Wave Splitting and Effective Boundary Conditions for Structural Elements

Abstract: The subject of this thesis is twofold. The first part concerns the wave splitting of structural elements. The second part concerns approximate boundary conditions for fluid-loaded thin plates.

Wave splitting is a mathematical tool for the formulation of scattering problems in the time domain that aims at solving inverse problems, e.g. material characterization and nondestructive testing. For the application of wave splitting and related techniques to mathematical models for structural elements, there are two situations where the usual approach is inadequate. These are when the original partial differential equation contains spatially dependent coefficients and when it is of higher order.

The first situation arises, for instance, in problems formulated in cylindrical coordinates. A wave splitting for this situation is presented, which also admits the reduction to dispersion free dynamics for the split variables. A direct problem is solved and numerical results are presented and compared to an FEM solution. The second situation arises in the modelling of beams and plates. The usual approach involve the decoupling of waves propagating in the same direction, which introduces an exponentially growing behaviour in the split variables. An alternative method is presented, which comprises only the directional decoupling and thereby resolves this problem. A numerical example is given, in which a wave field is propagated along a Timoshenko beam further than would be possible for the standard methods.

Approximate boundary conditions are derived for thin fluid-loaded elastic and porous plates. These are obtained by approximating the displacements in the plate by means of truncated series expansions in the thickness coordinate. The boundary conditions and the governing equations for the plate are used to eliminate all plate variables. This results in two partial differential equations relating the fluid pressures and normal displacement components on the two sides of the plate. Applications to several fluid, elastic and porous material combinations are presented numerically and compared to exact theory and classic plate theories.

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