Hostname: page-component-cb9f654ff-mnl9s Total loading time: 0 Render date: 2025-08-24T07:25:05.588Z Has data issue: false hasContentIssue false
  • English
  • Français

Numerical Analysis of Inverse Elasticity Problemwith Signorini's Condition

Part of: Partial differential equations, boundary value problems Special kinds of problems Numerical methods Partial differential equations, initial value and time-dependent initial-boundary value problems

Published online by Cambridge University Press:  05 October 2016

Cong Zheng ,
Xiaoliang Cheng  and
Kewei Liang
Cong Zheng*
Affiliation:
School of Mathematical Sciences, Zhejiang University, Hangzhou 310027, P.R. China
Xiaoliang Cheng*
Affiliation:
School of Mathematical Sciences, Zhejiang University, Hangzhou 310027, P.R. China
Kewei Liang*
Affiliation:
School of Mathematical Sciences, Zhejiang University, Hangzhou 310027, P.R. China
*
*Corresponding author. Email addresses:congzheng@zju.edu.cn (C. Zheng), xiaoliangcheng@zju.edu.cn (X. Cheng), matlkw@zju.edu.cn (K. Liang)
*Corresponding author. Email addresses:congzheng@zju.edu.cn (C. Zheng), xiaoliangcheng@zju.edu.cn (X. Cheng), matlkw@zju.edu.cn (K. Liang)
*Corresponding author. Email addresses:congzheng@zju.edu.cn (C. Zheng), xiaoliangcheng@zju.edu.cn (X. Cheng), matlkw@zju.edu.cn (K. Liang)
Get access

Abstract

An optimal control problem is considered to find a stable surface traction, which minimizes the discrepancy between a given displacement field and its estimation. Firstly, the inverse elastic problem is constructed by variational inequalities, and a stable approximation of surface traction is obtained with Tikhonov regularization. Then a finite element discretization of the inverse elastic problem is analyzed. Moreover, the error estimation of the numerical solutions is deduced. Finally, a numerical algorithm is detailed and three examples in two-dimensional case illustrate the efficiency of the algorithm.

Keywords

Optimal controlvariational inequalityinverse elastic problemfinite elementerror estimatesnumerical simulations

MSC classification

Secondary: 65M15: Error bounds 65N21: Inverse problems 65N22: Solution of discretized equations 74M15: Contact 74S05: Finite element methods

Information

Type
Research Article
Information
Communications in Computational Physics , Volume 20 , Issue 4 , October 2016 , pp. 1045 - 1070
Copyright
Copyright © Global-Science Press 2016 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Article purchase

Temporarily unavailable

References

[1] Atkinson, K. and Han, W., Theoretical Numerical Analysis: A Functional Analysis Framework 3rd edn, Springer, New York, 2009.Google Scholar
[2] Beck, J.V., Blackwell, B. and Clair, C.R.St., Inverse Heat Conduction: Ill-Posed Problems, Wiley-Interscience, New York, 1985.Google Scholar
[3] Bermudez, A. and Saguez, C., Optimal control of a Signorini problem, SIAM Journal on Control and Optimization, 25(1987), 576–582.Google Scholar
[4] Busby, H.R. and Trujillo, D.M., Numerical solution to a two-dimensional inverse heat conduction problem, Int. J. Numer. Meth. Eng. 21(1985), 349–359.CrossRefGoogle Scholar
[5] Chiou, W., Chen, C. and Lu, W., The Inverse Numerical Solutions of the Nonlinear Heat Transfer Problem in Electrical Discharge Machining, Numerical Heat Transfer, Part A: Applications 59(2011), 247–266.Google Scholar
[6] Eck, C., Jarušek, J. and Krbeč, M., Unilateral Contact Problems: Variational Methods and Existence Theorems, Chapman/CRC Press, New York, 2005.Google Scholar
[7] Glashoff, K. and Gustafson, S.A., Linear Optimization and Approximation: An Introduction to the Theoretical Analysis and Numerical Treatment of Semi-infinite Programs, Springer, New York, 1983.Google Scholar
[8] Grysa, K., Cialkowksi, M.J. and Kaminski, H., An inverse temperature field problem of the theory of thermal stresses, Nucl. Eng. Des. 64(1981), 169–184.Google Scholar
[9] Han, W., A Posteriori Error Analysis Via Duality Theory: With Applications in Modeling and Numerical Approximations, Springer, New York, 2005.Google Scholar
[10] Han, W. and Sofonea, M., Quasistatic Contact Problems in Viscoelasticity and Viscoplasticity, American Mathematical Society–International Press, 2002.Google Scholar
[11] Huang, C. and Shih, W., An inverse problem in estimating interfacial cracks in bimaterials by boundary element technique, Int. J. Numer. Meth. Eng. 45(1999), 1547–1567.Google Scholar
[12] Isakov, V., Inverse Problems for Partial Differential Equations 2nd edn, Springer, New York, 2006.Google Scholar
[13] Kaipio, J. and Somersalo, E., Statistical and Computational Inverse Problems, Springer, NewYork, 2005.Google Scholar
[14] Kikuchi, N. and Oden, J.T., Contact Problems in Elasticity: A Study of Variational Inequalities and Finite Element Methods, PA: SIAM, Philadelphia, 1998.Google Scholar
[15] Maniatty, A., Zabaras, N. and Stelson, K., Finite element analysis of some inverse elasticity problems, J. Eng. Mech. Diu. ASCE 115(1989), 1302–1316.Google Scholar
[16] Maniatty, A. and Zabaras, N., A method for solving inverse elastoviscoplastic problems, J. Eng. Mech. Dio. ASCE 115(1989), 2216–2231.Google Scholar
[17] Mura, T., A new NDT: Evaluation of plastic strains in bulk from displacements on surfaces, Mech. Res. Commun. 12(1985), 243–248.CrossRefGoogle Scholar
[18] Mura, T., Cox, B. and Gao, Z., Computer-aided nondestructive measurements of plastic strains from surface displacements, Computational Mechanics’86, Vol.I, Springer-Verlag, New York, 1986.Google Scholar
[19] Samarskii, A.A. and Vabishchevich, P.N., Numerical Methods for Solving Inverse Problems of Mathematical Physics, Walter de Gruyter, Berlin, 2007.Google Scholar
[20] Schnur, D.S. and Zabaras, N., Finite element solution of two-dimensional inverse elastic problems using spatial smoothing, Int. J. Numer. Meth. Eng. 30(1990), 57–75.Google Scholar
[21] Shillor, M., Sofonea, M. and Telega, J., Models and Variational Analysis of Quasistatic Contact, Springer, Berlin, 2004.Google Scholar
[22] Sofonea, M. and Matei, A., Mathematical Models in Contact Mechanics, Cambridge University Press, Cambridge, 2012.Google Scholar
[23] Tichonov, A. and Arsenin, V., Solution of ill-Posed Problems, John Wiley, New York, 1977.Google Scholar
[24] Wang, F., Han, W. and Cheng, X., Discontinuous Galerkin methods for solving Signorini problem, IMA J. Numer. Anal. 31(2011), 1754–1772.CrossRefGoogle Scholar
[25] Zeidler, E., Nonlinear Functional Analysis and Its Applications III. Variational Methods and Optimizations, Springer-Verlag, New York, 1985.Google Scholar