Mathematical and numerical modelling of piezoelectric sensors

The present work aims at proposing a rigorous analysis of the mathematical and numerical modelling of ultrasonic piezoelectric sensors. This includes the well-posedness of the final model, the rigorous justification of the underlying approximation and the design and analysis of numerical methods. More precisely, we first justify mathematically the classical quasi-static approximation that reduces the electric unknowns to a scalar electric potential. We next justify the reduction of the computation of this electric potential to the piezoelectric domains only. Particular attention is devoted to the different boundary conditions used to model the emission and reception regimes of the sensor. Finally, an energy preserving finite element / finite difference numerical scheme is developed; its stability is analyzed and numerical results are presented. Résumé. ... 1991 Mathematics Subject Classification. 35L05, 35A35, 73R05, 35A40. The dates will be set by the publisher. Introduction The present work has been achieved in the framework of a collaboration between the laboratory LIST of CEA Saclay and the Project Team POEMS which is common to INRIA, ENSTA and CNRS (UMR 7231), and motivated by an important application : non destructive testing by ultra-sound. The laboratory LIST is specialized in various aspects of non destructive testing, from both experimental and theoretical aspects, which includes numerical simulation which has now become a fundamental tool to understand and analyze the result of a non destructive testing experiment. The propagation of ultra-sonic elastic waves is often used to investigate the presence of defects in any manufactured item. In particular it is used for detecting defects (we mean a local heterogeneity, a geometrical imperfection, a crack, ...) inside metallic objects. As a fundamental and illustrative industrial application, we can cite the inspection of nuclear reactors. Of course, to construct a reliable and efficient simulation tool in ultra-sonic non destructive testing, it is important to master the numerical simulation of elastic waves, for instance efficient discretization techniques for solving elastodynamic equations in a very general context : heterogeneous, anisotropic elastic (possibly viscoelastic) media. Nowadays, there are many satisfactory highly accurate numerical methods for the resolution of elastodynamic equations. In particular, at Project POEMS, we have intensively developed the technique