Fast Global Optimization of Difficult Lennard-Jones Clusters

The minimization of the potential energy function of Lennard-Jones atomic clusters has attracted much theoretical as well as computational research in recent years. One reason for this is the practical importance of discovering low energy configurations of clusters of atoms, in view of applications and extensions to molecular conformation research; another reason of the success of Lennard Jones minimization in the global optimization literature is the fact that this is an extremely easy-to-state problem, yet it poses enormous difficulties for any unbiased global optimization algorithm.In this paper we propose a computational strategy which allowed us to rediscover most putative global optima known in the literature for clusters of up to 80 atoms and for other larger clusters, including the most difficult cluster conformations. The main feature of the proposed approach is the definition of a special purpose local optimization procedure aimed at enlarging the region of attraction of the best atomic configurations. This effect is attained by performing first an optimization of a modified potential function and using the resulting local optimum as a starting point for local optimization of the Lennard Jones potential.Extensive numerical experimentation is presented and discussed, from which it can be immediately inferred that the approach presented in this paper is extremely efficient when applied to the most challenging cluster conformations. Some attempts have also been carried out on larger clusters, which resulted in the discovery of the difficult optimum for the 102 atom cluster and for the very recently discovered new putative optimum for the 98 atom cluster.     

Developments in modelling of gas injection and slag foaming

Gas injection into metallurgical ladles has been an active area of CFD modelling for many years. Recent work with both Eulerian and Lagrangian frameworks is presented for bottom stirring in ladle and steelmaking electric furnace configurations. Comparison with water and liquid metal results shows that the Lagrangian models provide a better representation of the systems. Slag foaming is an important phenomenon in smelting–reduction processes and electric furnace steelmaking. The void fraction in the foam is generally greater than 0.9, a regime that has received considerably less attention than bottom stirring where the local void fraction is less than 0.1. Again, it was found, by comparison with experimental data, that Lagrangian models were generally preferable over Eulerian models.     

Self-consistent cluster theory of the electron spectra of alloys

A method of self-consistent cluster expansion for the mass operator of the Green's function with allowance for short-and longrange order in alloys is proposed. A self-consistent system of equations for the coherent potential and the mass operator is obtained in the approximation in which the contributions of processes involving electron scattering by clusters of three or more atoms are ignored. The contributions of the scattering processes to the Green's function of the alloy decrease with increasing number of particles in a cluster and can be estimated by means of a certain small parameter. Analytic and numerical investigations of the energy dependence of the density of single-electron states of a binary alloy are made for different values of the parameter of the short-range order.Institute of Metal Physics, Ukrainian Academy of Sciences; Kiev University. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 97, No. 2, pp. 304–319, November, 1993.     

Minimizing atomic configurations of short range pair potentials in two dimensions: crystallization in the Wulff shape

We investigate ground state configurations of atomic systems in two dimensions interacting via short range pair potentials. As the number of particles tends to infinity, we show that low-energy configurations converge to a macroscopic cluster of finite surface area and constant density, the latter being given by the density of atoms per unit volume in the triangular lattice. In the special case of the Heitmann–Radin sticky disc potential and exact ground states, we show that the macroscopic cluster has a (unique) Wulff shape. This is done by showing that the atomistic energy of crystalline configurations, after subtracting off a bulk part and re-scaling, Gamma-converges to a macroscopic anisotropic surface energy.     

Clustered equilibria of point vortices

Point vortex equilibria in which the vortices are arranged in clusters are examined. The vortex velocities in these configurations are all equal. Necessary conditions for their existence are established that relate the circulations within the clusters to the cluster radius. A method for generating these configurations by singular continuation is proved to be valid for the generic case. Finally, a partial analysis of exceptional cases is given and their connection to the existence of parametrized families of equilibria is described.     

Minimal interatomic distance in Morse clusters

In this paper we derive a lower bound, independent from the number of atoms N, for the minimal interatomic distances between atoms in a cluster whose total energy is modelled by means of the so called Morse potential. A similar result was previously proven for Lennard–Jones clusters but the proof can not be extended to Morse clusters. Besides the theoretical interest, the derivation of this lower bound is important for the definition of efficient procedures for the computation of the total energy of clusters with a large number of atoms.     

Modeling the Oscillations of the CO + O2 Reaction Rate in a Granular Catalyst Layer

A new point model is proposed for the oxidation reaction of CO on the surface of a Pd cluster. The model reflects the oxidation–reduction mechanism on the palladium surface and incorporates all the stages of the kinetic scheme from the previous three-component model. A new assumption based on a series of experimental facts claims that the adsorbed oxygen atoms may diffuse into deeper-lying layers of the crystal lattice and thus influence the processes in the adsorption layer due to the micro size of the palladium clusters. The new point system has been built into the distributed general model for a granular catalyst developed previously. As a result, for parameter values corresponding to experimental conditions we have managed for the first time to obtain a wide region of chaos and complex mixed modes, close to those observed in real experiments.     

Clusters of extremes: modeling and examples

We study clusters of threshold exceedances caused by dependence in time series. The clusters are defined as conglomerates containing consecutive threshold exceedances of the series separated by return intervals with consecutive non-exceedances. We derive asymptotic distributions of the cluster and inter-cluster sizes for processes with the extremal index equal to zero, the asymptotic expectation of the inter-cluster size and an exponential rate of convergence of the distribution tail of the return interval between clusters to the stable distribution tail. Distributions of the cluster and inter-cluster sizes of ARMAX, MM and AR(1) processes are obtained.     

Dynamics of a rate equation describing cluster-size evolution

In this paper, we show dynamics of Smoluchowski's rate equation which has been widely applied to studies of aggregation processes (i.e., the evolution of cluster-size distribution) in physics. We introduce dissociation in the rate equation while dissociation is neglected in previous works. We prove the positiveness of solutions of the equation, which is a basic guarantee for the effectiveness of the model since the possibility that some solution may be negative is excluded. For the case of cluster coalesce without dissociation, we show both the equilibrium uniqueness and the equilibrium stability under the condition that the monomer deposition stops. For the case that clusters evolve with dissociation and there is no monomer deposition, we show the equilibrium uniqueness and prove the equilibrium stability if the maximum cluster size is not larger than three while we show the equilibrium stability by numerical simulations if the maximum size is larger than three.     

Mathematical and physical modelling of steel flow and solidification in twin-roll/horizontal belt thin-strip casting machines

Near-net-shape casting technology is one of the most important research areas in the iron and steel industry today. Driving forces for the development of this technology include a reduction in the number of operations needed for conventionally produced strip. This is especially true of hot rolling operations. The consequent reduction in investment cost when considering new industrial facilities, makes near-net-shape casting operations extremely attractive from a commercial standpoint. Various processes for near-net-shape casting of steel are currently being developed around the world. Of these processes, twin-roll casting machines represent a major area of concentration. We believe that one of the main issues concerning the design of twin-roll casters is the metal delivery system and its effect on the homogeneity of solid shell formation, segregation and surface quality. In the present work, computational fluid dynamics has been used to study different metal delivery systems for twin-roll casting (TRC) and horizontal belt casting (HBC) operations. The METFLO code has been adapted to simulate three-dimensional turbulent fluid flows, heat transfer and solidification in these types of machines. The enthalpy–porosity technique was used to couple fluid flow and solidification phenomena. Two configurations for metal delivery system have been studied to date for TRC: one is a conventional tubular nozzle with horizontal outlets in the directions of the side dams; the other is a slot nozzle with a vertical inlet stream. These simulations have been applied to a pilot caster being studied in Canada, with a roll radius of 0.30 m, producing steel strips with thicknesses ranging from 4 to 7 mm, at relatively low roll speeds ranging between 4 and 12 m/min. Different positions and penetrations of the nozzles in the liquid pool have also been analysed. It has been shown that a tubular nozzle leads to the formation of a non-uniform solid shell along the roll width. In both configurations, a thicker solid shell is formed close to the roll edges, due to the presence of the side dams. In the case of HBC, computations have been made for an extended nozzle metal delivery system, and preliminary water modelling tests carried out to confirm the flow delivery concepts proposed. In addition, instantaneous heat flux measurements to simulated belt substrates have been performed for the horizontal casting of aluminum strip that show somewhat similar characteristics to those measured for steel in the pilot TRC, in terms of transient peaks and decays.     

Free-surface instabilities during electromagnetic shaping of liquid metals

Electromagnetic shaping of free surfaces of liquid metals is a well-known EPM technology used in a couple of metallurgic processes like cold crucible melting, semi-levitation, and electromagnetic slit sealing, among others. However, the stability of such free surfaces is the most important problem and stability control is crucial for success. Within this context we investigate experimentally the stability behavior of liquid metal free surfaces submitted to a high-frequency magnetic field. In this case, the induced Lorentz forces act as an electromagnetic pressure directly on the free surface of the liquid met al. We consider three experimental model configurations: (i) Sessile liquid metal drop (ii) liquid metal ring, and (iii) liquid metal disc. In each model experiment, upon increasing the feeding current beyond a certain threshold value, I_C, we observe that the initial surface contour becomes unstable resulting in (i) drop oscillations (ii) electromagnetic pinching and (iii) static disc deformations. In each configuration the threshold value depends in a similar manner on the frequency of the applied magnetic field. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Handling multicriteria preferences in cluster analysis

In the framework of multicriteria decision aid, a lot of interest has been devoted to sorting problems, in which the set of categories is pre-defined. Besides, preference oriented multicriteria clustering has received little attention. Usual geometric and related metrics are not well suited for this problem. Here, we propose a clustering method based on a valued indifference relation inspired by outranking methods. We suggest a method (based on comparing cluster centers and an average net flow score of clusters) to build a complete ranking of the set of clusters, that is, a way of defining a set of ordered categories for sorting purposes. The new approach performs very well in some examples.     

Distinguishing Different Types of Inhomogeneity in Neyman–Scott Point Processes

In this paper we introduce a graphical and formal approach to distinguishing different typed of inhomogeneity on Neyman–Scott point processes. The assumed types of inhomogeneity are (1) inhomogeneous cluster centers, (2) second order intensity reweighted stationarity, (3) location dependent scaling and a new type (4) growing clusters. The performance of the method is studied via a simulation study. This work has been motivated and illustrated by ecological studies of the spatial distribution of fish in an inland reservoir.     

一类爆炸凝结过程的数学模型

所建立的数学模型是由可数无穷多个彼此相互关联的非线性常微分方程所组成的自治系统,它刻划了在只有基本粒子与i-粒子(i≥1)进行碰撞反应的系统里,粒子增长过程中密度随时间的变化规律.本文研究了这一自治系统解的性质.     

Visualizing high density clusters in multidimensional data using optimized star coordinates

Multidimensional multivariate data have been studied in different areas for quite some time. Commonly, the analysis goal is not to look into individual records but to understand the distribution of the records at large and to find clusters of records that exhibit correlations between dimensions or variables. We propose a visualization method that operates on density rather than individual records. To not restrict our search for clusters, we compute density in the given multidimensional space. Clusters are formed by areas of high density. We present an approach that automatically computes a hierarchical tree of high density clusters. For visualization purposes, we propose a method to project the multidimensional clusters to a 2D or 3D layout. The projection method uses an optimized star coordinates layout. The optimization procedure minimizes the overlap of projected clusters and maximally maintains the cluster shapes, compactness, and distribution. The star coordinate visualization allows for an interactive analysis of the distribution of clusters and comprehension of the relations between clusters and the original dimensions. Clusters are being visualized using nested sequences of density level sets leading to a quantitative understanding of information content, patterns, and relationships.     

Efficient solutions for weight-balanced partitioning problems

We prove polynomial-time solvability of a large class of clustering problems where a weighted set of items has to be partitioned into clusters with respect to some balancing constraints. The data points are weighted with respect to different features and the clusters adhere to given lower and upper bounds on the total weight of their points with respect to each of these features. Further the weight-contribution of a vector to a cluster can depend on the cluster it is assigned to. Our interest in these types of clustering problems is motivated by an application in land consolidation where the ability to perform this kind of balancing is crucial.Our framework maximizes an objective function that is convex in the summed-up utility of the items in each cluster. Despite hardness of convex maximization and many related problems, for fixed dimension and number of clusters, we are able to show that our clustering model is solvable in time polynomial in the number of items if the weight-balancing restrictions are defined using vectors from a fixed, finite domain. We conclude our discussion with a new, efficient model and algorithm for land consolidation.     

The bi-objective fuzzy c-means cluster analysis for TSK fuzzy system identification

For conventional fuzzy clustering-based approaches to fuzzy system identification, a fuzzy function is used for cluster formation and another fuzzy function is used for cluster validation to determine the number and location of the clusters which define IF parts of the rule base. However, the different fuzzy functions used for cluster formation and validation may not indicate the same best number and location of the clusters. This potential disparity motivates us to propose a new fuzzy clustering-based approach to fuzzy system identification based on the bi-objective fuzzy c-means (BOFCM) cluster analysis. In this approach, we use the BOFCM function for both cluster formation and validation to simultaneously determine the number and location of the clusters which we hope can efficiently and effectively define IF parts of the rule base. The proposed approach is validated by applying it to the truck backer-upper problem with an obstacle in the center of the field.     

An overlapping cluster algorithm to provide non-exhaustive clustering

The partitioning clustering is a technique to classify n objects into k disjoint clusters, and has been developed for years and widely used in many applications. In this paper, a new overlapping cluster algorithm is defined. It differs from traditional clustering algorithms in three respects. First, the new clustering is overlapping, because clusters are allowed to overlap with one another. Second, the clustering is non-exhaustive, because an object is permitted to belong to no cluster. Third, the goals considered in this research are the maximization of the average number of objects contained in a cluster and the maximization of the distances among cluster centers, while the goals in previous research are the maximization of the similarities of objects in the same clusters and the minimization of the similarities of objects in different clusters. Furthermore, the new clustering is also different from the traditional fuzzy clustering, because the object–cluster relationship in the new clustering is represented by a crisp value rather than that represented by using a fuzzy membership degree. Accordingly, a new overlapping partitioning cluster (OPC) algorithm is proposed to provide overlapping and non-exhaustive clustering of objects. Finally, several simulation and real world data sets are used to evaluate the effectiveness and the efficiency of the OPC algorithm, and the outcomes indicate that the algorithm can generate satisfactory clustering results.     

Obviating some of the theoretical barriers of data envelopment analysis-discriminant analysis: an application in predicting cluster membership of customers

Data envelopment analysis-discriminant analysis (DEA-DA) has been used for predicting cluster membership of decision-making units (DMUs). One of the possible applications of DEA-DA is in the marketing research area. This paper uses cluster analysis to cluster customers into two clusters: Gold and Lead. Then, to predict cluster membership of new customers, DEA-DA is applied. In DEA-DA, an arbitrary parameter imposing a small gap between two clusters (η) is incorporated. It is shown that different η leads to different prediction accuracy levels since an unsuitable value for η leads to an incorrect classification of DMUs. We show that even the data set with no overlap between two clusters can be misclassified. This paper proposes a new DEA-DA model to tackle this issue. The aim of this paper is to illustrate some computational difficulties in previous DEA-DA approaches and then to propose a new DEA-DA model to overcome the difficulties. A case study demonstrates the efficacy of the proposed model.     

Three-dimensional manifestly Poincaré-invariant approach to the relativistic three-body problem

A three-dimensional manifestly Poincaré-invariant approach to the relativistic three-body problem is developed that satisfies the requirement of cluster separability and at the same time does not lead to so-called spurious states devoid of physical meaning. It is shown that these requirements make it possible to fix the form of the operators of the two-body interactions. The problem is solved with allowance for the dependence of the interaction operators on the spectral parameter. This dependence is a manifestation of the structure of the particles in the three-body system (i.e., it reflects the circumstance that the complete Hilbert space of state vectors of the system includes not only three-body configurations of the original particles) and leads to the appearance of certain factors in the cross sections of physical processes. Two alternative formulations of the method are investigated. In the first formulation, equations are written down for the amplitudes of transitions between free-particle states. In the second formulation, the states of interacting particles in the two-body scattering channels are used as complete orthogonal bases. Partial-wave expansions of the equations with respect to states with given total angular momentum of the system in the helicity basis are made.Institute of Nuclear Physics of the State University, Moscow. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 103, No. 2, pp. 200–232, May, 1995.