Approximate solutions to the half-space integral transport equation near a plane boundary

1980 ◽  
Vol 58 (9) ◽  
pp. 1291-1310 ◽  
Author(s):  
Michael S. Milgram
Keyword(s):  
Transport Equation ◽  
Half Space ◽  
Matrix Multiplication ◽  
Solution Space ◽  
Plane Geometry ◽  
Plane Boundary ◽  
Infinite Plane ◽  
Algebraic Equations ◽  
Linear Algebraic Equations ◽  
Integral Transport

A set of functions spanning the solution space of the integral transport equation near a boundary in semi-infinite plane geometry is obtained and used to reduce the problem to that of a system of linear algebraic equations. Expressions for the boundary angular flux are obtained by matrix multiplication, and the theory is extended to adjacent half-space problems by matching the angular flux at the boundary. Thus a unified theory is obtained for well-behaved arbitrary sources in semi-infinite plane geometry. Numerical results are given for both Milne's problem and the problem of constant production in adjacent half-spaces, and albedo problems in semi-infinite geometry. The solutions for the flux density are best near the boundary, and for the angular flux are best for angles near the plane of the boundary; it is conjectured that the theory will prove most useful when extended to arrays of finite slabs.

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 Fourier Expansion to Elastic Vector Wave Functions and Applications of Wave Bases to Scattering in Half-Space

2012 ◽  
Vol 28 (1) ◽  
pp. 19-39 ◽  
Author(s):  
P.-J. Shih ◽  
T.-J. Teng ◽  
C.-S. Yeh
Keyword(s):  
Boundary Conditions ◽  
Half Space ◽  
Fourier Expansion ◽  
Homogeneous Medium ◽  
Wave Functions ◽  
Basis Set ◽  
Plane Boundary ◽  
Surface Boundary ◽  
Infinite Plane ◽  
Vector Wave

ABSTRACTThis paper proposes a complete basis set for analyzing elastic wave scattering in half-space. The half-space is an isotropic, linear, and homogeneous medium except for a finite inhomogeneity. The wave bases are obtained by combining buried source functions and their reflected counter-waves generated from the infinite-plane boundary. The source functions are the vector wave functions of infinite-space. Based on the source functions expressed in the Fourier expansion form, the reflected counter-waves are easily obtained by solving the infinite-plane boundary conditions. Few representations adopt Wely's integration, but the Fourier expansion is developed from it and applied to decouple the angular-differential terms of the vector wave functions. In addition to the scattering of the finite inhomogeneity, the transition matrix method is extended to express the surface boundary conditions. For the numerical application in this paper, the P- and the SV- waves are assumed as the incoming fields. As an example, this paper computes stress concentrations around a cavity. The steepest-descent path method yielding the optimum integral paths is used to ensure the numerical convergence of the wave bases in the Fourier expansion. The resultant patterns from these approaches are compared with those obtained from numerical simulations.

Stress Distribution in Bonded Dissimilar Materials Containing Circular or Ring-Shaped Cavities

1965 ◽  
Vol 32 (4) ◽  
pp. 829-836 ◽  
Author(s):  
F. Erdogan
Keyword(s):  
Integral Equations ◽  
Complete Solution ◽  
Closed Form Solution ◽  
Form Solution ◽  
Fredholm Integral Equations ◽  
Infinite Plane ◽  
Algebraic Equations ◽  
Axially Symmetric ◽  
Linear Algebraic Equations ◽  
Penny Shaped Crack

The general problem of two semi-infinite elastic media with different properties bonded to each other along a plane and containing a series of concentric ring-shaped flat cavities is considered. Using the Green’s functions for the semi-infinite plane, the problem is formulated as a system of simultaneous singular integral equations. Closed-form solution of the corresponding dominant system with Cauchy kernels is given. To obtain the complete solution, a technique reducing the problem to solving a system of linear algebraic equations rather than a pair of Fredholm integral equations is outlined. The examples for which the contact stresses are plotted include the bonded media with an axially symmetric external notch subject to axial load or homogeneous temperature changes and the case of penny-shaped crack.

scholarly journals Dynamic Analysis of a Shallow Buried Tunnel Influenced by a Neighboring Semi-cylindrical Hill and Semi-cylindrical Canyon

2019 ◽  
Author(s):  
Xiaotang Lv ◽  
Cuiling Ma ◽  
Meixiang Fang
Keyword(s):  
Stress Concentration ◽  
Dynamic Analysis ◽  
Half Space ◽  
Dynamic Stress ◽  
Mixed Boundary ◽  
Scattered Wave ◽  
Dynamic Stress Concentration ◽  
Algebraic Equations ◽  
Sh Waves ◽  
Linear Algebraic Equations

This paper provides a dynamic analysis of the response of a subsurface cylindrical tunnel to SH waves influenced by a neighboring semi-cylindrical hill and semi-cylindrical canyon in half-space using complex functions. For convenience in finding a solution, the half-space is divided into two parts and the scattered wave functions are constructed in both parts. Then the mixed boundary conditions are satisfied by moving coordinates. Finally, the problem is reduced to solving a set of infinite linear algebraic equations, for which the unknown coefficients are obtained by truncation of the infinite set of equations. The effects of the incident angles and frequencies of SH waves, as well as of the radius of the tunnel, hill, and canyon on the dynamic stress concentration of the tunnel are studied. The results show that the hill and canyon have a significant effect on the dynamic stress concentration of the tunnel.

The Scattering of SH-Wave Caused by Subsurface Circular Cavities in a Layered Half-Space

2011 ◽  
Vol 488-489 ◽  
pp. 226-229
Author(s):  
Dong Ni Chen ◽  
Hui Qi ◽  
Yong Shi
Keyword(s):  
Half Space ◽  
Elastic Layer ◽  
Complex Function ◽  
Expansion Method ◽  
Elastic Half Space ◽  
Wave Functions ◽  
Large Radius ◽  
Algebraic Equations ◽  
Sh Wave ◽  
Linear Algebraic Equations

The scattering of SH-wave caused by the subsurface circular cavities in an elastic half-space covered with an elastic layer was discussed, which was based on the complex function method ,wave functions expansion method and big circular arc postulation method in which the circular boundary of large radius was used to approximate straight boundary of surface elastic layer. By the theory of Helmholtz, the general solution of the Biot’s wave function was achieved. Utilizing the complex series expansion technology and the boundary conditions, we could transform the present problem into the problem in which we needed to solve the infinite linear algebraic equations with unknown coefficients in wave functions. Finally, the dynamic stress concentration factors around the circular cavities were discussed in numerical examples.

Several Elliptical Punches on an Elastic Half Space

1986 ◽  
Vol 53 (2) ◽  
pp. 390-394 ◽  
Author(s):  
V. I. Fabrikant
Keyword(s):  
Half Space ◽  
General Theorem ◽  
Elastic Half Space ◽  
Algebraic Equations ◽  
Linear Algebraic Equations ◽  
Generalized Displacements

A general theorem is established which relates the resulting forces, acting on a set of arbitrary punches, with their generalized displacements through a system of linear algebraic equations. The theorem is applied to the case of arbitrarily located elliptical punches. Several specific examples are considered.

The Effect on Scattering of SH-Wave in a Layered Half-Space with the Subsurface Circular Cavities

2011 ◽  
Vol 194-196 ◽  
pp. 1908-1911
Author(s):  
Dong Ni Chen ◽  
Hui Qi ◽  
Yong Shi
Keyword(s):  
Half Space ◽  
Elastic Layer ◽  
Complex Function ◽  
Expansion Method ◽  
Elastic Half Space ◽  
Wave Functions ◽  
Large Radius ◽  
Algebraic Equations ◽  
Sh Wave ◽  
Linear Algebraic Equations

The scattering of SH-wave caused by the subsurface circular cavities in an elastic half-space covered with an elastic layer was discussed, which was based on the complex function method and wave functions expansion method. The solution of scattering of SH-wave was given by using circular boundary of large radius to approximate straight boundary of surface elastic layer. According to boundary conditions, we needed to solve the infinite linear algebraic equations with unknown coefficients in wave functions. Finally, the dynamic stress concentration factors around circular cavities were discussed in numerical examples.

XII.—Studies in Practical Mathematics. I. The Evaluation, with Applications, of a Certain Triple Product Matrix

1938 ◽  
Vol 57 ◽  
pp. 172-181 ◽  
Author(s):  
A. C. Aitken
Keyword(s):  
Quadratic Form ◽  
Matrix Multiplication ◽  
Matrix Product ◽  
Algebraic Equations ◽  
Product Matrix ◽  
Linear Algebraic Equations ◽  
Special Cases ◽  
The Matrix ◽  
Practical Mathematics ◽  
General Operation

The solution of simultaneous linear algebraic equations, the evaluation of the adjugate or the reciprocal of a given square matrix, and the evaluation of the bilinear or quadratic form reciprocal to a given form, are all special cases of a certain general operation, namely the evaluation of a matrix product H′A-1K, where A is square and non-singular, that is, the determinant | A | is not zero. (Matrix multiplication is like determinant multiplication, but exclusively row-into-column. The matrix H′ is obtained from H by transposition, that is, by changing rows into columns.) The matrices H′ and K may be rectangular. If A is singular, the reciprocal A-1 does not exist; and in such a case the product H′(adj A) K may be required. Arithmetically, the only difference in the computation of H′A-1K and H(adj A) K is that in the latter case a final division of all elements by | A | is not performed.

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