Showing papers on "Spherical shell published in 2001"
TL;DR: In this article, the subdivision shell elements of Cirak et al. were extended to the finite-deformation range, allowing for finite membrane and thickness stretching, as well as for large deflections and bending strains.
Abstract: We have extended the subdivision shell elements of Cirak et al. [18] to the finite-deformation range. The assumed finite-deformation kinematics allows for finite membrane and thickness stretching, as well as for large deflections and bending strains. The interpolation of the undeformed and deformed surfaces of the shell is accomplished through the use of subdivision surfaces. The resulting ‘subdivision elements’ are strictly C1-conforming, contain three nodes and one single quadrature point per element, and carry displacements at the nodes only. The versatility and good performance of the subdivision elements is demonstrated with the aid of a number of test cases, including the stretching of a tension strip; the inflation of a spherical shell under internal pressure; the bending and inflation of a circular plate under the action of uniform pressure; and the inflation of square and circular airbags. In particular, the airbag solutions, while exhibiting intricate folding patterns, appear to converge in certain salient features of the solution, which attests to the robustness of the method.
268 citations
TL;DR: In this article, the authors investigated the asymptotic properties of inertial modes confined in a spherical shell when viscosity tends to zero and showed that these attractors exist in bands of frequencies the size of which decreases with the number of reflection points of the attractor.
Abstract: We investigate the asymptotic properties of inertial modes confined in a spherical shell when viscosity tends to zero. We first consider the mapping made by the characteristics of the hyperbolic equation (Poincare's equation) satisfied by inviscid solutions. Charac- teristics are straight lines in a meridional section of the shell, and the mapping shows that, generically, these lines converge towards a periodic orbit which acts like an attrac- tor (the associated Lyapunov exponent is always negative or zero). We show that these attractors exist in bands of frequencies the size of which decreases with the number of reflection points of the attractor. At the bounding frequencies the associated Lyapunov exponent is generically either zero or minus infinity. We further show that for a given frequency the number of coexisting attractors is finite. We then examine the relation between this characteristic path and eigensolutions of the inviscid problem and show that in a purely two-dimensional problem, convergence towards an attractor means that the associated velocity field is not square-integrable. We give arguments which generalize this result to three dimensions. Then, using a sphere immersed in a fluid filling the whole space, we study the critical latitude singularity and show that the velocity field diverges as 1/ √ d, d being the distance to the characteristic grazing the inner sphere. We then consider the viscous problem and show how viscosity transforms singularities into internal shear layers which in general betray an attractor expected at the eigenfre- quency of the mode. Investigating the structure of these shear layers, we find that they are nested layers, the thinnest and most internal layer scaling with E 1/3 -scale, E being the Ekman number; for this latter layer, we give its analytical form and show its simi- larity to vertical 1 -shear layers of steady flows. Using an inertial wave packet traveling around an attractor, we give a lower bound on the thickness of shear layers and show how eigenfrequencies can be computed in principle. Finally, we show that as viscosity decreases, eigenfrequencies tend towards a set of values which is not dense in (0,2), contrary to the case of the full sphere ( is the angular velocity of the system). Hence, our geometrical approach opens the possibility of describing the eigenmodes and eigenvalues for astrophysical/geophysical Ekman numbers (10 −10 − 10 −20 ), which are out of reach numerically, and this for a wide class of containers.
182 citations
TL;DR: In this paper, the authors studied the ZW generation by thermal convection in rotating spherical shells using numerical calculations and found that in a thin shell (radius ratio 0.75) with stress-free boundaries, nearly 90% of the total kinetic energy is contained in ZW at Rayleigh number 106 and Taylor number 4.4 × l07.
Abstract: Zonal wind (ZW) generation by thermal convection in rotating spherical shells is studied using numerical calculations. Strong ZW accompany quasi-geostrophic, high Rayleigh number convection in shells with stress-free boundaries. In a thin shell (radius ratio 0.75) with stress-free boundaries, nearly 90% of the total kinetic energy is contained in the ZW at Rayleigh number 106 and Taylor number 4.4 × l07. The same parameters in a thicker shell produce weaker convection and weaker ZW. Rigid boundaries reduce the kinetic energy in the ZW to less than 20% of the total. The ZW are eastward (prograde) in the equatorial region and westward at higher latitudes, and are driven by Reynolds stresses associated with the convection. Episodes with strong ZW alternate with episodes of strong convection. Although far from the dynamical regime of Jupiter and Saturn, our results support the interpretation that the prograde equatorial jets on these planets originate from deep convection.
118 citations
03 Jul 2001
TL;DR: A 3-D spherical navigator echo technique for MRI that can measure rigid body motion in all six degrees of freedom simultaneously, in a single echo, by sampling a spherical shell in k-space is developed.
Abstract: We are developing a 3-D spherical navigator (SNAV) echo technique for MRI that can measure rigid body motion in all six degrees of freedom simultaneously, in a single echo, by sampling a spherical shell in k-space. MRI pulse sequences were developed to acquire varying amounts of data on such a shell. 3-D rotations of an imaged object simply rotate the data on this shell, and can be detected by registration of magnitude values. 3-D translations add phase shifts to the data on the shell, and can be detected with a weighted least squares fit to the phase differences at corresponding points. Data collected with a computer controlled motion phantom with known rotational and translational motions was used to evaluate the technique. The accuracy and precision of the technique depend on the sampling density, with roughly 1000 sample points necessary for accurate detection to within the error limits of the motion phantom. This number of samples can be captured in a single SNAV echo with a 3-D helical spiral trajectory. Motion detection in MRI with spherical navigator echoes is thus feasible and practical. Accurate motion measurements about all three axes, suitable for retrospective or prospective correction, can be obtained in a single pulse sequence.
80 citations
TL;DR: In this article, a detailed comparison between axisymmetric spherical shell and cylindrical geometry for use in mantle convection modeling is presented, where a rescaling of the mantle and core radii is adopted, such that the curvature of the cylinear model approximates that of the spherical Earth, the heat and mass transport properties turn out to be very similar.
61 citations
TL;DR: In this paper, the structural response of a composite shell structure intended as a model of an under-water vehicle for service in sea environment was investigated using both analytical expressions and non-linear numerical analysis.
Abstract: The paper is concerned with the structural response of a composite shell structure intended as a model of an under-water vehicle for service in sea environment. The main objective of the research is the prediction of the collapse pressure using both analytical expressions and linear or non-linear numerical analysis and the following comparison with the experimental pressure obtained in off-shore tests. The structure is composed of three basic parts with regular geometry: a cylindrical part (with the following geometrical properties: R/t=30.5, L/R=2 being the internal radius 305 mm, the length 610 mm and the thickness 10 mm) and two conical and spherical end-closures with the same thickness. The cylindrical shell was made up of 7 plies of E-glass woven roving with polyester resin. Various structural analyses were conducted before performing the experiment in the sea to verify the reliability of the analytical and numerical tools. Firstly the entire model was analysed to predict the nature of the collapse (material failure or elastic buckling) and it was stated that the collapse was due to elastic buckling of the cylindrical part. Consequently, the attention was focused on this component and approximation formulae for the evaluation of the linear buckling pressure of isotropic and composite cylindrical shells were used together with finite element models. Afterward the study was enlarged to consider the effects of the recorded geometric imperfections into a non-linear buckling analysis. The collapse pressures were compared to the design values derived from the available recommendations and to the experimental result obtained in an off-shore test (1.3 MPa).
60 citations
TL;DR: In this article, the authors investigated numerically the flow of an electrically conducting fluid confined in a spherical shell, with the inner sphere rotating, the outer sphere stationary, and a strong magnetic field imposed parallel to the axis of rotation, and showed that for sufficiently large rotation rates both the shear layer and the jet spawn a series of vortices encircling the tangent cylinder.
Abstract: We investigate numerically the flow of an electrically conducting fluid confined in a spherical shell, with the inner sphere rotating, the outer sphere stationary, and a strong magnetic field imposed parallel to the axis of rotation. It has previously been shown that the axisymmetric basic state depends strongly on the electromagnetic boundary conditions used, with insulating boundaries yielding a shear layer, but conducting boundaries yielding a counter–rotating jet, where in both cases these structures are located on the cylinder parallel to the imposed field and tangent to the inner sphere. Here we compute the non–axisymmetric instabilities of these basic states, and show that for sufficiently large rotation rates both the shear layer and the jet spawn a series of vortices encircling the tangent cylinder. Finally, we consider the fully three–dimensional nonlinear equilibration, and show that in the supercritical regime a secondary bifurcation occurs in which the number of vortices (for the shear layer) or vortex pairs (for the jet) is reduced by one.
58 citations
TL;DR: In this article, numerical solutions for fluid flows in precessing spherical cavities and spherical shells have been obtained and compared with earlier analytical expressions, and it is shown that the approximate validity of the analytical expressions extends further than could have been expected.
Abstract: Numerical solutions for fluid flows in precessing spherical cavities and spherical shells have been obtained and are compared with earlier analytical expressions. It is shown that the approximate validity of the analytical expressions extends further than could have been expected. The details of the flow structure exhibit significant departures, however, from the assumptions of the analytical theory. Standing waves are found in the case of sufficiently thin shells. Some results on instabilities of the basic flow are also discussed.
52 citations
TL;DR: In this article, columnar convection in a rapidly rotating spherical shell into a stably stratified fluid layer near the outer boundary is investigated, and an analytical expression for the penetration thickness is derived by considering perturbations to a stable rotating Boussinesq fluid in a semi-infinite region with the rotation axis tilted relative to the gravity axis.
51 citations
01 Jan 2001
TL;DR: In this paper, a model of the geodynamo that is based on the symmetries of velocity fields in a rotating spherical shell, and that contains such a cycle is presented.
Abstract: The Earth’s magnetic field is by and large a steady dipole, but its history has been punctuated by intermittent excursions and reversals. This is at least superficially similar to the behaviour of differential equations containing structurally stable heteroclinic cycles. We present a model of the geodynamo that is based on the symmetries of velocity fields in a rotating spherical shell, and that contains such a cycle. Patterns of excursions and reversals that resemble the geomagnetic record can be obtained by introducing small symmetry-breaking terms.
49 citations
TL;DR: In this article, a general non-axisymmetric exact analysis of the statics of a laminated piezoelectric hollow sphere is presented by using a state space method.
TL;DR: It is shown that contrast agents also support shell resonance responses in addition to the monopole response, which holds untapped potential for clinical ultrasound applications such as tissue perfusion studies and high frequency contrast agent imaging.
Abstract: Ultrasound contrast agents are encapsulated microbubbles, filled either with air or a higher weight molecular gas, ranging in size from 1 to 10 /spl mu/m in diameter. The agents are modeled as air-filled spherical elastic shells of variable thickness and material properties. The scattered acoustic field is computed from a modal series solution, and reflectivity and angular scattering are then determined from the computed fields for agents of various properties. We show that contrast agents also support shell resonance responses in addition to the monopole response, which has been the focus of previous contrast agent studies. Lamb waves appear to be the source of these additional responses. A shell or curvature Lamb wave generates dipole peaks in the 1- to 40-MHz range for 2.5 to 3.5 /spl mu/m radius agents with elastic properties approximating those of albumin protein. The inclusion of damping affects the lower frequency dipole peaks but is less important for responses occurring above approximately 30 MHz. Moreover, these responses hold untapped potential for clinical ultrasound applications such as tissue perfusion studies and high frequency contrast agent imaging.
TL;DR: In this paper, the problem of dynamic thermoelastic stresses in a spherical shell with fixed boundaries whose inner surface is subjected to a step jump in temperature is considered, and exact solutions for the temperature and displacement equations are obtained using the Laplace transform and eigenfunction expansion method, respectively.
Abstract: The problem of dynamic thermoelastic stresses in a spherical shell with fixed boundaries whose inner surface is subjected to a step jump in temperature is considered. The analysis is carried out in the uncoupled framework under the classical Lord and Shulman (1967) and Green and Lindsay (1972) formulations of thermoelasticity. Exact solutions for the temperature and displacement equations are obtained using the Laplace transform and eigenfunction expansion method, respectively. Of particular interest is the propagation and nature of discontinuities in the temperature, displacement, and stress fields that are possible under the two nonclassical theories. Exact expressions for the magnitudes of discontinuities in these quantities are also given, and shock waves are noted. Numerical results are presented graphically along with a comparison of the three models. In addition, special and limiting cases of the physical parameters are investigated and the application of this research to fuel vessels used in rocke...
TL;DR: In this paper, the problem of non-linear planetary dynamos with a prescribed two-or three-dimensional, time-dependent α is investigated using a finite element method, where magnetic fields are generated in a turbulent electrically conducting fluid spherical shell of constant electric conductivity surrounded by an electrically insulating uniform mantle.
TL;DR: In this article, the Casimir stress on a spherical shell in the de Sitter background for a massless scalar field satisfying Dirichlet boundary conditions on the shell is calculated.
Abstract: The Casimir stress on a spherical shell in the de Sitter background for a massless scalar field satisfying Dirichlet boundary conditions on the shell is calculated. The metric is written in conformally flat form. Although the metric is time dependent, no particles are created. The Casimir stress is calculated for inside and outside of the shell with different backgrounds corresponding to different cosmological constants. The detailed dynamics of the bubble depends on different parameters of the model. Specifically, bubbles with a true vacuum inside expand if the difference in the vacuum energies is small, otherwise they collapse.
TL;DR: In this article, the exact solution of the boundary (Ekman) layer flow associated with inertial r-modes has been derived for the case of a spherical container, and it was shown that the presence of an inner core has little influence on this dissipation.
Abstract: Recently, eigenmodes of rotating fluids, namely, inertial modes, have received much attention in relation to their destabilization when coupled to gravitational radiation within neutron stars. However, these modes have been known for a long time in fluid dynamics. We give a short account of their history and review our present understanding of their properties. Considering the case of a spherical container, we then give the exact solution of the boundary (Ekman) layer flow associated with inertial r-modes and show that previous estimations all underestimated the dissipation by these layers. We also show that the presence of an inner core has little influence on this dissipation. As a conclusion, we compute the window of instability in the temperature/rotation plane for a crusted neutron star when it is modeled by an incompressible fluid.
TL;DR: In this article, the free damped vibrations of sandwich shells of revolution were investigated using a zig-zag model and the vibration analysis under consideration of damping of cylindrical, conical and conical anisotropic structures.
Abstract: Using a zig-zag model the free damped vibrations of sandwich shells of revolution are investigated. As special cases the vibration analysis under consideration of damping of cylindrical, conical an...
TL;DR: In this article, a displacement separation technique is employed to simplify the basic equations of a piezoceramic body with radial inhomogeneity, and the controlling equations are finally reduced to an uncoupled second-order ordinary differential equation and a coupled system of three second order ordinary differential equations.
TL;DR: In this article, a preferred definition of the Rayleigh number is proposed that is particularly well-suited for geodynamo simulations, where the governing equations can be scaled in more than one way.
TL;DR: In this paper, the size distribution and surface condition of nano-crystalline diamond powder produced by shock detonation were predicted using the Debye equation, starting with a set of spherical diamond ball models.
Abstract: High-resolution transmission electron microscopy and electron nano-diffraction patterns provide useful information concerning the size distribution and surface condition of nano-crystalline diamond powder produced by shock detonation. Analysis of X-ray powder diffractometer data, using the Debye equation to predict the powder profile, starting with a set of spherical diamond ball models, allows the mean size to be obtained for bulk quantities of the same powders. The electron powder diffraction patterns for some diamond spheres, and also spherical shell, models of the surface structures were calculated using the Debye equation, and compared with experiment. It is suggested that the surface structure may become accessible to diffraction methods, if a single nano-crystal is irradiated and both the single crystal pattern plus the diffuse scattering are recorded, as a function of the orientation of that single crystal.
TL;DR: In this paper, the authors presented a comprehensive classification of all observed QPOs within the framework of the transition layer model using a large set of Rossi X-ray Timing Explorer (RXTE) data for Sco X-1.
Abstract: We present a comprehensive classification of all observed QPOs within the framework of the transition layer model using a large set of Rossi X-ray Timing Explorer (RXTE) data for Sco X-1 The model assumes an optically thin material along the observer's line of sight in the horizontal branch and an increasingly optically thick material while in the other two branches that is consistent with X-ray and radio observations and the disk transition layer model of QPOs We identify the ~ 6 Hz frequencies in the normal branch as acoustic oscillations of a spherical shell around the neutron star (NS) that is formed after radiation pressure near the Eddington accretion rate destroys the disk The size of the shell is on the order of one NS radii from the NS We also estimate the upper limit of Sco X-1's magnetic field to be 07 x 10^6 G at about one NS radii above the NS surface while in the horizontal X-ray branch
TL;DR: In this article, the bending behavior of fiber reinforced plastic laminated shells without a cutout has been studied using the finite element method based on a higher-order shear deformation theory.
TL;DR: In this paper, an exact analysis on the piezothermoelastic behavior of a pyroelectric spherical shell with arbitrary thickness subjected to prescribed temperatures at two spherical surfaces is presented.
Abstract: An exact analysis on the piezothermoelastic behavior of a pyroelectric spherical shell with arbitrary thickness subjected to prescribed temperatures at two spherical surfaces is presented in the article. At first, three displacement functions are introduced to simplify the three-dimensional equations of equilibrium and the electrostatic charge equation of a spherically isotropic pyroelectric body. By expanding the displacement functions as well as the electric potential in terms of spherical harmonic functions, these equations are further converted to an uncoupled Euler-type, second-order ordinary differential equation and a coupled system of three such equations. A general solution to the homogeneous ordinary differential equations is derived. The piezothermoelastic analysis of a pyroelectric spherical shell is then given and numerical illustrations are presented.
TL;DR: This study quantitatively investigates the intrinsic limitations in the assignment of a location to the equivalent dipole due to distributed electrical source and a method is devised to obtain the size of the distributed source during the cardiac cycle.
Abstract: Although the single equivalent point dipole model has been used to represent well-localised bio-electrical sources, in realistic situations the source is distributed. Consequently, position estimates of point dipoles determined by inverse algorithms suffer from systematic error due to the non-exact applicability of the inverse model. In realistic situations, this systematic error cannot be avoided, a limitation that is independent of the complexity of the torso model used. This study quantitatively investigates the intrinsic limitations in the assignment of a location to the equivalent dipole due to distributed electrical source. To simulate arrhythmic activity in the heart, a model of a wave of depolarisation spreading from a focal source over the surface of a spherical shell is used. The activity is represented by a sequence of concentric belt sources (obtained by slicing the shell with a sequence of parallel plane pairs), with constant dipole moment per unit length (circumferentially) directed parallel to the propagation direction. The distributed source is represented by N dipoles at equal arc lengths along the belt. The sum of the dipole potentials is calculated at predefined electrode locations. The inverse problem involves finding a single equivalent point dipole that best reproduces the electrode potentials due to the distributed source. The inverse problem is implemented by minimising the chi2 per degree of freedom. It is found that the trajectory traced by the equivalent dipole is sensitive to the location of the spherical shell relative to the fixed electrodes. It is shown that this trajectory does not coincide with the sequence of geometrical centres of the consecutive belt sources. For distributed sources within a bounded spherical medium, displaced from the sphere's centre by 40% of the sphere's radius, it is found that the error in the equivalent dipole location varies from 3 to 20% for sources with size between 5 and 50% of the sphere's radius. Finally, a method is devised to obtain the size of the distributed source during the cardiac cycle.
TL;DR: In this article, an exact static stress analysis of a multilayered elastic spherical shell (hollow sphere) completely based on three-dimensional elasticity for spherical isotropy is presented.
Abstract: This paper presents an exact static stress analysis of a multilayered elastic spherical shell(hollow sphere) completely based on three-dimensional elasticity for spherical isotropy.Two independent state equations are derived after introducing three displacement func-tions and two stress functions. In particular, a variable substitution technique is used toderive the state equations with constant coefficients. Matrix theory is then employed toobtain the relationships between the state variables at the upper and lower surfaces ofeach lamina. By virtue of the continuity conditions between two adjacent layers, a second-order linear algebraic equation and a fourth-order one about the boundary variables atthe inner and outer surfaces of a multilayered spherical shell are obtained. Numericalexamples are presented to show the effectiveness of the present method.@DOI: 10.1115/1.1343913#
Patent•
11 Jul 2001
TL;DR: In this paper, a movable spherical robot is disclosed, where the internal frame is linked to bearing frame via suspended axle and beval gears are driven by reversible motor to generate relative rotation between frame and spherical body, generating steering action of spherical body.
Abstract: A movable spherical robot is disclosed The perpendicularly fixed middle frame and bearing frame are in the spherical shell The internal frame is linked to bearing frame via suspended axle The beval gears perpendicular to each other are driven by reversible motor to generate relative rotation between frame and spherical body, generating steering action of spherical body The driven motor drivesinternal frame via synchronous belt-wheel to do circumference motion along synchronous toothed belt Said motor is remote controlled Its advantages are compact structure and high dynamic power to pass through obstacles
TL;DR: In this article, the velocity field is decomposed into a Rossby-wave (r-mode) component and a gravity-wave component, and their evolution and interaction is investigated.
Abstract: The solar tachocline is a differentially rotating shear layer that appears to be largely located in a stably stratified region of the solar interior, immediately below the convective envelope. Turbulence could be generated in this layer by overshooting convection and by shear instabilities. In order to study the nature of such turbulence, we introduce a three-dimensional nonlinear model of stably stratified fluid motions in a thin rotating spherical shell. Numerical solutions of the freely evolving system are presented for a variety of random initial conditions and parameter values. The velocity field is decomposed into a Rossby-wave (r-mode) component and a gravity-wave (g-mode) component, and their evolution and interaction is investigated. The two components are closely coupled when the rotation is rapid, with oscillatory exchanges of energy and wave modes that propagate toward the equator. Upscale kinetic energy transfer is produced by nonlinear interactions among r-modes and also among g-modes when the stratification is strong.
TL;DR: In this article, the authors presented a comprehensive classification of all observed quasi-periodic oscillations (QPOs) within the framework of the transition layer model using a large set of Rossi X-Ray Timing Explorer data for Scorpius X-1.
Abstract: We present a comprehensive classification of all observed quasi-periodic oscillations (QPOs) within the framework of the transition layer model using a large set of Rossi X-Ray Timing Explorer data for Scorpius X-1. The model assumes an optically thin material along the observer's line of sight in the horizontal branch and an increasingly optically thick material while in the other two branches that is consistent with X-ray and radio observations and the disk transition layer model of QPOs. We identify the ~6 Hz frequencies in the normal branch as acoustic oscillations of a spherical shell around the neutron star (NS) that is formed after radiation pressure near the Eddington accretion rate destroys the disk. The size of the shell is on the order of one NS radius from the NS. We also estimate the upper limit of Sco X-1's magnetic field to be 0.7 × 106 G at about one NS radius above the NS surface while in the horizontal X-ray branch.
TL;DR: In this article, an asymptotic solution is formulated for non-linear buckling of elastically restrained imperfect shallow spherical shells continuously supported on a nonlinear elastic foundation, which can be used easily to evaluate the effects of these factors on buckling behaviors.
Patent•
25 Sep 2001
TL;DR: In this article, a light weight fuel tank for a hydrogen powered airplane used as a platform for communication repeaters for communication services is described. Butler et al. describe an outer spherical shell member of a sandwich configuration surrounding an inner thin walled spherical shell members in which the liquid hydrogen is contained.
Abstract: A light weight fuel tank for a hydrogen powered airplane used as a platform for communication repeaters for communication services. An outer spherical shell member of a sandwich configuration surrounds an inner thin walled spherical shell member in which the liquid hydrogen is contained. A radial gap between the shell members is evacuated to a high vacuum. The facing surfaces of the shell members are coated with a low emissivity material. Electrical heaters are provided to control the evaporation rate of the hydrogen to match the fuel usage and to prevent icing during ascent and descent of the airplane.