Transient elastic waves in piezoelectric materials and their numerical discretization
Date
2018
Authors
Journal Title
Journal ISSN
Volume Title
Publisher
University of Delaware
Abstract
The work in this thesis concerns the analysis and numerical discretization of problems involving transient elastic waves moving through a solid with piezoelectric properties. Such solids are characterized by electric current being generated by the deformation of the solid, which in turn further deforms the solid. Mathematically this means that the elastic stress in the solid is defined by a coupling of the mechanical strain and an electric potential. Piezoelectricity is found naturally in some materials such as quartz and bone, but in applications most commonly takes the form of synthetic ceramics. These ceramics can be found in everyday items such as cell phone speakers and in such specialized equipment as diesel fuel injectors. ☐ The problems under consideration in this work include a wave-interaction problem where acoustic waves scatter off a piezoelectric solid as well as excite an elastic wave, an optimization problem in which deformation of a piezoelectric solid is controlled by a boundary condition on the electric flux, and an initial-boundary value problem involving a model consisting of a coupled system of hyperbolic, elliptic, and parabolic partial differential equations. We use and expand upon an abstract functional framework that applies some of the tools of semigroup theory to evolutionary partial differential equations. Using this framework we obtain stability bounds for continuous and semidiscretized versions of each problem as well as bounds on the error due to semidiscretization. Additionally, this work contains several numerical experiments that serve to reinforce and illustrate our theoretical results.
Description
Keywords
Pure sciences, Applied sciences, Elastic waves, Finite elements, Hyperbolic PDE, Numerical analysis, Optimal control, Piezoelectricy