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Relevant bibliographies by topics / Toroidal coordinates

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Academic literature on the topic 'Toroidal coordinates'

Author: Grafiati

Published: 5 June 2025

Last updated: 15 July 2025

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Journal articles on the topic "Toroidal coordinates"

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Nesnov, Dmitry V. "Field theory in normal toroidal coordinates." MATEC Web of Conferences 193 (2018): 03022. http://dx.doi.org/10.1051/matecconf/201819303022.

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Field theory is widely represented in spherical and cylindrical coordinate systems, since the mathematical apparatus of these coordinate systems has been thoroughly studied. Sources of field with more complex structures require new approaches to their study. The purpose of this research is to adapt the field theory referred to curvilinear coordinates and represent it in normal toroidal coordinates. Another purpose is to develop the foundations of geometric modeling with the use of computer graphics for visualizing the level surfaces. The dependence of normal toroidal coordinates on rectangular Cartesian coordinates and Lame coefficients is shown in this scientific paper. Differential characteristics of scalar and vector fields in normal toroidal coordinates are obtained: scalar and vector field laplacians, divergence, and rotation of vector field. The example shows the technique of modeling the field and its further computer visualization. The technique of reading the internal equation of the surface is presented and the influence of the values of the parameters on the shape of the surface is shown. For the first time, expressions of scalar and vector field characteristics in normal toroidal coordinates are obtained, the fundamentals of geometric modeling of fields with the use of computer graphics tools are developed for the purpose of providing visibility for their study.

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Guslienko, K. Y., and E. V. Tartakovskaya. "Hopf index of the toroidal magnetic hopfions in cylindrical and spherical dots." Low Temperature Physics 51, no. 6 (2025): 695–99. https://doi.org/10.1063/10.0036746.

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Topologically non-trivial 3D magnetization textures in the restricted curvilinear geometries are considered. 3D topological charges (the Hopf indices) are calculated for a particular case of 3D topological magnetic solitons, the toroidal hopfions in cylindrical and spherical ferromagnetic dots. The calculation method is based on the theory of toroidal hopfions developed within the classical field theory for infinite media. We exploited the property of the toroidal hopfions that the Hopf index density in any curvilinear coordinate system can be expressed as a Jacobian of the transformation from the toroidal coordinates to new coordinates, which match to the magnetic particle symmetry. The calculated Hopf indices are not integer and approach integer values increasing the dot sizes.

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LIU, YAN JANE, and GEORGE R. BUCHANAN. "FREE VIBRATION OF TRANSVERSELY ISOTROPIC SOLID AND THICK-WALLED TOROIDAL SHELLS." International Journal of Structural Stability and Dynamics 06, no. 03 (2006): 359–75. http://dx.doi.org/10.1142/s0219455406002027.

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The frequency of vibration of thick-walled toroidal shells is studied using a finite element formulation wherein the finite element is derived directly in toroidal coordinates. Hexagonal crystals of thallium and cadmium are used as representative transversely isotropic materials. The shell is assumed to be transversely isotropic with respect to the toroidal radial direction, and results based on that assumption are contrasted to a shell that is transversely isotropic with respect to the circumferential toroidal coordinate. It is established that an analysis based on a toroidal coordinate system is superior to an axisymmetric coordinate system and has some advantages over a commercial finite element code. Tables of results are presented that compare frequency of vibration for the above mentioned transversely isotropic materials and isotropic materials.

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László, István, André Rassat, P. W. Fowler, and Ante Graovac. "Topological coordinates for toroidal structures." Chemical Physics Letters 342, no. 3-4 (2001): 369–74. http://dx.doi.org/10.1016/s0009-2614(01)00609-1.

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Tsui, K. H. "Toroidal equilibria in spherical coordinates." Physics of Plasmas 15, no. 11 (2008): 112506. http://dx.doi.org/10.1063/1.3006340.

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Krisch, J. P., and E. N. Glass. "A space–time in toroidal coordinates." Journal of Mathematical Physics 44, no. 7 (2003): 3046. http://dx.doi.org/10.1063/1.1580999.

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McBain, G. D. "Plane poloidal-toroidal decomposition of doubly periodic vector fields. Part 1. Fields with divergence." ANZIAM Journal 47, no. 1 (2005): 21–38. http://dx.doi.org/10.1017/s1446181100009743.

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AbstractIt is shown how to decompose a three-dimensional field periodic in two Cartesian coordinates into five parts, three of which are identically divergence-free and the other two orthogonal to all divergence-free fields. The three divergence-free parts coincide with the mean, poloidal and toroidal fields of Schmitt and Wahl; the present work, therefore, extends their decomposition from divergence-free fields to fields of arbitrary divergence. For the representation of known and unknown fields, each of the five subspaces is characterised by both a projection and a scalar representation. Use of Fourier components and wave coordinates reduces poloidal fields to the sum of two-dimensional poloidal fields, and toroidal fields to the sum of unidirectional toroidal fields.

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Dubovik, V. M., and S. I. Vinitsky. "Toroidal coordinates for a three-body problem." Journal of Physics B: Atomic and Molecular Physics 20, no. 16 (1987): 4065–68. http://dx.doi.org/10.1088/0022-3700/20/16/017.

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Anghel, D. V., and A. T. Preda. "Quantized toroidal dipole eigenvalues in nano-systems." Journal of Physics: Conference Series 2090, no. 1 (2021): 012151. http://dx.doi.org/10.1088/1742-6596/2090/1/012151.

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Abstract The parity violation in nuclear reactions led to the discovery of the new class of toroidal multipoles. Since then, it was observed that toroidal multipoles are present in the electromagnetic structure of systems at all scales, from elementary particles, to solid state systems and metamaterials. The toroidal dipole T (the lowest order multipole) is the most common. This corresponds to the toroidal dipole operator T ^ in quantum systems, with the projections T ^ i (i = 1, 2, 3) on the coordinate axes. These operators are observables if they are self-adjoint, but, although it is commonly discussed of toroidal dipoles of both, classical and quantum systems, up to now no system has been identified in which the operators are self-adjoint. Therefore, in this paper we use what are called the “natural coordinates” of the T ^ 3 operator to give a general procedure to construct operators that commute with T ^ 3 . Using this method, we introduce the operators p ^ ( k ) , p ^ ( k 1 ) , and p ^ ( k 2 ) , which, together with T ^ 3 and L ^ 3 , form sets of commuting operators: ( p ^ ( k ) , T ^ 3 , L ^ 3 ) and ( p ^ ( k 1 ) , p ^ ( k 2 ) , T ^ 3 ) . All these theoretical considerations open up the possibility to design metamaterials that could exploit the quantization and the general quantum properties of the toroidal dipoles.

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Ambjorn, Jan, Zbigniew Drogosz, Jakub Gizbert-Studnicki, Andrzej Görlich, Jerzy Jurkiewicz, and Dániel Németh. "CDT Quantum Toroidal Spacetimes: An Overview." Universe 7, no. 4 (2021): 79. http://dx.doi.org/10.3390/universe7040079.

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Lattice formulations of gravity can be used to study non-perturbative aspects of quantum gravity. Causal Dynamical Triangulations (CDT) is a lattice model of gravity that has been used in this way. It has a built-in time foliation but is coordinate-independent in the spatial directions. The higher-order phase transitions observed in the model may be used to define a continuum limit of the lattice theory. Some aspects of the transitions are better studied when the topology of space is toroidal rather than spherical. In addition, a toroidal spatial topology allows us to understand more easily the nature of typical quantum fluctuations of the geometry. In particular, this topology makes it possible to use massless scalar fields that are solutions to Laplace’s equation with special boundary conditions as coordinates that capture the fractal structure of the quantum geometry. When such scalar fields are included as dynamical fields in the path integral, they can have a dramatic effect on the geometry.

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Dissertations / Theses on the topic "Toroidal coordinates"

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Hudson, Stuart Ronald. "Generalized magnetic coordinates for toroidal magnetic fields." Phd thesis, 1997. http://hdl.handle.net/1885/145321.

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Book chapters on the topic "Toroidal coordinates"

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D’haeseleer, William Denis, William Nicholas Guy Hitchon, James D. Callen, and J. Leon Shohet. "“Proper” Toroidal Coordinates." In Flux Coordinates and Magnetic Field Structure. Springer Berlin Heidelberg, 1991. http://dx.doi.org/10.1007/978-3-642-75595-8_10.

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D’haeseleer, William Denis, William Nicholas Guy Hitchon, James D. Callen, and J. Leon Shohet. "Toroidal Flux Coordinates." In Flux Coordinates and Magnetic Field Structure. Springer Berlin Heidelberg, 1991. http://dx.doi.org/10.1007/978-3-642-75595-8_6.

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D’haeseleer, William Denis, William Nicholas Guy Hitchon, James D. Callen, and J. Leon Shohet. "Conversion from Clebsch Coordinates to Toroidal Flux Coordinates." In Flux Coordinates and Magnetic Field Structure. Springer Berlin Heidelberg, 1991. http://dx.doi.org/10.1007/978-3-642-75595-8_7.

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Mestechkin M.M. "On magnetic field of ring current." In Atomic and Molecular Nonlinear Optics: Theory, Experiment and Computation. IOS Press, 2011. https://doi.org/10.3233/978-1-60750-742-0-364.

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The explicit expression for magnetic induction vector of a ring current at arbitrary point in toroidal coordinates is presented in terms of the full elliptic integral of the first kind and its derivative over integral modulus. There is a conjugated point for any observation point: it is an image of the latter in a sphere with the ring current as its equator. The conjugated point lies on the same vector and has the same integral modulus as the initial point.

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Conference papers on the topic "Toroidal coordinates"

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Kruisbrink, A. Ch H., and S. J. Pickering. "Analytical Modelling of a Fluid Coupling." In ASME Turbo Expo 2009: Power for Land, Sea, and Air. ASMEDC, 2009. http://dx.doi.org/10.1115/gt2009-59528.

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In this paper an analytical model is presented for fluid couplings in aero-engine applications. It can be used to predict the performance of fully filled as well as partially filled couplings in terms of a transmitted torque. As such it will lead to a predictive tool for design purposes. The model makes use of toroidal coordinates. This allows for the assessment of the mass flux and angular momentum flux within the entire toroid halves, formed by the driving unit (pump) and driven unit (turbine). In previous work only the fluxes at the coupling plane (between pump and turbine) could be evaluated, since cylindrical coordinates were used. The Euler equations in toroidal coordinates are used to obtain approximate solutions for the 2D pressure field within these toroid halves. Assuming that the pressure within an air cavity of a partially filled coupling is constant, the air-oil interface, flow regime (annular, stratified) and fill status are obtained from contours of constant pressure. In previous work the pressure distribution is not considered, except in criteria for the flow regimes, based on the centrifugal and vortex head in the coupling plane. The analytical model is validated. It shows a good agreement with torque measurements on a fully filled coupling, after a Reynolds dependency for the power loss coefficients is introduced. It shows a reasonable, qualitative agreement with CFD simulations on a partially filled coupling, after the solution for the pressure distribution is corrected for the variable vortex speed and radial velocity component. The analytical approach is efficient compared to CFD, which is very expensive in terms of CPU times, in particular for the two-phase flow in partially filled fluid couplings.

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Riley, Peregrine E. J., and Louis E. Torfason. "Workspace Cusp Locations, and Simplified Workspace Boundary Equations for General RRR Regional Structures of Manipulators." In ASME 1996 Design Engineering Technical Conferences and Computers in Engineering Conference. American Society of Mechanical Engineers, 1996. http://dx.doi.org/10.1115/96-detc/mech-1008.

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Abstract General, complex geometry forms of RRR regional structures are often avoided due to the presence of inner boundaries within the workspace which tend to complicate robot guidance. Despite the added complexity, certain RRR geometries may have useful applications as they contain large workspace regions where four alternate configurations may be used to reach a given spatial location. Cusp points often appear on the workspace boundaries of general RRR regional structures, and determining their precise location may be useful for both design and guidance purposes. A twelfth degree polynomial equation in the outer joint variable is derived which defines the location of non-trivial cusps in the workspace. A new closed form workspace boundary equation is derived in the outer joint variable and x coordinate of the toroidal surface generated by rotation of the two outer revolutes. If the outer joint variable is incremented, a quadratic in x is formed at each step which enables a very efficient determination of the workspace boundaries while also providing the coordinates of the boundary on the toroidal surface.

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Ruqiong, Li, Li Guangbu, Tan Yonghong, Xu Guo Ping, and Li Zhi. "Swept volumes of toroidal cutters using generating curves based on the polar coordinates." In 2010 IEEE 11th International Conference on Computer-Aided Industrial Design & Conceptual Design (CAIDCD 2010). IEEE, 2010. http://dx.doi.org/10.1109/caidcd.2010.5681966.

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Li, Tianyi, Aravinda Kar, and Ranganathan Kumar. "Uniform and Gaussian UV Light Intensity Distribution on Droplet for Selective Area Deposition of Particles." In ASME 2020 Fluids Engineering Division Summer Meeting collocated with the ASME 2020 Heat Transfer Summer Conference and the ASME 2020 18th International Conference on Nanochannels, Microchannels, and Minichannels. American Society of Mechanical Engineers, 2020. http://dx.doi.org/10.1115/fedsm2020-20464.

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Abstract Particle transport through Marangoni convection inside a sessile droplet can be controlled by the UV light distribution on the surface. The photosensitive solution changes the surface tension gradient on the droplet surface and can induce clockwise and counter-clockwise circulations depending on the incident light distribution. In this paper, the stream function in the sessile drop has been evaluated in toroidal coordinates by solving the biharmonic equation. Multiple primary clockwise and counter-clockwise circulations are observed in the droplet under various concentric UV light profiles. The downward dividing streamlines are expected to deposit the particles on the substrate, thus matching the number of deposited rings on the substrate with the number of UV light rings. Moffatt eddies appear near the contact line or centerline of the droplet either due to a sharp change in the UV light profile or because the illuminated region is away from them.

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Li, Tianyi, Aravinda Kar, and Ranganathan Kumar. "Concentration Distribution of Photosensitive Liquid in a Droplet Under UV Light." In ASME-JSME-KSME 2019 8th Joint Fluids Engineering Conference. American Society of Mechanical Engineers, 2019. http://dx.doi.org/10.1115/ajkfluids2019-5519.

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Abstract A semi-analytical solution for the concentration of photosensitive suspension is developed in a hemispherical droplet illuminated with UV laser. A biharmonic equation in stream function is analytically solved using toroidal coordinates and the velocity is then used to solve the mass transport equation for concentration. Flow pattern and photosensitive material concentration are affected by the peak location of the UV light intensity, which corresponds to a surface tension profile. When the laser beam is moved from the droplet center to its edge, a rotationally symmetric flow pattern changes from a single counter clockwise circulation to a circulation pair and finally to a single clockwise circulation. This modulation in the orientation of circulation modifies the concentration distribution of the photosensitive material. The distribution depends on both diffusion from the droplet surface as well as Marangoni convection. The region beneath the droplet surface away from the UV light intensity peak has low concentration, while the region near the downward dividing streamline has the highest concentration. When the UV light peak reaches the droplet edge, the concentration is high everywhere in the droplet.

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Leamy, Michael J., and Anthony A. DiCarlo. "Phonon Prediction in Toroidal Carbon Nanotubes Using a Continuum Finite Element Approach." In ASME 2007 International Design Engineering Technical Conferences and Computers and Information in Engineering Conference. ASMEDC, 2007. http://dx.doi.org/10.1115/detc2007-35907.

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This work develops a tensor-based, reduced-order shell finite element formulation used to predict the phonon behavior of toroidal carbon nanotubes (CNTs). Displacements referencing two covariant basis vectors lying in the toroid’s tangent space, and one basis vector orthogonal to the tangent space, capture the kinematics of the toroidal CNT. These basis vectors compose a curvilinear coordinate system. Although specific attention is on toroidal CNTs, the formulation can be quickly adapted to cylindrical or other curvilinear CNTs by appropriate replacement of the metric tensor components and Christoffel symbols. The finite element procedure originates from a variational statement (Hamilton’s Principle) governing virtual work from internal, external (not considered), and inertial forces. Internal virtual work is related to changes in atomistic potential energy accounted for by an interatomic potential computed at reference area elements. Small virtual changes in the displacements allow a global mass and stiffness matrix to be computed, and these matrices then allow phonons to be predicted via the general eigenvalue problem. Results are generated for example toroidal CNTs documenting zero-energy behavior (rigid body motion) and the lowest phonons, which include the expected breathing-like and bending-like phonons.

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Riznyk, Volodymyr. "Methods of Big Vector Data Processing Under Toroidal Coordinate Systems." In 2020 IEEE 15th International Conference on Computer Sciences and Information Technologies (CSIT). IEEE, 2020. http://dx.doi.org/10.1109/csit49958.2020.9321955.

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Lin, Bo, Baxi Chong, Yasemin Ozkan-Aydin, et al. "Optimizing coordinate choice for locomotion systems with toroidal shape spaces." In 2020 IEEE/RSJ International Conference on Intelligent Robots and Systems (IROS). IEEE, 2020. http://dx.doi.org/10.1109/iros45743.2020.9341476.

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Aras, Eyyup. "An Object-Space Based Machining Simulation in Milling: Part 2 — By Toroidal Surface." In ASME 2009 International Design Engineering Technical Conferences and Computers and Information in Engineering Conference. ASMEDC, 2009. http://dx.doi.org/10.1115/detc2009-87638.

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This is the second part of a two-part paper presenting an efficient parametric approach for updating the in-process workpiece represented by the Z-map. With the Z-map representation, the machining process can be simulated by intersecting z-axis aligned vectors with cutter swept envelopes. In this paper the vector-envelope intersections are formulized for the toroidal section of a fillet-end mill which may be oriented arbitrarily in space. For a given tool motion a toroidal surface generates more than one envelope. In NC machining because the torus is considered as one of the constituent parts of a fillet-end mill, only some parts of the torus envelopes, called contact envelopes, can intersect with Z-map vectors. In this paper an analysis is developed for separating the contact-envelopes from the non-contact ones. When a fillet-end mill has an orientation along the vertical z-axis of the Cartesian coordinate system, which happens in 2 1/2 and 3-axis machining, the number of intersections between a Z-map vector and the swept envelope of a toroidal section of the fillet-end mill is maximum one. For finding this single intersection point one of the numerical root finding methods, i.e. bisection, can be applied to the nonlinear function obtained from vector-envelope intersections. On the other hand when a fillet-end mill has an arbitrary orientation, the number of intersections can be more than one and therefore the numerical root finding methods cannot be applied directly. Therefore for addressing those multiple intersections, a system of non-linear equations in several variables, obtained by intersecting a Z-map vector with the envelope surface of the toroidal section of a fillet-end mill, is transformed into a single variable non-linear function. Then developing a nonlinear root finding analysis which guarantees the root(s) in the given interval, those intersections are obtained.

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