General Orthogonal Curvilinear Coordinates Research Articles

Exact solutions of steady Euler equations in two dimension by functional separation

Exact solutions of two dimensional Euler equations for incompressible fluid flows are found by the functional separation of variables. More specifically, two independent variables are considered in the Cartesian coordinates, polar coordinates and also general orthogonal curvilinear coordinates which include elliptic, parabolic and hyperbolic systems. Several new solutions are obtained and other classical solutions are rediscovered.

Analysis of natural fibre composites for aerospace structures

PurposeThe main idea is the comparison between composites including natural fibres (such as the linoleum fibres) and typical composites including carbon fibres or glass fibres. The comparison is proposed for different structures (plates, cylinders, cylindrical and spherical shells), lamination sequences (cross-ply laminates and sandwiches with composite skins) and thickness ratios. The purpose of this paper is to understand if linoleum fibres could be useful for some specific aerospace applications.Design/methodology/approachA general exact three-dimensional shell model is used for the static analysis of the proposed structures to obtain displacements and stresses through the thickness. The shell model is based on a layer-wise approach and the differential equations of equilibrium are solved by means of the exponential matrix method.FindingsIn qualitative terms, composites including linoleum fibres have a mechanical behaviour similar to composites including glass or carbon fibres. In terms of stress and displacement values, composites including linoleum fibres can be used in aerospace applications with limited loads. They are comparable with composites including glass fibres. In general, they are not competitive with respect to composites including carbon fibres. Such conclusions have been verified for different structure geometries, lamination sequences and thickness ratios.Originality/valueThe proposed general exact 3D shell model allows the analysis of different geometries (plates and shells), materials and laminations in a unified manner using the differential equilibrium equations written in general orthogonal curvilinear coordinates. These equations written for spherical shells degenerate in those for cylinders, cylindrical shell panels and plates by means of opportune considerations about the radii of curvature. The proposed shell model allows an exhaustive comparison between different laminated and sandwich composite structures considering the typical zigzag form of displacements and the correct imposition of compatibility conditions for displacements and equilibrium conditions for transverse stresses.

Differential Transform Method in General Orthogonal Curvilinear Coordinates

Many real world problems are governed by non-linear differential equations. These may be single or system of ordinary or partial differential equations. In practice, problems involving single differential equations are mostly solved using analytical procedures, while those with systems of equations are solved numerically. A semi-analytical procedure namely Differential Transform Method (DTM) obtained from Taylor series in Cartesian co-ordinates is being used to solve linear or nonlinear equations in practice. This paper introduces the Taylor series and DTM for general orthogonal curvilinear co-ordinates and focuses mainly on DTM in standard two-dimensional polar co-ordinates and three-dimensional cylindrical polar and spherical polar coordinates.

A closed-form 3D shell solution for multilayered structures subjected to different load combinations

Multilayered composite and sandwich plates and shells are typical aerospace structures. They introduce complicating effects such as in-plane and transverse anisotropy which lead to zigzag forms of displacement and interlaminar continuity problems. The present closed-form 3D shell solution allows the static analysis of simply-supported cross-ply laminated and sandwich plates, cylinders and cylindrical/spherical shell panels subjected to different harmonic load types. It is possible to consider transverse normal and transverse shear loads simultaneously or separately applied at the top and at the bottom of the considered structure. The present work extends the previous exact 3D shell model developed for the static analysis of plates and shells in the case of transverse normal load applied at the top or at the bottom of the investigated structure. This new extension is still based on the 3D equilibrium equations written in general orthogonal curvilinear coordinates. The obtained system is solved using simply supported boundary conditions, harmonic forms for loads and displacements, a general layer wise approach and the exponential matrix method for the solution of the differential equations in z. However, the load boundary conditions introduced in the proposed shell model have been opportunely modified in order to allow the combination of different transverse normal and transverse shear loads applied at the external surfaces. The new proposed benchmarks fill the gap present in the literature where the proposed 3D exact models always use a transverse normal load applied at the external surfaces. The present paper investigates the zigzag effects, the interlaminar continuity, the equilibrium and compatibility conditions, the load boundary conditions, the symmetry characteristics, the thickness ratio effect and the 3D behavior in laminated and sandwich plates and shells in the case of different load applications. The new proposed benchmarks will be fundamental for the validation of those new refined 2D shell models which want to capture all these features for different load types.

A general exact elastic shell solution for bending analysis of functionally graded structures

This new work proposes a three-dimensional (3D) exact shell model for the static analysis of simply-supported structures embedding Functionally Graded Material (FGM) layers when they are subjected to harmonic transverse normal loads. Results are proposed in terms of displacement and stress amplitudes through the thickness direction. One-layered FGM plates and cylinders, and sandwich cylindrical and spherical shell panels embedding an internal FGM core and external classical skins have been analyzed. Proposed results give a complete 3D description of FGM structures in terms of displacement and stress states. Such results can be used to validate new refined 2D shell models proposed in numerical or analytical form. Different geometries, lamination schemes, thickness ratios, materials and FGM laws through the thickness have been analyzed in order to have a general overview of the problem. The proposed 3D shell model uses the spherical 3D equilibrium equations developed in general orthogonal curvilinear coordinates. These equations automatically degenerate in those for cylindrical and plate structures via opportune considerations made about the radii of curvature. Equilibrium equations are solved in closed form considering simply supported boundary conditions and harmonic applied loads. The exponential matrix method has been employed to solve the second order partial differential equations in z. These equations have constant coefficients because of the introduction of opportune mathematical layers for the FGM description and for the curvature evaluation. A layer-wise approach has been identified with the direct imposition in the 3D shell model of equilibrium conditions for transverse stresses and compatibility conditions for displacements.

Exact three-dimensional static analysis of single- and multi-layered plates and shells

Abstract This new work proposes an exact three-dimensional static analysis of plates and shells. One-layered and multilayered isotropic, orthotropic, sandwich and composite structures are investigated in terms of displacements and in-plane and out-of-plane stresses through the thickness direction. Proposed structures are completely simply-supported and a transverse normal load is applied. The proposed method is based on the 3D equilibrium equations written using general orthogonal curvilinear coordinates which are valid for spherical shells. Cylindrical shell, cylinder and plate results are obtained as particular cases of 3D spherical shell equations. All the considered structures are analyzed without any geometrical approximation. The exact solution is possible because of simply-supported boundary conditions and harmonic form for applied loads. The shell solution is based on a layer-wise approach and the second order differential equations are solved using the redouble of variables and the exponential matrix method. A preliminary validation of the model is made using reference results in the literature. Thereafter, the proposed exact 3D shell solution is employed with confidence to provide results for one-layered and multilayered plates, cylinders, cylindrical shell panels and spherical shell panels. All these geometries are analyzed via a unified and general solution, and the obtained results can be used to validate future numerical methods proposed for plates and shells (e.g., the finite element method or the differential quadrature method). Proposed results allow to remark substantial features about the thickness of the structures, their geometry, the zigzag effects of displacements, the interlaminar continuity of displacements and transverse stresses, and boundary loading conditions for stresses.

Exponential matrix method for the solution of exact 3D equilibrium equations for free vibrations of functionally graded plates and shells

The present paper analyzes the convergence of the exponential matrix method in the solution of three-dimensional equilibrium equations for the free vibration analysis of functionally graded material structures. The three-dimensional equilibrium equations are written in general orthogonal curvilinear coordinates for one-layered and sandwich plates and shells embedding functionally graded material layers. The resulting system of second-order differential equations is reduced to a system of first-order differential equations redoubling the variables. This system is exactly solved using the exponential matrix method and harmonic displacement components. In the case of functionally graded material plates, the differential equations have variable coefficients because of the material properties which depend on the thickness coordinate z. For functionally graded material shells, the differential equations have variable coefficients because of both changing material properties and curvature terms. Several mathematical layers M can be introduced to approximate the curvature terms and the variable functionally graded material properties to obtain differential equations with constant coefficients. The exponential matrix is applied to solve the resulting system of partial differential equations with constant coefficients, where the used expansion has a very fast convergence ratio. The present work investigates the convergence of the proposed method related to the order N used for the expansion of the exponential matrix and to the number of mathematical layers M used for the approximation of curvature shell terms and variable functionally graded material properties. Both N and M values are analyzed for different geometries, thickness ratios, materials, functionally graded material laws, lamination sequences, imposed half-wave numbers, frequency orders, and vibration modes.

Curvature Approximation Effects in the Free Vibration Analysis of Functionally Graded Shells

The present work investigates the effects of the curvature terms in the three-dimensional (3D) equilibrium equations used for the free vibration analysis of functionally graded material (FGM) structures. The 3D equilibrium equations have been written in general orthogonal curvilinear coordinates which are valid for spherical shells. They automatically degenerate in those for cylindrical shells and plates considering one of the two radii of curvature and both radii of curvature equal to infinite, respectively. The approximation of curvature terms in the 3D equilibrium equations has been evaluated by means of frequency analyses. Results obtained via 3D equilibrium equations with exact geometry have been compared with those calculated via 3D equilibrium equations written with the approximation of the curvature terms. The effects of the curvature approximations depend on the thickness and curvature of the structures, on the materials, lamination sequences and FGM laws, on the frequency orders and vibration modes. The resulting system of second order partial differential equations has been reduced into a system of first order partial differential equations redoubling the variables. Therefore, the exponential matrix method has been employed using a layer wise approach. The final 3D equations have been solved in exact form considering harmonic displacement components and simply supported structures. The approximation of the curvature terms has been introduced in the 3D equilibrium shell equations. For numerical reasons, interlaminar continuity conditions and the top and bottom boundary and loading conditions have been written including the exact geometry. The introduction of curvature approximations only in the equilibrium equations is sufficient to obtain an exhaustive qualitative analysis of the importance of curvature terms in the free vibration problems for FGM structures.

2D and 3D shell models for the free vibration investigation of functionally graded cylindrical and spherical panels

The paper shows the free vibration investigation of simply supported functionally graded material (FGM) shells. Spherical and cylindrical shell geometries are investigated for two different material configurations which are one-layered FGM structures and sandwich structures embedding an internal FGM core. A three-dimensional (3D) exact shell model and different two-dimensional (2D) computational models are compared in terms of frequencies and vibration modes. The proposed numerical solutions are typical 2D finite elements (FEs), and classical and advanced generalized 2D differential quadrature (GDQ) solutions. High and low frequency orders are investigated for thin and thick simply supported shells. Vibration modes are fundamental to compare the 3D exact shell model and 2D numerical solutions. The 2D finite element results based on the classical Reissner–Mindlin theory are calculated using a typical commercial FE software. Classical and advanced GDQ 2D models use the generalized unified formulation. The 3D exact shell model uses the differential equations of equilibrium written in general orthogonal curvilinear coordinates and developed in layer-wise (LW) form. The differences between 2D numerical results and 3D exact results depend on the thickness ratio and geometry of the structure, the considered mode and the frequency order, the lamination sequence and materials.

3D exact and 2D generalized differential quadrature models for free vibration analysis of functionally graded plates and cylinders

The paper proposes a comparison between a three-dimensional (3D) exact solution and several two-dimensional (2D) numerical solutions. Numerical methods include classical 2D finite elements (FEs), and classical and refined 2D generalized differential quadrature (GDQ) solutions. The free vibration analysis of two different configurations of functionally graded material (FGM) plates and cylinders is proposed. The first configuration considers a one-layered FGM structure. The second one is a sandwich configuration with external classical skins and an internal FGM core. Low and high order frequencies are analyzed for thick and thin simply supported structures. Vibration modes are investigated to make a comparison between results obtained via the 2D numerical methods and those obtained by means of the 3D exact solution. The 3D exact solution is based on the differential equations of equilibrium written in general orthogonal curvilinear coordinates. This exact method is based on a layer-wise approach where the continuity of displacements and transverse shear/normal stresses is imposed at the interfaces between the layers of the structure. The 2D finite element results are obtained by means of a well-known commercial FE code. Classical and refined 2D GDQ models are based on a generalized unified approach which considers both equivalent single layer and layer-wise theories. The differences between 2D numerical solutions and 3D exact solutions depend on the considered mode, the order of frequency, the thickness ratio of the structure, the geometry, the embedded material and the lamination sequence.

Numerical and exact models for free vibration analysis of cylindrical and spherical shell panels

The present paper shows a comparison between classical two-dimensional (2D) and three-dimensional (3D) finite elements (FEs), classical and refined 2D generalized differential quadrature (GDQ) methods and an exact three-dimensional solution. A free vibration analysis of one-layered and multilayered isotropic, composite and sandwich cylindrical and spherical shell panels is made. Low and high order frequencies are analyzed for thick and thin simply supported structures. Vibration modes are investigated to make a comparison between results obtained via the FE and GDQ methods (numerical solutions) and those obtained by means of the exact three-dimensional solution. The 3D exact solution is based on the differential equations of equilibrium written in general orthogonal curvilinear coordinates. This exact method is based on a layer-wise approach, the continuity of displacements and transverse shear/normal stresses is imposed at the interfaces between the layers of the structure. The geometry for shells is considered without any simplifications. The 3D and 2D finite element results are obtained by means of a well-known commercial FE code. Classical and refined 2D GDQ models are based on a generalized unified approach which considers both equivalent single layer and layer-wise theories. The differences between 2D and 3D FE solutions, classical and refined 2D GDQ models and 3D exact solutions depend on several parameters. These include the considered mode, the order of frequency, the thickness ratio of the structure, the geometry, the embedded material and the lamination sequence.

AN EXACT 3D SOLUTION FOR FREE VIBRATIONS OF MULTILAYERED CROSS-PLY COMPOSITE AND SANDWICH PLATES AND SHELLS

A 3D free vibration analysis of multilayered structures is proposed. An exact solution is developed for the differential equations of equilibrium written in general orthogonal curvilinear coordinates. The equations consider a geometry for shells without simplifications and allow the analysis of spherical shell panels, cylindrical shell panels, cylindrical closed shells and plates. The method is based on a layer-wise approach, the continuity of displacements and transverse shear/normal stresses is imposed at the interfaces between the layers of the structures. Results are given for multilayered composite and sandwich plates and shells. A free vibration analysis is proposed for a number of vibration modes, thickness ratios, imposed wave numbers, geometries and multilayer configurations embedding isotropic and orthotropic composite materials. These results can also be used as reference solutions for plate and shell 2D models developed for the analysis of multilayered structures.

Exact 3D solutions and finite element 2D models for free vibration analysis of plates and cylinders

Abstract The paper proposes a comparison between classical two-dimensional (2D) finite elements (FEs) and an exact three-dimensional (3D) solution for the free vibration analysis of one-layered and multilayered isotropic, composite and sandwich plates and cylinders. Low and high order frequencies are analyzed for thick and thin simply supported structures. Vibration modes are investigated to make a comparison between results obtained via the finite element method and those obtained by means of the exact three-dimensional solution. The 3D exact solution is based on the differential equations of equilibrium written in general orthogonal curvilinear coordinates. This exact method is based on a layer-wise approach, the continuity of displacements and transverse shear/normal stresses is imposed at the interfaces between the layers of the structure. The geometry for shells is considered without any simplifications. The 2D finite element results are obtained by means of a well-known commercial FE code. The differences between 2D FE solutions and 3D exact solutions depend on the considered mode, the order of frequency, the thickness ratio of the structure, the geometry, the embedded material and the lamination sequence.

Three-Dimensional Exact Free Vibration Analysis of Spherical, Cylindrical, and Flat One-Layered Panels

The paper proposes a three-dimensional elastic analysis of the free vibration problem of one-layered spherical, cylindrical, and flat panels. The exact solution is developed for the differential equations of equilibrium written in orthogonal curvilinear coordinates for the free vibrations of simply supported structures. These equations consider an exact geometry for shells without simplifications. The main novelty is the possibility of a general formulation for different geometries. The equations written in general orthogonal curvilinear coordinates allow the analysis of spherical shell panels and they automatically degenerate into cylindrical shell panel, cylindrical closed shell, and plate cases. Results are proposed for isotropic and orthotropic structures. An exhaustive overview is given of the vibration modes for a number of thickness ratios, imposed wave numbers, geometries, embedded materials, and angles of orthotropy. These results can also be used as reference solutions to validate two-dimensional models for plates and shells in both analytical and numerical form (e.g., closed solutions, finite element method, differential quadrature method, and global collocation method).

Implementation of an Atmosphere–Ocean General Circulation Model on the Expanded Spherical Cube

A hydrodynamical kernel that drives both an atmospheric and oceanic general circulation model is implemented in general orthogonal curvilinear coordinates using the finite-volume method on the sphere. The finite-volume method naturally describes arbitrary grids, and use of the vector-invariant form of the momentum equations simplifies the generalization to arbitrary coordinates. Grids based on the expanded spherical cube of Rancic et al., which contain eight singular points, are used. At these singularities the grid is nonorthogonal. The combined use of vector-invariant equations and the finite-volume method is shown to avoid degeneracy at these singular points. The model is tested using experiments proposed by Williamson et al. and Held and Saurez. The atmospheric solutions are examined seeking evidence of the underlying grid in solutions and eddy statistics. A global ocean simulation is also conducted using the same code. The solutions prove to be accurate and free of artifacts arising from the cubic grid.

The Elastic–Viscous–Plastic Sea Ice Dynamics Model in General Orthogonal Curvilinear Coordinates on a Sphere—Incorporation of Metric Terms

Abstract A new discretization for the elastic–viscous–plastic (EVP) sea ice dynamics model incorporates metric terms to account for grid curvature effects in curvilinear coordinate systems. A fundamental property of the viscous–plastic ice rheology that is invariant under changes of coordinate system is utilized; namely, the work done by internal forces, to derive an energy dissipative discretization of the divergence of the stress tensor that includes metric terms. Comparisons of simulations using an older EVP numerical model with the new formulation highlight the effect of the metric terms, which can be significant when ice deformation is allowed to affect the ice strength.

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.

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.

Dirac equation in external fields: Separation of variables in curvilinear coordinates

The algebraic method of separation of variables in the Dirac equation proposed by one of the present authors [Gravitation and Electromagnetism (U.P., Minsk, 1989) Issue 4, p. 156 (in Russian)] is developed for the case of the most general interaction of the Dirac particle in an external field, taking into account scalar, vector, tensor, pseudovector, pseudoscalar, and gravitation connections. The present work, which concludes this series of papers entitled ‘‘Dirac equation in external fields’’ [J. Math. Phys. 32, 3184 (1991); 33, XXX (1992)] is dedicated to the investigation of the problem in the case of general orthogonal curvilinear coordinates, and allows the introduction of gravitation through the spinor connection and Lame’s functions. Special consideration is given to the cases, when the generalized Lame’s functions do not separate the variables multiplicatively (e.g., elliptic cylindrical, parabolic cylindrical, and oblate and prolate spheroidal coordinates). All the previous results and numerous results of other authors are particular cases of this investigation.

Relativistic trajectory equations for charged particles in general orthogonal curvilinear coordinates

The relativistic trajectory equations are given in orthogonal curvilinear coordinates for charged particles moving in arbitrary electrostatic and magnetic fields without restrictions. The trajectory equations for Cartesian and cylindrical coordinates are presented as special cases.