Orthogonal Coordinate System Research Articles

Stress–Strain State of Nonthin Spherical Shells of Variable Thickness under Localized Loads

Introduction. Spherical shells of variable thickness are widely used in many areas of modern engineering. This necessitates analyzing their stress–strain state. The strength analysis of such shells involves severe computational difficulties associated with the complexity of the original system of partial differential equations and their boundary conditions. The basic numerical methods for the stress–strain analysis of shells and recommendations on how to use them are outlined in [4, 6, 7]. Some classes of spherical shells are studied in [11, 14, 15]. In this paper, we will analyze the stress–strain state of locally loaded spherical shells with thickness varying in one or two coordinate directions and with different types of boundary conditions. The effect of the variation in the thickness at constant weight on the stress–strain state of shells will also be analyzed. Use will be made of a nonclassical problem statement and a refined straight-line model [1, 3, 5, 12, 13]. 1. Problem Formulation. Consider nonthin spherical shells with thickness varying in one or two coordinate directions. We will use a refined problem statement based on the straight-line hypothesis: normals to the coordinate surface remain straight and unstretched after deformation, but do not remain normal. To describe the shell, we choose an orthogonal coordinate system , , , where , are spherical coordinates, and is the normal (to the datum surface) coordinate. The accepted hypothesis suggests that the displacements of the shell can be represented as u ( , , ) ( , ) ( , ) u

General higher-order equivalent single layer theory for free vibrations of doubly-curved laminated composite shells and panels

The present paper provides a general formulation of a 2D higher-order equivalent single layer theory for free vibrations of thin and thick doubly-curved laminated composite shells and panels with different curvatures. The theoretical framework covers the dynamic analysis of shell structures by using a general displacement field based on the Carrera’s Unified Formulation (CUF), including the stretching and zig-zag effects. The order of the expansion along the thickness direction is taken as a free parameter. The starting point of the present general higher-order formulation is the proposal of a kinematic assumption, with an arbitrary number of degrees of freedom, which is suitable to represent most of the displacement field presented in literature. The main aim of this work is to determine the explicit fundamental operators that can be used not only for the Equivalent Single Layer (ESL) approach, but also for the Layer Wise (LW) approach. Such fundamental operators, expressed in the orthogonal curvilinear co-ordinate system, are obtained for the first time by the authors. The 2D free vibration shell problems are numerically solved using the Generalized Differential Quadrature (GDQ) and Generalized Integral Quadrature (GIQ) techniques. GDQ results are compared with recent papers in the literature and commercial codes.

A new laminated composite beam element based on eigenvalue problem

Abstract A new laminated composite beam element is developed based on the eigenvalue problem for the flexural and torsional analyses of beams with I- and channel-sections. Analytical technique is used to present the laminated composite beam theory considering the transverse shear and the restrained warping induced shear deformation based on an orthogonal Cartesian coordinate system. From the energy principle, the equilibrium equations are derived and the member stiffness matrix is determined using the force–displacement relations of the beam. In order to demonstrate the validity of this study, the deflections and the twist angles of composite beams with I- and channel-sections are presented and compared with the results from other researchers and the finite beam element using the Lagrange interpolation functions, and the detailed finite element analysis results using ABAQUS. Especially the flexural and torsional behavior and the effect of shear deformation are investigated with respect to the modulus ratio and the fiber angle change.

General compliance transformation relations for all seven crystal systems

For crystals, the compliance (s ij ) and the stiffness (c ij ) matrices are specified in the orthogonal coordinate systems (Y i ), which do not coincide with the crystal axes (X i ) commonly used except for cubic and orthorhombic crystal systems. Transformations have been done in this paper and the general compliance transformation relations from the orthogonal coordinate systems (Y i ) to the measurement systems (M i ) are given for all seven crystal systems. Accordingly, useful expressions for Young’s modulus E and Poisson’s ratio υ are also derived.

Low Speed Preconditioning Algorithm Used in the Process of Calculating Euler Equation

This paper describes a numerical method for solving the unsteady Euler equation at any speed. In the process of calculating Euler's equation, the control equation in orthogonal curvilinear coordinate system is discretized by the finite -volume scheme based on the center-difference method, and convection flux used Jameson central deference scheme was solved at every pseudo time step, and the Runge-Kutta method, dual-time algorithm and the implicit LU-SGS add preconditioning algorithm are used for time-marching. For obtaining the numerical solution of two-dimensional unsteady flow around a cylinder and the flow of the Convex Hull, finding that the calculated results agree well with literature values and theoretical analytical solution.

Cavitation shape of the three-dimensional slender at a small attack angle in a steady flow

In this paper, based on the boundary element method, the cavitation shape of the three-dimensional slender at a small attack angle in a steady flow is simulated through the iterative method, while Dirichlet boundary conditions and Neumann boundary conditions are satisfied in cavitation and slender respectively. The linear triangular elements are adopted and the control points are arranged in grid nodes. The velocity potential for cavity surface is determined through an iterative method in a local orthogonal coordinate system, and then the distribution of cavitation thickness can be determined by the boundary integral equation. To prevent the remeshing operation in the iterative process, the Lagrange interpolation method is used to determine the thickness at the end of cavity. The numerical results are in good agreement with the experimental data. The influence of those on cavitation shape of the three-dimensional slender are investigated, such as cavitation number, attack angle and cone angle. Numerical results show that the cavitation shape of the three-dimensional slender is asymmetric at an attack angle and is analogous to the cavitation stacking in the lee side. While with the decrease in the cavity number or the increase in the cone angle, the asymmetry for the cavity shape will be more serious.

Solutions of Helmholtz and Schrödinger Equations with Side Condition and Nonregular Separation of Variables

Olver and Rosenau studied group-invariant solutions of (generally nonlinear) partial differential equations through the imposition of a side condition. We apply a similar idea to the special case of finite-dimensional Hamiltonian systems, namely Hamilton-Jacobi, Helmholtz and time-independent Schr\"odinger equations with potential on $N$-dimensional Riemannian and pseudo-Riemannian manifolds, but with a linear side condition, where more structure is available. We show that the requirement of $N-1$ commuting second-order symmetry operators, modulo a second-order linear side condition corresponds to nonregular separation of variables in an orthogonal coordinate system, characterized by a generalized St\"ackel matrix. The coordinates and solutions obtainable through true nonregular separation are distinct from those arising through regular separation of variables. We develop the theory for these systems and provide examples.

Constitutive Model Based on the Multiplicative Decomposition of Deformation Gradient in Arbitrary Orthogonal Coordinate System

According to general crystallographic slip constitutive laws, the stress-strain analysis of nickel based single crystal superalloy (NBSCS) in arbitrary coordinate system were based on material coordinate system (SRCS method). Ignoring the other symmetry of NBSCS, only the symmetry of plane families {001} were considered, which bring considerable errors into the stress-strain analysis. In this paper, the variation regulation of micro-physical systematical property about this alloy in arbitrary directions was investigated. From the arrangement of atoms in different crystal planes, NBSCS has four symmetry plane families, {001}, {110}, {112} and {111}. According to these symmetry planes, three reference orthogonal coordinate systems were established. Based on these coordinate systems and the coordinate rotation method (TRCS method), the stress-strain relationship of crystallographic slip constitutive model in arbitrary coordinate system was established. Meanwhile elastic constants in arbitrary directions were obtained. Comparing the results of the tensile stress-strain curves obtained from TRCS method with that from SRCS method, it is found that by the TRCS method the elastoplastic stress-strain simulation error of the NBSCS could be effectively reduced.

Extension of Material Derivative Concept in Fluid Mechanics

The original definition of material derivative is changing rate relative to the time of a physical quantity which belongs to a fluid particle. But when studying fluid mechanics problems, need in certain cases to calculate the material derivative of some quantities which don’t belong to a fluid particle, such as calculating the material derivative of base vectors of orthogonal curvilinear coordinate system. The original definition of material derivative isn’t suitable for these quantities. Calculating material derivative of these quantities is not clear in concept. To solve this problem, thorough analysis about material derivative was made according to the original definition and calculation formula of material derivative. Results found that the concept of material derivative can be extended and so can apply to any quantities which don’t belong to a fluid particle and are defined in flow fields, including vector or scalar, physical quantities or non physical quantities.

Mathematical modeling of nonisothermal thin-film two-phase flow over permeable surfaces

The thermohydrodynamics of the nonisothermal film flow of two-phase media over permeable surfaces has been investigated. The mechanical conservation equations for the heterogeneous media have been formulated and simplified in an orthogonal system of coordinates for arbitrarily shaped surfaces, with the specific features of the flow taken into consideration. The problem has been solved using the flow rate isosurface method, and numerical calculations have been carried out.

STUDY ON INFLUENCE OF BAIMAO SANDBANK CONTROL PROJECT IN SOUTHERN BRANCH OF THE YANGTZE RIVER ESTUARY, CHINA

The author establishes the Yangtze estuary 2D tidal and sediment mathematical model based on the orthogonal curvilinear coordinate system. The erosion and siltation of engineering section under the hydrological conditions of flood season, dry season and the great flood after the Baimao Sand control engineering is calculated. Preliminarily expounds the correlation of Baimao Sand and Biandan Sand and the impact of Biandan Sand after the Baimao Sand control engineering is implemented.

Ultrasonic lateral modulation imaging, speckle reduction, and displacement vector measurements using simple single-beam scanning or plural crossed-beam scanning with new spectra frequency division processing methods

Ultrasonic lateral modulation imaging, speckle reduction, and displacement vector measurements using simple single-beam scanning or plural crossed-beam scanning with new spectra frequency division processing methods Chikayoshi Sumi, Yousuke IshiiDepartment of Information and Communication Sciences, Faculty of Science and Technology, Sophia University, Tokyo, JapanAbstract: The development of effective ultrasonic tissue displacement measurement methods increases the number of possible applications for various tissue displacement and strain measurements. These applications include measurements of spontaneous motions/deformations generated by heart motion; pulsations from phenomena such as blood flow (intracardiac, intravascular, and carotid); heart, blood vessel, and liver motion; and motion from artificial sources such as motions/deformations generated by applying static compression/stretching forces, vibration or acoustic radiation forces (breast and liver). For arbitrary orthogonal coordinate systems obtained using arbitrary transducer types (eg, linear, convex, sector, arc, or radial array types, or single aperture types with a mechanical scan), several lateral modulation (LM) methods (eg, scanning with plural crossed or steered beams over a region of interest) have been developed that can be used with new echo imaging methods for tissue displacement/deformation measurements. Specifically, by using such beamforming methods, in addition to highly accurate displacement vector and lateral displacement measurements, LM echo imaging with a high lateral carrier frequency and a high lateral resolution has been developed. Another new beamforming method, referred to as “a steering angle (ASTA) method,” ie, scanning with a defined steering angle, is also described. In addition to conventional non-steered-beam scanning (ie, a version of ASTA) and conventional steered-beam scanning with a variable steering angle (eg, sector, arc, radial scan), a simple, single-beam scanning method also permits the use of LM, which yields an accurate displacement vector measurement with fewer calculations than the original LM methods. This is accomplished by using a previously developed spectra frequency division method (SFDM). However, the lateral carrier frequency and the measurement accuracy acquired by using such a single-beam scanning method are lower than those achieved with the original LM scanning methods and should be increased (ie, by using a quasi-LM method). In this report, the effectiveness of the use of the new SFDMs is verified with experiments on agar phantoms, in which conventional non-steered, focused single-beam transmission/reception scanning is performed together with high-speed non-steered single plane-wave transmission and non-steered, focused single-beam reception scanning using a linear array-type transducer. For comparison, the original LMs, with their respective transmissions of crossed, steered focused beams and plane waves are also performed. Because the use of rectangular apodization functions (ie, no apodization) yields a larger bandwidth in a lateral direction than the effective use of parabolic functions with the original LM method, it is shown that disregarding the lateral low-frequency spectra yields useful quasi-LM echo imaging with a high lateral frequency, and further significantly increases the measurement accuracy of a displacement vector. In addition, when no apodization is used with the original version of LM, disregarding the low-frequency lateral spectra is effective. In addition, the interchangeability of cosine and sine modulations performed after completing beamforming can also be used for single-beam scanning as well as for the original LM scanning method. Specifically, the cosine and sine modulations, respectively, are used for LM and quasi-LM imaging and displacement vector measurements. It is concluded that the appropriate use of the new SFDMs with simple single-beam scanning or with simple plural crossed-beam scanning with no apodization can achieve almost the same accuracy as the original LM scanning method using plural crossed beams with the effective apodization. Another new application of SFDM is also described: an incoherent superposition of the frequency-divided spectra reduces speckles. The new methods will also be effective for other beamforming methods and with other types of transducers.Keywords: lateral modulation scanning, single-beam scanning, apodization, spectra frequency division method, low-frequency spectra, quasi-LM imaging, coherent superposition, incoherent superposition

The x-and-y-axes travelling salesman problem

The x-and-y-axes travelling salesman problem forms a special case of the Euclidean TSP, where all cities are situated on the x-axis and on the y-axis of an orthogonal coordinate system of the Euclidean plane. By carefully analyzing the underlying combinatorial and geometric structures, we show that this problem can be solved in polynomial time. The running time of the resulting algorithm is quadratic in the number of cities.

Theoretical Design and Analysis for ${\rm TE}_{20}$–${\rm TE}_{10}$ Rectangular Waveguide Mode Converters

From Maxwell's equations in an orthogonal curvilinear coordinates system, a strict derivation of mode coupling coefficients among TE <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n0</sub> rectangular waveguide modes is given, obtaining the general and explicit formulas of the coupling coefficients. Theoretical design methods and a useful set of expressions for several kinds of overmoded rectangular waveguides are reported, which enable engineers to quickly design these types of waveguide bends. Firstly, a 90° <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">H</i> -plane bend was investigated to change the wave propagating direction to its perpendicular direction for a certain purpose. Its transmission efficiency of TE <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">10</sub> mode is 99.9% at 9.5 GHz, with bandwidth (for transmission efficiency >; 95%) of 8-12 GHz. Secondly, a strict and explicit derivation of mode conversion efficiencies of two mode converters, i.e., dual-bend and tri-bend TE <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">20</sub> -to- TE <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">10</sub> converters, are proposed. Both of them have similar conversion efficiency of 99.9% at 8.5 GHz with bandwidths (for mode-conversion efficiency >; 95% ) of 7.98-9.07 and 8.09-8.97 GHz, respectively. All of the theoretical models presented in this paper are verified by the finite-element simulations. We also experimentally show the performances of these devices, which are sufficient to demonstrate the validity of the theoretical models.

Incorporating Human Ventricular Fiber Architecture in Patient‐Specific Computational Models

Patient‐specific models of the cardiac ventricles require a realistic representation of the myofiber architecture to accurately simulate mechanical and electrical properties of healthy and failing hearts. Diffusion tensor magnetic resonance imaging (DT‐MRI) techniques have made it possible to reconstruct the fiber anatomy of the ventricles. However, in vivo DT‐MRI scans are still impractical to obtain for individual patients. The aim of this study develops a method to register DT‐MRI measurements from an explanted human heart into an anatomical model and warping the result to patient‐specific anatomies using large deformation diffeomorphic mapping.An explanted human organ donor heart was subjected to a DT‐MRI scan. The raw tensor data was registered into an anatomical model of the ventricles by fitting the measured diffusion tensor components in log‐Euclidean space. A material coordinate transformation model defined a local orthogonal coordinate system representing the fiber, sheet, and sheet‐normal directions of the myocardium from the eigenvectors of a locally interpolated diffusion tensor. Diffeomorphic maps of the deformation F between the donor and patient models were computed and used to warp the donor fiber architecture into the patient models. The registration error between the eigenvectors of the measured tensor and its interpolated tensor was quantified by a scalar metric of axes overlap. It found 74% of the registered DT‐MRI data (112,419 of 152,651 voxels) had mean angular errors of 12.5°±7° (SD) in the fiber, 20°±11.5° (SD) in the sheet, and 18°±11.6° (SD) in the sheet normal axes.

Reducing the data: Analysis of the role of vascular geometry on blood flow patterns in curved vessels

Three-dimensional simulations of blood flow usually produce such large quantities of data that they are unlikely to be of clinical use unless methods are available to simplify our understanding of the flow dynamics. We present a new method to investigate the mechanisms by which vascular curvature and torsion affect blood flow, and we apply it to the steady-state flow in single bends, helices, double bends, and a rabbit thoracic aorta based on image data. By calculating forces and accelerations in an orthogonal coordinate system following the centreline of each vessel, we obtain the inertial forces (centrifugal, Coriolis, and torsional) explicitly, which directly depend on vascular curvature and torsion. We then analyse the individual roles of the inertial, pressure gradient, and viscous forces on the patterns of primary and secondary velocities, vortical structures, and wall stresses in each cross section. We also consider cross-sectional averages of the in-plane components of these forces, which can be thought of as reducing the dynamics of secondary flows onto the vessel centreline. At Reynolds numbers between 50 and 500, secondary motions in the directions of the local normals and binormals behave as two underdamped oscillators. These oscillate around the fully developed state and are coupled by torsional forces that break the symmetry of the flow. Secondary flows are driven by the centrifugal and torsional forces, and these are counterbalanced by the in-plane pressure gradients generated by the wall reaction. The viscous force primarily opposes the pressure gradient, rather than the inertial forces. In the axial direction, and depending on the secondary motion, the curvature-dependent Coriolis force can either enhance or oppose the bulk of the axial flow, and this shapes the velocity profile. For bends with little or no torsion, the Coriolis force tends to restore flow axisymmetry. The maximum circumferential and axial wall shear stresses along the centreline correlate well with the averaged in-plane pressure gradient and the radial displacement of the peak axial velocity, respectively. We conclude with a discussion of the physiological implications of these results.

X-FEM Applied to Three-Dimensional Curvilinear Crack Front

A three-dimensional extended finite element algorithm coupled with the level set method is developed for curvilinear crack front problems. This method does not need to force the discontinuities to be in conformity with the mesh. In order to compute the pointwise stress intensity factors along the curvilinear crack front, an orthogonal curvilinear coordinate system is constructed at the intersection point between the crack front and a surface of the crack front element. As the crack surface is expressed by the level set method, no explicit meshing of the crack geometry emerges on the crack front. This makes the domain integral more difficult. An independent grid of hexahedral cells around the crack front needs to be built as the virtual extension domain for accomplishing the domain integral. To assess the accuracy of the approach, an example with semi-elliptic crack surface is considered. The results show excellent agreement with the results calculated by the standard finite element method and those obtained from the literature.

Modeling and Numerical Simulation of Yield Viscoplastic Fluid Flow in Concentric and Eccentric Annuli

Numerical solution of yield viscoplastic fluid flow is hindered by the singularity inherent to the Herschel-Bulkley model. A finite difference method over the boundary-fitted orthogonal coordinate system is utilized to investigate numerically the fully developed steady flow of non-Newtonian yield viscoplastic fluid through concentric and eccentric annuli. The fluid rheology is described with the Herschel-Bulkley model. The numerical simulation based on a continuous viscoplastic approach to the Herschel-Bulkley model is found in poor accordance with the experimental data on volumetric flow rate of a bentonite suspension. A strict mathematical model for Herschel-Bulkley fluid flow is established and the corresponding numerical procedures are proposed. However, only the case of flow of a Herschel-Bulkley fluid in a concentric annulus is resolved based on the presumed flow structure by using the common optimization technique. Possible flow structures in an eccentric annulus are presumed, and further challenges in numerical simulation of the Herschel-Bulkley fluid flow are suggested.

The problem of temporal validity of reference coordinates in the context of reliability of the ETRS89 system realization in Poland

ABSTRACT The IAG (International Association of Geodesy) Sub-Commission for the European reference frame (EUREF) passed a resolution recommending application of the ETRF2000 frame as national realizations of the ETRS89 reference system during the XXth EUREF Symposium in Gävle (Sweden) in 2010. The PL-ETRF2000 system is comprised of EPN (EUREF Permanent Network) sites. Their coordinates and their temporal changes (velocities) were accurately determined on the basis of long-term GNSS observations. The transfer of the PLETRF2000 system onto the territory of Poland is realized by the ASG-EUPOS (Active Geodetic Network – European Position Determination System) permanent sites network. The ASG-EUPOS tasks include also system maintenance by continuous or periodical control of the coordinates and velocities constancy. According to the current Regulation of the Ministry of Administration and Digitization concerning geodesic, gravimetric and magnetic warps, the accuracy of sites of the fundamental warp (EPN sites belonging to ASG-EUPOS) may not be worse than 0.01 m for horizontal position and 0.02 m for geodesic height. It results in the necessity to monitor the performance of the reference sites for proper maintenance of the reference system. A new team for modeling deformations in Europe was established within the EUREF Working Group (The EUREF Working Group on Deformation Models) during the EUREF 2012 Symposium in Paris. One of its tasks is taking geo-kinematic models into consideration for national realizations of the ETRS89 and maximum usage of knowledge concerning the velocity field. The paper deals with the problem of temporal validity of the catalogue coordinates and the necessity of periodical updating them on the basis of velocities derived from long-term observations. Although Poland is located in a tectonically calm area and the intraplate velocities are small, some of the EPN and ASG-EUPOS sites have significant velocities and are subject to vertical movements. Lack of periodical updating of the coordinates realizing the reference system will cause lack of fulfillment of the assumed accuracy criterion. Analysis of horizontal and vertical velocities of sites belonging to the fundamental warp lead to assumption that two sites will exceed horizontal criterion in a period of time shorter than 10 years counting from the reference epoch (i.e. from 2011.0): KATO horizontal coordinates should be updated in 2014 and GWWL in 2021. Mores sites will exceed vertical criterion: BOGI (2021), GWWL (2015), KATO (2019), SWKI (2019) and ZYWI (2021). The vertical component updating will require of course changes of the horizontal component (coordinates are determined in the Cartesian orthogonal XYZ coordinate system and then recalculated for the ellipsoidal coordinates). Besides periodical coordinates updating it is recommended to conduct repeated processing for the entire network according to the EUREF standards for taking the current GNSS observations into consideration.

西江航道整治工程中的数值模型研究和应用

以西江四滩河道整治工程为例，建立了航道整治工程数值模型，模型采用正交贴体网格，耦合求解水动力场和悬移质浓度场，数值模拟了工程前后的流场变化和河床冲淤过程。经校准后，模拟效果较好，可以为整治工程提供定量预测，并为今后类似工作提供借鉴。 A model of Xijiang channel regulation project was developed, which used the curvilinear and orthogonal coordinate system, the primitive variable and suspended sediment concentration field was simulated by coupling method. After the model was calibrated, the modeling effect was good. The model constructed here is a powerful tool for those similar projects.