; rather, it corresponds to an extended or filamentary magnetic distribution as derived from Maxwell’s equations. We demonstrate that pairs of oppositely oriented metapoles (up/down) can, at large distances, produce magnetic fields resembling those of classical monopoles. A regularized formulation of the potential resolves singularities for the potential and the field. When applied in a bounded region, it yields finite field energy, enabling practical modeling applications. We propose that the metapole can serve as a conceptual and computational framework for representing large-scale magnetic field structures particularly where standard dipole-based models fall short. This construct may have utility in both geophysical and astrophysical contexts, and it provides a new tool for equivalent source modeling and magnetic field decomposition. Full article

In the framework of harmonic and Clifford analysis, specific distributions in Euclidean space of arbitrary dimension, which are of particular importance for theoretical physics, are once more thoroughly studied. Indeed, actions involving spherical coordinates, such as the radial derivative and multiplication and division by the radial distance, only make sense when considering so-called signumdistributions, that is, bounded linear functionals on a space of test functions showing a singularity at the origin. Introducing these signumdistributions, the actions of a whole series of scalar and vectorial differential operators on the distributions under consideration, lead to a number of surprising results, as illustrated by some examples in three-dimensional mathematical physics. Full article

As a major agricultural center in China, Henan Province is highly dependent on groundwater resources for its socioeconomic development. However, under the triple pressure of intensive agricultural irrigation, surging industrial water demand, and accelerating urbanization, the sustainable use of groundwater resources has become a key issue for regional development. This paper utilizes GRACE satellite data and the Global Land Data Assimilation System (GLDAS) assimilation model from 2003 to 2023 to invert alterations in terrestrial water storage (TWS) and groundwater storage (GWS) in Henan Province. We examine the factors influencing these changes and compare the spherical harmonic coefficient (SH) data with Mascon data, integrating precipitation and soil moisture data. Using the GRACE Mascon data as a reference, GWS in Henan Province exhibited a stable trend from January 2003 to October 2010, with a rate of −0.060 cm/month. From October 2010 to June 2020, GWS demonstrated a declining trend, with a rate of −0.121 cm/month. Conversely, from June 2020 to December 2023, GWS revealed a significant upward trend, with a rate of 0.255 cm/month. The TWS and GWS of the inverse performances of the Centre for Space Research (CSR) SH data and the CRS Mascon data exhibited a similar trend, albeit with differing values. Additionally, the precipitation data, soil moisture, and GLDAS data demonstrated significant seasonal variations, with a lag of approximately two months between changes in precipitation and GWS. Declining GWS could be related to climatic and anthropogenic factors. The changes in groundwater in Henan Province studied in this paper can provide a reference for the sustainable utilization of groundwater resources in the region. Full article

A long-wavelength geoidal geometry reflects mainly lateral density variations in the Earth’s mantle, with the most pronounced features of the Indian Ocean Geoid Low and the West Pacific and North Atlantic Geoid Highs. Despite this spatial pattern being clearly manifested in the global geoidal geometry determined from gravity-dedicated satellite missions, the gravitational signature of the deep mantle could be refined by modelling and subsequently removing the gravitational contribution of lithospheric geometry and density structure. Nonetheless, the expected large uncertainties in available lithospheric density models (CRUST1.0, LITHO1.0) limit, to some extent, the possibility of realistically reproducing the gravitational signature of the deep mantle. To address this issue, we inspect an alternative approach. Realizing that the gravity geopotential field (i.e., gravity potential) is smoother than its gradient (i.e., gravity), we apply the integral operator to geopotential and then investigate the spatial pattern of this functional (i.e., radially integrated geopotential). Results show that this mathematical operation enhances a long-wavelength signature of the deep mantle by filtering out the gravitational contribution of the lithosphere. This finding is explained by the fact that in the definition of this functional, spherical harmonics of geopotential are scaled by the factor 1/n (where n is the degree of spherical harmonics), thus lessening the contribution of higher-degree spherical harmonics in the radially integrated geopotential. We also demonstrate that further enhancement of the mantle signature in this functional could be achieved based on modelling and subsequent removal of the gravitational contribution of lithospheric geometry and density structure. Full article

The G-modified Helmholtz equation is a partial differential equation that allows gravity intensity g to be expressed as a series of spherical harmonics, with the radial distance r raised to irrational powers. In this study, we consider a non-rotating triaxial ellipsoid parameterized by the geodetic latitude φ and geodetic longitude λ, and eccentricities e_e, e_x, e_y. On its surface, the value of gravity potential has a constant value, defining a level triaxial ellipsoid. In addition, the gravity intensity is known on the surface, which allows us to formulate a Dirichlet boundary value problem for determining the gravity intensity as a series of spherical harmonics. This expression for gravity intensity is presented here for the first time, filling a gap in the study of triaxial ellipsoids and spheroids. Given that the triaxial ellipsoid has very small eccentricities, a first order approximation can be made by retaining only the terms containing e_e² and e_x². The resulting expression in spherical harmonics contains even degree and even order harmonic coefficients, along with the associated Legendre functions. The maximum degree and order that occurs is four. Finally, as a special case, we present the geometrical degeneration of an oblate spheroid. Full article

Electrically charged particles present in this layer of the Earth’s atmosphere can alter radio waves, such as those from GPS, Galileo, or BeiDou, resulting in non-estimated errors with respect to the available navigation models for the end user. For most positioning algorithms based in sequential filters, this effect is translated into a slow convergence towards a solution around the decimeter error level. If we consider that the ionosphere’s effect varies based on the user’s location and solar activity due to the atmosphere particle composition, it becomes clear that a global accurate model, valid across wide areas accounting for different seasons and timespans, is, at the very least, quite challenging. The focus of this paper is the demonstration of a global ionosphere model designed to improve the positioning accuracy of the end user through the estimation of ionospheric corrections to the broadcasted navigation message. Mathematically, this method is based on a spherical harmonic expansion model. This approach has the advantage of reducing the dependency from a highly densified station network where the ionosphere delay must be constantly estimated in dozens of locations, in favor of a simplified model that barely needs to be adjusted with a limited set of real-time data (around 40 stations). In this case, GMV’s global station network was used, which comprises geodetic-grade receivers tracking the signal in open-sky locations around the globe. The global ionospheric model is configured to process signals from GPS and Galileo constellations. To evaluate the performances of this model on the final user position estimation, several precise point positioning (PPP) solutions were computed at different locations. The results were compared with PPP solutions calculated without ionospheric corrections at the same stations. The goal of this paper is to show the significant performance improvement observed with the implementation of the global model. Full article

To address the limited inversion accuracy caused by low-fidelity data in satellite gravimetry, this study proposes a data preprocessing framework based on iterative residual correction. Utilizing Level-1B observations from the Gravity Recovery and Climate Experiment Follow-On (GRACE-FO) satellite (January 2020), outliers were systematically detected and removed, while data gaps were compensated through spline interpolation. Experimental results demonstrate that the proposed method effectively mitigates data discontinuities and anomalous perturbations, achieving a significant improvement in data quality. Furthermore, a 60-order Earth gravity field model derived via the energy balance approach was validated against contemporaneous models published by the University of Texas Center for Space Research (CSR), German Research Centre for Geosciences (GFZ), and Jet Propulsion Laboratory (JPL). The results reveal a two-order-of-magnitude enhancement in inversion precision, with model accuracy improving from 10⁻⁶–10⁻⁷ to 10⁻⁸–10⁻⁹. This method provides a robust solution for enhancing the reliability of gravity field recovery in satellite-based geodetic missions. Full article

The multipole expansion on the basis of Spherical Harmonics is a multifaceted mathematical tool utilized in many disciplines of science and engineering. Regarding physics, in electromagnetism, the multipole expansion is exclusively focused on the scalar potential, $U\left(\mathit{r}\right)$, defined only in the so-called inside, $U_{in}\left(\mathit{r}\right)$, and outside, $U_{out}\left(\mathit{r}\right)$, spaces, separated by the middle space wherein the source resides, for both dielectric and magnetic materials. Intriguingly, though the middle space probably encloses more physics than the inside and outside spaces, it is never assessed in the literature, probably due to the rather complicated mathematics. Here, we investigate the middle space and introduce the multipole expansion of the scalar potential, $U_{mid}\left(\mathit{r}\right)$, in this, until now, unsurveyed area. This is achieved through the complementary superposition of the solutions of the inside, $U_{in}\left(\mathit{r}\right)$, and outside, $U_{out}\left(\mathit{r}\right)$, spaces when carefully adjusted at the interface of two appropriately defined subspaces of the middle space. Importantly, while the multipole expansion of $U_{in}\left(\mathit{r}\right)$ and $U_{out}\left(\mathit{r}\right)$ satisfies the Laplace equation, the expression of the middle space, $U_{mid}\left(\mathit{r}\right)$, introduced here satisfies the Poisson equation, as it should. Interestingly, this is mathematically proved by using the method of variation of parameters, which allows us to switch between the solution of the homogeneous Laplace equation to that of the nonhomogeneous Poisson one, thus completely bypassing the standard method in which the multipole expansion of ${|\mathit{r}-{\mathit{r}}^{\prime }|}^{-1}$ is used in the generalized law of Coulomb. Due to this characteristic, the notion of $U_{mid}\left(\mathit{r}\right)$ introduced here can be utilized on a general basis for the effective calculation of the scalar potential in spaces wherein sources reside. The proof of concept is documented for representative cases found in the literature. Though here we deal with the static and quasi-static limit of low frequencies, our concept can be easily developed to the fully dynamic case. At all instances, the exact mathematical modeling of $U_{mid}\left(\mathit{r}\right)$ introduced here can be very useful in applications of both homogeneous and nonhomogeneous, dielectric and magnetic materials. Full article

The gradual deterioration of underground water infrastructure requires constant condition monitoring to prevent catastrophic failures, reduce leaks, and avoid costly unexpected repairs. However, given the large scale and tight budgets of water utilities, it is essential to implement strategies for optimal selection and deployment of monitoring technologies. This article introduces a unified framework and methods for optimally selecting condition monitoring technologies while locating their deployment at the most vulnerable pipe segments. The approach is underpinned by an R-E-R-A-V (Redundant, Established, Reliable, Accurate, and Viable) principle and asset management concepts. The proposed method is supported by a thorough review of assessment and monitoring technologies, as well as common sensor placement approaches. The approach selects optimal technology using a combination of technology readiness levels and SFAHP (Spherical Fuzzy Analytic Hierarchy Process). Optimal placement is achieved with a k-Nearest Neighbors (kNN) model tuned with minimal topological and physical pipeline system features. Feature engineering is performed with OPTICS (Ordering Points to Identify the Clustering Structure) by evaluating the pipe segment vulnerability to failure-prone areas. Both the optimal technology selection and placement methods are integrated through a proposed algorithm. The optimal placement of monitoring technology is demonstrated through a modified benchmark network (Net3). The results reveal an accurate model with robust performance and a harmonic mean of precision and recall of approximately 65%. The model effectively identifies pipe segments requiring monitoring to prevent failures over a period of 11 years. The benefits and areas of future exploratory research are explained to encourage improvements and additional applications. Full article

This work shows a three-dimensional (3D) layerwise model for static and free vibration analyses of functionally graded piezomagnetic materials (FGPM) spherical shell structures where magnetic and elastic fields are completely coupled. The 3D magneto-elastic governing equations for spherical shells are made of the three equations of equilibrium in three-dimensional form and the three-dimensional divergence equation for the magnetic induction. Governing equations are written in the orthogonal mixed curvilinear reference system ($\alpha$, $\beta$, z) allowing the analysis of several curved and flat geometries (plates, cylindrical shells and spherical shells) thanks to proper considerations of the radii of curvature. The static cases, actuator and sensor configurations and free vibration investigations are proposed. The resolution method uses the imposition of the Navier’s harmonic forms in the two in-plane directions and the exponential matrix methodology in the transverse normal direction. Single-layered and multilayered simply-supported FGPM structures have been investigated. In order to understand the behavior of FGPM structures, numerical values and trends along the thickness direction for displacements, stresses, magnetic potential, magnetic induction and free vibration modes are proposed. In the results section, a first assessment phase is proposed to demonstrate the validity of the formulation and to fix proper values for the convergence of results. Therefore, a new benchmark section is presented. Different cases are proposed for several material configurations, load boundary conditions and geometries. The possible effects involved in this problem (magneto-elastic coupling and effects related to embedded materials and thickness values of the layers) are discussed in depth for each thickness ratio. The innovative feature proposed in the present paper is the exact 3D study of magneto-elastic coupling effects in FGPM plates and shells for static and free vibration analyses by means of a unique and general formulation. Full article

The relationship between ionospheric research and global navigation satellite systems (GNSS) can be analysed through two approaches. The direct approach utilises ionospheric models to mitigate its influence, while the indirect approach leverages GNSS data to model ionospheric parameters. This study presents an indirect approach in which the total electron content (TEC), a fundamental parameter for ionospheric conditions, is modelled as a harmonic function using spherical harmonic (SH) expansion. Station-specific (local) and regional ionospheric models are developed by decomposing ionospheric influence into deterministic and stochastic components. GNSS data from seven evenly distributed stations in Serbia were used to estimate TEC coefficients. Local models were provided in the ION format as SH coefficients, allowing TEC determination at any epoch, while regional models had a ${0.5}^{\circ }\times {0.5}^{\circ }$ spatial and 2 h temporal resolution. The TEC root mean square (RMS) values ranged from 0.2 to 0.5 TECU (total electron content unit), with a mean of 0.3 TECU. Validation against global ionospheric maps showed agreement within 5.0 TECU. The impact of the SH expansion degree and order on TEC values was also analysed. These results refine regional ionospheric modelling, improving GNSS positioning accuracy in Serbia and beyond. Full article

Recent studies have identified a near six-year oscillation (SYO) in Global Navigation Satellite Systems (GNSS) surface displacements, with a degree 2, order 2 spherical harmonic (SH) pattern and retrograde motion. The cause is uncertain, with proposals ranging from deep Earth to near-surface sources. This study investigates the SYO and possible causes from surface loading. Considering the irregular spatiotemporal distribution of GNSS data and the variety of contributors to surface displacements, we used synthetic experiments to identify optimal techniques for estimating low degree SH patterns. We confirm a reported retrograde SH degree 2, order 2 displacement using GNSS data from the same 35 stations used in a previous study for the 1995–2015 period. We also note that its amplitude diminished when the time span of observations was extended to 2023, and the retrograde dominance became less significant using a larger 271-station set. Surface loading estimates showed that terrestrial water storage (TWS) loads contributed much more to the GNSS degree 2, order 2 SYO, than atmospheric and oceanic loads, but TWS load estimates were highly variable. Four TWS sources—European Centre for Medium-Range Weather Forecasts Reanalysis 5 (ERA5), Modern-Era Retrospective analysis for Research and Applications (MERRA), Global Land Data Assimilation System (GLDAS), and Gravity Recovery and Climate Experiment (GRACE/GRACE Follow-On)—yielded a wide range (24% to 93%) of predicted TWS contributions with GRACE/GRACE Follow-On being the largest. This suggests that TWS may be largely responsible for SYO variations in GNSS observations. Variations in SYO GNSS amplitudes in the extended period (1995–2023) were also consistent with near surface sources. Full article

The spherical harmonics method ($P_N$ method) is often used for solving the radiative transport equation in terms of analytical functions. A severe and unsolved problem in this context was the evaluation of the angle-resolved radiance near sources and boundaries, which is a serious limitation of this method in view of concrete applications, e.g., in biomedical optics for investigating the different types of optical microscopy, within NIR spectroscopy, such as for the determination of ingredients in foods or in pharmaceuticals, and within physics-based rendering. In this article, we report on a hybrid method that enables accurate evaluation of the angle-resolved radiance directly at the boundary of an anisotropically scattering medium, avoiding the problems of the traditional $P_N$ methods. The derived integral equation needed for the realization of the hybrid $P_N$ method is formally valid for an arbitrary convex bounded medium. The proposed approach can be evaluated with practically the same computational effort as the traditional $P_N$ method while being far more accurate. Full article

Ionospheric tomography, an effective method for reconstructing 3-D electron density, is traditionally pictured by 3-D IED (ionospheric electron density) slices to express ionospheric disturbances, which may overlook the critical information in 3-D spherical manifold space. Here, we develop a novel visualization framework that integrates tomography reconstruction with a spherical latitude–longitude grid system, enabling the comprehensive characterization of 3-D IED dynamic evolution in 3-D manifold spherical space. Through this method, we visualized two cases: the Hualien earthquake on 2 April 2024 and the geomagnetic storm on 24 April 2023. The results demonstrate the evolution of the electron density during earthquake and geomagnetic storms in the real 3-D space, showing that seismic events induce bottom-up IED negative anomalies localized near epicenters, while geomagnetic storms trigger top-down depletion processes, with IED propagating from higher altitudes in the real 3-D manifold space. Compared to the conventional slice, our visualization model can visualize the characteristics, with the coverage area being observed to increase with the altitude within the same geospatial coordinates. This framework can advance the identification of ionosphere anomalies by enabling the precise differentiation of anomaly sources. This work bridges gaps in geospatial modeling by harmonizing ionospheric tomography with Earth system grids, offering a feasible solution for analyzing multi-scale ionospheric phenomena. Full article