We consider the problem of avoided crossings (level repulsion) in phononic crystals and suggest a computationally efficient strategy to distinguish them from normal cross points. This process is essential for the correct sorting of the phononic bands and, subsequently, for the accurate determination of mode continuation, group velocities, and emergent properties which depend on them such as thermal conductivity. Through explicit phononic calculations using generalized Rayleigh quotient, we identify exact locations of exceptional points in the complex wavenumber domain which results in level repulsion in the real domain. We show that in the vicinity of the exceptional point the relevant phononic eigenvalue surfaces resemble the surfaces of a 2 by 2 parameter-dependent matrix. Along a closed loop encircling the exceptional point we show that the phononic eigenvalues are exchanged, just as they are for the 2 by 2 matrix case. However, the behavior of the associated eigenvectors is shown to be more complex in the phononic case. Along a closed loop around an exceptional point, we show that the eigenvectors can flip signs multiple times unlike a 2 by 2 matrix where the flip of sign occurs only once. Finally, we exploit these eigenvector sign flips around exceptional points to propose a simple and efficient method of distinguishing them from normal crosses and of correctly sorting the band-structure. Our proposed method is roughly an order-of-magnitude faster than the zoom-in method and correctly identifies > 96% of the cases considered. Both its speed and accuracy can be further improved and we suggest some ways of achieving this. Our method is general and, as such, would be directly applicable to other eigenvalue problems where the eigenspectrum needs to be correctly sorted.
In this paper we use level set topology optimization to reveal novel phononic crystal topologies which give rise to metamaterial properties including negative and singular effective properties. The level set formalism has been developed on the basis of polynomial functions, the locations of whose zeros control the distribution of material phases. This significantly reduces the number of design variables involved and allows us to search very large design spaces using global optimization techniques. Optimization process reveals that a 2-phase unit cell in which one of the phases is simultaneously lighter and stiffer than the other results in dynamic behavior which has all the attendant characteristics of a locally resonant composite. This behavior is further explored through the use of mode shape analysis. Results presented in this paper are also an example of how purely computational techniques can illuminate novel physical phenomenon.
In this paper we present a Graphical Processing Unit (GPU) accelerated variational formulation for fast phononic band-structure calculations. The thousands of parallel threads available on GPUs massively reduce the time taken to assemble the phononic eigenvalue problem for arbitrarily complex unit cells. The computation times then become bounded by the eigenvalue solution and if only a few eigenvalues are desired then the computation becomes linear in complexity. Since most of the current applications of phononic crystals require the calculation of only the first few eigenvalues, the GPU acceleration scheme presented in this paper promises to facilitate the solutions to currently tough phononic problems such as 3-D optimization and inverse solutions. The parallelization scheme and GPU application presented in this paper are not limited to the variational scheme used but can be easily extended to other phononic algorithms such as the Plane Wave Expansion method and also to the general eigenvalue solutions of elastodynamics.
Three fundamental variational principles used for solving elastodynamic eigenvalue problems are studied within
the context of elastic wave propagation in periodic composites (phononics). We study the convergence of the
eigenvalue problems resulting from the displacement Rayleigh quotient, the stress Rayleigh quotient and the
mixed quotient. The convergence rates of the three quotients are found to be related to the continuity and
differentiability of the density and compliance variation over the unit cell. In general, the mixed quotient
converges faster than both the displacement Rayleigh and the stress Rayleigh quotients, however, there exist
special cases where either the displacement Rayleigh or the stress Rayleigh quotient shows the exact same
convergence as the mixed-method. We show that all methods converge faster for smoother material property
variations, but when density variation is rough, the difference between the mixed quotient and stress Rayleigh
quotient is higher and similarly, when compliance variation is rough, the difference between the mixed quotient
and displacement Rayleigh quotient is higher. Since eigenvalue problems such as those considered in this paper
tend to be highly computationally intensive, it is expected that these results will lead to fast and efficient
algorithms in the areas of phononics and photonics.
In this paper we present a method to design composites which are acoustically impedance matched with a homogeneous
medium at a desired frequency. We use dynamic homogenization of layered elastic composites to calculate their
effective acoustic impedance. It is shown that the microstructure of a layered composite can be designed so that its
acoustic impedance matches the impedance of the homogeneous medium at the desired frequency. As a result, the
reflection at the interface of such a composite with the homogeneous medium is minimized. Transfer matrix calculation
and finite element modeling of wave propagation through a layered periodic composite sandwiched between two
homogenous media are done. It is observed that at the design frequency where the composite has matched impedance
with homogenous media the reflection at the interfaces is almost zero.
In this paper we show that the bandstructure of a periodic elastic composite, in addition to being dependent upon the
micro-constituents and their microarchitecture, may also be controlled by changing the temperature. The essential idea
is to fabricate a periodic composite with constituent materials which have temperature dependent elastic properties. As
temperature is changed, such a composite is expected to exhibit a bandstructure which changes with the temperature
dependent properties of its micro-constituents. For our purpose, we use polyurea and steel to make a 1-D periodic
composite. Ultrasonic measurements are done on the sample from 0.5 kHz to 1.5 MHz under changing temperature
and the change in the second passband is studied. It is observed that the change in the bandstructure is significant when
the temperature is changed from -50°C to 50°C. Experimental results are compared with the theoretical calculations
and it is shown that good agreement exists for the observed bandstructure.
Central to the idea of metamaterials is the concept of dynamic homogenization which seeks to define frequency
dependent effective properties for Bloch wave propagation. While the theory of static effective property calculations
goes back about 60 years, progress in the actual calculation of dynamic effective properties for periodic
composites has been made only very recently. Here we discuss the explicit form of the effective dynamic constitutive
equations. We elaborate upon the existence and emergence of coupling in the dynamic constitutive relation
and further symmetries of the effective tensors.
A method for homogenization of an elastic composite with periodic microstructure is presented, focusing on the
Floquet-type elastic waves. The resulting homogenized frequency-dependent elasticity and mass-density then
automatically satisfy the overall conservation laws and by necessity produce the exact dispersion relations. The
method is used to calculate the dynamic effective parameters for a layered composite by using the exact solution.
This paper presents numerical results on the dynamic behavior of continuously welded rails (CWR) subjected to a static
axial stress. The results quantify the sensitivity of guided waves to stress variations and could be potentially used to
estimate the stress level in CWR or alternatively the rail Neutral Temperature (stress free rail temperature). This work
represents the initial concept phase of a research and development study funded by the Federal Railroad Administration. The ultimate objective of this study is to develop and test a prototype system that uses non-contact dynamic sensing to measure in-situ rail stress in motion, to determine rail Neutral Temperatures (NT) and the related Incipient Buckling Risks in CWR.
This article theoretically studies the symmetry characteristics of Rayleigh-Lamb guided waves in nonlinear,
isotropic plates. It has been known that the nonlinearity driven double harmonic in Lamb waves does not
support antisymmetric motion. However the proof of this has not been obvious. Moreover, little is known
on nonlinearity driven Lamb harmonics higher than double. These gaps were here studied by the method of
perturbation coupled with wavemode orthogonality and forced response. This reduced the nonlinear problem
to a forced linear problem which was subsequently investigated to formulate an energy level constraint as the
defining factor for the absence of antisymmetry at any order of higher harmonic. This constraint was then
used to explain the reason behind the absence of antisymmetric Lamb waves at the double harmonic. Further,
it was shown that antisymmetric motion is prohibited at all the higher-order even harmonics, whereas all the
higher order odd harmonics allow both symmetric and antisymmetric motions. Finally, experimental results
corroborating theoretical conclusions are presented.
Many bridges, including 90% of the California inventory, are post-tensioned box-girders concrete structures.
Prestressing tendons are the main load-carrying components of these and other post-tensioned structures. Despite their
criticality, much research is needed to develop and deploy techniques able to provide real-time information on the level
of prestress in order to detect dangerous stress losses. In collaboration with Caltrans, UCSD is investigating the
combination of ultrasonic guided waves and embedded sensors to provide both prestress level monitoring and defect
detection capabilities in concrete-embedded PS tendons.
This paper presents a technique based on nonlinear ultrasonic guided waves in the 100 kHz - 2 MHz range for
monitoring prestress levels in 7-wire PS tendons. The technique relies on the fact that an axial stress on the tendon
generates a proportional radial stress between adjacent wires (interwire stress). In turn, the interwire stress modulates
nonlinear effects in ultrasonic wave propagation through both the presence of finite strains and the interwire contact. The
nonlinear ultrasonic behavior of the tendon under changing levels of prestress is monitored by tracking higher-order
harmonics at (nω) arising under a fundamental guided-wave excitation at (ω). Experimental results will be presented to
identify (a) ranges of fundamental excitations at (ω) producing maximum nonlinear response, and (b) optimum lay-out of
the transmitting and the receiving transducers within the test tendons. Compared to alternative methods based on linear
ultrasonic features, the proposed nonlinear ultrasonic technique appears more sensitive to prestress levels and more
robust against changing excitation power at the transmitting transducer or changing transducer/tendon bond conditions.
It has recently been demonstrated theoretically and experimentally that Green's functions (impulse
responses) can be estimated from coherent processing of random vibrations using only passive
sensors studies in various applications (ultrasonics, acoustic, seismic...). This article investigates the
passive-only estimation of coherent guided waves waves (DC-500 kHz) in an aluminum plate of
thickness comparable to aircraft fuselage and wing panels. Furthermore these passively
reconstructed waveforms can also be used for damage detection in the plate similarly to
conventional active testing. Based on this study, passive structural health monitoring techniques for
aircraft panels can be developed using random vibrations.
Ultrasonic guided wave testing necessitates of quantitative, rather than qualitative, information on flaw size, shape
and position. This quantitative diagnosis ability can be used to provide meaningful data to a prognosis algorithm for
remaining life prediction, or simply to generate data sets for a statistical defect classification algorithm. Quantitative
diagnostics needs models able to represent the interaction of guided waves with various defect scenarios. One such
model is the Global-Local (GL) method, which uses a full finite element discretization of the region around a flaw to
properly represent wave diffraction, and a suitable set of wave functions to simulate regions away from the flaw.
Displacement and stress continuity conditions are imposed at the boundary between the global and the local regions.
In this paper the GL method is expanded to take advantage of the Semi-Analytical Finite Element (SAFE) method in
the global portion of the waveguide. The SAFE method is efficient because it only requires the discretization of the
cross-section of the waveguide to obtain the wave dispersion solutions and it can handle complex structures such as
multilayered sandwich panels. The GL method is applied to predicting quantitatively the interaction of guided waves
with defects in aluminum and composites structural components.