A Diagonally Dominant Solution for the Cylinder End Problem

1981 ◽  
Vol 48 (4) ◽  
pp. 876-880 ◽  
Author(s):  
T. D. Gerhardt ◽  
Shun Cheng
Keyword(s):  
Infinite System ◽  
Linear Equations ◽  
Infinite Cylinder ◽  
Algebraic Equations ◽  
Precise Calculation ◽  
Linear Algebraic Equations ◽  
Present Solution ◽  
Diagonally Dominant ◽  
Expansion Coefficients

An improved elasticity solution for the cylinder problem with axisymmetric torsionless end loading is presented. Consideration is given to the specification of arbitrary stresses on the end of a semi-infinite cylinder with a stress-free lateral surface. As is known from the literature, the solution to this problem is obtained in the form of a nonorthogonal eigenfunction expansion. Previous solutions have utilized functions biorthogonal to the eigenfunctions to generate an infinite system of linear algebraic equations for determination of the unknown expansion coefficients. However, this system of linear equations has matrices which are not diagonally dominant. Consequently, numerical instability of the calculated eigenfunction coefficients is observed when the number of equations kept before truncation is varied. This instability has an adverse effect on the convergence of the calculated end stresses. In the current paper, a new Galerkin formulation is presented which makes this system of equations diagonally dominant. This results in the precise calculation of the eigenfunction coefficients, regardless of how many equations are kept before truncation. By consideration of a numerical example, the present solution is shown to yield an accurate calculation of cylinder stresses and displacements.

Diffraction by a set of three parallel impedance half-planes with the one amidst located in the opposite direction

2002 ◽  
Vol 80 (8) ◽  
pp. 893-909 ◽  
Author(s):  
G Çinar ◽  
A Büyükaksoy
Keyword(s):  
Fourier Transform ◽  
Opposite Direction ◽  
Infinite System ◽  
Plane Waves ◽  
Mode Matching ◽  
Algebraic Equations ◽  
Matching Method ◽  
Linear Algebraic Equations ◽  
Expansion Coefficients ◽  
The One

The problem of diffraction of plane waves by a set of three parallel half-planes with different surface impedances on upper and lower faces where the one in the middle is placed in the opposite direction, is solved by the mode-matching method where available, and by Fourier-transform technique elsewhere. The solution includes two independent Wiener–Hopf equations each involving an infinite number of expansion coefficients that satisfy an infinite system of linear algebraic equations. PACS No.: 41.20J

Sound Radiation from the Perforated End of a Lined Duct

2019 ◽  
Vol 105 (4) ◽  
pp. 591-599 ◽  
Author(s):  
Burhan Tiryakioglu
Keyword(s):  
Infinite System ◽  
Normal Modes ◽  
Mode Matching ◽  
Algebraic Equations ◽  
Matching Method ◽  
Linear Algebraic Equations ◽  
Matching Technique ◽  
Specific Impedance ◽  
Expansion Coefficients ◽  
The Fourier Transform

Radiation of sound wave through a lined duct with perforated end is analyzed rigorously. The problem considered is axisymmetric. By using the Fourier transform technique in conjunction with the Mode Matching method, the related boundary value problem is formulated as a Wiener-Hopf (W-H) equation. The Mode-Matching technique allows us to express the field component defined in the waveguide region in terms of normal modes. The solution involves a set of infinitely many expansion coefficients satisfying an infinite system of linear algebraic equations. The numerical solution of this system is obtained for different parameters of the problem such as the surface impedances, specific impedance of the perforated screen and their effects on the radiation phenomenon are shown graphically.

scholarly journals On the Solubility of Linear Algebraic Equations (Contd.)

1913 ◽  
Vol 12 ◽  
pp. 137-138
Author(s):  
John Dougall
Keyword(s):  
Linear Equations ◽  
Homogeneous System ◽  
Algebraic Equations ◽  
Similar Reasoning ◽  
Null Solution ◽  
Linear Algebraic Equations ◽  
The Given ◽  
Homogeneous Linear

A system of n non-homogeneous linear equations in n variables has one and only one solution if the homogeneous system obtained from the given system by putting all the constant terms equal to zero has no solution except the null solution.This may be proved independently by similar reasoning to that given for Theorem I., or it may be deduced from that theorem. We follow the latter method.

scholarly journals Two Contact Problems for Annular Rigid Stamp on an Elastic Half Space

2018 ◽  
Vol 17 (6) ◽  
pp. 458-464
Author(s):  
S. V. Bosakov
Keyword(s):  
Functional Equation ◽  
Half Space ◽  
Bessel Functions ◽  
Infinite System ◽  
Annular Plate ◽  
Contact Problems ◽  
Elastic Half Space ◽  
Contact Stresses ◽  
Algebraic Equations ◽  
Linear Algebraic Equations

The paper presents solutions of two contact problems for the annular plate die on an elastic half-space under the action of axisymmetrically applied force and moment. Such problems usually arise in the calculation of rigid foundations with the sole of the annular shape in chimneys, cooling towers, water towers and other high-rise buildings on the wind load and the load from its own weight. Both problems are formulated in the form of triple integral equations, which are reduced to one integral equation by the method of substitution. In the case of the axisymmetric problem, the kernel of the integral equation depends on the product of three Bessel functions. Using the formula to represent two Bessel functions in the form of a double row on the works of hypergeometric functions Bessel function, the problem reduces to a functional equation that connects the movement of the stamp with the unknown coefficients of the distribution of contact stresses. The resulting functional equation is reduced to an infinite system of linear algebraic equations, which is solved by truncation. Under the action of a moment on the annular plate  die, the distribution of contact stresses is searched as a series by the products of the Legendre attached functions with a weight corresponding to the features in the contact stresses at the die edges. Using the spectral G. Ya. Popov ratio for the ring plate, the problem is again reduced to an infinite system of linear algebraic equations, which is also solved by the truncation method. Two examples of calculations for an annular plate die on an elastic half-space on the action of axisymmetrically applied force and moment are given. A comparison of the results of calculations on the proposed approach with the results for the round stamp and for the annular  stamp with the solutions of other authors is made.

scholarly journals Diffraction of the field of vertical electric dipole on the spiral conductive sphere in the presence of a cone

2018 ◽  
Keyword(s):  
Hilbert Space ◽  
Electric Dipole ◽  
Infinite System ◽  
Matrix Operator ◽  
Algebraic Equations ◽  
Problem Statement ◽  
Vertical Electric Dipole ◽  
Linear Algebraic Equations ◽  
Method Of Regularization ◽  
The Matrix

The problem of diffraction of a vertical electric dipole field on a spiral conductive sphere and a cone has been solved. By the method of regularization of the matrix operator of the problem, an infinite system of linear algebraic equations of the second kind with a compact matrix operator in Hilbert space $\ell_2$ is obtained. Some limiting variants of the problem statement are considered.

scholarly journals Numerical solution of a weakly singular integral equation by the method of orthogonal polynomials

2021 ◽  
pp. 98-103
Author(s):  
Sergei M. Sheshko
Keyword(s):  
Integral Equation ◽  
Approximate Solution ◽  
Singular Integral Equation ◽  
Orthogonal Polynomials ◽  
Numerical Solution ◽  
Singular Integral ◽  
Algebraic Equations ◽  
Linear Algebraic Equations ◽  
Expansion Coefficients ◽  
Analytical Expressions

A scheme is constructed for the numerical solution of a singular integral equation with a logarithmic kernel by the method of orthogonal polynomials. The proposed schemes for an approximate solution of the problem are based on the representation of the solution function in the form of a linear combination of the Chebyshev orthogonal polynomials and spectral relations that allows to obtain simple analytical expressions for the singular component of the equation. The expansion coefficients of the solution in terms of the Chebyshev polynomial basis are calculated by solving a system of linear algebraic equations. The results of numerical experiments show that on a grid of 20 –30 points, the error of the approximate solution reaches the minimum limit due to the error in representing real floating-point numbers.

scholarly journals On the economical solution method for a system of linear algebraic equations

2004 ◽  
Vol 2004 (4) ◽  
pp. 377-410 ◽  
Author(s):  
Jan Awrejcewicz ◽  
Vadim A. Krysko ◽  
Anton V. Krysko
Keyword(s):  
Differential Equations ◽  
Boundary Value ◽  
Linear Equations ◽  
Three Dimensional ◽  
Boundary Element Methods ◽  
Minimal Amount ◽  
Algebraic Equations ◽  
Linear Algebraic Equations ◽  
Point Of Intersection ◽  
Stationary Heat Transfer

The present work proposes a novel optimal and exact method of solving large systems of linear algebraic equations. In the approach under consideration, the solution of a system of algebraic linear equations is found as a point of intersection of hyperplanes, which needs a minimal amount of computer operating storage. Two examples are given. In the first example, the boundary value problem for a three-dimensional stationary heat transfer equation in a parallelepiped inℝ3is considered, where boundary value problems of first, second, or third order, or their combinations, are taken into account. The governing differential equations are reduced to algebraic ones with the help of the finite element and boundary element methods for different meshes applied. The obtained results are compared with known analytical solutions. The second example concerns computation of a nonhomogeneous shallow physically and geometrically nonlinear shell subject to transversal uniformly distributed load. The partial differential equations are reduced to a system of nonlinear algebraic equations with the error ofO(hx12+hx22). The linearization process is realized through either Newton method or differentiation with respect to a parameter. In consequence, the relations of the boundary condition variations along the shell side and the conditions for the solution matching are reported.

The Problem of an Elastic Stiffener Bonded to a Half Plane

1971 ◽  
Vol 38 (4) ◽  
pp. 937-941 ◽  
Author(s):  
F. Erdogan ◽  
G. D. Gupta
Keyword(s):  
Mechanical Properties ◽  
Integral Equation ◽  
Contact Problem ◽  
Half Plane ◽  
Elastic Constants ◽  
Infinite System ◽  
Stress Singularity ◽  
Algebraic Equations ◽  
Numerical Example ◽  
Linear Algebraic Equations

The contact problem of an elastic stiffener bonded to an elastic half plane with different mechanical properties is considered. The governing integral equation is reduced to an infinite system of linear algebraic equations. It is shown that, depending on the value of a parameter which is a function of the elastic constants and the thickness of the stiffener, the system is either regular or quasi-regular. A complete numerical example is given for which the strength of the stress singularity and the contact stresses are tabulated.

scholarly journals Radiation analysis of sound waves from semi-infinite coated pipe

2018 ◽  
Vol 18 (1) ◽  
pp. 92-111 ◽  
Author(s):  
Burhan Tiryakioglu ◽  
Ahmet Demir
Keyword(s):  
Fourier Transform ◽  
Sound Waves ◽  
Algebraic Equations ◽  
Total Field ◽  
Hopf Equation ◽  
Waveguide Modes ◽  
Linear Algebraic Equations ◽  
Pipe Radius ◽  
Expansion Coefficients ◽  
The Fourier Transform

An analytical solution is presented for the problem of radiation of sound waves from a semi-infinite circular cylindrical coated pipe which is partially lined from inside. By stating the total field in duct region in terms of normal waveguide modes (Dini’s series) and using the Fourier transform technique elsewhere, we obtain a Wiener–Hopf equation whose solution involving three sets of infinitely many unknown expansion coefficients satisfying three systems of linear algebraic equations. This system is solved numerically and the influence of some parameters (pipe radius, impedances, extension, etc.) on the radiation phenomenon is displayed graphically.

Unsteady double-diffusive buoyancy-induced flow in a triangular solar collector shape enclosure with corrugated bottom wall

2017 ◽  
Vol 27 (6) ◽  
pp. 1282-1303 ◽  
Author(s):  
M.M. Rahman ◽  
Sourav Saha ◽  
Satyajit Mojumder ◽  
Khan Md. Rabbi ◽  
Hasnah Hasan ◽  
...  
Keyword(s):  
Mass Transfer ◽  
Solar Collector ◽  
Linear Equations ◽  
Bottom Wall ◽  
Algebraic Equations ◽  
Weighted Residual Method ◽  
Content Type ◽  
Linear Algebraic Equations ◽  
Non Linear ◽  
Induced Flow

Purpose The purpose of this investigation is to determine the nature of the flow field, temperature distribution and heat and mass transfer in a triangular solar collector enclosure with a corrugated bottom wall in the unsteady condition numerically. Design/methodology/approach Non-linear governing partial differential equations (i.e. mass, momentum, energy and concentration equations) are transformed into a system of integral equations by applying the Galerkin weighted residual method. The integration involved in each of these terms is performed using Gauss’ quadrature method. The resulting non-linear algebraic equations are modified by the imposition of boundary conditions. Finally, Newton’s method is used to modify non-linear equations into the linear algebraic equations. Findings Both the buoyancy ratio and thermal Rayleigh number play an important role in controlling the mode of heat transfer and mass transfer. Originality/value Calculations are performed for various thermal Rayleigh numbers, buoyancy ratios and time periods. For each specific condition, streamline contours, isotherm contours and iso-concentration contours are obtained, and the variation in the overall Nusselt and Sherwood numbers is identified for different parameter combinations.

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