Interpolation and computer based models.
Gavrila, Gabriela ; Oanta, Emil
1. INTRODUCTION
Each age was influenced by the means employed to calculate the data
in the technical and scientific projects. At present, the computer is
the main instrument involved in all the aspects regarding research,
development and education. In addition to fast and reliable results, the
knowledge integration is an important effect of the computer based
approaches which must be considered in the development of the models in
engineering. Numerical methods are widely used and a particular and
important aspect is the approximation of different values, operation
designated as interpolation, which is employed in most branches of
science.
2. GENERAL ASPECTS REGARDING THE
INTERPOLATION
Interpolation or approximation is one of the most important methods
in computer based investigations.
Interpolation is closely connected with other numerical methods.
For instance, in order to compute the coefficients of the functions for
a polynomial approximation it is required to solve of a system of
equations. On the other hand, other numerical methods use approximation
in order to accelerate the convergence or as an alternative computing
method. An accurate approximation of the initial solution is important
in any iterative method. Integration of the experimental data may be
done either by using numerical methods, or by using interpolation
followed by a direct integration of the functions resulted at the
previous stage. All the customized computing methods must be assessed in
terms of stability, efficiency and sensitivity.
Interpolation is also closely related to the computing instrument.
One can notice the progress with respect to the methods of approximation
employed 30 years ago when the computer was an instrument used by a
handful of researchers and the modern, flexible and 'popular'
nowadays methods. If years ago the semi-empirical or empirical formulae
were used, together with graphs for different coefficients employed in
design, at present many software applications have facilities regarding
the trend line or the approximation of different data, by using
graphical colourful representations which offer a facile overview
regarding the results.
Interpolation is also required in more complex models not only as
simple instruments, numerical analysts being asked to develop customized
libraries which are used by the rest of the members of the research
team.
3. INTERPOLATION IN ENGINEERING
Experimental projects require instruments to process the acquired
data and to represent them in a relevant way in order to reveal the
phenomenon in question and possible influences.
Figure 1 presents the variation of equivalent stresses in the block
of cylinders of an internal combustion engine. Interpolation is
necessary in order to locate the peaks and to relate them to the
phenomenon in the engine. As it can be noticed, the customized method to
interpolate the data is more accurate than the default method.
[FIGURE 1 OMITTED]
[FIGURE 2 OMITTED]
[FIGURE 3 OMITTED]
The original data acquisition chain uses strain gages bonded to the
cylinder block in some locations considered relevant for the phenomenon.
The influence of the steady-state temperature must be corrected by the
use of the polynomial recommended by the manufacturer. The results are
checked by the use of the values resulted from an interpolation study
(figure 2).
Figure 3 presents the facilities employed for interpolation in some
computer aided design applications. As it can be noticed, the accuracy
of the interpolation may be controlled by the use of some parameters.
4. CONCLUSION
The interpolation problem is important for practical reasons. The
knowhow and the computing methods are implemented in several software
applications but the user is not allowed to perform certain calculi because of the lack of flexibility of the applications. Thus, the
authors present some of the automatic data processing methods used for
interpolation together with their weak points, in order to emphasize the
fact that customized solutions are always necessary, the numerical
analysis being an important component in research.
Even the interpolation is a facility offered in several
applications, it is still needed a method to offer to the researchers
not only data in terms of points located on the interpolated curve, but
also as coefficients employed to define the set of interpolated curves.
These requirements are fulfilled if original numerical methods libraries
are developed and tested. Complex research projects rely on advanced
computing methods useful in all the analytic, experimental and numerical
approaches or sub-models.
5. ACKNOWLEDGEMENT
Several ideas presented in the paper regarding the computer based
models are the direct result of the research and development activity
performed in the framework of the "Computer Aided Advanced Studies
in Applied Elasticity from an Interdisciplinary Perspective" ID1223
scientific research project, under the supervision of the National
University Research Council (CNCSIS), Romania (Oanta & al., 2007 d).
6. REFERENCES
Gavrila, G. (2001). Numerical Derivation Methods,
"Babes-Bolyai" University of Cluj-Napoca, Faculty of
Mathematics and Computer Science, Department of Applied Mathematics,
Chair of Numerical and Statistical Computing, Scientific essay in the
framework of the PhD thesis "Contributions to the study of the
linear approximation methods", under the supervision of Prof. PhD.
Alexandru Lupas and of Prof. PhD. Octavian Agratini
Gavrila, G. (2004). Generalization of the Gauss Quadrature Formulae, "Babes-Bolyai" University of Cluj-Napoca, Faculty of
Mathematics and Computer Science, Department of applied mathematics,
Chair of Numerical and Statistical Computing, Scientific essay in the
framework of the PhD thesis "Contributions to the study of the
linear approximation methods", under the supervision of Prof. PhD.
Alexandru Lupas and of Prof. PhD. Octavian Agratini
Gavrila, G. (2004). s-Orthogonal Polynomials,
"Babes-Bolyai" University of Cluj-Napoca, Faculty of
Mathematics and Computer Science, Department of Applied Mathematics,
Chair of Numerical and Statistical Computing, Scientific essay in the
framework of the PhD thesis "Contributions to the study of the
linear approximation methods", under the supervision of Prof. PhD.
Alexandru Lupas and of Prof. PhD. Octavian Agratini
Gavrila, G. (2005). Study of the Optimality of the Quadrature
Formulae, "Babes-Bolyai" University of Cluj-Napoca, Faculty of
Mathematics and Computer Science, Department of applied mathematics,
Chair of Numerical and Statistical Computing, Scientific essay in the
framework of the PhD thesis "Contributions to the study of the
linear approximation methods", under the supervision of Prof. PhD.
Alexandru Lupas and of Prof. PhD. Octavian Agratini
Oanta, E. (2001), Study of the strains and stresses in the block of
cylinders of the naval internal combustion engines, PhD Thesis, 442
pages, 'Cum laude', Field of science: Mechanical Engineering,
2001, 'Politehnica' University of Bucharest, Promoter Acad.
Prof. Dr. H. C. Constantin Arama, Member of the Romanian Science Academy
Oanta, E. (2007). Virtual Reality Original Instrument Employed in
Crises Management, Proceedings of the 12th International Congress of the
International Maritime Association of the Mediterranean (IMAM2007),
Varna, Bulgaria, 2-6 September 2007, Maritime Industry, Ocean
Engineering and Costal Resources--Editors: Guedes Soares & Kolev,
2008 Taylor & Francis Group, London, ISBN 978-0-415-45523-7, Vol. 2,
pg. 1095-1102
Oanta, E. (2007). Numerical methods and models applied in economy,
PhD Thesis, 468 pages, Field of science: Informatics, Academy of
Economical Studies of Bucharest, promoter Prof. Mat. Ec. Ioan Odagescu
Oanta, E.; Panait, C.; Nicolescu, B.; Dinu, S.; Hnatiuc, M.;
Pescaru, A.; Nita, A.; Gavrila, G. (2007-2010). Computer Aided Advanced
Studies in Applied Elasticity from an Interdisciplinary Perspective,
ID1223 Scientific Research Project, under the supervision of the
National University Research Council (CNCSIS), Romania
Oanta, E. (2009). Applied Elasticity Computer Models in Automatic
Design, The 2nd International Multi-Conference on Engineering and
Technological Innovation: IMETI 2009, July 10-13, Organized by
International Institute of Informatics and Systemics, Orlando, Florida,
USA, paper accepted
