24 - Systems of Orthogonal Coordinates

The variation-difference method is a convenient numerical method for shells of complex forms. It is enough when only cinematic boundary conditions are satisfied because the method is based on the principle of Lagrange. Another advantage of the variation-difference method is the better opportunity to create computer programs based on it. For shell analysis in orthogonal coordinate system as well as for shell analysis in principal curvatures the system of equations describing stress-strain state can be simplified. In this paper the difference between analysis in orthogonal coordinate system and analysis in principal curvatures of the surface is considered. The main distinction of the analysis of shells in orthogonal curvilinear coordinate system is the necessity of determination of components which include curvature of torsion of coordinate lines. The addition of these components in the equations of the theory of shells for the coordinate system in principal curvatures gives possibility to analyze shells in common orthogonal coordinate system. In this article shell analysis in orthogonal coordinate system is applied to shells based on normal cyclic surfaces.

we had established the velocity and Acceleration in some of orthogonal curvilinear coordinates such as oblate spheroidal, prolate spheroidal, parabolic cylindrical, parabolic and second torridal coordinates for application in Mechanics, in this paper we proceed to derive expression for the instantaneous velocity and acceleration in rotational oblate spheroidal coordinates

According to Committee on Programs for Advanced Study of Mathematics and Science in American High Schools [1] (Gollub et al. (eds.) in Learning and Understanding. Improving Advanced Study of Mathematics and Science in U.S. Hight Schools. National Academic Press, Washington, 2002), “the primary goal of advanced study in any discipline should be for students to achieve a deep conceptual understanding of the disciplines content”. It is undoubted that abstraction is one of the skills that teachers wish to improve in their students, but, how can teachers take advantage of technological resources, such as CAS or DGS, as help in their classes in undergraduate courses? One concept, whose importance is both theoretical and practical, corresponds to the coordinate transformation, in particular orthogonal coordinate systems. We can use trigonometric constructions to find the transformation equations, namely, the algorithm for transforming a Cartesian system into other coordinate system, as cylindrical or spherical coordinates. Not only, if we add the knowledge and some techniques from Linear Algebra, we can motivate new mathematical properties, but also we will increase considerably the abstract reasoning and symbolic calculation. We know that visualization helps intuitive understanding. Therefore, we propose using CAS and DGS to show how a triad, of basis vectors, is continuously changing direction, keeping the norm vector without change, and how this match visualization with the reasoning from theories of linear algebra.

In this paper the equation of equilibrium for a nonhomogeneous isotropic elastic solid under shear has been solved in rectangular Cartesian coordinates as well as in cylindrical polar coordinates. The modulus of rigidity of the material is assumed to vary in lateral as well as vertical directions. As an example, the above solution has been used to solve the problem of a Griffith crack in an infinite solid under shear.

New nonlocal models of plates and shells based on Legendre’s polynomial series expansion have been developed here. The 3-D dynamic equations of the nonlocal elasticity have been presented in an orthogonal system of coordinates. For the development of 2-D models of plates and shells the curvilinear system of coordinates related to the middle surface of the shell has been used along with special hypothesizes based on assumptions that take into account the fact that the plate and shells are thin. Higher order theory is based on the expansion of the 3-D equations of the nonlocal theory of elasticity into Fourier series in terms of Legendre polynomials. The stress and strain tensors, as well as vectors of displacements have been expanded into Fourier series in terms of Legendre polynomials with respect to thickness. Thereby, all equations of the nonlocal theory of elasticity have been transformed to the corresponding equations for the Legendre polynomials coefficients. Then, in the same way as in the classical theory of elasticity, a system of differential equations in terms of displacements with initial and boundary conditions for the Legendre polynomials coefficients has been obtained. All equations for higher order theory of nonlocal plates in Cartesian and polar coordinates as well as for cylindrical and spherical shells in coordinates related to the shells geometry have been developed and presented here in detail. The obtained equations can be used for calculating stress-strain and for modelling thin walled structures in macro, micro and nano scale when taking into account size dependent and nonlocal effects.

The measurement of plasma displacement is one of the most basic diagnostic tools in the study of plasma equilibrium and control in a toroidal magnetic confinement configuration. During pulse discharge, the eddy current induced in the vacuum vessel and shell will produce an additional magnetic field at the plasma boundary, which will have a significant impact on the measurement of plasma displacement using magnetic probes. In the newly built Keda Torus eXperiment (KTX) reversed field pinch device, the eddy current in the composite shell can be obtained at a high spatial resolution. This device offers a new way to determine the plasma displacement for KTX through the multipole moment expansion of the eddy current, which can be obtained by unique probe arrays installed on the inner and outer surfaces of the composite shell. In an ideal conductor shell approximation, the method of multipole moment expansion of the poloidal eddy current for measuring the plasma displacement in toroidal coordinates, is more accurate than the previous method based on symmetrical magnetic probes, which yielded results in cylindrical coordinates. Through an analytical analysis of many current filaments and numerical simulations of the current distribution in toroidal coordinates, the scaling relation between the first moment of the eddy current and the center of gravity of the plasma current is obtained. In addition, the origin of the multipole moment expansion of the eddy current in KTX is retrieved simultaneously. Preliminary data on the plasma displacement have been collected using these two methods during short pulse discharges in the KTX device, and the results of the two methods are in reasonable agreement.

The mean-field induction equation ∂ t B ¯ − η Δ B ¯ = ∇ × F is considered in a conducting volume V , where B ¯ is the mean magnetic field, ∂ t is rate of change and η is magnetic diffusivity. Using the Green's function method and the second-order correlation approximation (SOCA), or the nearly axisymmetric methods of Braginskii [Self excitation of a magnetic field during the motion of a highly conducting fluid. Sov. Phys. JETP 1964, 20], then the electromotive force F is F = α ⋅ B ¯ . This work consists of two parts. Part I: The following antidynamo theorem (ADT) is derived: if there is no generation of azimuthal F from azimuthal B ¯ , that is 1 ϕ ⋅ α ⋅ 1 ϕ = α ϕϕ = 0 , where 1 ϕ is the unit vector in the ϕ direction, ( s , ϕ , z ) cylindrical polar coordinates, then an axisymmetric magnetic field ( B ¯ ( s , z ) ) will decay. Firstly, the magnetic field in meridional planes is shown to decay to zero. Then the azimuthal component of the magnetic field is shown to decay. As a weighted measure of the magnetic energy, ‖ b ‖ 2 = ∫ V b 2 d V , where b = B ¯ ( s , ϕ ) ⋅ 1 ϕ / s , is considered. The resulting ‖ b ‖ 2 magnetic energy analysis demonstrates that; for α = α ( s , z ) , and α ϕϕ = 0 , once the meridional field has decayed, diffusion decreases energy to more than account for the inductive contributions due to α , and, consistent with the α ϕϕ = 0 ADT, the field decays. Numerical results and field plots using the model α = s 1 z 1 ϕ , illustrate the interaction mechanisms responsible for the diffusive dominance as induction is increased. Using the SOCA and Green's function analysis, an explicit formulation for α ϕϕ is derived. Thus, physical mechanisms for the generation of α ϕϕ are established. Conditions are produced for which α ϕϕ = 0 , including a conductor filling all space with zero mean flow and co-axisymmetric perturbation flow and mean magnetic field. Part II: The Eulerian approach of Braginskii (1964), where the fields are analysed as perturbations from axisymmetry, is extended to compressible velocity fields for appropriate stellar and planetary dynamos. The hybrid Euler–Lagrange approach of Soward and Roberts [Eulerian and Lagrangian means in rotating, magnetohydrodynamic flows II. Braginsky's nearly axisymmetric dynamo. Geophys. Astrophys. Fluid Dyn. 2014, 108], following Soward [A kinematic theory of large magnetic Reynolds number dynamos. Phil. Trans. R. Soc. A 1972, 272], is continued to compressible flow and non-isochoric transformations in cylindrical polar coordinates, producing results that can be used for more general, higher-order approximations. These compressible extensions produce an α ϕϕ component and other effects for a reformulation of the problem into new effective mean, magnetic and velocity fields. Each of these approaches provides insight into mechanisms responsible for generating this critical α ϕϕ component. Common to all these approaches is the induction mechanism generated by the non-zero mean helicity of the meridional perturbation velocity field. Conclusions for non-magnetic stars are proposed and implications for hidden dynamos are drawn.

In this paper, the drift-diffusion numerical technique is discussed for the simulation of photovoltaic applications. The necessary differential equations, which are the Poisson and electron and hole charged particle continuity equations are explained, and the coupling between the above equations is discussed. Thereafter, the different physical processes involved in the simulation of thin film silicon photovoltaics are exploited. Then, all the transport properties related specifically to silicon and at a material temperature of 300 K such as the mobilities and diffusion coefficients for electrons and holes, the intrinsic silicon concentration, and generation and recombination mechanisms are identified from the literature. Finally, the formulation of the drift-diffusion model using finite elements in two-dimensional Cartesian and two-dimensional cylindrical axisymmetric coordinates is deployed to be used in the simulation of thin film photovoltaic applications. (7 pages)

The paper considers the regularity problem on three-dimensional incompressible Navier-Stokes equations in general orthogonal curvilinear coordinate systems. We establish one regularity criteria of the weak solutions involving only in a vorticity component ω 3 and one a priori estimate on the solution that H 3 u 3 L ∞ 0 , T ; L p ℝ 3 is bounded for 1 ≤ p ≤ ∞ to three-dimensional incompressible Navier-Stokes equations in orthogonal curvilinear coordinate systems. These extent greatly the corresponding results on axisymmetric cylindrical flow.

In Part I, an analytical examination of electromagnetic boundary conditions at the interface between two different media is presented in an orthogonal curvilinear coordinate system which conforms locally and instantaneously to the interface. Subsequently, an original natural coordinate system is introduced to resolve the uncertainty in the choice of an orthogonal curvilinear coordinate system. This natural coordinate system is also appeared to reduce the number of the nonvanishing discontinuity relations, and to determine local and instantaneous natural aspects of the e.m. boundary conditions problem.

Compared with 8-node and 4-node isoparametric element, the 9-node isoparametric element is more flexible and higher rate of convergence. Based on state-space techniques, the 9-node isoparametric element of piezothermoelastic material under orthogonal curvilinear coordinate system is established. Finally, the example shows the practicability and accuracy of the 9-node isoparametric element of piezothermoelastic material under orthogonal curvilinear coordinate system.

MESH GENERATION BY CONFORMAL AND QUASICONFORMAL MAPPINGS

By modifying the Rodi assumption to take account of the influence of flow curvature, a new curvature modified algebraic stress model (CMASM) is developed from the second moment closure in the generalized curvilinear coordinate system. And the explicit form of the ASM, a new curvature modified nonlineark-e model (CMNKE), is derived in the orthogonal curvilinear coordinate system. This new nonlineark-e model is further validated by a numerical simulation of a two-dimensional U-type turnaround duct flow. The results show that the CMNKE can effectively capture the main characteristic of this curvature flow and simulate the damping effect of the shear stress by a convex curvature and the enhancing effect by a concave curvature. So, this model is a rational and effective simplification to the second moment closure.

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For a particular finite elastic material, the governing fourth order nonlinear partial differential equations for plane strain deformations are shown to admit a new first integral, which, together with the constraint of incompressibility, gives rise to a second order problem. By an appropriate transformation of variables, the second order problem can be reduced to a single Monge-Ampere equation. Remarkably, this latter equation admits a linearization to the standard Helmholtz equation, so that the possibility arises for the determination of numerous exact finite elastic deformations. Of particular importance is that one of the parameters arising in the linearization turns out to be the physical angle E involved in standard cylindrical polar coordinates, and therefore these exact solutions might be particularly relevant to problems concerned with sectors of cylinders. Known first integrals of the governing fourth order equations are summarized in cylindrical polar coordinates, and new solutions of the form 0 = g(0) are obtained. Finally, an alternative approach is suggested and the procedure is illustrated with two examples. Corresponding results for the two separate topics of the plane stress theory of highly elastic thin sheets and axially symmetric deformations are given in Part II of the paper.