scholarly journals Examination of the Iterative Series-Expansion Method for Quantitative Texture Analysis

1995 ◽  
Vol 23 (2) ◽  
pp. 115-129 ◽  
Author(s):  
D. Raabe
Keyword(s):  
Series Expansion ◽  
Linear Equations ◽  
Three Dimensional ◽  
Expansion Method ◽  
Orientation Space ◽  
First Case ◽  
Pole Figures ◽  
Polycrystalline Aggregates ◽  
Analytical Tools ◽  
Series Expansion Method

Three-dimensional orientation distributions of grains in polycrystalline aggregates are referred to as crystallographic textures. Commonly, they are computed from two-dimensional centro-symmetric pole figures by employment of series expansion techniques or so called direct inversion methods. Both approaches lead to inaccuracies which are due to the absence of the odd coefficients and by truncation errors in the first case and to the under-determination of the set of linear equations combining cells in the pole figures and in the three-dimensional orientation space in the second case. For both types of calculation methods various correction procedures were suggested. In case of the series expansion methods the introduction of the non-negativity condition was reported to considerably improve the obtained solution. However, before large series of experimental data can be processed by such a method, its reliability has to be checked by use of analytical tools. Hence, in the present study a recently introduced iterative series-expansion method which accounts for the non-negativity condition is examined by use of standard functions.

The iterative series-expansion method for quantitative texture analysis. II. Applications

1992 ◽  
Vol 25 (2) ◽  
pp. 259-267 ◽  
Author(s):  
M. Dahms
Keyword(s):  
Distribution Function ◽  
Series Expansion ◽  
Expansion Method ◽  
Orientation Distribution ◽  
Polycrystalline Materials ◽  
Pole Density ◽  
General Outline ◽  
Positivity Condition ◽  
Pole Figures ◽  
Series Expansion Method

The orientation distribution function (ODF) of the crystallites of polycrystalline materials can be calculated from experimentally measured pole density functions (pole figures). This procedure, called pole-figure inversion, can be achieved by the series-expansion method (harmonic method). As a consequence of the (hkl)-({\bar h}{\bar k}{\bar l}) superposition, the solution is mathematically not unique. There is a range of possible solutions (the kernel) that is only limited by the positivity condition of the distribution function. The complete distribution function f(g) can be split into two parts, \tilde {f}(g) and \tildes {f}(q), expressed by even- and odd-order terms of the series expansions. For the calculation of the even part \tilde {f}(g), the positivity condition for all pole figures contributes essentially to an `economic' calculation of this part, whereas, for the odd part, the positivity condition of the ODF is the essential basis. Both of these positivity conditions can be easily incorporated in the series-expansion method by using several iterative cycles. This method proves to be particularly versatile since it makes use of the orthogonality and positivity at the same time. In the previous paper in this series [Dahms & Bunge (1989) J. Appl. Cryst. 22, 439–447] a general outline of the method was given. This, the second part, gives details of the system of programs used as well as typical examples showing the versatility of the method.

A Method for Aerodynamic Design of Blades in Quasi-Three-Dimensional Calculation of Turbomachines

1988 ◽  
Vol 110 (2) ◽  
pp. 181-186 ◽  
Author(s):  
Zhengming Wang
Keyword(s):  
Inverse Problem ◽  
Turbine Blade ◽  
Series Expansion ◽  
Thickness Distribution ◽  
Three Dimensional ◽  
Expansion Method ◽  
Aerodynamic Design ◽  
Blade Profile ◽  
The Mean ◽  
Series Expansion Method

A special inverse problem is formulated in which the shape of the mean streamline and the circumferential thickness distribution of the profile are given. On the basis of the series expansion method on a selected streamline, in quasi-three-dimensional aerodynamic design, the blade profile thickness is automatically fulfilled by computer. Six radial sections of a turbine blade are designed by this method.

scholarly journals The Use of a Quadratic Form for the Determination of Non-negative Texture Functions

1983 ◽  
Vol 6 (1) ◽  
pp. 1-19 ◽  
Author(s):  
P. Van Houtte
Keyword(s):  
Series Expansion ◽  
Quadratic Forms ◽  
Expansion Method ◽  
Experimental Methods ◽  
Classical Analysis ◽  
Pole Figures ◽  
Zero Range ◽  
Euler Space ◽  
Series Expansion Method

The classical analysis of measured pole figures of textured polycrystals by the series expansion method does not necessarily produce a non-negative texture function. The main reason for this is, that the method is unable to find the terms of odd rank l of the series expansion.A new method is proposed, which introduces the non-negativity condition into the series expansion method by the use of quadratic forms. The method is found to be successful when treating sharp textures, which have a considerable zero range in Euler space. The preliminary determination of this zero range by experimental methods is however not necessary.

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