Skip to main content
SpringerLink
Log in

An Invariant Method of Fundamental Solutions for the Cauchy Problem in Two-Dimensional Isotropic Linear Elasticity

  • Published:
Journal of Scientific Computing Aims and scope Submit manuscript

Abstract

In this paper, we propose a numerical algorithm based on the method of fundamental solutions (MFS) for the Cauchy problem in two-dimensional linear elasticity. Through the use of the double-layer potential function, we give the invariance property for a problem with two different descriptions. In order to adapt this invariance property, we give an invariant MFS to satisfy this invariance property, i.e., formulate the MFS with an added constant and an additional constraint. The method is combining the Newton method and classical Tikhonov regularization with Morozov discrepancy principle to solve the inverse Cauchy problem. Some examples are given for numerical verification on the efficiency of the proposed method. The numerical convergence, accuracy, and stability of the method with respect to the the number of source points and the distance between the pseudo-boundary and the real boundary of the solution domain, and decreasing the amount of noise added into the input data, respectively, are also analysed with some examples.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9

Similar content being viewed by others

References

  1. Alessandrini, G., Rondi, L., Rosset, E., Vessella, S.: The stability for the Cauchy problem for elliptic equations. Inverse Probl. 25, 1–47 (2009)

  2. Andrieux, S., Baranger, T.N.: An energy error-based method for the resolution of the Cauchy problem in 3D linear elasticity. Comput. Methods Appl. Mech. Eng. 197, 902–920 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  3. Baranger, T.N., Andrieux, S.: An optimization approach for the Cauchy problem in linear elasticity. Struct. Multidiscip. Optim. 35, 141–152 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  4. Bilotta, A., Turco, E.: A numerical study on the solution of the Cauchy problem in elasticity. Int. J. Solids Struct. 46, 4451–4477 (2009)

    Article  MATH  Google Scholar 

  5. Calderon, A.P.: Uniquess in the Cauchy problem for partial differential equations. Am. J. Math. 80, 16–36 (1958)

    Article  MATH  MathSciNet  Google Scholar 

  6. Comino, L., Marin, L., Gallego, R.: An alternating iterative algorithm for the Cauchy problem in anisotropic elasticity. Eng. Anal. Bound. Elem. 31, 667–682 (2007)

    Article  MATH  Google Scholar 

  7. Delvare, F., Cimetiére, A., Hanus, J.L., Bailly, P.: An iterative method for the Cauchy problem in linear elasticity with fading regularization effect. Comput. Methods Appl. Mech. Eng. 199, 3336–3344 (2010)

    Article  MATH  Google Scholar 

  8. Durand, B., Delvare, F., Bailly, P.: Numerical solution of Cauchy problem in linear elasticity in axisymmetric situations. Int. J. Solids Struct. 48, 3041–3053 (2011)

    Article  Google Scholar 

  9. Fairweather, G., Karageorghis, A.: The method of fundamental solutions for elliptic boundary value problems. Adv. Comput. Math. 9, 69–95 (1998)

    Article  MATH  MathSciNet  Google Scholar 

  10. Isakov, V.: Inverse Problems for Partial Differential Equations. Springer, New York (1998)

    Book  MATH  Google Scholar 

  11. Karageorghis, A., Lesnic, D., Marin, L.: A survey of applications of the MFS to inverse problems. Inverse Probl. Sci. Eng. 19(3), 309–336 (2011)

    Article  MATH  MathSciNet  Google Scholar 

  12. Kirsch, A.: An Introduction to the Mathematical Theory of Inverse Problems. Springer, New York (1996)

    Book  MATH  Google Scholar 

  13. Kozlov, V.A., Maz’ya, V.G., Fomin, A.V.: An iterative method for solving the Cauchy problem for elliptic equations, Zh. Vych. noi Mat. Mat. Fiziki 31 64–74. English translation. USSR Computational Mathematics and Mathematical Physics 31, 45–52 (1991)

  14. Koya, T., Yeih, W.C., Mura, T.: An inverse problem in elasticity with partially overspecified boundary conditions. II. Numerical details. Trans. ASME J. Appl. Mech. 60, 591–606 (1993)

    Article  Google Scholar 

  15. Linkov, A.M.: Boundary Integral Equations in Elasticity Theory. Springer, Russia (2002)

    Book  MATH  Google Scholar 

  16. Liu, C.S.: To solve the inverse Cauchy problem in linear elasticity by a novel Lie-group integrator. Inverse Probl. Sci. Eng. 22, 641–671 (2014)

    Article  MATH  MathSciNet  Google Scholar 

  17. Maniatty, A., Zabaras, N., Stelson, K.: Finite element analysis of some elasticity problems. J. Mech. Eng. Div. ASCE 115, 1302–1316 (1989)

    Google Scholar 

  18. Maniatty, A., Zabaras, N.: Investigation of regularization parameters and error estimating in inverse elasticity problems. Int. J. Numer. Methods Eng. 37, 1039–1052 (1994)

    Article  MATH  Google Scholar 

  19. Marin, L., Elliott, L., Ingham, D.B., Lesnic, D.: Boundary element method for the Cauchy problem in linear elasticity. Eng. Anal. Bound. Elem. 25, 783–793 (2001)

    Article  MATH  Google Scholar 

  20. Marin, L., Elliott, L., Ingham, D.B., Lesnic, D.: Boundary element regularization methods for solving the Cauchy problem in linear elasticity. Inverse Probl. Eng. 10, 335–357 (2002)

    Article  Google Scholar 

  21. Marin, L., Elliott, L., Ingham, D.B., Lesnic, D.: An iterative boundary element algorithm for a singular Cauchy problem in linear elasticity. Comput. Mech. 28, 479–488 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  22. Marin, L., Háo, D.N., Lesnic, D.: Conjugate gradient-boundary element method for the Cauchy problem in elasticity. Q. J. Mech. Appl. Math. 55, 227–247 (2002)

    Article  MATH  Google Scholar 

  23. Marin, L., Lesnic, D.: Boundary element solution for the Cauchy problem in linear elasticity using singular value decomposition. Comput. Methods Appl. Mech. Eng. 191, 3257–3270 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  24. Marin, L., Lesnic, D.: Regularized boundary element solution for an inverse boundary value problem in linear elasticity. Commun. Numer. Methods Eng. 18, 817–825 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  25. Marin, L., Lesnic, D.: The method of fundamental solutions for the Cauchy problem in two-dimensional linear elasticity. Int. J. Solids Struct. 41, 3425–3438 (2004)

    Article  MATH  Google Scholar 

  26. Marin, L., Lesnic, D.: Boundary element-Landweber method for the Cauchy problem in linear elasticity. IMA J. Appl. Math. 18, 817–825 (2005)

    MathSciNet  Google Scholar 

  27. Marin, L.: A meshless method for solving the Cauchy problem in threedimensional elastostatics. Comput. Math. Appl. 50(1–2), 73–92 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  28. Marin, L.: The minimal error method for the Cauchy problem in linear elasticity. Numerical implementation for two-dimensional homogeneous isotropic linear elasticity. Int. J. Solids Struct. 46, 957–974 (2009)

    Article  MATH  Google Scholar 

  29. Marin, L., Johansson, B.T.: Relaxation procedures for an iterative MFS algorithm for the stable reconstruction of elastic fields from Cauchy data in two-dimensional isotropic linear elasticity. Int. J. Solids Struct. 47, 3462–3479 (2010)

    Article  MATH  Google Scholar 

  30. Marin, L., Johansson, B.T.: A relaxation method of an alternating iterative algorithm for the Cauchy problem in linear isotropic elasticity. Comput. Methods Appl. Mech. Eng. 199, 3179–3196 (2010)

    Article  MATH  MathSciNet  Google Scholar 

  31. Martin, T.J., Haldermann, J.D., Dulikravich, G.S.: An inverse method for finding unknown surface tractions and deformations in elastostatics. Comput. Struct. 56, 825–835 (1995)

    Article  MATH  Google Scholar 

  32. Schnur, D., Zabaras, N.: Finite element solution of two-dimensional elastic problems using spatial smoothing. Int. J. Numer. Methods Eng. 30, 57–75 (1999)

    Article  Google Scholar 

  33. Sun, Y., Ma, F., Zhang, D.: An integral equations method combined minimum norm solution for 3D elastostatics Cauchy problem. Comput. Methods Appl. Mech. Eng. 271, 231–252 (2014)

    Article  MATH  MathSciNet  Google Scholar 

  34. Turco, E.: A boundary elements approach to identify static boundary conditions in elastic solids from stresses at internal points. Inverse Probl. Eng. 7, 309–333 (1999)

    Article  Google Scholar 

  35. Turco, E.: An effective algorithm for reconstructing boundary conditions in elastic solids. Comput. Methods Appl. Mech. Eng. 190, 3819–3829 (2001)

    Article  MATH  Google Scholar 

  36. Zabaras, N., Morellas, V., Schnur, D.: Spatially regularized solution of inverse elasticity problems using the BEM. Commun. Appl. Numer. Methods 5, 547–553 (1989)

    Article  MATH  Google Scholar 

Download references

Acknowledgments

The very constructive comments and suggestions made by the reviewers, which have enriched this paper, are to be acknowledged. Finally, we express our gratitude to the anonymous referees for pointing out a few confusing expressions in the original manuscript, and pointing to future research.

Author information

Authors and Affiliations

  1. College of science, Civil Aviation University of China, Tianjin, 300300, China

    Yao Sun

  2. School of Mathematics, Jilin University, Changchun, 130012, China

    Fuming Ma

  3. College of Computer Science and Technology, Jilin University, Changchun, 130012, China

    Xu Zhou

Authors
  1. Yao Sun
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Fuming Ma
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Xu Zhou
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Yao Sun.

Additional information

The research was supported by the fundamental Research Funds for the Central Universities (No: 3122014K012) and the NSFC (Nos: 11371172, 11271159, 11401574, 61403395).

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Sun, Y., Ma, F. & Zhou, X. An Invariant Method of Fundamental Solutions for the Cauchy Problem in Two-Dimensional Isotropic Linear Elasticity. J Sci Comput 64, 197–215 (2015). https://doi.org/10.1007/s10915-014-9929-7

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10915-014-9929-7

Keywords

Search

Navigation