Date of Award

Spring 1-1-2017

Document Type

Dissertation

Degree Name

Doctor of Philosophy (PhD)

First Advisor

Mahmoud I. Hussein

Second Advisor

John A. Evans

Third Advisor

Carlos A. Felippa

Fourth Advisor

Matthew A. Glaser

Fifth Advisor

Per-Gunnar J. Martinsson

Abstract

The band structure is a frequency/energy versus wave vector/momentum relationship that fundamentally describes the nature of wave motion in a periodic medium. It is immensely valuable for predicting and understanding the properties of electronic, photonic, and phononic materials, and is typically computed numerically. For materials with large unit cells, such as nanostructured supercells for example, band-structure computation is very costly. This inhibits the ability to feasibly analyze new material systems with potentially extraordinary properties. This thesis describes a novel unit-cell model-reduction technique for band-structure calculations that is capable of lowering computational costs by one or two orders of magnitude with practically insignificant loss of accuracy.

This new methodology, termed \textit{Bloch mode synthesis}, is based on unit-cell modal analysis. It begins from a free-boundary unit-cell model. Before periodic boundary conditions are applied, this free unit cell behaves as though it has been cut out from its periodic surroundings. A truncated set of normal mode shapes is then used to compactly represent the interior portion of the unit cell while retaining nearly all of the dynamically important information. A Ritz basis for the unit cell is formed by combining the interior modes with a second set of modes that preserves the flexibility needed to enforce a Bloch wave solution in the unit cell. Residual mode enhancement and interface modal reduction improve performance further. With this highly reduced model, Bloch boundary conditions corresponding to waves of any directions and wavelength can be applied to very quickly obtain the band structure.

Bloch mode synthesis is derived in the context of elastic wave propagation in phononic crystals and metamaterials, but the framework is also well suited for other types of waves. It shows particular promise in speeding up electronic structure calculations –-- a central problem in computational materials science that lies at the heart of property determination for numerous applications including semiconductors, superconductors, photovoltaics, thermoelectrics, lasers, and light emitting diodes. It also shows promise for predicting thermal properties of nanophononic materials. Thermal conductivity calculations require the full-spectrum band structure, which are obtained by modifying the Bloch mode synthesis formulation to incorporate high-frequency information in the basis.

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