Boundary reconstruction in two-dimensional functionally graded materials using a regularized MFS

CMES - Computer Modeling in Engineering and Sciences

Volumn 46, Issue 3, 2009, Pages 221-253

  Marin, Liviu ^a  

^a INSTITUTE OF SOLID MECHANICS   (Romania)

Author keywords

Functionally graded materials (FGMs); Inverse geometric problem; Method of fundamental solutions (MFS); Regularization

Indexed keywords

BOUNDARY RECONSTRUCTION; CAUCHY DATA; INVERSE GEOMETRIC PROBLEMS; L-CURVE CRITERION; METHOD OF FUNDAMENTAL SOLUTIONS; METHOD OF FUNDAMENTAL SOLUTIONS (MFS); NUMERICAL RECONSTRUCTION; OPTIMAL VALUES; REGULARIZATION PARAMETERS; TIKHONOV REGULARIZATION METHOD; TWO-DIMENSIONAL DOMAIN;

BEAMS AND GIRDERS; CRACK INITIATION; FRACTURE; TWO DIMENSIONAL;

FUNCTIONALLY GRADED MATERIALS;

EID: 76449098701     PISSN: 15261492     EISSN: None     Source Type: Journal    
DOI: None     Document Type: Article

References (61)

1. Ahmadabadi, M. N.; Arab, M.; Ghaini, F. M. M. (2009): The method of fundamental solutions for the inverse space-dependent heat source problem. Engineering Analysis with Boundary Elements, vol.33, pp. 1231-1235.
2. Chen, L. Y.; Chen, J. T.; Hong, H. K.; Chen, C. H. (1995): Application of Cesàro mean and the L-curve for the dconvolution problem. Soil Dynamics Earthquake Engineering, vol.14, pp. 361-373.
3. Chen, C.W.; Young, D. L.; Tsai, C. C.; Murugesan, K. (2005): The method of fundamental solutions for inverse 2D Stokes problems. Computational Mechanics, vol.37, pp. 2-14. (Pubitemid 41521401)
4. Cho, H. A.; Golberg, M. A.; Muleshkov, A. S.; Li, X. (2004): Trefftz methods for time dependent partial differential equations. CMC: Computers, Materials & Continua, vol.1, pp. 1-37.
5. Dong, C. F.; Sun, F. Y.; Meng, B. Q. (2007): A method of fundamental solutions for inverse heat conduction problems in an anisotropic medium. Engineering Analysis with Boundary Elements, vol.31, pp. 75-82.
6. Engl, H. W.; Hanke, M.; Neubauer, A. (2000): Regularization of Inverse Problems. Kluwer Academic, Dordrecht.
7. Fairweather, G.; Karageorghis, A. (1998): The method of fundamental solutions for elliptic boundary value problems. Advances in Computational Mathematics, vol.9, pp. 69-95. (Pubitemid 128403959)
8. Fairweather, G.; Karageorghis, A.; Martin, P. A. (2003): The method of fundamental solutions for scattering and radiation problems. Engineering Analysis with Boundary Elements, 27, pp. 759-769.
9. Fam, G. S. A.; Rashed, Y. F. (2009): The method of fundamental solutions applied to 3D elasticity problems using a continuous collocation scheme. Engineering Analysis with Boundary Elements, vol.33, pp. 330-341.
10. Gill, P. E.; Murray, W.; Wright, M. H. (1981): Practical Optimization. Academic Press, London.
11. Golberg, M. A.; Chen, C. S. (1999): The method of fundamental solutions for potential, Helmholtz and diffusion problems. In: M. A. Golberg (ed.) Boundary Integral Methods: Numerical and Mathematical Aspects, WIT Press and Computational Mechanics Publications, Boston, pp. 105-176.
12. Hadamard, J. (1923): Lectures on Cauchy Problem in Linear Partial Differential Equations. Yale University Press, New Haven.
13. Hanke, M. (1996): Limitations of the L-curve method in ill-posed problems. BIT, vol.36, pp. 287-301.
14. Hansen, P. C. (1998): Rank-Deficient and Discrete Ill-Posed Problems: Numerical Aspects of Linear Inversion. SIAM, Philadelphia.
15. Hon, Y. C.; Li, M. (2008): A computational method for inverse free boundary determination problem. International Journal for Numerical Methods in Engineering, vol.73, 1291-1309.
16. Hon, Y. C.; Wei, T. (2004): A fundamental solution method for inverse heat conduction problems. Engineering Analysis with Boundary Elements, vol.28, pp. 489-495.
17. Hon, Y. C.; Wei, T. (2005): The method of fundamental solutions for solving multidimensional heat conduction problems. CMES: Computer Modeling in Engineering & Sciences, vol.13, pp. 219-228.
18. Hon, Y. C.; Wu, Z. (2000): A numerical computation for inverse boundary determination problem. Engineering Analysis with Boundary Elements, vol.24, 599-606.
19. Hsieh, C. K.; Kassab, A. J. (1986): A general method for the solution of inverse heat conduction problem with partially unknown system geometries. International Journal of Heat and Mass Transfer, vol.29, 47-58.
20. Huang, C. C.; Kassab, B. H. (1997): An inverse geometry problem for identifying irregular boundary configurations. International Journal of Heat and Mass Transfer, vol.40, 2045-2053.
21. Huang, C. C.; Tsai, C. C. (1997): A transient inverse two-dimensional geometry problem in estimating time-dependent irregular boundary configurations. International Journal of Heat and Mass Transfer, vol.41, 1707-1718.
22. Jin, B. T.; Marin, L. (2007): The method of fundamental solutions for inverse source problems associated with the steady-state heat conduction. International Journal for Numerical Methods in Engineering, vol.69, pp. 1570-1589.
23. Jin, B. T.; Zheng, Y. (2006): A meshless method for some inverse problems associated with the Helmholtz equation. Computer Methods in Applied Mechanics and Engineering, vol.195, pp. 2270-2280.
24. Kunisch, K.; Zou, J. (1998): Iterative choices of regularization parameters in linear inverse problems. Inverse Problems, vol.14, pp. 1247-1264.
25. Lesnic, D.; Berger, J. R.; Martin, P. A. (2002): A boundary element regularization method for the boundary determination in potential corrosion damage. Inverse Problems in Engineering, vol.10, pp. 163-182.
26. Ling, L.; Takeuchi, T. (2008): Boundary control for inverse Cauchy problems of the Laplace equations. CMES: Computer Modeling in Engineering & Sciences, vol.29, pp. 45-54.
27. Liu, C.-S. (2008a): Solving an inverse Sturm-Liouville problem by a Lie-group method. Boundary Value Problems, vol.2008, Article ID 749865.
28. Liu, C.-S. (2008b): Identifying time-dependent damping and stiffness functions by a simple and yet accurate method. Journal of Sound and Vibration, vol.318, pp. 148-165.
29. Liu, C.-S. (2008c): A Lie-group shooting method for simultaneously estimating the time-dependent damping and stiffness coefficients. CMES: Computer Modeling in Engineering & Sciences, vol.27, pp. 137-149.
30. Liu, C.-S. (2008d): A time-marching algorithm for solving non-linear obstacle problems with the aid of an NCP-function. CMC: Computers, Materials & Continua, vol.8, pp. 53-65.
31. Liu, C.-S. (2008e): A fictitious time integration method for two-dimensional quasilinear elliptic boundary value problems. CMES: Computer Modeling in Engineering & Sciences, vol.33, pp. 179-198.
32. Liu, C.-S. (2008f): A fictitious time integration method for solving m-point boundary value problems. CMES: Computer Modeling in Engineering & Sciences, vol.39, pp. 125-154.
33. Liu, C.-S.; Atluri, S. N. (2008a): A fictitious time integration method (FTIM) for solving mixed complementarity problems with applications to non-linear optimization. CMES: Computer Modeling in Engineering & Sciences, vol.34, pp. 155-178.
34. Liu, C.-S.; Atluri, S. N. (2008b): A novel fictitious time integration method for solving the discretized inverse Sturm-Liouville problems, for specified eigenvalues. CMES: Computer Modeling in Engineering & Sciences, vol.36, pp. 261-285.
35. Liu, C.-S.; Atluri, S. N. (2009): A fictitious time integration method for the numerical solution of the Fredholm integral equation and for numerical differentiation of noisy data, and its relation with the filter theory. CMES: Computer Modeling in Engineering & Sciences, vol.41, pp. 243-261.
36. Liu, C.-S.; Chang, J. R.; Chang, K. H.; Chen, Y. W. (2008): Simultaneously estimating the time-dependent damping and stiffness coefficients with the aid of vibrational data. CMC: Computers, Materials & Continua, vol.7, pp. 97-107.
37. Marin, L. (2005a): A meshless method for solving the Cauchy problem in threedimensional elastostatics. Computers and Mathematics with Applications, vol.50, pp. 73-92.
38. Marin, L. (2005b): Numerical solutions of the Cauchy problem for steady-state heat transfer in two-dimensional functionally graded materials. International Journal of Solids and Structures, vol.42, pp. 4338-4351.
39. Marin, L. (2005c): A meshless method for the numerical solution of the Cauchy problem associated with three-dimensional Helmholtz-type equations. Applied Mathematics and Computation, vol.165, pp. 355-374. (Pubitemid 40567737)
40. Marin, L. (2006): Numerical boundary identification for Helmholtz-type equations. Computational Mechanics, vol.39, pp. 25-40.
41. Marin, L. (2008): The method of fundamental solutions for inverse problems associated with the steady-state heat conduction in the presence of sources. CMES: Computer Modeling in Engineering & Sciences, vol.30, pp. 99-122.
42. Marin, L.; Lesnic, D. (2003): BEM first-order regularisation method in linear elasticity for boundary identification. Computer Methods in Applied Mechanics and Engineering, vol.192, pp. 2059-2071.
43. Marin, L.; Lesnic, D. (2004): The method of fundamental solutions for the Cauchy problem in two-dimensional linear elasticity. International Journal of Solids and Structures, vol.41, pp. 3425-3438.
44. Marin, L.; Lesnic, D. (2005a): The method of fundamental solutions for the Cauchy problem associated with two-dimensional Helmholtz-type equations. Computers & Structures, vol.83, pp. 267-278. (Pubitemid 40072215)
45. Marin, L.; Lesnic, D. (2005b): The method of fundamental solutions for inverse boundary value problems associted with the two dimentional biharmonic equation. Mathematical and Computers in Modelling, vol.42, pp. 261-278.
46. Marin, L.; Karageorghis, A.; Lesnic, D. (2009): The MFS for boundary identification in two-dimensional harmonic problems, submitted.
47. Mathon, R.; Johnston, R. L. (1977): The approximate solution of elliptic boundary value problems by fundamental solutions. SIAM Journal on Numerical Analysis, vol.14, pp. 638-650.
48. Mera, N. S. (2005): The method of fundamental solutions for the backward heat conduction problem. Inverse Problems in Science and Engineering, vol.13, pp. 79-98.
49. Mera, N. S.; Lesnic; D. (2005): A three-dimensional boundary determination problem in potential corrosion damage. Computational Mechanics, vol.36, pp. 129-138.
50. Morozov, V. A. (1966): On the solution of functional equations by the method of regularization. Doklady Mathematics, vol.7, pp. 414-417.
51. Numerical Algorithms Group Library Mark 21 (2007). NAG(UK) Ltd, Wilkinson House, Jordan Hill Road, Oxford, UK.
52. Park, H. M.; Ku, J. H. (2001): Shape identification for natural convection problems. Communications in Numerical Methods in Engineering, vol.17, pp. 871-880.
53. Peneau, S.; Jarny, Y.; Sarda, A. (1966): Isotherm Shape Identification for a Two-Dimensional Heat Conduction Problem. In: H. D. Bui, M. Tanaka, M. Bonnet, H. Maigre, E. Luzzato and M. Reynier (eds.) Inverse Problems in Engineering Mechanics, Balkema, Rotterdam, pp. 47-53.
54. Shigeta, T.; Young, D. L. (2009): Method of fundamental solutions with optimal regularization techniques for the Cauchy problem of the Laplace equation with singular points. Journal of Computational Physics, vol.228, pp. 1903-1915.
55. Suresh, S.; Mortensen, A. (1998): Fundamentals of Functionally Graded Materials. IOM Communications Ltd., London.
56. Tikhonov, A. N.; Arsenin, V. Y. (1986): Methods for Solving Ill-Posed Problems. Nauka, Moscow.
57. Vogel, C. R. (1996): Non-convergence of the L-curve regularization parameter selection method. Inverse Problems, vol.12, pp. 535-547.
58. Wahba, G. (1977): Practical approximate solutions to linear operator equations when the data are noisy. SIAM Journal on Numerical Analysis, vol.14, pp. 651-667.
59. Yan, L.; Fu, C.-L.; Yang, F.-L. (2008): The method of fundamental solutions for the inverse heat source problem. Engineering Analysis with Boundary Elements, vol.32, pp. 216-222.
60. Young, D. L.; Tsai, C. C.; Chen, C. W.; Fan, C. M. (2008): The method of fundamental solutions and condition number analysis for inverse problems of Laplace equation. Computers and Mathematics with Applications, vol.55, pp. 1189-1200.
61. Zeb, A.; Ingham, D. B.; Lesnic, D. (2008): The method of fundamental solutions for a biharmonic boundary determination, Computational Mechanics, vol.42, pp. 371-379.