Review of Curvilinear Coordinates

2021 ◽  
pp. 9-24
Author(s):  
Woon-Shing Yeung ◽  
Ruey-Jen Yang
Keyword(s):  
Curvilinear Coordinates

Certain problems with the application of stochastic diffusion processes for description of chemical engineering phenomena; Diffusion equations in curvilinear coordinates

1991 ◽  
Vol 56 (3) ◽  
pp. 602-618
Author(s):  
Vladimír Kudrna
Keyword(s):  
Heat Transfer ◽  
Mass Transport ◽  
Probability Density ◽  
Chemical Engineering ◽  
Diffusion Processes ◽  
Curvilinear Coordinates ◽  
Diffusion Equations ◽  
Parabolic Partial Differential Equations ◽  
Spherical Coordinates ◽  
Transformation Procedure

Parabolic partial differential equations used in chemical engineering for the description of mass transport and heat transfer and analogous relationship derived in stochastic processes theory are given. A standard transformation procedure is applied, allowing these relations to be generally written in curvilinear coordinates and particular expressions for cylindrical and spherical coordinates to be derived. The relation between the probability density for the position of a discernible particle and the concentration of a set of such particles is discussed.

XVI.—On Triple Systems of Surfaces and Non-Orthogonal Curvilinear Coordinates

1927 ◽  
Vol 46 ◽  
pp. 194-205 ◽  
Author(s):  
C. E. Weatherburn
Keyword(s):  
Curvilinear Coordinates ◽  
Triple Systems ◽  
Orthogonal Curvilinear Coordinates ◽  
Orthogonal Systems ◽  
Considerable Detail

The properties of “triply orthogonal” systems of surfaces have been examined by various writers and in considerable detail; but those of triple systems generally have not hitherto received the same attention. It is the purpose of this paper to discuss non-orthogonal systems, and to investigate formulæ in terms of the “oblique” curvilinear coordinates u, v, w which such a system determines.

A New Nonlinear Higher-Order Shear Deformation Theory for Nonlinear Vibrations of Laminated Shells

2010 ◽  
Author(s):  
M. Amabili ◽  
J. N. Reddy
Keyword(s):  
Shear Deformation ◽  
Deformation Theory ◽  
Nonlinear Vibrations ◽  
Circular Cylindrical Shell ◽  
Shear Deformation Theory ◽  
Higher Order ◽  
Curvilinear Coordinates ◽  
Geometrically Nonlinear ◽  
Geometric Imperfections ◽  
Laminated Shells

A consistent higher-order shear deformation nonlinear theory is developed for shells of generic shape; taking geometric imperfections into account. The geometrically nonlinear strain-displacement relationships are derived retaining full nonlinear terms in the in-plane displacements; they are presented in curvilinear coordinates in a formulation ready to be implemented. Then, large-amplitude forced vibrations of a simply supported, laminated circular cylindrical shell are studied (i) by using the developed theory, and (ii) keeping only nonlinear terms of the von Ka´rma´n type. Results show that inaccurate results are obtained by keeping only nonlinear terms of the von Ka´rma´n type for vibration amplitudes of about two times the shell thickness for the studied case.

Solution of Navier’s Equation in Cylindrical Curvilinear Coordinates

1978 ◽  
Vol 45 (4) ◽  
pp. 812-816 ◽  
Author(s):  
B. S. Berger ◽  
B. Alabi
Keyword(s):  
Fourier Transform ◽  
Finite Difference ◽  
Boundary Conditions ◽  
Half Space ◽  
Coordinate Transformation ◽  
Numerical Results ◽  
Three Dimensions ◽  
Curvilinear Coordinates ◽  
Rectangular Region

A solution has been derived for the Navier equations in orthogonal cylindrical curvilinear coordinates in which the axial variable, X3, is suppressed through a Fourier transform. The necessary coordinate transformation may be found either analytically or numerically for given geometries. The finite-difference forms of the mapped Navier equations and boundary conditions are solved in a rectangular region in the curvilinear coordinaties. Numerical results are given for the half space with various surface shapes and boundary conditions in two and three dimensions.

Non-standard finite-differences schemes for reaction–diffusion equations in curvilinear coordinates

2009 ◽  
Vol 33 (1) ◽  
pp. 277-286 ◽  
Author(s):  
Jose Alvarez-Ramirez ◽  
Francisco J. Valdes-Parada ◽  
Jesus Alvarez ◽  
J. Alberto Ochoa-Tapia
Keyword(s):  
Finite Differences ◽  
Reaction Diffusion ◽  
Reaction Diffusion Equations ◽  
Curvilinear Coordinates ◽  
Diffusion Equations

Simulation of Autofrettage in Curvilinear Coordinates

PAMM ◽  
2014 ◽  
Vol 14 (1) ◽  
pp. 437-438
Author(s):  
Wolfgang H. Müller ◽  
Felix A. Reich
Keyword(s):  
Curvilinear Coordinates

IZOGRID - A new tool for setting up orthogonal curvilinear grids for oceanic modeling

2021 ◽  
Author(s):  
Alexander Shchepetkin
Keyword(s):  
Iterative Procedure ◽  
Cartesian Grid ◽  
Reference Points ◽  
Curvilinear Coordinates ◽  
Grid Spacing ◽  
Curvilinear Grids ◽  
Curvilinear Grid ◽  
Curvilinear Contour ◽  
Grid Points ◽  
Grid Nodes

<p>Virtually all modern structured-grid ocean modeling codes are written in orthogonal curvilinear coordinates in horizontal directions, yet the overwhelming majority of modeling studies are done using very simple grid setups - mostly rectangular patches of Mercator grids rotated to proper orientation.  Furthermore, in communities like ROMS, we even observe decline in both interest and skill of creating curvilinear grids over long term.  This is caused primarily by the dissatisfaction with the existing tools and procedures for grid generation due to inability to achieve acceptable level of orthogonality errors.  Clearly, this causes underutilization of full potential of the modeling codes.</p><p>To address these issues, a new algorithm for constructing orthogonal curvilinear grids on a sphere for a fairly general geometric shape of the modeling region is implemented as a compile-once - use forever software package.  Theoretically one can use Schwartz-Christoffel conformal transform to project a curvilinear contour onto rectangle, then draw a Cartesian grid on it, and, finally, apply the inverse transform (the one which maps the rectangle back to the original contour) to the Cartesian grid in order to obtain the orthogonal curvilinear grid which fits the contour.  However, in the general case, the forward transform is an iterative algorithm of Ives and Zacharias (1989), and it is not easily invertible, nor it is feasible to apply it to a two-dimensional object (grid) as opposite to just one-dimensional (contour) because of very large number of operations.  To circumvent this, the core of the new algorithm is essentially based on the numerical solution of the inverse problem by an iterative procedure - finding such distribution of grid points along the sides of curvilinear contour, that the direct conformal mapping of it onto rectangle turns this distribution into uniform one along each side of the rectangle.  Along its way, this procedure also finds the correct aspect ratio, which makes it possible to automatically chose the numbers of grid points in each direction to yield locally the same grid spacing in both horizontal directions.  The iterative procedure itself turns out to be multilevel - i.e. an iterative loop built around another, internal iterative procedure.  Thereafter, knowing this distribution, the grid nodes inside the region are obtained solving a Dirichlet elliptic problem.  The latter is fairly standard, except that we use "mehrstellenverfahren" discretization, which yields fourth-order accuracy in the case of equal grid spacing in both directions.  The curvilinear contour is generated using splines (cubic or quintic) passing through the user-specified reference points, and, unlike all previous tools designed for the same purpose, it guarantees by the construction to yield the exact 90-degree angles at the corners of the curvilinear perimeter of grid.</p><p>Overall, with the combination of all the new features, it is shown that it is possible to achieve very small, previously unattainable level of orthogonality errors, as well as make it isotropic -- local distances between grid nodes in both directions are equal to each other.</p>

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