A priori error analysis of two force-based atomistic/continuum models of a periodic chain

The force-based quasicontinuum (QCF) approximation is a non-conservative atomistic/continuum hybrid model for the simulation of defects in crystals. We present an a priori error analysis of the QCF method, applied to a one-dimensional periodic chain, that is valid for an arbitrary interaction range, large deformations, and takes coarse-graining into account. Our main tool in this analysis is a new concept of atomistic stress. Moreover, we formulate a new atomistic/continuum coupling mechanism based on coupling stresses instead of forces and extend the a priori analysis to this new method. We show that the new method has several theoretical advantages over the original QCF method.     

Complete atomistic lattices are classification lattices

We prove that each complete atomistic lattice G is isomorphic to the lattice of classification systems of an appropriate complete atomistic lattice L. This implies an affirmative solution to a problem raised by S. Radeleczki in 2002.     

Some Structure Theorems for Atomistic Algebraic Lattices

We prove that any atomistic algebraic lattice is a direct product of subdirectly irreducible lattices iff its congruence lattice is an atomic Stone lattice. We define on the set A(L) of all atoms of an atomistic algebraic lattice L a relation R as follows: for a, b A(L), (a, b) R ? θ(0, a) ∧ θ(0, b) ≠ ?_{Con L} . We prove that Con L is a Stone lattice iff R is transitive and we give a characterization of Cen (L) using R. We also give a characterization of weakly modular atomistic algebraic lattices.     

Microstructure Evolution during Nanoindentation by the Quasicontinuum Method

A new energy-based quasicontinuum formulation is presented which is based on sampling the crystal energy at carefully-chosen lattice sites and which allows for efficiently bridging from the atomistic to the continuum length scale. The presented technique is applied to experiments of nanoindentation whose microstructure-induced size effects can now be studied with full atomistic detail at the micrometer scale without the necessity of phenomenological material models. (© 2013 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

A note on atomistic weak congruence lattices

It is proved that a codistributive element in an atomistic algebraic lattice has a complement, implying that kernels of the related homomorphisms coincide. Some applications to weak congruence lattices of algebras are presented. In particular, necessary and sufficient conditions under which the weak congruence lattice of an algebra is atomistic are given.     

A higher-order gradient theory for modeling of the vibration behavior of single-wall carbon nanocones

A higher-order gradient theory is used to investigate the free vibration characteristics of single-wall carbon nanocones (SWCNCs). This atomistic–continuum model simulates SWCNCs at the atomistic level and links the deformation of the crystal lattice structure to that of the continuum field. The dependence of vibration frequencies of SWCNCs on apex angles, heights and top radii, as well as constraints, is studied under a developed mesh-free computational framework based on moving Kriging interpolation. It is found that the proposed model gives a good prediction of the MD simulation and Timoshenko beam model. Several kinds of SWCNCs were investigated and the results reveal that the apex angle markedly affects the vibration frequency. It is observed that the fundamental frequency increases as the top radius increases, until it reaches a critical value. The critical top radii are largely dependent on the constraints at the ends of the SWCNCs. It is also observed that for SWCNCs with different apex angles, the same fundamental frequency is obtained by an appropriate combination of height and top radius. As the top radius continues to increase, the change of fundamental frequency becomes smaller and smaller.     

Congruence-preserving extensions of finite lattices to sectionally complemented lattices

In 1962, the authors proved that every finite distributive lattice can be represented as the congruence lattice of a finite sectionally complemented lattice. In 1992, M. Tischendorf verified that every finite lattice has a congruence-preserving extension to an atomistic lattice. In this paper, we bring these two results together. We prove that every finite lattice has a congruence-preserving extension to a finite sectionally complemented lattice.

     

Solitary Waves and Their Linear Stability in Nonlinear Lattices

Solitary waves in a general nonlinear lattice are discussed, employing as a model the nonlinear Schrödinger equation with a spatially periodic nonlinear coefficient. An asymptotic theory is developed for long solitary waves, which span a large number of lattice periods. In this limit, the allowed positions of solitary waves relative to the lattice, as well as their linear stability properties, hinge upon a certain recurrence relation which contains information beyond all orders of the usual two‐scale perturbation expansion. It follows that only two such positions are permissible, and of those two solitary waves, one is linearly stable and the other unstable. For a cosine lattice, in particular, the two possible solitary waves are centered at a maximum or minimum of the lattice, with the former being stable, and the analytical predictions for the associated linear stability eigenvalues are in excellent agreement with numerical results. Furthermore, a countable set of multi‐solitary‐wave bound states are constructed analytically. In spite of rather different physical settings, the exponential asymptotics approach followed here is strikingly similar to that taken in earlier studies of solitary wavepackets involving a periodic carrier and a slowly varying envelope, which underscores the general value of this procedure for treating multiscale solitary‐wave problems.     

Aggregation on a Finite Lattice

In this paper a result of B. Leclerc and B. Monjardet concerning meet-projections in finite congruence-simple atomistic lattices is generalized. We prove that the result remains valid for any finite tolerance-simple lattice; moreover, we extend it to a type of subdirect product of such lattices, introducing the notion of a generalized oligarchy.     

Memory and a priori best strategy in complex adaptive systems

We employ an agent‐based model to show that memory and the absence of an a priori best strategy are sufficient for self‐segregation and clustering to emerge in a complex adaptive system with discrete agents that do not compete over a limited resource nor contend in a winner‐take‐all scenario. An agent starts from a corner of a two‐dimensional lattice and aims to reach a randomly selected site in the opposite side within the shortest possible time. The agent is isolated during the course of its journey and does not interact with other agents. Time‐bound obstacles appear at random lattice locations and the agent must decide whether to challenge or evade any obstacle blocking its path. The agent is capable of adapting a strategy in dealing with an obstacle. We analyze the dependence of strategy‐retention time with strategy for both memory‐based and memory‐less agents. We derive the equality spectrum to establish the environmental conditions that favor the existence of an a priori best strategy. We found that memory‐less agents do not polarize into two opposite strategy‐retention time distributions nor cluster toward a center distribution. © 2004 Wiley Periodicals, Inc. Complexity 9: 41–46, 2004     

Complements in lattices of varieties and equational theories

We investigate various weak conditions ensuring that a lattice be complemented. Using these general results in connection with a famous result due to Lampe, we show that the lattice of all equational theories containing a fixed theory must be complemented if it is lower semicomplemented, thereby answering in the affirmative a question raised by Volkov and Vernikov. Moreover, such a lattice must be a finite Boolean algebra if it has one of the following properties: upper or lower sectionally complemented; incomparably complemented; lower semicomplemented and lower semimodular; or atomistic and upper semimodular.Presented by R. Freese.     

Finite Intervals in the Lattice of Topologies

We discuss the question whether every finite interval in the lattice of all topologies on some set is isomorphic to an interval in the lattice of all topologies on a finite set – or, equivalently, whether the finite intervals in lattices of topologies are, up to isomorphism, exactly the duals of finite intervals in lattices of quasiorders. The answer to this question is in the affirmative at least for finite atomistic lattices. Applying recent results about intervals in lattices of quasiorders, we see that, for example, the five-element modular but non-distributive lattice cannot be an interval in the lattice of topologies. We show that a finite lattice whose greatest element is the join of two atoms is an interval of T ₀-topologies iff it is the four-element Boolean lattice or the five-element non-modular lattice. But only the first of these two selfdual lattices is an interval of orders because order intervals are known to be dually locally distributive.     

Analysis of a modified two-lane lattice model by considering the density difference effect

In this paper, a modified lattice hydrodynamic model of traffic flow is proposed by considering the density difference between leading and following lattice for two-lane system. The effect of density difference on the stability of traffic flow is examined through linear stability analysis and shown that the density difference term can significantly enlarge the stability region on the phase diagram. To describe the phase transition of traffic flow, the Burgers equation and mKdV equation near the critical point are derived through nonlinear analysis. To verify the theoretical findings, numerical simulation is conducted which confirms that traffic jam can be suppressed efficiently by considering the density difference effect in the modified lattice model for two-lane traffic.     

Data-driven estimation of atomistic support for continuum stress using the Gaussian mixture model

The notion of stress being an inherent continuum concept has been a matter of discussion at the atomistic level. The atomistic stress measure at a given spatial position contains a space averaging volume over nearby atoms to provide an averaged macroscopic stress measure. Previous work on atomistic stress measures introduce the characteristic length as an a priori given parameter. In this contribution we learn the characteristic length directly from the atomistic data itself. Central to our proposed approach is the grouping of atoms with highly similar values of position and stress into the same atomistic sub-population. We hypothesise that atoms with similar values for position and stress are those atoms which harbour the greatest influence over each other and therefore should be contained within the same space averaging volume. Consequently the characteristic length can be computed directly from the discovered sub-populations by averaging over the maximum extent of each sub-population. We motivate the Gaussian mixture model (GMM) as a principled probabilistic method of estimating the similarity between atoms within position-stress space. The GMM parameters are learnt from the atomistic data using the Expectation Maximization (EM) algorithm. To form a parsimonious representation of the dataset we regularise our model using the Bayesian Information Criterion (BIC) which maintains a balance between too few and too many atomistic sub-populations. We use the GMM to segment the atoms into homogeneous sub-populations based on the probability of each atom belonging to a particular sub-population. Thorough evaluation is conducted on a numerical example of an edge dislocation in a single crystal. We derive estimates of the space averaging volume which are in very close agreement to the corresponding analytical solution. (© 2012 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

A new lattice model of two-lane traffic flow with the consideration of optimal current difference

In this paper, a new lattice model of traffic flow is proposed with the consideration of the optimal current difference for two-lane system. The linear stability condition is derived through linear stability analysis, which shows that the optimal current difference term can improve the stability of traffic flow. The mKdV equation is obtained through nonlinear analysis. Thus the space of traffic flow is divided into three regions: the stable region, the metastable region and the unstable region respectively. Moreover, numerical simulation confirms that the traffic jam can be suppressed efficiently by considering the optimal current difference effect in extended lattice model of two-lane traffic flow.     

Largest extension of a finite convex geometry

We present a new embedding of a finite join-semidistributive lattice into a finite atomistic join-semidistributive lattice. This embedding turns out to be the largest extension, when applied to a finite convex geometry.In Celebration of the Sixtieth Birthday of Ralph N. McKenzieReceived September 18, 2002; accepted in final form September 29, 2003.     

A representation theorem for CAC lattices

Each CAC (complete atomistic coatomistic) lattice is isomorphic to the lattice of left annihilators of a O-simple Rees matrix semigroup.     

Sampling colourings of the triangular lattice

We show that the Glauber dynamics on proper 9‐colourings of the triangular lattice is rapidly mixing, which allows for efficient sampling. Consequently, there is a fully polynomial randomised approximation scheme (FPRAS) for counting proper 9‐colourings of the triangular lattice. Proper colourings correspond to configurations in the zero‐temperature anti‐ferromagnetic Potts model. We show that the spin system consisting of proper 9‐colourings of the triangular lattice has strong spatial mixing. This implies that there is a unique infinite‐volume Gibbs distribution, which is an important property studied in statistical physics. Our results build on previous work by Goldberg, Martin and Paterson, who showed similar results for 10 colours on the triangular lattice. Their work was preceded by Salas and Sokal's 11‐colour result. Both proofs rely on computational assistance, and so does our 9‐colour proof. We have used a randomised heuristic to guide us towards rigourous results. © 2011 Wiley Periodicals, Inc. Random Struct. Alg., 40, 501–533, 2012     

Tunneling of the hard‐core model on finite triangular lattices

We consider the hard‐core model on finite triangular lattices with Metropolis dynamics. Under suitable conditions on the triangular lattice sizes, this interacting particle system has 3 maximum‐occupancy configurations and we investigate its high‐fugacity behavior by studying tunneling times, that is, the first hitting times between these maximum‐occupancy configurations, and the mixing time. The proof method relies on the analysis of the corresponding state space using geometrical and combinatorial properties of the hard‐core configurations on finite triangular lattices, in combination with known results for first hitting times of Metropolis Markov chains in the equivalent zero‐temperature limit. In particular, we show how the order of magnitude of the expected tunneling times depends on the triangular lattice sizes in the low‐temperature regime and prove the asymptotic exponentiality of the rescaled tunneling time leveraging the intrinsic symmetry of the state space.     

Embedding Finite Lattices into Finite Biatomic Lattices

For a class C of finite lattices, the question arises whether any lattice in C can be embedded into some atomistic, biatomic lattice in C. We provide answers to the question above for C being, respectively,– the class of all finite lattices;– the class of all finite lower bounded lattices (solved by the first author's earlier work);– the class of all finite join-semidistributive lattices (this problem was, until now, open).We solve the latter problem by finding a quasi-identity valid in all finite, atomistic, biatomic, join-semidistributive lattices but not in all finite join-semidistributive lattices.