Fundamental solution method for inverse source problem of plate equation
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摘要: 主要考察弹性薄板在规则外力作用下的振动模型.在给定外力源项随时间变化模式的情况下,通过对薄板局部区域一段时间的振动位移观测数据,来反演外力大小的问题,也就是通常所谓的弹性薄板反源问题.给出了弹性薄板反源解的唯一性定理,并推导出板方程的基本解.取基本解方法和Tikhonov正则化方法的精髓,在简谐模式源项作用的情况下,构造了一套算法来反解源项.对Euler-Bernoulli杆和Kirchhoff-Love板的数值算例表明,无论源项是否光滑,测量是否带有误差,基本解方法都因其较好的计算效果,有着广泛的适用性.
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关键词:
- Kirchhoff-Love板 /
- Euler-Bernoulli杆 /
- 弹性 /
- 反源问题 /
- 基本解方法 /
- Tikhonov正则化方法 /
- 无网格方法
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. -
[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 simplysupported laminated cylindrical panels via the meshfree kpRitz 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 PetrovGalerkin 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 PetrovGalerkin(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 EulerBernoulli 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 twodimensional 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 spacewisedependent 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 steadystate 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 spacewisedependent 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 onedimensional 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 twodimensional 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.
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