Eine elektro-mechanische XFEM-Formulierung für Vorwärtsprobleme und inverse Probleme
Final Report Abstract
In this research project, we developed an inverse method based on the extended finite element method (XFEM) to detect inclusions and flaws in piezo-electric materials. Therefore, a coupled XFEM formulation for piezo-electric materials was developed and implemented first. The method exploited different enrichments (Heaviside and crack tip enrichment) for cracks and inclusions (modified abs-enrichment). The method was subsequently verified by comparison to analytical solutions and results of other computational methods in the literature. Moreover, the method was extended from 2D to 3D. To detect inclusions (cracks were modelled as holes in our inverse analysis), a level set based inverse analysis was devised. The regularized Hamilton-Jacobi equation was solved to find the new positions of the inclusions or holes. Most importantly, the solution was performed over the same discretization. Only the parts of the discretization influenced by the enrichment functions needed to be modified so that optimal computational efficiency was ensured. We also tested the influence of noise on the results. Finally, our approach was tested and verified by numerous computational examples in 2D and later on in 3D. To detect inclusions and cracks in a piezo-electric solid, results from another computational method was exploited. Most importantly, we avoided inverse crimes, i.e. by not using the same method to generate this experimental data. We subsequently showed that we were able to detect the inclusions and cracks for different noise level and initial conditions.
Publications
- Detection of flaws in piezoelectric structures using XFEM, International Journal of Numerical Methods in Engineering, 2013, 96(6), 373 - 389
Nanthakumar S.S., Lahmer T., Rabczuk T.
- Detection of multiple flaws in Piezoelectric Structures using XFEM and Level sets, Computer Methods in Applied Mechanics and Engineering, 2014, 275, 98-112
Nanthakumar S.S., Lahmer T., Rabczuk T.
(See online at https://doi.org/10.1016/j.cma.2014.03.001) - Detection of material interfaces using a regularized level set method in piezoelectric structures. Inverse Problems in Science and Engineering, 24:1, 153-176, 2016
Nanthakumar S., Lahmer T., Zhuang X., Zi G., Rabczuk T.
(See online at https://doi.org/10.1080/17415977.2015.1017485)
