Numerical Calculation of Axial Divergence Profiles

1991 ◽  
Vol 35 (A) ◽  
pp. 611-616
Author(s):  
R. A. Coyle
Keyword(s):  
Numerical Calculation ◽  
Analytical Solution ◽  
Ray Tracing ◽  
Exact Analytical Solution ◽  
Bragg Angle ◽  
Flat Specimen ◽  
The Third ◽  
Tracing Techniques

AbstractThe diffractometer instrumental profiles convoluted by Alexander from aberration profiles calculated using the methods developed Easterbrook, Pike and Wilson include two profiles which are not amenable to exact analytical solution. These two profiles are generated by the axial divergence and flat specimen aberrations. Using ray tracing techniques the exact profiles of these two aberrations have been computed for the Philips PW1050 and Scintag DA5 diffractometers. sample profiles at 2θ = 28°, 90° and 152° have been convoluted with the third asymmetric profile, that due to specimen transparency, and plots of the resultant profiles are shown. Shifts of the centroids and the peaks of the profiles from the true Bragg angle have also been computed and plotted over the range 2θ = 0°-180°.

On numerically solving the complex eikonal equation using real ray-tracing methods: A comparison with the exact analytical solution

Geophysics ◽  
2012 ◽  
Vol 77 (4) ◽  
pp. T109-T116 ◽  
Author(s):  
Václav Vavryčuk
Keyword(s):  
Analytical Solution ◽  
Ray Tracing ◽  
Eikonal Equation ◽  
Exact Analytical Solution ◽  
Simple Type ◽  
The Real ◽  
S Waves ◽  
Complex Ray ◽  
Constant Gradient ◽  
Complex Eikonal

The exact analytical solution of the complex eikonal equation describing P- and S-waves radiated by a point source situated in a simple type of isotropic viscoelastic medium was ascertained. The velocity-attenuation model is smoothly inhomogeneous with a constant gradient of the square of the complex slowness. The resultant traveltime is complex; its real part describes the wave propagation and its imaginary part describes the attenuation effects. The solution was further used as a reference solution for numerical tests of the accuracy and robustness of two approximate ray-tracing approaches solving the complex eikonal equation: real elastic ray tracing and real viscoelastic ray tracing. Numerical modeling revealed that the real viscoelastic ray tracing method is unequivocally preferable to elastic ray tracing. It is more accurate and works even in situations when the elastic ray tracing fails. Also, the ray fields calculated by the real viscoelastic ray tracing are excellently reproduced even in the case when the elastic ray tracing yields completely distorted results. Compared with complex ray tracing, which is limited to simple types of media, the real viscoelastic ray tracing offers a fast and computationally straightforward procedure for calculating complex traveltimes in complicated 3D inhomogeneous attenuating structures.

scholarly journals Stress strain behavior research of triangular dam using analytical and numerical methods

2018 ◽  
Vol 251 ◽  
pp. 04043
Author(s):  
Maria Kozelskaya ◽  
Daria Donskova ◽  
Vera Ulianskaya ◽  
Pavel Shvetsov
Keyword(s):  
Finite Element Method ◽  
Finite Element ◽  
Numerical Calculation ◽  
Analytical Solution ◽  
Inverse Method ◽  
The Finite Element Method ◽  
Strain Behavior ◽  
The Third ◽  
Comparison Of The Results ◽  
Numerical And Analytical Methods

The article deals with the calculation of triangular dams by numerical and analytical methods. The analytical solution is performed by a semi-inverse method. The stress function is taken as a polynomial of the third degree. Numerical calculation is performed using the finite element method. A comparison of the results obtained by the two methods is performed.

scholarly journals The exact analytical solution of the dual-phase-lag two-temperature bioheat transfer of a skin tissue subjected to constant heat flux

2020 ◽  
Vol 10 (1) ◽  
Author(s):  
Hamdy M. Youssef ◽  
Najat A. Alghamdi
Keyword(s):  
Heat Flux ◽  
Analytical Solution ◽  
Exact Analytical Solution ◽  
Constant Heat Flux ◽  
Bioheat Transfer ◽  
Skin Tissue ◽  
Phase Lag ◽  
Dual Phase ◽  
Constant Heat ◽  
Two Temperature

Abstract This work is dealing with the temperature reaction and response of skin tissue due to constant surface heat flux. The exact analytical solution has been obtained for the two-temperature dual-phase-lag (TTDPL) of bioheat transfer. We assumed that the skin tissue is subjected to a constant heat flux on the bounding plane of the skin surface. The separation of variables for the governing equations as a finite domain is employed. The transition temperature responses have been obtained and discussed. The results represent that the dual-phase-lag time parameter, heat flux value, and two-temperature parameter have significant effects on the dynamical and conductive temperature increment of the skin tissue. The Two-temperature dual-phase-lag (TTDPL) bioheat transfer model is a successful model to describe the behavior of the thermal wave through the skin tissue.

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