8th European Workshop On Structural Health Monitoring (EWSHM 2016)
5-8 July 2016, Spain, Bilbao
Right Now in Bilbao 07:04 Sat 14
14:00 Wednesday 6. Jul - A2
Robust mixed finite element simulations of interactions of ultrasonic waves with cracks
Abstract »We are faced with interactions of ultrasonic waves with cracks in thin plates. These are important means in structural health monitoring (SHM) of aircrafts. This is a joint work with Honeywell International, s.r.o., in Brno, Czech Republic. In the first part of our contribution we are inspired by a traditional SHM computational approach that relies on Lamb waves. They are certain eigenmodes, surviving at large distance, of the elastic wave equation in an unbounded plate. In order to justify our finite element simulations we validate the models in terms of Lamb dispersive (frequency-velocity) curves. In 2d it turns out that membranes and Kirchhoff plates imitate well the low frequency case of the lowest-order shear and bending modes, respectively. We switch to 3d displacement finite elements enhanced with higher-order ansatz functions in the thickness direction. These imitate properly the shear-horizontal modes, however, they suffer from the locking effect in case of the Lamb modes. Finally, we end up with 3d mixed elastic elements, which are referred to as tangential-displacements and normal-normal-stresses (TD-NNS), recently proposed and analyzed by Astrid Pechstein (born Sinwel) and Joachim Schoeberl at TU Vienna. We show that these new elements cover the whole spectrum of elastic waves including higher-order Lamb waves. In the second part we introduce the construction of the mixed TD-NNS hexahedral elements and its hybridization resulting in a robust displacement ansatz. The wave equation is discretized by an unconditionally stable Newmark scheme. This results in sequential solution to linear systems of equations the matrix of whose is, fortunately, mass-dominant. When the Cholesky decomposition becomes demanding to use we employ the conjugate gradients preconditioned with a vertical-edge-based smoother with 3 smoothing steps. The relative precision 1e-9 is achieved in 4 PCG iterations.
AuthorsLukas, DaliborLukas, Dalibor
Dalibor Lukas (CZ, 1976) is an associate professor at the Department of Applied Mathematics at the Faculty of Electrical Engineering and Computer Science, VSB-Technical University of Ostrava, Czech Republic. He is also an external specialist in the National Supercomputing Center IT4Innovations. His research includes numerical analysis and optimal design in electromagnetism, acoustics, and mechanics by means of finite and boundary elements accelerated by domain decomposition, multigrid, or parallel hierarchical matrices. The research is mostly driven by industrial applications. Currently, in a project with Honeywell International, he is concerned with structural health monitoring of aircrafts while relying on intensive simulations of interaction of elastic waves with cracks in thin structures. He is a vice-chair of EU-MATHS-IN.CZ.
VSB-Technical University of Ostrava
Department of Applied Mathematics, FEECS