Showing papers on "Spherical shell published in 2022"

An improved pseudo spherical shell algorithm for vector radiative transfer

Abstract: • An improved pseudo-spherical shell (IPSS) algorithm has been developed. • The IPSS algorithm is highly accurate for the Earth atmosphere. • This work will greatly improve the remote sensing of environment in polar regions. The radiative transfer solution in a plane-parallel geometry is a good approximation for many applications in the Earth or other planetary systems as the Earth’s radius is quite large ( ∼ 6371 km). The plane-parallel geometry is however problematic in polar regions where the solar zenith angle is usually large ( > 60 ∘ ) and the spherical shell effect is significant. One simple solution is the so-called pseudo-spherical shell (PSS) approximation, which treats the solar beam attenuation exactly along the nadir in the spherical shell atmosphere while keeping the plane parallel geometry for multiple scattering calculation. The PSS approximation improves the solution for intermediately large solar zenith angles, though the error is still large for large viewing zenith angles. In order to further improve the treatment of a spherical shell geometry, we have developed an improved pseudo spherical shell (IPSS) approximation. In the method, we used the following techniques: I.) The single scattering solution is solved exactly for the spherical shell atmosphere; II.) The multiple to single scattering solution ratio is solved using the plane-parallel geometry with our radiative transfer code based on successive order of scattering method; III.) The ratio of the multiple to single scattering solution is assumed to be the same for both the plane parallel and spherical shell geometry. We tested the performance of IPSS with two benchmark cases involving the Rayleigh scattering matrix. If the Rayleigh optical thickness is 0.25, the error is smaller than 1% for most of the viewing directions ( < 70 ∘ ). If the Rayleigh optical thickness is 1.0, the error is bounded within ± 2%. The error does not show obvious dependence on the viewing zenith angle. Our newly developed IPSS scheme is highly accurate and can be used in the remote sensing applications of the polar regions.

Analysis of the Acceleration Response Spectra of Single-Layer Spherical Reticulated Shell Structures

Abstract: In recent years, earthquake disasters have seriously damaged nonstructural components, so it is necessary to study their seismic performance. However, the existing scholarly research mainly concentrates on multistorey and high-rise buildings, and there are still deficiencies in the analysis of the seismic performance of the nonstructural components in large-span structures under seismic action. In this paper, the acceleration responses of a single-layer spherical reticulated shell structure are compared with those described in the current seismic design codes of the nonstructural components, and it is found that the current codes are not fully applicable to the seismic design of the nonstructural components in reticulated shell structures. The calculation formulas of the acceleration response spectra of single-layer spherical shell nodes are theoretically derived, and the shell node acceleration response spectra are affected by higher-order modes, orthogonal horizontal seismic input directions, and the membrane stiffness of the shell nodes. The variations in the acceleration responses of the shell nodes with node position and rise-to-span ratio are analysed, and a design method for the equivalent seismic action of the nonstructural components in a single-layer spherical reticulated shell with a roofing system is proposed.

Buckling of viscoelastic spherical shells

Abstract: Viscoelastic spherical shells exhibit a wide range of time/rate-dependent buckling behaviors when subjected to pressure. For certain loadings, buckling can even occur after a significant time delay, termed creep buckling. To gain a thorough understanding of the nonlinear time-dependent buckling behavior of viscoelastic spherical shells, this work develops an analytical model employing the small-strain, moderate-rotation shell theory combined with a linearly viscoelastic material law. Numerical results are presented for axisymmetric spherical shells with geometric imperfections for two types of loading: a prescribed rate of volume change and a prescribed pressure that remains constant after it is applied. The first type reveals the rate-dependent behavior of viscoelastic buckling while the constant pressure loading is used to quantify creep buckling phenomena. The results show that viscoelasticity and loading rates play important roles in the load-carrying behavior of these shells, and the results for the constant pressure loading reveal an unexpected and important connection between the short-time elastic buckling limit and the long-time creep buckling limit. An imperfection sensitivity map is constructed for the constant pressure loading showing three regimes with qualitatively different behaviors: near-instantaneous buckling, creep buckling and no buckling.

OUP accepted manuscript

Abstract: SUMMARY We present investigations of rapidly rotating convection in a thick spherical shell geometry relevant to planetary cores, comparing results from quasi-geostrophic (QG), 3-D and hybrid QG-3D models. The 170 reported calculations span Ekman numbers, Ek, between 10−4 and 10−10, Rayleigh numbers, Ra, between 2 and 150 times supercritical and Prandtl numbers, Pr, between 10 and 10−2. The default boundary conditions are no-slip at both the ICB and the CMB for the velocity field, with fixed temperatures at the ICB and the CMB. Cases driven by both homogeneous and inhomogeneous CMB heat flux patterns are also explored, the latter including lateral variations, as measured by Q*, the peak-to-peak amplitude of the pattern divided by its mean, taking values up to 5. The QG model is based on the open-source pizza code. We extend this in a hybrid approach to include the temperature field on a 3-D grid. In general, we find convection is dominated by zonal jets at mid-depths in the shell, with thermal Rossby waves prominent close to the outer boundary when the driving is weaker. For the thick spherical shell geometry studied here the hybrid method is best suited for studying convection at modest forcing, $Ra \le 10 \, Ra_c$ when Pr = 1, and departs from the 3-D model results at higher Ra, displaying systematically lower heat transport characterized by lower Nusselt and Reynolds numbers. We find that the lack of equatorially-antisymmetric motions and z-correlations between temperature and velocity in the buoyancy force contributes to the weaker flows in the hybrid formulation. On the other hand, the QG models yield broadly similar results to the 3-D models, for the specific aspect ratio and range of Rayleigh numbers explored here. We cannot point to major disagreements between these two data sets at Pr ≥ 0.1, with the QG model effectively more strongly driven than the hybrid case due to its cylindrically averaged thermal boundary conditions. When Pr is decreased, the range of agreement between the hybrid and 3-D models expands, for example up to $Ra \le 15 \, Ra_c$ at Pr = 0.1, indicating the hybrid method may be better suited to study convection in the low Pr regime. We thus observe a transition between two regimes: (i) at Pr ≥ 0.1 the QG and 3-D models agree in the studied range of Ra/Rac while the hybrid model fails when $Ra\gt 15\, Ra_c$ and (ii) at Pr = 0.01 the QG and 3-D models disagree for $Ra\gt 10\, Ra_c$ while the hybrid and 3-D models agree fairly well up to $Ra \sim 20\, Ra_c$. Models that include laterally varying heat flux at the outer boundary reproduce regional convection patterns that compare well with those found in similarly forced 3-D models. Previously proposed scaling laws for rapidly rotating convection are tested; our simulations are overall well described by a triple balance between Coriolis, inertia and Archimedean forces with the length-scale of the convection following the diffusion-free Rhines-scaling. The magnitude of Pr affects the number and the size of the jets with larger structures obtained at lower Pr. Higher velocities and lower heat transport are seen on decreasing Pr with the scaling behaviour of the convective velocity displaying a strong dependence on Pr. This study is an intermediate step towards a hybrid model of core convection also including 3-D magnetic effects.

Influence of Centrifugal Buoyancy in Thermal Convection within a Rotating Spherical Shell

Abstract: The dynamo action, which is of importance in the study of the geomagnetism mechanism, is considered to be caused by the convection structure formed inside a rotating spherical shell. This convection structure elongated in the rotation axis is generated by the action of both heat and rotation on the fluid inside a spherical shell. In this study, we analyzed thermal convection in such a rotating spherical shell and attempted to understand the phenomenon of this convective structure. It is known that each value of the Prandtl number, the Ekman number and the Rayleigh number and their balance are important for the generation of such convective structure. We fixed these three parameters and considered the effect of centrifugal buoyancy as the Froude number additionally. To investigate how the effects of centrifugal buoyancy affect the convective structure, we carried out both three-dimensional numerical simulations and linear stability analyses. In particular, we focused on the transition from axisymmetric flow to non-axisymmetric flow having wavenumbers in the toroidal direction and investigated both growth rate and phase velocity of the disturbance. It was found that axisymmetric flow tends to be maintained as the effect of centrifugal buoyancy increases.

Magnetoconvection in a rotating spherical shell in the presence of a uniform axial magnetic field

Abstract: ABSTRACT We report simulations of thermal convection and magnetic-field generation in a rapidly-rotating spherical shell, in the presence of a uniform axial magnetic field of variable strength. We consider the effect of the imposed field on the critical parameters (Rayleigh number, azimuthal wavenumber and propagation frequency) for the onset of convection, and on the relative importance of Coriolis, buoyancy and Lorentz forces in the resulting solutions. The imposed field strength must be of order one (corresponding to an Elsasser number of unity) to observe significant modifications of the flow; in this case, all the critical parameters are reduced, an effect that is more pronounced at small Ekman numbers. Beyond onset, we study the variations of the structure and properties of the magnetically-modified convective flows with increasing Rayleigh numbers. In particular, we note the weak relative kinetic helicity, the rapid breakdown of the columnarity, and the enhanced heat transport efficiency of the flows obtained for imposed field strengths of order one. Furthermore, magnetic and thermal winds drive a significant zonal flow in this case, which is not present with no imposed field or with stronger imposed fields. The mechanisms for magnetic field generation (particularly the lengthscales involved in the axisymmetric field production) vary with the strength of the imposed field, with three distinct regimes being observed for weak, order one, and stronger imposed fields. In the last two cases, the induced magnetic field reinforces the imposed field, even exceeding its strength for large Rayleigh numbers, which suggests that magnetically-modified flows might be able to produce large-scale self-sustained magnetic field. These magnetoconvection calculations are relevant to planets orbiting magnetically active hosts, and also help to elucidate the mechanisms for field generation in a strong-field regime.

Coupled vibration analysis of partially liquid-filled cylindrical shell considering free surface sloshing

Abstract: In this paper, a semi-analytical method is proposed to analyze the dynamic behavior of horizontal cylindrical shells partially filled with liquid, considering the sloshing effect of the free surface. Two coordinate systems are set at the midpoint of the free liquid surface and the geometric center of the cylindrical shell’s cross-section, respectively. The internal fluid is an inviscid, irrotational, and incompressible fluid. The liquid potential functions which satisfy the Laplace function are described in the anti-symmetrical/symmetrical forms based on the liquid surface coordinate system. Meanwhile, the coupled motion functions of the shell are established on the structural coordinate system using the Flugge shell theory. Through the continuous condition on the internal wet surface, the coupled system’s governing equations are achieved and solved by the coordinate transformation and the Galerkin method. The fluid sloshing frequencies and the coupled vibration frequencies of the shell are simultaneously obtained in this coupled model. The accuracy of this method is verified by the published data and the finite element method. Furthermore, the influences of the coupled system’s parameters on the shell’s natural frequencies and sloshing frequencies are discussed, and the coupling effect is revealed between the shell’s vibration and the sloshing of the free surface.

Mechanics under pressure of gold nanoparticle supracrystals: the role of the soft matrix

Abstract: We report on High Pressure Small Angle X-ray Scattering (HP-SAXS) measurements on 3D face-centered cubic (FCC) supracrystals (SCs) built from spherical gold nanoparticles (NPs). Dodecane-thiol ligands are grafted on the surface and ensure the stability of the gold NPs by forming a protective soft layer. Under a hydrostatic pressure of up to 12 GPa, the SC showed a high structural stability. The bulk elastic modulus of the SC was derived from the HP-SAXS measurements. The compression of the SC undergoes two stages: the first one related to the collapse of the voids between the NPs followed by the second one related to the compression of the soft matrix which gives a major contribution to the mechanical behavior. By comparing the bulk modulus of the SC to that of dodecane, the soft matrix appears to be less compressible than the crystalline dodecane. This effect is attributed to a less optimized chain packing under pressure compared to the free chains, as the chains are constrained by both grafting and confinement within the soft matrix. We conclude that these constraints on chain packing within the soft matrix enhance the stability of SCs under pressure.

Linear and geometric algebra approaches for sphere and spherical shell intersections in Rn

Abstract: Many practical problems involve sphere intersections. Examples include but are not limited to estimations using the Global Positioning System (GPS), data science applications and 3D protein structure determination. Motivated by practical situations, where radii of spheres are not known precisely, we consider what happens when a spherical shell must be included in the intersection. We present and compare two approaches for this problem: one uses linear algebra and the other is based on conformal geometric algebra (CGA). The theoretical development is illustrated with some numerical examples, where it is possible to note the main advantage of CGA compared to the linear algebra approach: even in dimensions higher than three, CGA naturally preserves the geometric intuition of the problem.

Active buckling of pressurized spherical shells : Monte Carlo Simulation

Abstract: We study the buckling of pressurized spherical shells by Monte Carlo simulations in which the detailed balance is explicitly broken -- thereby driving the shell active, out of thermal equilibrium. Such a shell typically has either higher (active) or lower (quiescent) fluctuations compared to one in thermal equilibrium depending on how the detailed balance is broken. We show that for the same set of elastic parameters, a shell that is not buckled in thermal equilibrium can be buckled if turned active. Similarly, a shell that is buckled in thermal equilibrium can unbuckle if turned quiescent. Based on this result, we suggest that it is possible to experimentally design microscopic elastic shells whose buckling can be optically controlled.

Compression of a pressurized spherical shell by a spherical or flat probe

Abstract: Measuring the mechanical properties of cells and tissues often involves indentation with a sphere or compression between two plates. Different theoretical approaches have been developed to retrieve material parameters (e.g., elastic modulus) or state variables (e.g., pressure) from such experiments. Here, we extend previous theoretical work on indentation of a spherical pressurized shell by a point force to cover indentation by a spherical probe or a plate. We provide formulae that enable the modulus or pressure to be deduced from experimental results with realistic contact geometries, giving different results that are applicable depending on pressure level. We expect our results to be broadly useful when investigating biomechanics or mechanobiology of cells and tissues.

Elastic–Plastic Equilibrium of a Hollow Ball Made of Inhomogeneous Ideal-Plastic Material

Abstract: The article discusses the solution to the elastoplastic problem of the development of the stress–strain state in an inhomogeneous thick-walled spherical shell. It is assumed that the shell material is ideally plastic. The inhomogeneity of the material consists in the change in the modulus of elasticity E and the yield stress \(\sigma _{T}\) along the thickness of the radius, which is described by power functions with three constants. The problem is solved in a centrally symmetric setting. Three options are considered: (1) plastic deformations occur near the inner surface of the shell, (2) plastic deformations occur between two surfaces of the shell, (3) an infinite array with a spherical cavity is considered.

On the stability of hyperelastic spherical and cylindrical shells subjected to external pressure using a numerical approach

Abstract: In this paper, the incremental equilibrium equations and corresponding boundary conditions for the isotropic, hyperelastic and incompressible shells are derived and then employed in order to analyze the behavior of spherical and cylindrical shells subjected to external pressure. The generalized differential quadrature (GDQ) method is utilized to solve the eigenvalue problem that results from a linear bifurcation analysis. The results are in full agreement with the previously obtained results and the effects of thickness and mode number are studied on the shell’s stability. For the spherical and cylindrical shells of arbitrary thickness which are subjected to external hydrostatic pressure, the symmetrical buckling takes place at a value of [Formula: see text] which depends on the geometric parameter [Formula: see text] and the mode number [Formula: see text], where [Formula: see text] and [Formula: see text] are the undeformed inner and outer radii, respectively, and [Formula: see text] is the ratio of the deformed inner radius to the undeformed inner radius.

Design and analysis of a spherical robot based on reaction wheel stabilization

Abstract: In this paper, we present the design of a spherical robot with a high spherical shell to drive-unit mass ratio. A drive unit and a novel shell structure design for the spherical robot are proposed, and reaction wheels are used to reduce the robot’s oscillations. Dynamic models of the spherical robot are established for linear motion and static state by using Lagrange equation. The stability of the robot is studied via simulations in different, such as accelerated state, uniform state, and so on. The results demonstrate the ability of the reaction wheel structure to enhance the robot’s stability.