【作者】 周正平；

【导师】 罗松南；

【作者基本信息】 湖南大学 ， 工程力学， 2004， 硕士

【摘要】 研究了波在非均匀损伤介质中的传播及波动方程的反演。首先讨论了波在非均匀损伤介质中传播的正问题，根据已知介质的性质和入射的波，求出接收处波的响应。由于介质的非均匀性，在数学上表现为控制方程为高阶变系数微分方程。对于二次曲线变化的非均匀损伤介质中波的传播，求出了利用特殊函数表示的解析形式解。对于一般的非均匀损伤介质中波的传播，不便于用解析解表示，可利用数值法求解。把非均匀损伤区域进行分层处理，利用边界条件和层间的连续性条件求解波动方程式。对于谐波形式的稳态波场，利用波在不同介质分界面上的反射和透射系数，推出了波在分层介质中的传播的传递矩阵，利用传递矩阵法，加上定解条件，直接求出接收端处接收到的稳态波场。对于瞬态波场，利用傅立叶变换，把瞬态波展开成稳态波场的叠加，再运用分层介质中稳态波传播的分析结果，求出各个稳态波的响应结果，然后再运用傅立叶逆变换，把结果反变换到时域中去，求出瞬态波场的解。对非均匀损伤介质中波传播的解析解和数值解都进行了算例分析，讨论了非均匀损伤介质的不同损伤分布、不同损伤度、不同损伤区域长度对波传播的影响。 在波传播的正问题的基础上，利用所求得的接收波的响应和入射波，对损伤介质的性质进行反演。由于反问题的非线性和不适定性，在反演过程中，采用Newton迭代法，把原来的非线性问题化为线性问题；采用Tikhonov正则化方法求解，使问题的不适定性得以解决。在正则化过程中，采用L曲线来确定正则参数，解决了正则参数的选取这一核心问题。通过实例分析表明，上述反问题算法是非常有效的。更多还原

【Abstract】 The theory of waves propagation in inhomogeneous damaged zone and inversion of wave motion equations are studied. By the property of media and incident wave to seek of response wave, the problems of wave propagation in inhomogeneous damaged zone media are studied at first. Due to inhomogeneous in physics, the control equation is high-order differential equation with varied coefficient. For wave propagation in a quadratic curve inhomogeneous damaged zone, analytical solutions expressed special functions are gained. But to wave propagation in common inhomogeneous damaged zone, its analytical solutions can not be gained easily and the numerical methods can be applied. Dispersing inhomogeneous zone into layered zones and using boundary and continuous conditions between two layers, wave motion equation is solved. The harmonic wave propagation in inhomogeneous zone is studied, transmissive coefficient and reflective coefficient at the interface between two layers are gained, and transmission array of wave propagation is gained in layered media. Based on transmission array and boundary conditions, the response is obtained. On the base of harmonic solution, the solutions of transitory wave propagation in inhomogeneous zone can be obtained. By using Fourier transform, the transitory wave can be expressed as sum of steady wave. Appling solutions of steady wave propagation in inhomogeneous damaged zone, the response of each corresponding frequency is gained. At last, using of inverse Fourier transform, the result is transmitted to time and the solution of transitory wave propagation is gained. The result of numerical solutions are compared with the result of analytical solutions for wave propagation in inhomogeneous damaged zone. The influence of damaged degree, length of damaged zone and distributing of damage on wave propagation are studied.On the base of forward problem of wave propagation in inhomogeneous damaged zone, using the received and incident wave, the properties of media are inversed. Due to the ill-posed of inversion problems, using Newton iterative method at the course of inversion, the nonlinear equations is transformed into linear equations. Tikhonov’s regularization method is applied for solving linear ill-posed problems. The regularization parameters are keys and L-curve method is used to fix on regularization parameter in regularization. The numerical example of inversion shows the effectiveness of this inversed method.更多还原