288
Views
3
CrossRef citations to date
0
Altmetric
Articles

To solve the inverse Cauchy problem in linear elasticity by a novel Lie-group integrator

Pages 641-671 | Received 06 Jun 2012, Accepted 22 Sep 2013, Published online: 24 Oct 2013

References

  • Tessler A, Spangler JL. A least-square variational method for full-field reconstruction of elastic deformations in shear-deformable plates and shells. Comput. Meth. Appl. Mech. Eng. 2005;194:327–339.
  • Yeih WC, Koya T, Mura T. An inverse problem in elasticity with partially overspecified boundary conditions. I. Theoretical approach. ASME. J. Appl. Mech. 1993;60:595–600.
  • Koya T, Yeih WC, Mura T. An inverse problem in elasticity with partially overspecified boundary conditions. II. Numerical details. ASME. J. Appl. Mech. 1993;60:601–606.
  • Kozlov VA, Maz’ya VG, Fomin AV. An iterative method for solving the Cauchy problem for elliptic equations. Comput. Math. Math. Phys. 1991;31:45–52.
  • Marin L, Elliott L, Ingham DB, Lesnic D. Boundary element method for the Cauchy problem in linear elasticity. Eng. Anal. Bound. Elem. 2001;25:783–793.
  • Marin L, Lesnic D. Regularized boundary element solution for an inverse boundary value problem in linear elasticity. Commun. Numer. Meth. Eng. 2002;18:817–825.
  • Comino L, Marin L, Gallego R. An alternating iterative algorithm for the Cauchy problem in anisotropic elasticity. Eng. Anal. Bound. Elem. 2007;31:667–682.
  • Ellabib A, Nachaoui A. An iterative approach to the solution of an inverse problem in linear elasticity. Math. Comput. Simul. 2008;77:189–201.
  • Marin L, Johansson BT. A relaxation method of an alternating iterative algorithm for the Cauchy problem in linear isotropic elasticity. Comput. Meth. Appl. Mech. Eng. 2010;199:3179–3196.
  • Andrieux S, Baranger TN. An energy error-based method for the resolution of the Cauchy problem in 3D linear elasticity. Comput. Meth. Appl. Mech. Eng. 2008;197:902–920.
  • Baranger TN, Andrieux S. An optimization approach for the Cauchy problem in linear elasticity. Struct. Multidisc. Optim. 2008;35:141–152.
  • Marin L, Lesnic D. Boundary element solution for the Cauchy problem in linear elasticity using singular value decomposition. Comput. Meth. Appl. Mech. Eng. 2002;191:3257–3270.
  • Marin L. Reconstruction of boundary data in two-dimensional isotropic linear elasticity from Cauchy data using an iterative MFS algorithm. CMES: Comput. Model. Eng. Sci. 2010;60:221–245.
  • Marin L, Lesnic D. Boundary element-Landweber method for the Cauchy problem in linear elasticity. IMA J. Appl. Math. 2005;18:817–825.
  • 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. 2009;46:957–974.
  • Fu Z, Chen W, Zhang C. Boundary particle method for Cauchy inhomogeneous potential problems. Inverse. Probl. Sci. Eng. 2011;20:189–207.
  • Lin J, Chen W, Wang F. A new investigation into regularization techniques for the method of fundamental solutions. Math. Compu. Simul. 2011;81:1144–1152.
  • Chen W, Gu Y. An improved formulation of singular boundary method. Adv. Appl. Math. Mech. 2012;4:543–558.
  • Gu Y, Chen W, He XQ. Singular boundary method for steady-state heat conduction in three dimensional general anisotropic media. Int. J. Heat. Mass. Transfer. 2012;55:4837–4848.
  • Liu C-S, Kuo CL. A spring-damping regularization and a novel Lie-group integration method for nonlinear inverse Cauchy problems. CMES: Comput. Model. Eng. Sci. 2011;77:57–80.
  • Liu C-S, Kuo CL, Liu D. The spring-damping regularization method and the Lie-group shooting method for inverse Cauchy problems. CMC: Comput. Mater. Continua. 2011;24:105–123.
  • Liu C-S, Chang CW. A novel mixed group preserving scheme for the inverse Cauchy problem of elliptic equations in annular domains. Eng. Anal. Bound. Elem. 2012;36:211–219.
  • Fung YC, Tong P. Classical and computational solid mechanics. Singapore: World Scientific; 2001.
  • Essers JA. New fast super-dashpot time-dependent techniques for the numerical simulation of steady flows-I. Compu. Fluids. 1980;8:351–368.
  • Engl HW, Hanke M, Neubauer A. Regularization of inverse problems. Dordrechet: Kluwer Academic Publishers; 1996.
  • Liu C-S. A dynamical Tikhonov regularization for solving ill-posed linear algebraic systems. Acta Appl. Math. 2013;123:285–307.
  • Liu C-S. Group preserving scheme for backward heat conduction problems. Int. J. Heat Mass Transfer. 2004;47:2567–2576.
  • Liu C-S. Cone of non-linear dynamical system and group preserving schemes. Int. J. Non-Linear Mech. 2001;36:1047–1068.
  • Liu C-S, Chang CW, Chang JR. Past cone dynamics and backward group preserving schemes for backward heat conduction problems. CMES: Comput. Model. Eng. Sci. 2006;12:67–81.
  • Bellman RE, Casti J. Differential quadrature and long-term integration. J. Math. Anal. Appl. 1971;34:235–238.
  • Bellman RE, Kashef BG, Casti J. Differential quadrature: a technique for the rapid solution of nonlinear partial differential equations. J. Comp. Phys. 1972;10:40–52.
  • Liu C-S, Atluri SN. A highly accurate technique for interpolations using very high-order polynomials, and its applications to some ill-posed linear problems. CMES: Comput. Model. Eng. Sci. 2009;43:253–276.
  • Shen YH, Liu C-S. A new insight into the differential quadrature method in solving 2-D elliptic PDEs. CMES: Comput. Model. Eng. Sci. 2010;71:157–178.

Reprints and Corporate Permissions

Please note: Selecting permissions does not provide access to the full text of the article, please see our help page How do I view content?

To request a reprint or corporate permissions for this article, please click on the relevant link below:

Academic Permissions

Please note: Selecting permissions does not provide access to the full text of the article, please see our help page How do I view content?

Obtain permissions instantly via Rightslink by clicking on the button below:

If you are unable to obtain permissions via Rightslink, please complete and submit this Permissions form. For more information, please visit our Permissions help page.