Orthogonal Curvilinear Coordinates Research Articles

THERMOELASTIC STRESS OF MULTILAYER CYLINDRICAL BODIES

An effective solution of the boundary value and boundary contact problems of thermal stress of elastic one- and multilayer bodies bounded by the coordinate surfaces of generalized cylindrical coordinates 𝜌 , f , z ( 𝜌 , f are orthogonal curvilinear coordinates on the plane and z is a linear coordinate) is given. The body occupies the domain z = { 𝜌 0 < 𝜌 < 𝜌 1 , f 0 < f < f 1 , 0 < z < z 1 } ; and a thermal disturbance is defined on the free of stress planar parts z = 0 and z = z 1 of the boundary surface, while homogeneous conditions of either symmetry or antisymmetry type are given for the remaining part of the boundary. The elastic body is assumed to be nonhomogeneous along z and transversally isotropic with the plane of isotropy z = const . The transversally isotropic layers of the multilayer body make contact along the planes z = const .

The operator ∇ in orthogonal curvilinear coordinates

It is possible to define a single expression for the operator∇ in orthogonal curvilinear coordinates that can fitall common differential operators and vectorial identities.

Dynamic Analysis of Anisotropic Cylindrical Shells Containing Flowing Fluid

This paper deals with the study of dynamic behavior of anisotropic cylindrical shells, based on refined shell theory, subjected simultaneously to an internal and external fluid. In the present theory, the transverse shear deformation effect is taken into account, therefore, the equations of motion are determined with displacements and transverse shear as independent variables. The solution is divided into three parts: In Section 2, the displacement functions are derived from the exact solution of refined shell equations based on orthogonal curvilinear coordinates. The mass and stiffness matrices of each structural element are derived by exact analytical integration. In Section 3, the velocity potential, Bernoulli’s equation and impermeability condition have been applied to the shell fluid interface to obtain an explicit expression for fluid pressure which yields three forces (inertial, centrifugal, Coriolis). Numerical examples are given in Section 4 for the free vibration of laminated composite and isotropic materials for both open and closed circular cylindrical shells. Reasonable agreement is found with other theories and experiments.

Shear deformation in dynamic analysis of anisotropic laminated open cylindrical shells filled with or subjected to a flowing fluid

The free vibration of anisotropic laminated composite, as well as isotropic open or closed, cylindrical shells submerged in and subjected simultaneously to an internal and external incompressible, inviscid fluid are discussed on the basis of a refined shell theory in which transverse shear deformation and rotary inertia effects are taken into account. The shell may be uniform or non-uniform in the circumferential direction. In this approach, displacements and rotations of the shell and the dynamic pressure of the fluid are modeled by a hybrid finite-element method. The displacement functions are derived from the exact solution of refined shell equations based on orthogonal curvilinear coordinates. The velocity potential and Bernoulli's equation have been used to describe an expression for fluid pressure which yields three forces (inertial, centrifugal and Coriolis) of the moving fluid. The mass, stiffness and damping matrices due to the fluid effect can be obtained by an analytical integration of the fluid pressure over the liquid element. Extensive results are given of computations carried out to illustrate the theory and dynamic behavior of open and closed cylindrical shells partially or completely filled with liquid, as well as subjected to a flowing fluid. A satisfactory agreement is seen between the numerical results predicted by the present theory and the results of existing available other theories.

3D Numerical Model for Pearl River Estuary

A 3D numerical model with an orthogonal curvilinear coordinate in the horizontal direction and sigma coordinate in the vertical direction has been developed. This model is based on POM (Princeton Ocean Model). In this model a second moment turbulence closure submodel is embedded and the stratification caused by salinity and temperature is considered. Furthermore, to adapt to estuary locations where the flow pattern is complex, the horizontal time differencing is implicit with the use of a time-splitting method instead of the explicit method in POM. This model is applied to the Pearl River estuary, which is the largest river system in South China, with Hong Kong at the eastern side of its entrance. The computation is verified and calibrated with field measurement data. The computed results mimic the field data well.

Three-dimensional finite element for thick shells of general shape

When the thickness of a shell increases in comparison with other dimensions as seen in arch dams, pressure vessels and shelter roofs, the effect of the unit deformations and shear deformations along the thickness have to be taken into account for achieving an acceptable approximation. This however, may entail the use of three dimensional elasticity solutions in developing suitable finite elements. In the present study, a twenty node isoparametric finite element is developed for analysis of thick shells having a general shape by using the three-dimensional theory of elasticity in orthogonal curvilinear coordinates. Shell geometric properties and their derivatives, which are necessary for obtaining a displacement-strain matrix, are determined by means of the shape functions. The accuracy of the presented formulation is assessed by comparing the numerical results with those given in the three dimensional elasticity solutions as well as with those obtained by the use of three dimensional finite elements.

Vibration of Truncated Conical Cylinders of Crystal Class 6/mmm

A nine-node Lagrangian finite element is formulated in an axisymmetric conical coordinate sys tem and used to study the vibration of truncated hollow cones. The governing elasticity equations (strain- displacement) are derived using the generalized expressions for orthogonal curvilinear coordinates. The transformation for hexagonal material constants from a Cartesian coordinate system to the conical coordi nate system is illustrated and results are presented for titanium and beryllium. Results for frequency and mode shape are presented in tabular form and graphically for hexagonal materials of class 6/mmm for a vari ety of truncated cone parameters. Vibration frequencies for isotropic cones are compared with similar results in the existing literature.

Conductive–radiative coupling in an absorbing–emitting axisymmetric medium

A modeling of simultaneous radiation–conduction in axisymmetric semi-transparent motionless bodies (hereafter labeled stm) is proposed. The stm is an emitting–absorbing medium with gray spectral radiative properties. Its interfaces are smooth with specular reflection; they are assumed to be opaque or transparent. The two-dimensional modeling is based on a double-grid representation: (i) a physical one corresponding to curvilinear coordinates adapted to the stm body shape, in which the radiative source is calculated by means of a ray-tracing technique and the formal solution of the radiative transfer equation (RTE), and (ii) an abstract one with an orthogonal grid for which the enthalpy equation is solved by a finite differences scheme. The nonlinear discretized equations are solved iteratively by passing from one to the other representation. Results are presented for the cases of a finite length cylinder and a truncated paraboloid both fitted with prescribed temperatures at the ending black cross sections, the lateral surface being opaque or transparent, and subjected to an imposed uniform temperature. In the case of the cylinder, results are compared with those obtained by a discrete ordinates method in curvilinear orthogonal coordinates.

Basic relations of the theory of nonlinear dynamic deformation of high-strength polymer films

We present basic relations of the theory of nonlinear dynamic deformation of rigid (high-strength) polymeric films. On the basis of the experimental results, we formulate the determining relations for the elastic and plastic stages of deformation. We introduce a measure of the stress-strain state and obtain the equations of motion for arbitrary curvilinear orthogonal coordinates in the case of large displacements and low strains and squares of the angles of rotation.

On the divergence-form finite-difference approximation to the momentum advection in curvilinear orthogonal coordinates

The traditional finite-difference schemes for the dynamical equations in curvilinear orthogonal coordinates have a basic flaw: they conserve only energy, but not momentum. A finite-difference approximation to these equations is suggested that conserves both energy and momentum.

Application of a hyperbolic grid generation technique to a conformal PML implementation

Recently proposed anisotropic material-based conformal perfectly matched layer (PML) formulations are based on general orthogonal curvilinear coordinates and, consequently, their practical application to a conformal finite-difference time-domain (FDTD) method requires body-conformal orthogonal grids. In this work, we present an application of a two-dimensional (2-D) hyperbolic grid generation scheme for a numerical implementation of a conformal PML in conjunction with Janaswamy and Liu's FDTD method. Numerical results show that our hyperbolic grid-based conformal PML implementation provides an efficient absorbing boundary condition (ABC) for a body-conformal FDTD simulation involving complex geometries.

THE EXACT FINITE-DIFFERENCE SCHEME FOR VECTOR BOUNDARY‐VALUE PROBLEMS WITH PIECE‐WISE CONSTANT COEFFICIENTS

We will consider the exact finite‐difference scheme for solving the system of differential equations of second order with piece‐wise constant coefficients. It is well‐known, that the presence of large parameters at first order derivatives or small parameters at second order derivatives in the system of hydrodynamics and magnetohydrodynamics (MHD) equations (large Reynolds, Hartmann and others numbers) causes additional difficulties for the applications of general classical numerical methods. Thus, important to work out special methods of solution, the so‐called uniform converging computational methods. This gives a basis for the development of special monotone finite vector‐difference schemes with perturbation coefficient of function‐matrix for solving the system of differential equations. Special finite‐difference approximations are constructed for a steady‐state boundary‐value problem, systems of parabolic type partial differential equations, a system of two MHD equations, 2‐D flows and MHD‐flows equations in curvilinear orthogonal coordinates.

Deformable curved magnetic surfaces

We study curved magnetic surfaces in the context of soft condensed matter. Specifically, we consider classical Heisenberg spins on elastically deformable curved geometries in orthogonal curvilinear coordinates such as a cylinder, a torus, etc. We find that a mismatch of length scales (geometrical frustration) in the presence of magnetic solitons leads to an elastic soliton (deformation) on the magnetic surfaces. We illustrate the results on (i) a circular cylinder with either spin anisotropy or external magnetic field or multiple solitons, (ii) an elliptic cylinder and (iii) a torus section. Our results are applicable to microtubules and vesicles (spheroidal or toroidal) comprised of magnetic organic materials such as magnetic polymers.

Coping with Curvilinear Coordinates

A systematic method to generate expressions appearing in physics and engineering and valid in curvilinear orthogonal coordinates is presented. The method, which is called ‘the method of local cartesian coordinates’ achieves the same as ‘the method of moving axes’ presented in Love [1] but with a smaller effort and with more familiar mathematical tools. The main idea is: if we have an expression valid in rectangular cartesian coordinates, a corresponding expression for curvilinear orthogonal coordinates can be formed with simple steps. Some general expressions are generated, but the method can be used equally well to produce the desired formulas directly in specific cases. For this purpose, polar coordinates are employed extensively as a demonstration example.

From Ramond fermions to Lamé equations for orthogonal curvilinear coordinates

We show how Ramond free neutral Fermi fields lead to a τ-function theory of BKP type which describes iso-orthogonal deformations of systems of orthogonal curvilinear coordinates. We also provide a vertex operator representation for the classical Ribaucour transformation.

On coupled modes theory of two-dimensional wave motion in elastic waveguides with slowly varying parameters in curvilinear orthogonal coordinates

Coupled modes theory, as applied to 2-D wave propagation in range-dependent isotropic elastic waveguides in curvilinear orthogonal coordinates, is proposed. The model consists of one or more curvilinear layers with interfaces and outer boundaries departing slowly from the coordinate surfaces. The material parameters are supposed to be slowly dependent on the longitudinal coordinate. Four-dimensional vector-valued functions with the displacements, rotation, and divergence as the components are shown to be a convenient form of wave field description to extend the coupled modes theory from liquid to elastic waveguides. For this case, the cross section operator is defined and the generalized bi-orthogonality property of its eigenfunctions is established. Using the generalized bi-orthogonality, the coupling equations are derived. Wave propagation in a tube which narrows slowly to a conic rod and in a disk of radially dependent thickness are considered as the examples. The differences between the results obtained by the coupled modes theory and those the multiple-scales theory leads to are discussed. The theory modification for dealing with a cutoff phenomenon and the general algorithm of computation of the wave field excited by a point source are considered.

Analytical derivation of a conformal perfectly matched absorber for electromagnetic waves

We present an analytical derivation of a 3-D conformal perfectly matched layer (PML) for mesh termination in general orthogonal curvilinear coordinates. The derivation is based on the analytic continuation to complex space of the normal coordinate to the mesh termination. The resultant fields in the complex space do not obey Maxwell's equations. However, it is demonstrated that, through simple field transformations, a new set of fields can be introduced so that they obey Maxwell's equations for an anisotropic medium with properly chosen constitutive parameters depending on the local radii of curvature. The formulation presented here recovers, as particular cases, the previously proposed Cartesian, cylindrical, and spherical PMLs. A previously employed anisotropic (quasi-) PML for conformal terminations is shown to be the large radius of curvature approximation of the anisotropic conformal PML derived herein. © 1998 John Wiley & Sons, Inc. Microwave Opt Technol Lett 17: 231–236, 1998.

On Free Vibrations of Fiber Reinforced Doubly Curved Panels, Part 1: Formulation/Convergence Study

This paper presents a Ritz-type numerical scheme for the analysis of doubly curved laminated open panels. The fundamental strain-displacement relations and energy expressions are developed in orthogonal curvilinear coordinates. Higher-order shear deformation theory and the effects of rotary inertia are included in the formulation. The displacement fields are prescribed by Bezier surface patches and the procedure to implement the boundary conditions in this context is also described. The numerical method is developed such that any arbitrary open panel bounded by four curved edges can be analyzed. Two examples namely: cantilevered cross-ply cylindrical and spherical panels are used to demonstrate the convergence of the solution procedure. Bezier surface patches formed by the eighth order polynomials yield good values of the natural frequencies.

On Free Vibrations of Fiber Reinforced Doubly Curved Panels, Part 2: Applications

The applications of a Ritz-type numerical scheme, in which the displacement fields are prescribed by Bezier surface patches, are presented for the analysis of doubly curved laminated open panels. The fundamental strain-displacement relations and energy expressions are developed in orthogonal curvilinear coordinates. The higher-order shear deformation theory and the effects of rotary inertia are considered in the formulation. Good comparisons of the results are obtained for a class of open panels. For example, values of the natural frequencies of open cylindrical and spherical panels made of isotropic material are compared with the results from the finite element analysis. Cases of cantilevered and simply supported angle-ply laminated cylindrical panel and a fully clamped isotropic conical panel are also examined for comparison with the available sources in the literature. In addition, the natural frequencies are presented for angle-ply laminated circular cylindrical, conical and spherical panels and the influence of the fiber orientation on the fundamental frequency is also examined for the angle ply having one, two [φ/−φ] and four [φ/−φ/φ/−φ] laminae arrangements.

Hybrid grid—a specialized mesh system for full three-dimensional numerical simulation in natural waters

A specialized refined grid system for simulating full three-dimensional natural waters, named hybrid grid system by the authors, is reported in the paper. This system is composed of part orthogonal body-fitted curvilinear coordinates in horizontal directions and part Cartesian coordinates in the vertical direction. In the horizontal plan the orthogonal body-fitted coordinate (BFC) system exactly fits shore lines, in the vertical direction the blockage coefficients approximate the bottom topography. The main principle and particulars of numerical mesh generation have been stated in detail, the established nonsimplified governing equations of fluid movement and temperature transport under the present grid system are also presented. An example of the generation of the hybrid grid system for a 90° curved river reach has been provided, the discretization of governing equations and corresponding result analyses of the numerical computation will be presented in more detail in another paper.