Regularization of Parameter Problems for Dynamic Beam Models

The field of inverse problems is an area in applied mathematics that is of great importance in several scientific and industrial applications. Since an inverse problem is typically founded on non-linear and ill-posed models it is a very difficult problem to solve. To find a regularized solution it is crucial to have a priori information about the solution. Therefore, general theories are not sufficient considering new applications. In this thesis we consider the inverse problem to determine the beam bending stiffness from measurements of transverse dynamic displacement. Of special interest is to localize weak parts, i.e. parts with reduced bending stiffness. For the mathematical modelling we use the Euler-Bernoulli beam equation in the frequency domain. Different regularization strategies are considered in the developing of time-efficient methods. For the problem to localize weak parts we first consider the Euler-Bernoulli beam equation with piecewise constant material parameters. Regularized solutions are obtained by including the a priori information that there is a weak part in the modelling. The constant bending stiffness in the weak part and in the surrounding areas are found together with the coordinates of the weak part by minimizing an error-functional depending on the difference between measured and modelled transverse displacement. Errors due to noise are reduced by using a frequency near an eigen frequency for the measurement. The problem to determine continuously varying beam bending stiffness is a much harder problem. A numerical efficient method without non-linear regularization, is developed for the simpler problem to determine the string refractive index from measurement of the transverse displacement, and then generalized for the beam problem. To avoid non-linear regularization the problem is transformed to the problem of solving a Fredholm integral equation of the first kind followed by a simpler division operation. The linear regularization problems, for the string and for the beam, are solved by the method of Tikhonov together with the discrepancy principle of Morozov. To localize weak parts from the reconstructed bending stiffness a recently developed convex image segmentation method is considered. A great advantage with this method is that spurious details due to noise can be removed. The entire procedure for reconstructing the bending stiffness and localizing weak parts is much simpler compared with the method based on non-linear and non-convex optimization considered first for the problem with piecewise constant material parameters.