scholarly journals Bessel Equation in the Semiunbounded Intervalx∈[x0,∞]: Solving in the Neighbourhood of an Irregular Singular Point

2016 ◽  
Vol 2016 ◽  
pp. 1-7
Author(s):  
Qing-Hua Zhang ◽  
Jian Ma ◽  
Yuanyuan Qu
Keyword(s):  
Fourier Series ◽  
Solution Method ◽  
Formal Solution ◽  
Solution Procedure ◽  
Compatibility Conditions ◽  
Bessel Equation ◽  
Irregular Solution ◽  
Complex Constant ◽  
Irregular Singular Point ◽  
Coefficient Property

This study expresses the solution of the Bessel equation in the neighbourhood ofx=∞as the product of a known-form singular divisor and a specific nonsingular function, which satisfies the corresponding derived equation. Considering the failure of the traditional irregular solution constructed with the power series, we adopt the corrected Fourier series with only limited smooth degree to approximate the nonsingular function in the interval[x0,∞]. In order to guarantee the series’ uniform convergence and uniform approximation to the derived equation, we introduce constraint and compatibility conditions and hence completely determine all undetermined coefficients of the corrected Fourier series. Thus, what we found is not an asymptotic solution atx→∞(not to mention a so-called formal solution), but a solution in the interval[x0,∞]with certain regularities of distribution. During the solution procedure, there is no limitation on the coefficient property of the equation; that is, the coefficients of the equation can be any complex constant, so that the solution method presented here is universal.

Computational contact mechanics analysis of laterally graded orthotropic half-planes

2017 ◽  
Vol 14 (2) ◽  
pp. 145-154 ◽  
Author(s):  
Onur Arslan
Keyword(s):  
Contact Mechanics ◽  
Solution Method ◽  
Solution Procedure ◽  
Engineering Applications ◽  
Linear Solution ◽  
Content Type ◽  
Analytical Technique ◽  
Computational Contact Mechanics ◽  
Mechanics Analysis ◽  
Contact Response

Purpose Frictional sliding contact problems between laterally graded orthotropic half-planes and a flat rigid stamp are investigated. The presented study aims at guiding engineering applications in the prediction of the contact response of orthotropic laterally graded members. Design/methodology/approach The solution procedure is based on a finite element (FE) approach which is conducted with an efficient FE analysis software ANSYS. The spatial gradations of the orthotropic stiffness constants through the horizontal axis are enabled utilizing the homogeneous FE approach. The Augmented Lagrangian contact algorithm is used as an iterative non-linear solution method in the contact analysis. Findings The accuracy of the proposed FE solution method is approved by using the comparisons of the results with those computed using an analytical technique. The prominent results indicate that the surface contact stresses can be mitigated upon increasing the degree of orthotropy and positive lateral gradations. Originality/value One can infer from the literature survey that, the contact mechanics analysis of orthotropic laterally graded materials has not been investigated so far. In this study, an FE method-based computational solution procedure for the aforementioned problem is addressed. The presented study aims at guiding engineering applications in the prediction of the contact response of orthotropic laterally graded members. Additionally, this study provides some useful points related to computational contact mechanics analysis of orthotropic structures.

Numerical Investigation of a Centrifugal Compressor for Supercritical CO2 Cycles

2020 ◽  
Author(s):  
Renan Emre Karaefe ◽  
Pascal Post ◽  
Marwick Sembritzky ◽  
Andreas Schramm ◽  
Francesca di Mare ◽  
...  
Keyword(s):  
Experimental Data ◽  
Thermodynamic Properties ◽  
Flow Field ◽  
Numerical Simulations ◽  
Centrifugal Compressor ◽  
Supercritical Co2 ◽  
Solution Method ◽  
Three Dimensional ◽  
Solution Procedure ◽  
Test Loop

Abstract In this work, the performance characteristics and the flow field of a centrifugal compressor operating with supercritical CO2 are investigated by means of three-dimensional CFD. The considered geometry is based on main dimensions of the centrifugal compressor installed in the supercritical CO2 compression test-loop operated by Sandia National Laboratories. All numerical simulations are performed with a recently developed in-house hybrid CPU/GPU compressible CFD solver. Thermodynamic properties are computed through an efficient and accurate tabulation technique, the Spline-Based Table Look-Up Method (SBTL), particularly optimised for the applied density-based solution procedure. Numerical results are compared with available experimental data and accuracy as well as potentials in computational speedup of the solution method in combination with the SBTL are evaluated in the context of supercritical CO2 turbomachinery.

scholarly journals On Step Approximation for Roseau's Analytical Solution of Water Waves

2011 ◽  
Vol 2011 ◽  
pp. 1-20 ◽  
Author(s):  
Chia-Cheng Tsai ◽  
Tai-Wen Hsu ◽  
Yueh-Ting Lin
Keyword(s):  
Transfer Matrix ◽  
Water Waves ◽  
Water Wave ◽  
Solution Method ◽  
Linear Equations ◽  
Solution Procedure ◽  
Water Wave Scattering ◽  
Scattering Effects ◽  
Wave Problems ◽  
Marching Method

An indirect eigenfunction marching method (IEMM) is developed to provide step approximations for water wave problems. The bottom profile is in terms of successive flat shelves separated by abrupt steps. The marching conditions are represented by the horizontal velocities at the steps in the solution procedure. The approximated wave field can be obtained by solving a system of linear equations with unknown coefficients which represents the horizontal velocities under a proper basis. It is also demonstrated that this solution method can be exactly reduced to the transfer-matrix method (TM method) for a specific setting. The combined scattering effects of a series of steps can be described by a single two-by-two transfer matrix for connecting the far-field behaviors of both sides for this method. The solutions obtained by the IEMM are basicallyexactfor water wave problems considering step-like bottoms. Numerical simulations were performed to validate the present and commonly used methods. Furthermore, it also shows that the solutions obtained by the IEMM converge very well to Roseau's analytical solutions for both mild and steep curved bottom profiles. The present method improves the converges of the TM method for solving water wave scattering over steep bathymetry.

A Simultaneous Variable Solution Procedure for Laminar and Turbulent Flows in Curved Channels and Bends

2000 ◽  
Vol 122 (3) ◽  
pp. 552-559 ◽  
Author(s):  
Jianrong Wang ◽  
Siamack A. Shirazi
Keyword(s):  
Experimental Data ◽  
Numerical Solution ◽  
Numerical Method ◽  
Solution Method ◽  
Stokes Equations ◽  
Turbulence Models ◽  
Solution Procedure ◽  
Navier Stokes ◽  
Navier Stokes Equations ◽  
Curved Channels

Direct Numerical Simulation of turbulent flow requires accurate numerical techniques for solving the Navier-Stokes equations. Therefore, the Navier-Stokes equations in general orthogonal and nonorthogonal coordinates were employed and a simultaneous variable solution method was extended to solve these general governing equations. The present numerical method can be used to accurately predict both laminar and turbulent flow in various curved channels and bends. To demonstrate the capability of this numerical method and to verify the method, the time-averaged Navier-Stokes equations were employed and several turbulence models were also implemented into the numerical solution procedure to predict flows with strong streamline curvature effects. The results from the present numerical solution procedure were compared with available experimental data for a 90 deg bend. All of the turbulence models implemented resulted in predicted velocity profiles which were in agreement with the trends of experimental data. This indicates that the solution method is a viable numerical method for calculating complex flows. [S0098-2202(00)01803-4]

scholarly journals A New Method to Solve Interval Transportation Problems

2020 ◽  
pp. 802-811
Author(s):  
Abdul Quddoos ◽  
Ummey Habiba
Keyword(s):  
Comparative Study ◽  
Transportation Problem ◽  
Solution Method ◽  
Fuzzy Programming ◽  
Solution Procedure ◽  
Programming Technique ◽  
Left Limit ◽  
Objective Model ◽  
The Cost ◽  
Single Objective

transportation problem (ITP) in which the cost-coefficients of the objective function, source and destination parameters are all in the form of interval. In this paper, the single objective interval transportation problem is transformed into an equivalent crisp bi-objective transportation problem where the left-limit and width of the interval are to be minimized. The solution to this bi-objective model is then obtained with the help of fuzzy programming technique. The proposed solution procedure has been demonstrated with the help of a numerical example. A comparative study has also been made between the proposed solution method and the method proposed by Das et al.(1999) .

A New Solution Method for the Contact Mechanics of Graded Coatings

2010 ◽  
Author(s):  
S. J. Chidlow ◽  
M. Teodorescu ◽  
N. D. Vaughan
Keyword(s):  
Fourier Transform ◽  
Fourier Series ◽  
Stress Function ◽  
Solution Method ◽  
Inverse Fourier Transform ◽  
Elastic Solid ◽  
Airy Stress Function ◽  
Graded Coatings ◽  
Contour Plots ◽  
Selection Of

This paper attempts to solve analytically for the stresses present in a graded elastic solid resulting from pressure applied to its surface by computing the Airy stress function. The horizontal dimensions of the solid are assumed finite and hence we form the solution of the stress function as a Fourier series rather than an inverse Fourier transform. Finally, a selection of contour plots is presented to exhibit the behavior of this new model.

Determining the Sub-Surface Stresses in a Graded-Elastic Solid Using Fourier Series Decomposition

2011 ◽  
Author(s):  
Stewart Chidlow ◽  
Mircea Teodorescu ◽  
Nick Vaughan
Keyword(s):  
Fourier Series ◽  
Shear Modulus ◽  
Solution Method ◽  
Analytic Solution ◽  
Elastic Solid ◽  
Local Deformation ◽  
Elastic Substrate ◽  
Micro Scale ◽  
Surface Stresses ◽  
Layered Solid

This paper describes a fully analytic solution method for the displacements and sub-surface stresses within a graded elastic layered solid. This method can be utilised to predict the local deformation of nano or micro-scale depositions under contacting conditions. The solid consists of two distinct layers which are considered to be perfectly bonded and comprise of a graded elastic coating whose shear modulus varies exponentially with the depth coordinate and an infinitely deep homogeneously elastic substrate. The solution given in this paper is generic and easily utilised to solve real problems as it requires only known physical characteristics of the solid under study and an applied surface pressure. As a result, this model is very cheap to use and can be easily integrated into tribological codes to predict local deflections.

scholarly journals On the Borel summability of divergent solutions of the heat equation

1999 ◽  
Vol 154 ◽  
pp. 1-29 ◽  
Author(s):  
D. A. Lutz ◽  
M. Miyake ◽  
R. Schäfke
Keyword(s):  
Heat Equation ◽  
Heat Kernel ◽  
Formal Solution ◽  
Cauchy Data ◽  
Borel Summability ◽  
Stripe Domain ◽  
Integral Expression ◽  
Irregular Singular Point ◽  
The Cauchy Problem ◽  
Continuation Property

AbstractIn recent years, the theory of Borel summability or multisummability of divergent power series of one variable has been established and it has been proved that every formal solution of an ordinary differential equation with irregular singular point is multisummable. For partial differential equations the summability problem for divergent solutions has not been studied so well, and in this paper we shall try to develop the Borel summability of divergent solutions of the Cauchy problem of the complex heat equation, since the heat equation is a typical and an important equation where we meet diveregent solutions. In conclusion, the Borel summability of a formal solution is characterized by an analytic continuation property together with its growth condition of Cauchy data to infinity along a stripe domain, and the Borel sum is nothing but the solution given by the integral expression by the heat kernel. We also give new ways to get the heat kernel from the Borel sum by taking a special Cauchy data.

Computer Analysis of Three-Dimensional Turbulent Flows around Ships' Hulls

1980 ◽  
Vol 194 (1) ◽  
pp. 239-248 ◽  
Author(s):  
N. C. Markatos ◽  
M. R. Malin ◽  
D. G. Tatchell
Keyword(s):  
Differential Equations ◽  
Coordinate System ◽  
Turbulent Flows ◽  
Solution Method ◽  
Initial Conditions ◽  
Three Dimensional ◽  
Grid Method ◽  
Solution Procedure ◽  
The Body ◽  
Cross Sectional

This paper describes a general solution method for three-dimensional, steady, turbulent flows around long, smoothly-shaped bodies, of arbitrary and varying cross-sectional shape. The particular example considered here concerns the flow around the hull of a ship, but the method can equally well be applied to other, similarly shaped bodies such as an aircraft fuselage, or a submarine. Moreover, the basic non-orthogonal grid method described can also be applied to internal flows in irregular shaped passages, or to the prediction of flows around bodies in ducts. The mathematical model consists of the partial differential equations for continuity and three components of momentum, along with a two-equation model of turbulence, and proper modelling of the ship's hull. The solution method utilizes a non-orthogonal coordinate system in the plane normal to the axis of the body, which has one coordinate surface coinciding with the hull surface. This coordinate system is flexible and is easily modified to enable the calculation procedure to handle bulbous ships' hulls, which are of great importance in modern ship design. The differential equations involved are solved numerically after provision of the proper boundary and initial conditions. The solution procedure is a unique one, called ‘partially-parabolic’, as first used by Pratap and Spalding (1). Solutions are presented for flow around ships' hulls, which demonstrate the physical realism of the achieved results and the potential of the present method.

scholarly journals A Novel Analytical Solution Method for Constraint Forces of the Kinematic Pair and Its Applications

2015 ◽  
Vol 2015 ◽  
pp. 1-8 ◽  
Author(s):  
Changjian Zhi ◽  
Sanmin Wang ◽  
Yuantao Sun ◽  
Bo Li
Keyword(s):  
Solution Method ◽  
Solution Procedure ◽  
Dynamics Analysis ◽  
Kinematic Pair ◽  
Constraint Forces ◽  
Constraint Force ◽  
Spatial Mechanism ◽  
Observation Method ◽  
Crank Mechanism ◽  
Mechanism Theory

Constraint forces of the kinematic pair are the basis of the kinematics and dynamics analysis of mechanisms. Exploring the solution method for constraint forces is a hot issue in the mechanism theory fields. Based on the observation method and the theory of reciprocal screw system, the solution method of reciprocal screw system is improved and its solution procedures become easier. This method is also applied to the solution procedure of the constraint force. The specific expressions of the constraint force are represented by the reciprocal screw system of twist. The transformation formula of twist under different coordinates is given and it make the expression of the twist of kinematic pair more facility. A slider-crank mechanism and a single loop spatial RUSR mechanism are taken as examples. It confirms that this method can be used to solve the constraint force of the planar and spatial mechanism.

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