The numerical solution of certain integral equations with non-integrable kernels arising in the theory of crack propagation and elastic wave diffraction

1969 ◽  
Vol 265 (1163) ◽  
pp. 353-381 ◽  
Keyword(s):  
Integral Equation ◽  
Elastic Wave ◽  
Numerical Solution ◽  
Wave Diffraction ◽  
Crack Surface ◽  
Relative Displacement ◽  
Method Of Solution ◽  
Set Up ◽  
Prestressed State ◽  
Numerical Method Of Solution

An infinite elastic medium is initially at rest in a prestressed state of plane- or anti-plane strain. At time t = 0 a plane crack comes into existence which occupies a strip parallel to the y axis and whose width varies in time. Assuming that the components of the traction are known on the crack surface it is possible to set up an integral equation on the area of the crack for the relative displacement across the crack. Although the kernel of this integral equation is non-integrable a method is found for discretizing it and a numerical method of solution is carried out. The results, which in some cases are the solutions of diffraction problems, are presented graphically.

ON FLAME PROPAGATION IN EXPLOSIVE MIXTURES OF GASES: I. GENERAL THEORY

1956 ◽  
Vol 34 (3) ◽  
pp. 313-323 ◽  
Author(s):  
Robert Sandri
Keyword(s):  
Temperature Distribution ◽  
Flame Propagation ◽  
Energy Equation ◽  
Approximation Formula ◽  
Flame Velocity ◽  
System Of Differential Equations ◽  
Flame Zone ◽  
Method Of Solution ◽  
Set Up ◽  
Numerical Method Of Solution

The system of differential equations of flame propagation is set up and discussed. It is shown that, without any major influences being neglected, the energy equation can be reduced to the form[Formula: see text]with the boundary conditions[Formula: see text]Some qualities of the solutions of this equation are discussed and a simple numerical method of solution is described. The flame velocity V0 is found as an eigenvalue of the energy equation. The temperature distribution in the flame zone can then be found by an ordinary quadrature. Further, an approximation formula for finding V0 directly is derived[Formula: see text]where F(η) is proportional to [Formula: see text]and has a maximum for η = ηm.

scholarly journals Numerical solution of a boundary value problem for a loaded parabolic equation with nonlocal boundary conditions

2020 ◽  
pp. 15-28
Author(s):  
В.М. Абдуллаев
Keyword(s):  
Parabolic Equation ◽  
Boundary Value Problem ◽  
Boundary Conditions ◽  
Numerical Solution ◽  
Boundary Value ◽  
Nonlocal Boundary ◽  
Nonlocal Boundary Conditions ◽  
Test Problems ◽  
Method Of Solution ◽  
Numerical Method Of Solution

В работе с использованием метода прямых исследуется численное решение краевой задачи относительно нагруженного параболического уравнения с нелокальными краевыми условиями. Получены расчетные формулы и приводится алгоритм для решения задачи. Приведены результаты численного решения двух тестовых задач, иллюстрирующие эффективность предложенного подхода In the work, we propose a numerical method of solution to the boundary-value problem with respect to the loaded parabolic equation with nonlocal boundary conditions. We have obtained formulas and derived an algorithm for the solution of the problem. We provide the results of numerical solution to two test problems, which illustrates the efficiency of the approach proposed.

A Numerical Solution of the Dynamic Characteristics of an Aerostatic, Porous Thrust Bearing Having a Uniform Film Subjected to Linear Axial Load Variations

1977 ◽  
Vol 19 (3) ◽  
pp. 122-127 ◽  
Author(s):  
R. Taylor
Keyword(s):  
Numerical Solution ◽  
Pressure Distribution ◽  
Dynamic Characteristics ◽  
Axial Load ◽  
Thrust Bearing ◽  
Vibrational Motion ◽  
Load Variations ◽  
Method Of Solution ◽  
Air Film ◽  
Numerical Method Of Solution

A numerical method of solution is presented for the pressure distribution in the uniform air film of a porous thrust bearing in which the top plate is subjected to vertical vibrational motion. From this pressure distribution it is possible to compute the bearing's dynamic characteristics, the knowledge of which is essential for the design of these bearings if unstable operating regions are to be avoided.

Numerical Solution of Constant-Q Fractional Order Integral Modeling

2013 ◽  
Vol 690-693 ◽  
pp. 1821-1825
Author(s):  
Xiu Li ◽  
Hui Li Han
Keyword(s):  
Integral Equation ◽  
Plane Wave ◽  
Numerical Solution ◽  
Quality Factor ◽  
Fractional Order ◽  
Chebyshev Polynomial ◽  
Integral Model ◽  
Set Up ◽  
Quality Factor Q ◽  
Fractional Order Integral

In this paper fractional order integral model is set up about quality factor. Complex rectangular and Chebyshev polynomial are utilized to approximate the fractional order integral equation. Moreover, plane wave is used to calculate quality factor-Q. The numerical results show that Chebyshev polynomial approximation is more suitable to describe integral modeling of quality factor.

Numerical Methods for Solving Physically Nonlinear Problems for Inhomogeneous Thick-Walled Shells

2017 ◽  
Vol 865 ◽  
pp. 325-330 ◽  
Author(s):  
Vladimir I. Andreev ◽  
Lyudmila S. Polyakova
Keyword(s):  
Numerical Method ◽  
Nonlinear Problems ◽  
Successive Approximations ◽  
Method Of Successive Approximations ◽  
Deformation Diagram ◽  
Method Of Solution ◽  
Properties Of Materials ◽  
Physical Fields ◽  
Cylinders And Spheres ◽  
Numerical Method Of Solution

The paper proposes the numerical method of solution the problems of calculation the stress state in thick-walled cylinders and spheres from physically nonlinear inhomogeneous material. The urgency of solved problem due to the change of mechanical properties of materials under the influence of different physical fields (temperature, humidity, radiation, etc.). The deformation diagram describes the three-parameter formula. The numerical method used the method of successive approximations. The results of numerical calculation are compared with the test analytical solutions obtaining the authors with some restrictions on diagram parameters. The obtained results can be considered quite satisfactory.

Optimal Performance of Ships Under Combined Power and Sail

1982 ◽  
Vol 26 (03) ◽  
pp. 209-218
Author(s):  
John S. Letcher
Keyword(s):  
Numerical Solution ◽  
Economic Model ◽  
Analytic Solutions ◽  
Simultaneous Equations ◽  
Simultaneous Optimization ◽  
Fixed Cost ◽  
University Of Michigan ◽  
Cost Rate ◽  
Optimal Setting ◽  
Set Up

A simplified hydrodynamic and economic model is developed to describe the operation of a ship equipped with both sails and engine. In the range of light-to-moderate winds in which use of the engine is likely to be economical, the vessel is described by a characteristic speed, a characteristic fixed-cost rate, and five dimensionless parameters (four hydrodynamic, one economic). The model includes simultaneous optimization of three control variables: sail lift, throttle setting, and course angle; optimal setting of variable draft devices can be included optionally. Although no analytic solutions are attained, the simultaneous equations expressing minimization of cost per mile made good are set up, and a general algorithm is given for numerical solution of these problems. As an illustrative example, numerical values are worked out for the 30,000-dwt square-rigged bulk cargo ship from the 1975 University of Michigan study.

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