This dissertation introduces the idea of an equivalent continuous medium (ECM) that has the same impedance as that of an unbounded discrete periodic medium. Contrary to existing knowledge, we constructively show that it is indeed possible to achieve perfect matching for periodic and discrete media. We present analytical results relating the propagation characteristics of periodic media and the corresponding ECM, leading to the development of numerical methods for wave propagation in these media. In this dissertation, we present the main idea of ECM and apply it, with mixed results, to seemingly different problems requiring effective numerical methods for modeling wave propagation in unbounded media.
An immediate application of ECM is in developing absorbing boundary conditions (ABCs) for wave propagation in unbounded discrete media. Using the idea of ECM, and building on class of continuous ABCs called perfectly matched discrete layers (PMDL), we propose a new class of discrete ABCs called discrete PMDL and develop frequency domain formulations that are shown to be superior to continuous ABCs.
Another application that is explored in this dissertation is the design of interface conditions for concurrent coupling of two-scale wave propagation models, e.g. Atomistic-to-Continuum (AtC) coupling. We propose a domain-decomposition (DD) approach and develop accurate interface conditions that are critical for the concurrent coupling of the two-scale models. It turns out that time-domain discrete ABCs are key to the the accuracy of these interface conditions. Since discrete PMDL is well-posed and accurate for the model problem, we build on it to propose an efficient and accurate interface condition for two-scale wave propagation models. Although many open problems remain with respect to implementation, we believe that the proposed DD based approach is a good first step towards achieving efficient coupling of two-scale wave propagation models.
Time-domain discrete PMDL can also be very useful for numerical simulation of dynamics of crystal lattices, for both MD simulations and AtC coupling. A critical problem that needs to be resolved in this context is the well-posedness of the discrete PMDL formulation. The proposed ECM facilitates direct extension of well-posedness criteria from continuous ABCs to discrete ABCs, and thus to discrete PMDL. This criterion is then used to develop a well-posed discrete PMDL formulation for a diatomic lattice, a simple but representative complex lattice, which is numerically verified to be stable. The methodology can be extended to more complex lattices, but at the expense of increased algebraic complexity, and perhaps significant increase in computational cost.
Time-harmonic discrete PMDL can play an important role in numerical simulation of wave propagation in continuous periodic media, e.g. photonic crystals. ECM provides an alternate viewpoint for developing effective ABCs for periodic media. For instance, using ECM we derive an alternative expression for the exact Dirichlet-to-Neumann (DtN) operator of an unbounded periodic waveguide. While it is not yet clear if this alternative expression is computationally advantageous, it is nevertheless an interesting alternative. More significantly, we build on the existing idea of recursive doubling to propose a new and efficient method, based on discrete PMDL, to compute the DtN operator of an unbounded periodic waveguide. Compared to the original method, applicable only to dissipative media, the proposed method is more general and applicable to both dissipative and non-dissipative media.
Based on the observations, it appears that discrete PMDL, and more generally the idea of equivalent continuous media, offers additional flexibility towards developing accurate methods for modeling wave propagation in a variety of application problems.
|Advisor:||Guddati, Murthy N.|
|School:||North Carolina State University|
|School Location:||United States -- North Carolina|
|Source:||DAI-B 76/07(E), Dissertation Abstracts International|
|Subjects:||Applied Mathematics, Geophysics, Acoustics|
|Keywords:||Atomistic-continuum coupling, Discrete ABCs, Domain decomposition, Interface conditions, Periodic ABCs, Wave propogation|
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