Modeling of wave phenomena in heterogeneous elastic solids
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This dissertation addresses the analysis of the classical problem in con tinuum mechanics of wave propagation through heterogeneous elastic media. The class of waves that are considered are stress waves propagating through linearly elastic media with highly oscillatory material properties. This work provides an approach which resolves a classical open problem: the accurate characterization of interfacial stresses in highly heterogeneous media through which stress waves propagate. This is accomplished using an extension of the theory and methodology of adaptive modeling to complex sesquilinear forms. A general abstract notion of residual based a posteriori error analysis is presented, which makes possible the development of a mathematical framework for the mathematical modeling and numerical analysis of this elastodynamic problem. viii The notion of hierarchical modeling is ﬁrst applied to the derivation of computable and reliable estimates of the modeling error in a speciﬁc quantity of interest: the average stress on a subdomain in the elastic body. The estimate is subsequently employed in a goal-oriented adaptive modeling algorithm that is introduced for solving wave propagation in heterogeneous media. To control the error due to geometric dispersion, the algorithm solves the wave problem in the complex frequency domain by iteratively adapting the mathematical material model until the error estimate meets a preset tolerance. The algo rithm is applicable to elastic materials with arbitrary microstructure and does not require geometric periodicity. A number of one-dimensional steady-state and transient examples are investigated, which demonstrate the application of an adaptive modeling algorithm and the reliability and accuracy of the error estimate. A new Discontinuous Galerkin Method (DGM) is presented to numer ically solve the wave equation in the frequency domain. Well-posedness and convergence of the formulation is proved for the case of a Reaction-Diﬀusion type model problem. One- and two-dimensional numerical veriﬁcations are shown. The general abstract framework of a posteriori error analysis is then again applied, but now to the new DGM formulation of the wave equation to derive an estimate of the numerical approximation error in the quantity of in terest. An hp-adaptive algorithm for numerical error control is introduced and numerical results are presented for one-dimensional steady state applications.