Integral and coupled integral-volume methods for transient problems in wave-structure interaction
Date
2016
Authors
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Journal ISSN
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Publisher
University of Delaware
Abstract
We study the discretization of the equations describing the propagation of elastic waves through a homogeneous medium and the transient interaction between acoustic waves traveling on free space and interacting with scatterers with different sorts of elastic properties. ☐ In the case of plane elastodynamic waves a simple and efficient method based on deltaBEM is developed for scatterers with smooth boundaries. We provide mathematical and numerical evidence that the proposed discretization is the only choice of the one-parameter family of discretizations proposed in [31] that provides third order consistency for the operators of linear elasticity. ☐ The remaining chapters of the thesis study the scattering of acoustic waves by obstacles with linearly elastic, piezoelectric, and thermoelastic behavior. The proposed formulations use a boundary potential representation of the scattered acoustic wave, resulting in systems of boundary integral equations, in the case of a homogeneous elastic scatterer, or coupled integro-differential systems for non homogeneous solids. The analysis is done in the Laplace domain, following [6, 82] where the systems are semi-discretized in space and the well-posedness of the continuous and semidiscrete problems is proven simultaneously. ☐ The equations are fully discretized using second order multi-step Convolution Quadrature [91] for time evolution. We prove that the resulting fully discrete methods obtained with BDF2-CQ are of second order and provide explicit dependence of the error constants with respect to time. Numerical evidence is given to show that Trapezoidal Rule Convolution Quadrature yields a second order method as well.