Characterization of Acoustic Waves in Multi-Layered Structures
Oleg Pykhteev · mediaTUM – the media and publications repository of the Technical University Munich (Technical University Munich) · 2011
The work is devoted to the modeling of acoustic waves in multi-layered structures surrounded by a fluid and consisting of different kinds of materials including piezoelectric materials and composite multilayers.It consists of three parts.The first part describes the modeling of an acoustic sensor by the finite element method.The existence and uniqueness of a time-harmonic solution are rigorously established under physically appropriate assumptions.It is shown that the elimination of nonzero eigenvalues of the arising Helmholtz-like system can be achieved by introducing an additional term that causes the damping effect near the boundary.We also establish the well-posedness of the discretized problem and the convergence of Ritz-Galerkin solutions to the solution of the exact problem.Besides, we derive a domain decomposition schema for the numerical treatment of the problem.Finally, the results of 3D simulations are presented.The second part of the work presents a semi-analytical method for the fast characterization of plane acoustic waves in multi-layered structures.The method identifies plane waves that can possibly exist in a given structure, determines their velocities, and computes the dispersion relation curves.It handles multi-layered structures composed of an arbitrary number of layers made of different material types.The influence of a surrounding fluid or dielectric medium can also be taken into account.The software implementing this approach is presented.The third part investigates a number of issues of the homogenization theory for linear systems of elasticity.The results presented here are exploited in the previous part for the modeling of acoustic waves in composite materials called multilayers.Such materials consist of huge number of very thin periodic alternating sublayers.In this part we rigorously derive the limiting equations in the general three-dimensional case by the two-scale method and establish an error estimate for the case where the right-hand side is in L 2 .The homogenization of laminated structures is considered as a special case.For this case an explicit formula for the elasticity tensor of the homogenized material is derived.iii