The finite-element time-domain method for elastic band-structure calculations

[EN] The finite-element time-domain method for elastic band-structure calculations is presented in this paper. The method is based on discretizing the appropriate equations of motion by finite elements, applying Bloch boundary conditions to reduce the analysis to a single unit cell, and conducting a simulation using a standard time-integration scheme. The unit cell is excited by a wide-band frequency signal designed to enable a large number of modes to be identified from the time-history response. By spanning the desired wave-vector space within the Brillouin zone, the band structure is then robustly generated. Bloch mode shapes are computed using the well-known concept of modal analysis, especially as implemented in an experimental setting. The performance of the method is analyzed in terms of accuracy, convergence, and computation time, and is compared to the finite-difference time-domain method as well as to a direct finite-element (FE) solution of the corresponding eigenvalue problem. The proposed method is advantageous over FD-based methods for unit cells with complex geometries, and over direct FE in situations where the formulation of an eigenvalue problem is not straightforward. For example, the new method makes it possible to accurately solve a time-dependent Bloch problem, such as the case of a complex unit cell model of a topological insulator where an internal fluid flow or other externally controlled physical fields are present. (C) 2018 Elsevier B.V. All rights reserved.A.C. is grateful for the support of Programa de Ayudas de Investigacion y Desarrollo (PAID) and Programa de Movilidad e Internacionalizacion Academica (PMIA-2013) of the UPV. This research was partially funded by the funded by the Ministerio de Economia e Innovacion (MINECO), Spain through project FIS2015-65998-C2-2-P, and partially funded by the National Science Foundation (NSF), USA through grant number 1538596. The authors acknowledge Dr. Noe Jimenez for fruitful discussions.Cebrecos, A.; Krattiger, D.; Sánchez Morcillo, VJ.; Romero García, V.; Hussein, MI. (2019). The finite-element time-domain method for elastic band-structure calculations. Computer Physics Communications. 238:77-87. https://doi.org/10.1016/j.cpc.2018.12.016S778723

Fast Band-Structure Computation for Phononic and Electronic Waves in Crystals

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

Anisotropic dissipation in lattice metamaterials

Plane wave propagation in an elastic lattice material follows regular patterns as dictated by the nature of the lattice symmetry and the mechanical configuration of the unit cell. A unique feature pertains to the loss of elastodynamic isotropy at frequencies where the wavelength is on the order of the lattice spacing or shorter. Anisotropy may also be realized at lower frequencies with the inclusion of local resonators, especially when designed to exhibit directionally non-uniform connectivity and/or cross-sectional geometry. In this paper, we consider free and driven waves within a plate-like lattice−with and without local resonators−and examine the effects of damping on the isofrequency dispersion curves. We also examine, for free waves, the effects of damping on the frequency-dependent anisotropy of dissipation. Furthermore, we investigate the possibility of engineering the dissipation anisotropy by tuning the directional properties of the prescribed damping. The results demonstrate that uniformly applied damping tends to reduce the intensity of anisotropy in the isofrequency dispersion curves. On the other hand, lattice crystals and metamaterials are shown to provide an excellent platform for direction-dependent dissipation engineering which may be realized by simple changes in the spatial distribution of the damping elements

Interface reduction for Hurty/Craig-Bampton substructured models: Review and improvements

The Hurty/Craig-Bampton method in structural dynamics represents the interior dynamics of each subcomponent in a substructured system with a truncated set of normal modes and retains all of the physical degrees of freedom at the substructure interfaces. This makes the assembly of substructures into a reduced-order system model relatively simple, but means that the reduced-order assembly will have as many interface degrees of freedom as the full model. When the full-model mesh is highly refined, and/or when the system is divided into many subcomponents, this can lead to an unacceptably large system of equations of motion. To overcome this, interface reduction methods aim to reduce the size of the Hurty/Craig-Bampton model by reducing the number of interface degrees of freedom. This research presents a survey of interface reduction methods for Hurty/Craig-Bampton models, and proposes improvements and generalizations to some of the methods. Some of these interface reductions operate on the assembled system-level matrices while others perform reduction locally by considering the uncoupled substructures. The advantages and disadvantages of these methods are highlighted and assessed through comparisons of results obtained from a variety of representative linear FE models.Accepted Author ManuscriptShip Hydromechanics and StructuresDynamics of Micro and Nano System