Parameter identification for elastic wave equations
In many industrial productions fibre composites are being used more and more often and also in ever more diverse setups. For example in air plane construction carbon reinforced fibre composites are of great importance. These high-performane materials come at a high price. Hence, especially from an economic point of view, it is important to detect damages as soon as possible before greater parts of a composite have to be replaced.
Traditional materials such as steel show a very nice behavior when going from elastic to plastic deformation and final breaking: They do it slowly and noticably. Fibre composites however behave very differently. It cannot be seen by the naked eye whether the fibres inside the composite have become detached, when fissures have grown around them, and delimination has occurred. Therefore, the development of robust and stable structual health monitoring (SHM) systems for individual building parts is required to quickly assess of (concealed) damage such as holes, fissures and delamination. To this end, a network of piezo-ceramical surface sensors is integrated into the supervised structure which localize faulty areas via Lamb waves. These waves that are actuated at specific sites propgate through the structure and are being reflected and deflected until they hit the sensors.
The relation between the shape of these signals and the material''s properties can be modelled through a parameter identification problem based on the initial boundary problem for the anisotropic, elastic wave equation. By the spatial distribution of the material properties structual damage in the fibre composites can be identified.
Moreover, two constraint optimization problems can be set up that describe mathematically the damage detection problem and contain the initial boundary problem as auxiliary constraint. This research topic is then on the further development and analysis of this model and also on the implementation of the developed solution procedures.
The project is funded by the DFG (SCHU 1978/4-2) and is done in cooperation with the Helmut-Schmidt-Universität Hamburg and the DLR in Braunschweig.
Here you can download the entire software package for solving the appropriate forward problem as well as the inverse problem from preprint "Identifying the stored energy of a hyperelastic structure by using an attenuated Landweber method" by J. Seydel and T. Schuster (2017).