The Dirichlet Problem on Wave Propagation in a 2D Exterior Cracked Domain without Compatibility Conditions at the Tips of the Cracks

2011 ◽  
Author(s):  
P. A. Krutitskii ◽  
Theodore E. Simos ◽  
George Psihoyios ◽  
Ch. Tsitouras ◽  
Zacharias Anastassi
Keyword(s):  
Wave Propagation ◽  
Dirichlet Problem ◽  
Compatibility Conditions

scholarly journals The 2D Dirichlet Problem for the Propagative Helmholtz Equation in an Exterior Domain with Cracks and Singularities at the Edges

2012 ◽  
Vol 2012 ◽  
pp. 1-18 ◽  
Author(s):  
P. A. Krutitskii
Keyword(s):  
Weak Solution ◽  
Dirichlet Problem ◽  
Integral Representation ◽  
Helmholtz Equation ◽  
Exterior Domain ◽  
Index Zero ◽  
Compatibility Conditions ◽  
Unique Classical Solution ◽  
Asymptotic Formulae ◽  
Solution Gradient

The Dirichlet problem for the 2D Helmholtz equation in an exterior domain with cracks is studied. The compatibility conditions at the tips of the cracks are assumed. The existence of a unique classical solution is proved by potential theory. The integral representation for a solution in the form of potentials is obtained. The problem is reduced to the Fredholm equation of the second kind and of index zero, which is uniquely solvable. The asymptotic formulae describing singularities of a solution gradient at the edges (endpoints) of the cracks are presented. The weak solution to the problem may not exist, since the problem is studied under such conditions that do not ensure existence of a weak solution.

scholarly journals On the high order convergence of the difference solution of Laplace’s equation in a rectangular parallelepiped

Filomat ◽  
2018 ◽  
Vol 32 (3) ◽  
pp. 893-901 ◽  
Author(s):  
Adiguzel Dosiyev ◽  
Ahlam Abdussalam
Keyword(s):  
Approximate Solution ◽  
Dirichlet Problem ◽  
Laplace Equation ◽  
Grid Point ◽  
Order Convergence ◽  
Compatibility Conditions ◽  
Second Derivatives ◽  
Boundary Functions ◽  
High Order Convergence ◽  
The Difference

The boundary functions ?j of the Dirichlet problem, on the faces ?j, j = 1,2,..., 6 of the parallelepiped R are supposed to have seventh derivatives satisfying the H?lder condition and on the edges their second, fourth and sixth order derivatives satisfy the compatibility conditions which result from the Laplace equation. For the error uh-u of the approximate solution uh at each grid point (x1,x2,x3), a pointwise estmation O(?h6) is obtained, where ?= ?(x1,x2,x3) is the distance from the current grid point to the boundary of R; h is the grid step. The solution of difference problems constructed for the approximate values of the first and pure second derivatives converge with orders O(h6 ?ln h?) and O(h5+?), 0 < ? < 1, respectivly.

scholarly journals The Dirichlet Problem for the 2D Laplace Equation in a Domain with Cracks without Compatibility Conditions at Tips of the Cracks

2012 ◽  
Vol 2012 ◽  
pp. 1-20 ◽  
Author(s):  
P. A. Krutitskii
Keyword(s):  
Dirichlet Problem ◽  
Classical Solution ◽  
Laplace Equation ◽  
Boundary Data ◽  
Exterior Domains ◽  
Compatibility Conditions ◽  
Closed Curves ◽  
Asymptotic Formulae ◽  
Well Posed ◽  
Solution Gradient

We study the Dirichlet problem for the 2D Laplace equation in a domain bounded by smooth closed curves and smooth cracks. In the formulation of the problem, we do not require compatibility conditions for Dirichlet's boundary data at the tips of the cracks. However, if boundary data satisfies the compatibility conditions at the tips of the cracks, then this is a particular case of our problem. The cases of both interior and exterior domains are considered. The well-posed formulation of the problem is given, theorems on existence and uniqueness of a classical solution are proved, and the integral representation for a solution is obtained. It is shown that weak solution of the problem does not typically exist, though the classical solution exists. The asymptotic formulae for singularities of a solution gradient at the tips of the cracks are presented.

Electron Microscopy of Shock Loaded Polycrystalline Beryllium

1974 ◽  
Vol 32 ◽  
pp. 506-507 ◽  
Author(s):  
J. M. Galbraith ◽  
L. E. Murr ◽  
A. L. Stevens
Keyword(s):  
Electron Microscopy ◽  
Wave Propagation ◽  
Structural Change ◽  
Single Crystal ◽  
Diffraction Pattern ◽  
Shock Loading ◽  
Slip Systems ◽  
Compression Tests ◽  
X Ray Diffraction ◽  
X Ray

Uniaxial compression tests and hydrostatic tests at pressures up to 27 kbars have been performed to determine operating slip systems in single crystal and polycrystal1ine beryllium. A recent study has been made of wave propagation in single crystal beryllium by shock loading to selectively activate various slip systems, and this has been followed by a study of wave propagation and spallation in textured, polycrystal1ine beryllium. An alteration in the X-ray diffraction pattern has been noted after shock loading, but this alteration has not yet been correlated with any structural change occurring during shock loading of polycrystal1ine beryllium.This study is being conducted in an effort to characterize the effects of shock loading on textured, polycrystal1ine beryllium. Samples were fabricated from a billet of Kawecki-Berylco hot pressed HP-10 beryllium.

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