Shells analysis in orthogonal curvilinear coordinate system with variation-difference method

24 - Systems of Orthogonal Coordinates

The geometry and methods of forming right conoids are considered in the article. Right conoids are formed by the movement of a rectilinear generatrix perpendicular to a fixed straight line – the axis of the conoid, along which the rectilinear generatrix moves. When moving, the rectilinear generatrix rotates according to a given law around the directrix curve. Usually, the generatrix line is connected to some reference curve that the generatrix line touches. The coordinate grid along the generatrix lines is also connected with the reference curve. In this case, a non-orthogonal coordinate system of a right conoid is formed. Coefficients of quadratic forms of surfaces with a non-orthogonal coordinate system, usually are complex. Methods for analyzing shells with a non-orthogonal coordinate system of the middle surface of the shell are becoming more complicated. The article considers the possibility of forming right conoids with an orthogonal coordinate system, including with directrix reference curves. A vector equation is given for specifying right conoids with an orthogonal nonconjugate coordinate system. Formulas of coefficients of quadratic forms characterizing the internal geometry and curvature of right conoidal surfaces in space with the help of this vector equation are obtained. Based on the vector equations of the surface assignment, drawings of right conoids with various geometric parameters are constructed in the MathCAD program. The difference in the construction of right conoids with orthogonal and non-orthogonal curved coordinate systems is clearly demonstrated, so with different methods of specifying the angles of inclination of rectilinear generatrix of surfaces.

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.

UDC 539.3 Investigations of stability of deformable bodies carried out from the three-dimensiona l elasticity theory point of view [3, 4] have shown that the application of classical theories of plates and shells to thin-walled construction elements consisting of modern anisotropic materials does not always produce reliable results. Yet in many cases [3] a refinement of the theory which takes into account the effects of cross shear determines satisfactorily the critical loads. Though sufficient attention has been focused on the stability of multilayered isotropic plates and shells with a poor shear rigidity [1, 5, 6, 9], constructions with anisotropic properties characteristic for composite materials have not been sufficiently investigated. It is also noted that under specific relations between the elastic characteristics of the fillers and of the carrier layers some effects may also appear which are related to cross compression. 1. Equations for neutral equilibrium of multilayer plates and shells can effectively be obtained by linearizing the nonlinear equations for large deflections (of the Ks type). To this end a multilayer shell is considered of the total thickness h = hi + h2 in an orthogonal curvilinear coordinate system (xl, x2, z) subjected to normal loads q§ and q-(xi) applied to the upper z = +h 2 and the lower z = -h 1 surfaces, respectively. The coordinate surface (xl, x2) is identical with the reduction surface z = 0, its coordinate lines being the lines of principal curvature. Similarly as in [7, 8] the following rules are adopted for changes along the thickness of the layered shell in the cross tangential stresses crj~ or in the normal one ors3 and in the cross strain e33:

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.

A new numerical solver is developed for a shallow-water flow model to simulate meandering river flows. This solver can investigate the flows in the curved river channel, by means of the adaptive CIP-Soroban (CIP-S) scheme in a cylindrical coordinate system. Time development of water velocity and free-surface level is computed by the orthogonal curvilinear coordinate system without any transformation of the governing equations, and the advection term is calculated by the high-accuracy CIP scheme. The numerical grid points of the waterfront line at the river bank are moved and tracked by the kinematic condition so that the grid points are always located on the line, and the rearrangement of the positions of the girds is easily conducted by the original CIP-S method. From the verification of this solver by the pure advection problem and by the numerical solution of the meandering river flows, in comparison with the CIP-S method in a Cartesian coordinate system and with the boundary fitted coordinate (BFC) method, it is shown that the proposed numerical solver reasonably predicts the main flow profile and water surface elevation and will be a promising numerical method as one of the practical solutions.

The deformation problem of elasticity theory with regard to nonlinear deformations is examined. The expressions of deformations through displacements in the orthogonal curvilinear coordinate system are recorded. The relations for finite deformations in cylindrical and polar coordinate systems are derived. Physical relations for finite deformations and corresponding generalized stresses are recorded.

Tensor analysis and an orthogonal curvilinear coordinate system have been used to derive a complete set of equations for piezo-magneto-elastic analysis of a functionally graded (FG) thick shell of revolution with variable thickness and curvature. The mentioned structure can be subjected to mechanical, electrical and magnetic fields. It was assumed that all material properties (mechanical, electrical and magnetic properties) change functionally throughout the three axis of employed coordinate system. Kinetic and potential energies of the system have been evaluated in order to constitute the functional of the system. Final partial differential equations of the system can be derived by using minimization of the energy functional with respect to five employed functions of the system. For validation, the obtained differential equations have been reduced to two previously studied problems i.e. functionally graded piezoelectric materials and functionally graded piezomagnetic cylinders. Furthermore, numerical results are evaluated for a case study.

A pedagogically pleasing and mnemonically useful method for expressing the velocity and acceleration vectors in orthogonal systems of coordinates is shown. The method relies on expressing the time derivatives of the unit vectors in matrix form. It is also noted that the spatial derivatives of the unit vectors obey a similar rule.

An Explicit Fourth-Order Orthogonal Curvilinear Staggered-Grid FDTD Method for Maxwell's Equations

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.

This paper investigates the globally dynamic stabilizing effects of the geometry of the domain at which the flow locates and of the geometrical structure of the finite-energy solutions to three-dimensional (3D) incompressible Navier-Stokes and Euler systems. We have established the global existence and uniqueness of the smooth solution to the Cauchy problem for 3D incompressible Navier-Stokes and Euler equations for a class of smooth large initial data in orthogonal curvilinear coordinate systems, which has no smallness assumption on initial data for the Cartesian coordinate system. Moreover, we have also established the existence, uniqueness, and exponential decay rate of the global strong solution to the initial-boundary-value problem for 3D Navier-Stokes equations for a class of smooth large initial data and a class of special bounded domain in orthogonal curvilinear coordinate systems. Regarding application, some new classes of large-amplitude global smooth solutions with very complex geometric structures, and that are either non-axisymmetric or non-helical in $ \mathbb{R}^3 $, have been established for 3D incompressible Navier-Stokes and Euler equations. This is the first result on the global existence and uniqueness of the large smooth solution to 3D incompressible Navier-Stokes and Euler equations in general orthogonal curvilinear coordinate systems.

An incremental variational formulation is presented for large deflection analysis of elastic‐plastic continuum by considering the energy consistency of a nonlinear multivariable discrete system based upon incompatible trial functions. The functional is given in orthogonal curvilinear coordinate system and the initial stress procedure has been introduced. Equilibrium imbalance correction is included to prevent drifting of the solution during the incremental process. As an initial application of the suggested formulation, a hybrid stress laminated shell element, in which an energy constraint is introduced to improve the element performance, has been developed to analyze the through‐thickness plastic behavior and the progressive cracking in RC plates and shells. It is of quadrilateral shape with only four corner nodes and can be used for both shallow and deep laminated shells with satisfactory numerical characteristics. Example problems are also included to illustrate the element performance.

The state equations for orthotropic, doubly curved shells are established in an orthogonal curvilinear coordinate system. Simplifying hypotheses about displacement models or stress distribution that were assumed in early work are not introduced in this paper. The continuity conditions of stresses and displacements at the interfaces between the contacting layers are satisfied, and analytical solutions for statics and dynamics of the simply supported, doubly curved shells with ortho‐tropic layers are presented by means of the Cayley‐Hamilton theorem. For any number of layers considered, the problem reduces to solving a set of linear algebraic equations with three unknowns. A unified analytical solution is given for thin, moderately thick, and thick laminated shells. The solution satisfied all the equations of elasticity, and each of nine elastic constants is considered. Arbitrary precision of the desired order can be obtained. Numerical results are given to compare with solutions available in the literature.