Lattice Size Effects Research Articles

Using binary-stiffness beams within mechanical neural-network metamaterials to learn

This work introduces the concept of applying binary-stiffness beams within a lattice to achieve a mechanical neural-network (MNN) metamaterial that learns its behaviors and properties with prolonged exposure to unanticipated ambient loading scenarios. Applying such beams to MNN metamaterials greatly increases their learning speed and simplifies the actuation demands, control circuitry, and optimization algorithms required by previously proposed concepts. A binary-stiffness beam design is proposed that uses principles of constraint manipulation and stiffness cancelation to achieve two switchable and discrete states of stiffness (i.e. binary stiffness) along its axis. The beam achieves a near-zero low-stiffness state and a large difference in stiffness between its high and low-stiffness states, which are both shown to be desirable attributes for learning mechanical behaviors. Simulations are conducted to characterize the effect of lattice size, the difference in stiffness between the constituent beam’s high and low-stiffness states, the magnitude of its low-stiffness state, and the number of simultaneously learned behaviors on MNN learning using binary-stiffness beams. Thus, this work provides a necessary step toward enabling practical artificial intelligent metamaterials.

Mechanical neural networks: Architected materials that learn behaviors.

Aside from some living tissues, few materials can autonomously learn to exhibit desired behaviors as a consequence of prolonged exposure to unanticipated ambient loading scenarios. Still fewer materials can continue to exhibit previously learned behaviors in the midst of changing conditions (e.g., rising levels of internal damage, varying fixturing scenarios, and fluctuating external loads) while also acquiring new behaviors best suited for the situation at hand. Here, we describe a class of architected materials, called mechanical neural networks (MNNs), that achieve such learning capabilities by tuning the stiffness of their constituent beams similar to how artificial neural networks (ANNs) tune their weights. An example lattice was fabricated to demonstrate its ability to learn multiple mechanical behaviors simultaneously, and a study was conducted to determine the effect of lattice size, packing configuration, algorithm type, behavior number, and linear-versus-nonlinear stiffness tunability on MNN learning as proposed. Thus, this work lays the foundation for artificial-intelligent (AI) materials that can learn behaviors and properties.

Phase-field simulations on the frequency-dependent evolution of nano-magnetic domains and hysteresis loops of ferromagnetic Terfenol-D

Ferromagnetic materials are intelligent materials that exhibit large magnetic-field-induced strains and stress-induced magnetization upon application of magnetic fields and/or mechanical stress. Their hysteresis behavior also exhibits frequency dependence which is closely related to the evolution of nano magnetic domains. In this paper, a phase-field model of nano-domain evolution in two-dimensional ferromagnetic nanostructures is established based on the time-dependent Ginzburg-Landau kinetic theory. The model is applied to study the domain evolution and hysteresis loops of Terfenol-D at different frequencies. The obtained results are consistent with the trend of existing experiments. In addition, the effect of lattice size of the nano-structure on the hysteresis loops of Terfenol-D is also demonstrated. It shows that the larger the lattice area, the more there are nano vortices and magnetic energy loss. The effect of temperature on the spontaneous magnetization is also investigated. It is found that, as temperature increases, the stability of nano-domain walls weakens and the degree of vortices decreases. This phase-field simulation establishes the direct connection between the nano-scale domain phenomena and the macroscopic hysteresis loops. It clearly demonstrates that the nano-vortex structure is the main cause of energy loss, and that this phase-field method is an effective approach to uncover the underlying mechanisms of hysteresis behavior in ferromagnetic Terfenol-D.

Consequences of adsorbate-adsorbate interactions for apparent kinetics of surface catalytic reactions

Lateral adsorbate interactions at catalyst surfaces are known to influence adsorption energies and reaction rates. Lattice-based kinetic Monte Carlo (kMC) simulations are able to capture these influences, but such models are typically parameterized for a specific reaction network and catalyst surface. Here we report kMC simulations to probe the influence of lateral interactions on simulated rates, rate orders, apparent activation energies, and Sabatier plots. We construct a simple, two-step reaction network involving a single adsorbate and rate-limiting diatomic dissociation, employ a generic repulsive lateral interaction model consistent with known adsorbate–adsorbate interactions on metal surfaces, and a rate model consistent with known Brønsted-Evans-Polyani relationships for diatomic dissocations. We juxtapose reaction kinetics over a wide range of reaction conditions and catalyst binding energies, as a function of interaction strength. We find that at a given zero-coverage binding energy and external conditions, adsorbate coverage decreases monotonically with increasing interaction strength, but absolute rates can vary linearly or nonlinearly. Interactions flatten the Sabatier volcano and shift the maximum towards stronger binding. Influences on apparent rate orders and activation energies are modest and are sensitive to interaction-induced adsorbate ordering. Model predictions are sensitive to lattice size effects at high coverages. The results, modelled for a simple reaction system, highlight the generic consequences of lateral interactions and guidance for identifying their signatures in observed kinetics.

Exact Solution of Non-Hermitian Systems with Generalized Boundary Conditions: Size-Dependent Boundary Effect and Fragility of the Skin Effect.

Systems with non-Hermitian skin effects are very sensitive to the imposed boundary conditions and lattice size, and thus an important question is whether non-Hermitian skin effects can survive when deviating from the open boundary condition. To unveil the origin of boundary sensitivity, we present exact solutions for one-dimensional non-Hermitian models with generalized boundary conditions and study rigorously the interplay effect of lattice size and boundary terms. Besides the open boundary condition, we identify the existence of non-Hermitian skin effect when one of the boundary hopping terms vanishes. Apart from this critical line on the boundary parameter space, we find that the skin effect is fragile under any tiny boundary perturbation in the thermodynamic limit, although it can survive in a finite size system. Moreover, we demonstrate that the non-Hermitian Su-Schreieffer-Heeger model exhibits a new phase diagram in the boundary critical line, which is different from either open or periodical boundary case.

Effect of Lattice Size on the Critical Temperature of Ising Nanofilms and Nanotubes on the Square Lattice: Cellular Automata Simulation

Lattice size dependency of critical temperature of Ising nanofilms and nanotubes on the square lattice have been investigated Using Cellular Automata simulation approach. The obtained values of critical temperature for single-layer nanofilm on the infinite size of lattice is equal to 2.36, the corresponding value for single-layer nanofilm on the honeycomb lattice was 1.45. The effects of length and diameter of nanotube on the critical temperature were also studied.

Adlayer structure and lattice size effects on catalytic rates predicted from KMC simulations: NO oxidation on Pt(111).

Repulsive and/or attractive interactions between surface adsorbates have an important effect on the structure of the adsorbate layer and consequently on the rate of heterogeneous catalytic reactions. Thus, developing reaction models that take into account adsorbate-adsorbate interactions is crucial for making accurate predictions of the catalytic rate and surface coverage during reaction. In the present work, we employ kinetic Monte Carlo simulation to model the catalytic NO oxidation on Pt (111), adopting a cluster expansion (CE) Hamiltonian approach for treating the aforementioned interactions. We investigate CEs of increasing complexity, ranging from pairwise 1st nearest neighbor to long-range and many-body terms. We show that energetic models incorporating solely short-range interactions result in ordered adlayer structures, which are disrupted by anti-phase boundaries and defective regions when the size of the periodic lattice is non-commensurate to the structure of the stable adlayer. We find that O2 dissociates on sites located in these defective regions, which are predominantly responsible for the activity, and the predicted catalytic rate is strongly depended on the lattice size. Such effects are absent when employing non-periodic lattices, whereon the catalytic activity appears more intense on edges/corner sites. Finally, inclusion of long-range interactions in the model Hamiltonian induces relative disorder in the adsorbate layer, which is ascribed to the "softening" of the repulsive interactions between adspecies. Under these circumstances, the distribution of activation energies for O2 dissociation is broader as compared to short-range interaction models and on this basis we explain the disparate catalytic rate predictions when using different CEs.

Size effects in lattice structures and a comparison to micropolar elasticity

This work models periodic thin walled cellular materials using either a discrete beam element lattice model or nominally equivalent micropolar elastic model. When the material properties of cellular materials are tested, small samples may have different apparent material properties than large samples. This difference is known in the literature as a size effect. The predictions of size effects from discrete and micropolar simulations are examined. Although the literature explains size effects arising from the effects simulated in the micropolar model, the micropolar model typically under predicts the stiffness due to size effects seen in the lattice model by an order of magnitude. The lattice size effects are examined for patterns and the size effect patterns found can be explained by the shape of the free edges, and by the specifics of how material is distributed within the material domain. These are causes for size effects that are not captured in the micropolar model.

Large scale discriminative training of hidden Markov models for speech recognition

This paper describes, and evaluates on a large scale, the lattice based framework for discriminative training of large vocabulary speech recognition systems based on Gaussian mixture hidden Markov models (HMMs). This paper concentrates on the maximum mutual information estimation (MMIE) criterion which has been used to train HMM systems for conversational telephone speech transcription using up to 265 hours of training data. These experiments represent the largest-scale application of discriminative training techniques for speech recognition of which the authors are aware. Details are given of the MMIE lattice-based implementation used with the extended Baum-Welch algorithm, which makes training of such large systems computationally feasible. Techniques for improving generalization using acoustic scaling and weakened language models are discussed. The overall technique has allowed the estimation of triphone and quinphone HMM parameters which has led to significant reductions in word error rate for the transcription of conversational telephone speech relative to our best systems trained using maximum likelihood estimation (MLE). This is in contrast to some previous studies, which have concluded that there is little benefit in using discriminative training for the most difficult large vocabulary speech recognition tasks. The lattice MMIE-based discriminative training scheme is also shown to out-perform the frame discrimination technique. Various properties of the lattice-based MMIE training scheme are investigated including comparisons of different lattice processing strategies (full search and exact-match) and the effect of lattice size on performance. Furthermore a scheme based on the linear interpolation of the MMIE and MLE objective functions is shown to reduce the danger of over-training. It is shown that HMMs trained with MMIE benefit as much as MLE-trained HMMs from applying model adaptation using maximum likelihood linear regression (MLLR). This has allowed the straightforward integration of MMIE-trained HMMs into complex multi-pass systems for transcription of conversational telephone speech and has contributed to our MMIE-trained systems giving the lowest word error rates in both the 2000 and 2001 NIST Hub5 evaluations.

Size distribution of percolating clusters on cubic lattices

We present Monte Carlo simulations of site percolation on cubic lattices with size L. The cut-off function of the size distribution at the characteristic size s* can be described by a Gaussian for site occupations (p) lower than the critical value for percolation (pc) while for p>pc it is better described by a stretched exponential. Finite lattice size effects depend on both e = (pc-p)/pc and L and cannot be eliminated by normalization with the distribution at e = 0. We give a general expression of the cut-off function valid for any e sufficiently small and L sufficiently large. Using this general expression, we calculate the finite lattice size effects on the weight average (sw) and z-average (sz) sizes which are in good agreement with values directly obtained from the simulations.

Cascade collision of Fe-atom caused by low energy He-incidence and effect of temperature to the type of defect

A new phenomenon, the focusing cascade, was found in the molecular dynamical model, when a low energy He-atom was injected into bcc-iron crystal. In this paper, we present the details of the cascade and discuss the type of defects which are related significantly to the lattice temperature. We also discuss the character of the lattice, i.e. the effect of lattice size, that of temperature, lattice vibration, sputtering, etc.

Gauge boson masses in the 3D, SU(2) gauge-Higgs model

We study gauge boson propagators in the symmetric and symmetry broken phases of the 3D, SU(2) gauge-Higgs model. Correlation functions for the gauge fields are calculated in Landau gauge. They are found to decay exponentially at large distances leading to a non-vanishing mass for the gauge bosons. We find that the W-boson screening mass drops in the symmetry broken phase when approaching the critical temperature. In the symmetric phase the screening mass stays small and is independent of the scalar-gauge coupling (the hopping parameter). Numerical results coincide with corresponding calculations performed for the pure gauge theory. We find m w = 0.35(1) g 2 T in this phase which is consistent with analytic calculations based on gap equations. This is, however, significantly smaller than masses extracted from gauge-invariant vector boson correlation functions. As internal consistency check we also have calculated correlation functions for gauge-invariant operators leading to scalar and vector boson masses. Finite lattice size effects have been systematically analyzed on lattices of size L 2 × L z with L = 4 – 24 and L z = 16 – 128.

Oscillatory patterns in an autocatalytic reaction ring network simulated by a probabilistic cellular automaton

A two dimensional probabilistic cellular automaton is presented. It simulates a reaction-diffusion process occurring in biomembranes during active transport described by an autocatalytic ring network. The automaton suffers a transition from a completely disordered state to a state with well defined oscillations and spatial inhomogeneous patterns by changing the parameters. The different dynamic states have been characterized by means of Fourier transforms of concentration time evolution and through the definition of an agglomeration parameter. Vacancy concentration and lattice size effects are analyzed.

Thermodynamic properties of a two-dimensional nearest-neighbor interacting lattice gas

Utilizing transfer matrices obtained from the shift operator matrix (SOM) method, we determine the coverage, entropy and heat capacity for a system of simple, indistinguishable. nearest-neighbor interacting particles distributed on an M × ∞ ( M = 2,3,4,...) cylindrical square-cell lattice in equilibrium with a gaseous environment. The lattice coverage and entropy show clearly the nature and position of a critical point as well as the presence of a coexistence line. The heat capacity calculations demonstrate the effect of lattice size (i.e. M) on the position and evolution of the phase transition at a coverage of one half.

Self-assembly of surfactant liquid crystalline phases by Monte Carlo simulation

Three-dimensional microstructures of surfactant–water–oil systems self-assemble in a Monte Carlo lattice model, as shown here. The microstructures that form depend on the volume ratios of oil, water, and surfactant, and on the length of the surfactant, and on the ratio R of the length of the oil-loving to the water-loving portion. For R=1 we find lamellar phases when the surfactant is mixed with equal amounts of oil and water. The lamellar spacing increases as the surfactant concentration is lowered. In the presence of water only, as the surfactant concentration is lowered the microstructure evolves from lamellar to broken lamellar to cylinders to spheroids. This progression is found to be independent of lattice size for lattices as large as 40×40×40. For R=3, the progression seems to be replaced by a progression from lamellae to regular bicontinuous structures to cylinders, although we are not yet confident that this latter progression is independent of lattice size effects. The Monte Carlo technique can be used to study a wide variety of interesting phenomena, from micelle size and shape transitions to packing transitions and phase behavior to interfacial properties in the presence of surfactant.

A phenomenology of QCD on a small lattice

Weak-coupling methods, appropriate to a small lattice, reproduce many features of the QCD spectrum derived on a 6 3 × 10 lattice using Monte Carlo techniques and offer some insight into finite lattice size effects.

Finite size effects for the non-interacting Fermi system at finite temperature and finite density

From the euclidean form of the partition function for a non-interaction dense Fermi system placed on a finite lattice we evaluate its energy density and particle density, using different forms of lattice action. Comparing them with the corresponding well-known results in the continuum, we obtain a measure of finite lattice size effects.

Finite size effects in euclidean lattice thermodynamics for non-interacting bose and fermi systems

In the Monte Carlo simulation of QCD, the euclidean form of the partition function is evaluated on a finite lattice. We use this method to calculate the partition function for non-interacting Bose and Fermi fields. Here the expressions on the lattice can be evaluated in closed form and the continuum limit is well-known; this provides us with a measure for finite lattice size effects in such approaches.

Analysis of the Cubic Crystal-Field Parameters for Gd3+ in the Alkaline Earth Fluorides

The optical spectra of the alkaline-earth fluorides doped with Ge3+ (MeF2:Gd3+, where Me represents Ca, Sr, or Ba) at octahedral sites are analyzed to determine the effect of lattice size on the cubic crystal-field parameters. The point-charge model predicts a linear dependence of the first-order crystal-field splitting on B04 which in turn varies inversely with the Gd3+–F− separation to the fifth power (R−5). The slopes of plots of log b40 vs logR yield values of − 2.1, − 2.7, and 0 for the S7 / 28, P7 / 26, and P5 / 26 multiplets, respectively. This apparent discrepancy is resolved by the inclusion of crystal-field interactions in all orders between the S8 and 6(P, I, D) multiplets. The log(A40〈r4〉) vs logR plot is found to be − 1.8 which agrees with the EPR result to within experimental error. The disagreement with the point-charge model is examined.