基本解方法求解简谐外力作用下的Kirchhoff-Love板反源问题

Fundamental Solution Method for the Inverse Source Problem for Kirchhoff-Love Plate Under Harmonic Load

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.

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