Shape Identification in Nonlinear Boundary Problems Solved bby Pies Method

Zieniuk, Eugeniusz; Bołtuć, Agnieszka; Szerszeń, Krzysztof

doi:10.2478/ama-2014-0003

Abstract

The paper presents the strategy for identifying the shape of defects in the domain defined in the boundary value problem modelled by the nonlinear differential equation. To solve the nonlinear problem in the iterative process the PIES method and its advantages were used: the efficient way of the boundary and the domain modelling and global integration. The identification was performed using the genetic algorithm, where in connection with the efficiency of PIES we identify the small number of data required to the defect’s definition. The strategy has been tested for different shapes of defects.

References

1. Ameen M. (2001), Boundary element analysis. Theory and programming, Alpha Science International Ltd.
Search in Google Scholar
2. Aparicio N. D., Pidcock M. K. (1996), The boundary inverse problem for the Laplace equation in two dimensions, Inverse Problems, 12, 565-577.10.1088/0266-5611/12/5/003
Search in Google Scholar
3. Bołtuć A., Zieniuk E. (2011a), Modeling domains using Bézier surfaces in plane boundary problems defined by the Navier-Lame equation with body forces, Engineering Analysis with Boundary Elements, 35, 1116-1122.10.1016/j.enganabound.2011.04.005
Search in Google Scholar
4. Bołtuć A., Zieniuk E. (2011b), PIES in problems of 2D elasticity with body forces on polygonal domains, J. Theor. Appl. Mech., 49/2, 369-384.
Search in Google Scholar
5. Cerrolaza M., Annicchiarico W., Martinez M. (2000), Optimization of 2D boundary element models using B-splines and genetic algorithms, Engineering Analysis with Boundary Elements, 24, 427-440.10.1016/S0955-7997(00)00006-0
Search in Google Scholar
6. Cholewa R., Nowak A. J., Bialecki R. A., Wrobel L. C. (2002), Cubic Bézier splines for BEM heat transfer analysis of the 2-D continuous casting problems, Computational Mechanics 28, 282-290.10.1007/s00466-001-0288-5
Search in Google Scholar
7. Durodola J. F., Fenner R. T. (1990), Hermitian cubic boundary elements for two-dimensional potential problems, International Journal for Numerical Methods in Engineering, 30/5, 1051-1062.10.1002/nme.1620300507
Search in Google Scholar
8. Farin G. (1990), Curves and Surfaces for computer Aided Geometric Design, Academic Press Inc., San Diego.10.1016/B978-0-12-460515-2.50020-2
Search in Google Scholar
9. Foley J., van Dam A., Feiner S., Hughes J., Phillips R. (1994), Introduction to Computer Graphics, Addison-Wesley.
Search in Google Scholar
10. Goldberg D.E. (1989), Genetic Algorithms in Search, Optimization and Machine Learning, Addison-Wesley Publishing Company, Massachusetts.
Search in Google Scholar
11. Ligget J. A., Salmon J. R. (1981), Cubic spline boundary element, International Journal for Numerical Methods in Engineering, 17, 543-556.10.1002/nme.1620170405
Search in Google Scholar
12. Liu G.R., Han X. (2003), Computational inverse techniques in nondestructive evaluation, CRC Press LLC.10.1201/9780203494486
Search in Google Scholar
13. Michalewicz Z. (1996), Genetic Algorithms + Data Structures = Evolution Programs, Springer-Verlag, Berlin.10.1007/978-3-662-03315-9
Search in Google Scholar
14. Nowak I., Nowak A. J., Wrobel L. C. (2002) Identification of phase change front by Bézier splines and BEM, International Journal of Thermal Sciences, 41, 492-499.10.1016/S1290-0729(02)01342-X
Search in Google Scholar
15. Rus G., Gallego R. (2002), Optimization algorithms for identification inverse problems with the boundary element method, Engineering Analysis with Boundary Elements, 26, 315-327.10.1016/S0955-7997(02)00008-5
Search in Google Scholar
16. Sen D. (1995), A cubic-spline boundary integral method for twodimensional free-surface flow problems. International Journal for Numerical Methods in Engineering, 38/11, 1809-1830.10.1002/nme.1620381103
Search in Google Scholar
17. Tikhonov A.N., Arsenin V.Y. (1977), Solution of Ill-posed problems, John Wiley & Sons, New York.
Search in Google Scholar
18. Wall M. (1996), GAlib: A C++ Library of Genetic Algorithm Components, version 2.4, Mechanical Engineering Department, Massachusetts Institute of Technology (http://lancet.mit.edu/ga/).
Search in Google Scholar
19. Zhu T., Zhang J., Atluri N. (1998), A meshless local boundary integral equation (LBIE) method for solving nonlinear problems, Computational Mechanics, 22, 174-186.10.1007/s004660050351
Search in Google Scholar
20. Zieniuk E. (2007), Modelling and effective modification of smooth boundary geometry in boundary problems using B-spline curves, Engineering with Computers, 23/1, 39-48.10.1007/s00366-006-0040-z
Search in Google Scholar
21. Zieniuk E., Bołtuć A. (2006), Bézier curves in the modeling of boundary geometry for 2D boundary problems defined by Helmholtz equation, Journal of Computational Acoustics, 14/3, 353-367.10.1142/S0218396X06003098
Search in Google Scholar
22. Zieniuk E., Bołtuć A. (2010), Iterative solution of linear and nonlinear boundary problems using PIES, Lecture Notes in Computer Science, 6067, 136-145.10.1007/978-3-642-14390-8_15
Search in Google Scholar
23. Zienkiewicz O. (1977), The Finite Element Methods, McGraw-Hill, London.
Search in Google Scholar

Shape Identification in Nonlinear Boundary Problems Solved bby Pies Method

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