Fundamental solution method for inverse source problem of plate equation

GUZhi-jie; TAN Yong-ji

doi:10.3879/j.issn.1000-0887.2012.12.004

Volume 33 Issue 12

Turn off MathJax

Article Contents

Article Navigation > Applied Mathematics and Mechanics > 2012 > 33(12): 1411-1430

GUZhi-jie, TAN Yong-ji. Fundamental solution method for inverse source problem of plate equation[J]. Applied Mathematics and Mechanics, 2012, 33(12): 1411-1430. doi: 10.3879/j.issn.1000-0887.2012.12.004

Citation:

PDF( 3930 KB)

Fundamental solution method for inverse source problem of plate equation

doi: 10.3879/j.issn.1000-0887.2012.12.004

School of Mathematical Sciences, Fudan University, Shanghai 200433, P. R. China

Received Date: 2012-02-10
Rev Recd Date: 2012-05-10
Publish Date: 2012-12-15

Abstract

Abstract

The elastic plate vibration model is studied under the external force. The size of the source term by the given mode of the source and some observations from the body of the plate is determined over a time interval, which is referred to be an inverse source problem of a plate equation. The uniqueness theorem for this problem is stated, and the fundamental solution to the plate equation is derived. In the case that the plate is driven by the harmonic load, the fundamental solution method (FSM) and the Tikhonov regularization technique are used to calculate the source term. Numerical experiments of the Euler-Bernoulli beam and the Kirchhoff-Love plate show that the FSM can work well for practical use, no matter the source term is smooth or piecewise.
- Kirchhoff-Love plate,
- Euler-Bernoulli beam,
- elastic,
- inverse source problem,
- fundamental solution method(FSM),
- Tikhonov regularization method,
- meshless numerical method

FullText(HTML)

References(31)

References

[1]	Yang Y, Lim C W. A new nonlocal cylindrical shell model for axisymmetric wave propagation in carbon nanotubes[J]. Advanced Science Letters, 2011, 4(1): 121-131.
[2]	Zhao X, Ng T Y, Liew K M. Free vibration of two-side simplysupported laminated cylindrical panels via the meshfree kpRitz method[J].International Journal of Mechanical Sciences, 2004, 46(1): 123-142.
[3]	Zhou D, Lo S H, Cheung Y K. 3-D vibration analysis of annular sector plates using the Chebyshev-Ritz method[J].Journal of Sound and Vibration, 2009, 320(1/2): 421-437.
[4]	Liu Y, Hon Y C, Liew K M. A meshfree Hermite-type radial point interpolation method for Kirchhoff plate problems[J].International Journal for Numerical Methods in Engineering, 2006, 66(7): 1153-1178.
[5]	Kurpa L, Pilgun G, Amabili M. Nonlinear vibrations of shallow shells with complex boundary: R-functions method and experiments[J].Journal of Sound and Vibration, 2007, 306(3/5): 580-600.
[6]	Qian L F, Batra R C, Chen L M. Static and dynamic deformations of thick functionally graded elastic plates by using higher-order shear and normal deformable plate theory and meshless local PetrovGalerkin method[J].Composites Part B: Engineering, 2004, 35(6/8): 685-697.
[7]	Sladek J, Sladek V, Wen P H, Aliabadi M H. Meshless local PetrovGalerkin(MLPG) method for shear deformable shells analysis[J].Chinese Journal of Mechanical Engineering, 2006, 13(2): 103-117.
[8]
[9]	Krys’ko V A, Papkova I V, Soldatov V V. Analysis of nonlinear chaotic vibrations of shallow shells of revolution by using the wavelet transform[J].Mechanics of Solids, 2010, 45(1): 85-93.
[10]	李善倾, 袁鸿. 简支梯形底扁球壳自由振动问题的准Green函数方法[J]. 应用数学和力学, 2010, 31(5): 602-608.(LI Shan-qing, YUAN Hong. Quasi-Green’s function method for free vibration of simply-supported trapezoidal shallow spherical shell[J].Applied Mathematics and Mechanics(English Edition), 2010, 31(5): 635-642.)
[11]	Michaels J E, Pao Y H. The inverse source problem for an oblique force on an elastic plate[J]. Journal of the Acoustical Society of America, 1985, 77(6): 2005-2011.
[12]	Li S M, Miara B, Yamamoto M. A Carleman estimate for the linear shallow shell equation and an inverse source problem[J].Discrete and Continuous Dynamical Systems, 2009, 23(1/2): 367-.
[13]
[14]	Alves C, Silvestre A L, Takahashi T. Solving inverse source problems using observability applications to the EulerBernoulli plate equation[J].SIAM Journal on Control and Optimization, 2009, 48(3): 1632-1659.
[15]	Wang Y H. Global uniqueness and stability for an inverse plate problem[J].Journal of Optimization Theory and Applications, 2007, 132(1): 161-173.
[16]	Yang C Y. The determination of two heat sources in an inverse heat conduction problem[J]. International Journal of Heat and Mass Transfer, 1999, 42(2): 345-356.
[17]	Yang C Y. Solving the twodimensional inverse heat source problem through the linear least-squares error method[J].International Journal of Heat and Mass Transfer, 1998, 41(2): 393-398.
[18]	Fatullayev A G. Numerical solution of the inverse problem of determining an unknown source term in a heat equation[J].Mathematical and Computers in Simulation, 2002, 58(3): 247-253.
[19]	Le N C, Lefevre F. A parameter estimation approach to solve the inverse problem of point heat sources identification[J].International Journal of Heat and Mass Transfer, 2004, 47(4): 827-841.
[20]	Johansson B T, Lesnic D. A variational method for identifying a spacewisedependent heat source[J].IMA Journal of Applied Mathematics, 2007, 72(6): 748-760.
[21]	Yan L, Fu C L, Yang F L. The method of fundamental solutions for the inverse heat source problem[J].Engineering Analysis With Boundary Elements, 2008, 32(3): 216-222.
[22]	Jin B T, Marin L. The method of fundamental solutions for inverse source problems associated with the steadystate heat conduction[J].International Journal for Numerical Method in Engineering, 2007, 69(8): 1570-1589.
[23]	Yan L, Yang F L, Fu C L. A meshless method for solving an inverse spacewisedependent heat source problem[J].Journal of Computational Physics, 2009, 228(1): 123-136.
[24]	Johansson B T, Lesnic D, Reeve T. A method of fundamental solutions for the onedimensional inverse Stefan problem[J].Applied Mathematical Modelling, 2011, 35(9): 4367-4378.
[25]	Chen C W, Young D L, Tsai C C. The method of fundamental solutions for inverse 2D Stokes problems[J].Computational Mechanics, 2005, 37(1):2-14.
[26]	Marin L, Lesnic D. The method of fundamental solutions for inverse boundary value problems associated with the twodimensional biharmonic equation[J].Mathematical and Computer Modelling, 2005, 42(3/4): 261-278.
[27]	Alves C J S, Colaco M J, Leitao V M A, Martins N F M , Orlande H R B, Roberty Ni C. Recovering the source term in a linear diffusion problem by the method of fundamental solutions[J].Inverse Problems in Science and Engineering, 2008, 16(8): 1005-1021.
[28]	Love A E H. On the small free vibrations and deformations of elastic shells[J]. Philosophical Transactions of the Royal Society A, 1888, 179: 491-549.
[29]	Alves C J S. On the choice of source points in the method of fundamental solutions[J]. Engineering Analysis With Boundary Elements, 2009, 33(12): 1348-1361.
[30]	Kim S M, McCullough B F. Dynamic response of plate on viscous Winkler foundation to moving loads of varying amplitude[J].Engineering Structure, 2003, 25(9): 1179-1188.
[31]	Liew K M, Han J B, Xiao Z M. Differential quadrature method for Mindlin plates on Winkler foundations[J]. International Journal of Mechanical Sciences, 1996, 38(4): 405-421.