Showing papers in "computational methods in science and technology in 2004"

Tetratic phase in the planar hard square system

Abstract: System of hard squares in two dimensions (2D) has been studied by Monte Carlo simulations. The simulations indicate that the isotropic fluid phase in this system does not freeze into a 2D 'crystalline' phase (of square lattice and quasi-long-range translational order) but transforms into an intermediate phase with the quadratic quasi-long-range orientational order (of coupled molecular axes and intermolecular bonds) and the translational order decaying faster than algebraically. The equation of state and the specific heat of the system are surprisingly well reproduced by smoothed version of the free volume theory in the whole density range. K e y w o r d s : melting in two dimensions, liquid crystals, hard convex body, equation of state, quasi-long range order 1. I N T R O D U C T I O N Studies of simple, well defined models are the key to understand complex behaviours observed in real systems. A well known example of this approach is the hard sphere system which has been used for modelling liquids and their freezing [1]. Existing in nature liquid crystalline and plastic crystalline states, exhibiting intermediate order between crystals and liquids stimulated development of models with anisotropic hard molecules. It has been shown by computer simulations that purely geometrical interactions can lead to various phases whose existence is related to molecular anisotropy. Amongst them are various kinds of liquid crystals [2-4] and plastic crystals [2, 5, 6], which have been known in nature for long time. Recently, there were also found examples of new kinds of order, like the cubatic phase in the system of cut spheres [7, 8] or the degenerate crystal of hard dimmers [9, 10]. New phases have been shown to exist in systems of multi-rod molecules of zero volume [7]. Such molecules, being generalization of the Onsager 's rods [11], allow for modelling generalized nematics, i.e. fluids with orientational order of any point group symmetry with an inversion centre [12]. Because the volume of real molecules is larger than zero, in contrast to the model multi-rods, a question arises if exotic, i.e. non-axial, nematics can be obtained in systems of hard body molecules of positive volume (and positive second virial coefficient). Cut spheres suggest * Corresponding author: e-mail: kww@rose.man.poznan.pl (Rec. 4 December 2004) 1 Institute of Molecular Physics, Polish Academy of Sciences Smoluchowskiego 17/19, 60-179 Poznań, Poland 2 FOM Institute for Atomic and Molecular Physics Postbus 41883, 1009 DB Amsterdam, The Netherlands K . W . W O J C I E C H O W S K I 1 * A N D D . F R E N K E L 2 TETRATIC PHASE IN THE PLANAR HARD SQUARE SYSTEM? 2 3 6 K. W. Wojciechowski and D. Frenkel positive answer to this question in three dimensions (3D). In two dimensions (2D) the answer to such a question is not known and it is one of problems addressed in the present paper. Very simple 2D molecules, one can consider as candidates for mesogens of the exotic nematics, are the regular polygons. Such bodies represent special cases of convex bodies for which many general results has been obtained [13]; in particular the second virial coefficient is known exactly [14]. Amongst the regular polygons, these of a small number of sides (triangle, square, etc), the most different from the circle, seem to offer best chances to generate mesophases of symmetry different from the axial one. In the present paper we consider the square. In such a case the symmetry of the hypothetical exotic liquid crystalline phase should be the same as the symmetry of the molecule. (In the case of triangles one should expect hexagonal symmetry of a liquid crystalline phase, if such a phase would be stable.) The square 'molecule' is a limiting case of hard rectangular bodies for which some higher virial coefficients have been recently obtained [15] and whose phase diagram has been studied recently [16]. Hard squares are interesting not only as a potential mesogen for liquid crystalline phases. Studies of anisotropic particles in 2D can, in general, throw a new light on still controversial problem of 2D melting [17, 18] and possible kinds of order in 2D. Difference of topology between three-dimensional (3D) and two-dimensional (2D) systems allows for qualitative differences in the nature of order and melting transition in three and two dimensions. 2D \"crystals\" with short range interactions between molecules cannot exhibit any true long-range translational order at positive temperature [19-22] because the long-wave phonons lead to logarithmic increase of positional fluctuations of the molecules. The decay of translational correlations in 2D has a power law form and leads to ordering known as quasi-long range order. (In one dimension the fluctuations are proportional to the size of the system and, hence, destroy any crystalline order completely. The correlations decay in the exponential way, what is typical for fluids.) This is in contrast with crystals in 3D for which periodicity, implied by the long range translational order, is one of the most fundamental properties. The periodicity of 3D crystals is lost only at melting, and the melting itself is, according to the general experience, a first-order phase transition. The lack of a true long-range translational order in 2D crystals does not imply, however, as one might conclude from the theory proposed by Kosterlitz and Touless [23], that the melting transition between a low temperature (high density) 2D solid and a high temperature (low density) fluid is continuous. Computer simulations strongly indicated first order character of melting in 2D model systems with short range interactions (for references see the review papers in Refs. [24-27]). Barker and Henderson calculated explicitly the positional fluctuations in the hard disc system and concluded that they are quantitatively very small even for systems of macroscopic size [24]. Such observations might suggest that the nature of melting of 2D systems with \"physical\", i.e. short range, interactions is not much different from that observed in 3D systems. Simulations of large Lennard-Jones system [28] showed, however, that it is not so. No two-phase coexistence in the standard meaning was found in the melting region in this system. On the contrary, the Tetratic Phase in the Planar Hard Square System? 2 3 7 system appeared to be quite homogeneous in this region [28, 29]. Moreover, in the intermediate region between fluid and 'crystal', a quasi-long-range order of bonds connecting the nearest neighbouring particles has been found [28]. It is generally accepted that in the absence of the translational long-range order, the longrange orientational order of bonds connecting the nearest neighbouring particles is that which is lost at melting of 2D crystals. In the theory of Kosterlitz and Thouless [23] the process responsible for destruction of this order is the dislocations unbinding [23]. Halperin and Nelson [30] noticed that dislocation unbinding alone cannot lead to isotropic fluid. They proposed a second transition, dislocations unbinding, as the final step in the 2D melting. According to their theory, generalized by Young [31] (and further referred to as the KTHNY theory), a 2D crystal may melt via two second-order transitions. In such a case the 2D crystal is separated from the fluid by a new phase with quasi-long-range orientational order and without translational order. Computer simulations did not support such a scenario for isotropic particles with short-range interactions [25-27]. On the other hand, some experiments performed with liquid crystalline layers indicated relevance of the two-stage melting to such systems [32]. Kleinert proposed a certain lattice model explaining qualitatively this situation [33]. In the free energy expansion he considered an additional term which he related to molecular anisotropy [33]. Janke and Kleinert [34] studied this model and observed either a single first-order transition or two continuous transitions, depending on the value of a coefficient at the new term. It is not obvious, however, if and when the expansion proposed by Kleinert is applicable to real systems. Although his results indicate that molecular anisotropy is required for two-stage melting, they cannot be seen as a definite argument that the molecular anisotropy is sufficient for such a melting scheme. Hence, a question arises if the molecular anisotropy itself can lead to a two-stage melting in two dimensional systems and, if yes, how large should it (the molecular anisotropy) be. In general, translational-rotational coupling present in systems of anisotropic particles may lead to qualitatively new orderings, absent in the case of isotropic particles. In the case of square particles one cannot exclude a priori neither an orientationally disordered (plastic) crystal nor orientationally ordered (liquid crystalline) fluid. In the latter case the molecular anisotropy may, in principle, lead to any of three possibilities: (i) a phase with molecular orientational (MO) order and without bond orientational (BO) order (nematic phase of 4-fold symmetry), (ii) a phase with a BO order and without a MO order, and if the molecular orientations couple to the bond-bond orientations (iii) a phase (further referred to as tetratic) with both (4-fold) MO and BO order. The first two cases, with no coupling between molecular orientations and bonds do not seem to be plausible in the case of hard squares, if one takes into account results of the Refs. [35, 36]: systems with non-separable interaction potential are expected to exhibit a coupling between the molecular orientations and the bond orientations. At present the most efficient way to check if any of the possibilities mentioned above holds in the case of the squares are Monte Carlo simulations. 2 3 8 K. W. Wojciechowski and D. Frenkel One more reason for which the squares constitute theoretically attractive system is a strong degeneracy of

A tabu search strategy for finding low energy structures of proteins in hp-model*

Abstract: HP-model is one of the most successful and well-studied simplified lattice models of protein folding. It uses mathematical abstraction of proteins for hiding many aspects of the folding process and works as hypothesis generator. Due to the NP-hardness results of the protein folding problem many approximation algorithms, have been used to solve it. In the paper, the method for finding low energy conformations of proteins, based on the tabu search strategy, has been proposed. The algorithm has been extensively tested and the tests showed its very good performance.

Molecular modelling of the deformation mechanisms acting in auxetic silica

Abstract: 118 A. Alderson et al. auxetic [13], and 69% of the cubic elemental metals and some fcc rare gas solids are auxetic when stretched along the [110] off-axis direction [14]. One of the first attempts to design materials displaying negative Poisson's ratio behaviour at the molecular level was based on an idealised 2D system of hard cyclic hexamers [15]. This was followed by the modelling of auxetic behaviour in molecular networks based on geometries known to lead to auxetic behaviour at the macroscale, such as the design of nanoscale macrocyclic hydrocarbons based on the macroscopic re-entrant honeycomb geometry known to lead to auxetic behaviour [8]. Molecular Mechanics simulations indicated auxetic behaviour in a range of idealised zeolitic cage nanostructures [16]. Combined Molecular Mechanics and Monte Carlo simulations were used in a preliminary investigation to show that the presence of a negative Poisson's ratio enabled selective loading of neopentane and benzene guest molecules in the host zeolite MFI nanostructure through the application of an external stress in one specific direction [16]. In order to develop this area further a detailed understanding of the mechanisms and geometries necessary to realise auxetic behaviour at the nanoscale is required. In this paper, Fig. 1. Tetrahedral framework unit cell for α-cristobalite showing tetrahedral rotation axes (solid arrows) and geometrical parameters. Filled circles are silicon atoms; empty circles are oxygen atoms hedron consisting of an O atom at each of the four corners surrounding a central Si atom. including foams [3], honeycombs [8], microporous polymers [9, 10] and fibre-reinforced composites [11, 12]. At the molecular scale, a number of single-crystal materials are known to exhibit auxetic reported. The data provide greater insight into the mechanisms likely to be operating for a range of different uniaxial and pressure loading conditions. 2 . N A N O S T R U C T U R E O F α C R I S T O B A L I T E Molecular Modelling of the Deformation Mechanisms Acting in Auxetic Silica 119 The structure consists of a framework of corner-sharing SiO4 tetrahedra in which each O atom is shared between two adjacent tetrahedra. The tetragonal primitive unit-cell (space group P41212) contains 4 tetrahedra (Fig. 1). 3 . M O L E C U L A R M E C H A N I C S M O D E L S The Cerius Molecular Modelling software (Accelrys) was employed on a Silicon Graphics 02 workstation. The starting structure was as provided within the Cerius structure database derived from experimental data. The modelling protocols for the structure and mechanical properties simulations were as described in detail in Ref. [16]. The stiffness matrix C was computed from the second derivative of the energy expression, and the on-axis Poisson's ratios and other elastic constants were obtained directly from the compliance matrix, S = C 1 . Based on our previous work [16], the BKS [17] and Burchart [18] force-fields were employed in the structure and mechanical properties simulations. These force-fields were developed specifically for silicas and aluminophosphates. The BKS force-field treats interatomic interactions as ionic, with parameterisation based on both experimental and ah initio data. The Burchart force-field assumes the frameworks are largely covalent and interactions are parameterised using experimental data. Whilst these force-fields have previously [16] been agreement with the experimentally-determined undeformed unit-cell lengths, and is in particularly good agreement with the values of the transverse unit cell dimensions (X1 and X2, see Fig. 1). The greatest discrepancy between the predicted and experimental values occurs for the BKS predicted value of X3, which is ~6% lower than the experimental value. The predicted values are similar to those from the pair potential calculations of Keskar and Chelikowsky [19] who reported predicted values of X1 = 4.96 Å and X3 = 6.68 Å. they did predict low positive values and also gave reasonable agreement with the experimental on-axis Young's moduli. The BKS force-field was previously found to predict a negative average polycrystalline isotropic aggregate Poisson's ratio, as also calculated from the experimental single-crystal elastic constants. Structure and mechanical properties simulations were performed for uniaxial loading along each of the mutually orthogonal principal axes x1, x2 and x3 and also for hydrostatic pressure loading. Loads were applied in the range -2GPa to 2GPa (i.e. compressive and tensile loads were considered).

The use of auxetic materials in smart structures

Abstract: Abstract: This paper presents a study of the implications of using auxetic materials in the design of smart structures. By using auxetic materials as core and piezoelectric actuators as face layers to provide control forces, the problem of the shape control of sandwich beams is analyzed under loading conditions. The mechanical model is based on the shear deformable theory for beams and the linear theory of piezoelectricity. The numerical solution of the model is based on superconvergent (locking-free) finite elements for the beam theory, using Hamilton's principle. The optimal voltages of the piezo-actuators for shape control of a cantilever beams with classical and auxetic material are determined by using a genetic optimization procedure. Related numerical solutions of static problems demonstrate the role of auxetic material in the deformation, shape control and stress distribution of the beam and related twodimensional composite elastic structures.

Slowness surfaces and energy focusing patterns of auxetic cubic media

Abstract: We study properties of slowness surfaces and energy focusing patterns of cubic elastic media. We restricted ourselves to the region of the stability triangle where Poisson’s ratio σ P of the specimen stretched in the [001] direction and measured for [100] is negative, i.e. we consider all cubic auxetic materials. We study properties of surfaces and energy focusing patterns for all elastic auxetic media characterized by σ P = !1/3. 1. INTRODUCTION Recently there is considerable interest in auxetics – elastic materials with negative Poisson’s ratio (cf. [1-3]). Generally the calculation of Poisson’s ratio is complicated for directions oblique to the crystal axes [4]. Therefore, in our previous papers [5, 6] we restricted ourselves to Poisson’s ratio for [001] stretch measured for [100] lateral direction and established the region of the stability triangle (ST) (cf. [6]) where the mentioned Poisson’s ratio is negative. We shall underline that the characteristics of particular region of auxeticity depends on the choice of the stretch direction n and the direction m of measurement [7]. To indicate this dependence we shall use notation nm-auxeticity region of ST. The present paper is devoted to the study of geometrical properties of slowness surfaces of long wavelength acoustic phonons [8, 9] in the [001][100]-auxeticity region (for simplicity we shall use a short notation zx-Poisson’s ratio and zx-auxeticity region). We also obtained the energy focusing patterns [8, 9] which are images of these surfaces under the mapping induced by phonon focusing, which is mathematically known as the Gauss maps (cf. [10]). These maps are obtained in time-of-flight experiments with ballistic phonon beams [8, 9]. 2. CHOICE OF ELASTICITY PARAMETERS 2.1. Partition of the stability triangle Within the framework of theory of elastic media, in a chosen Cartesian coordinate system, the phase c and group velocities v, as well as polarization vectors e of long wavelength acous-

Numerical and experimental study of auxetic closed-cell foams

Abstract: The procedures of fabrication and testing of auxetic foams with closed cells based on foaming a liquid substance and by joining microspheres are discussed. Physically , to obtain an auxetic structure, bending rigidity of elastic rods, plates and shells should strongly depend on the initial curvature. The cells of small size are found mostly to hold their original shape. Large ones show relatively low rigidity , and would get deformed similarly to thin-walled shells when compressed with a possibility of losing stability. Thus, the volumetric compression of a foamed material is mainly realized at the expense of decreased free volume of large cells. Separation of cells according to deformation levels is found to cause auxetic elastic behavior in converted closed cells foams. Technologically, to obtain this auxetics we proposed a two-stage process. It includes the formation of concave cell structure by a permanent volumetric compression of the initial material just after foaming in the solidification state under the action of a liquid or gas. High plasticity of foam materials in this stage allow s us to obtain the re-entrant structure of cells. To obtain a material with non-convex cells we used mostly a gas or liquid under pressure as a forming instrument. After cooling the foam material shows the property of elastic (reversible) deformation. I he homogeneity and isotropy of Poisson's ratio of obtained auxetics are caused by a uniform distribution of the gas or liquid pressure on the sample surface. Some problems of Poisson's ratio minimization for foam materials we have solved by the finite element analysis.

On representations of coefficients in implicit interval methods of runge-kutta type

Abstract: Interval methods for solving the initial value problem in floating-point interval arithmetic give solutions in the form of intervals which contain all possible numerical errors (see [3] or [4]). The estimations of diameters of interval-solutions are possible on account of (see [5]): • the minimization (with respect to the coefficients) of the interval extension of the principal part of the approximation error (see e.g. [6]), • the minimization of some constants which occur in the estimation of interval-solutions (see [1] and [2]) • the coefficients of the particular methods which have exact representations in the computer. In this paper we consider the last of these cases for one-, twoand three-stage implicit interval methods of Runge-Kutta type. The paper is organized as follows. Section 2 contains the implicit classical methods of Runge-Kutta type for solving the initial value problem. In Section 3 the implicit interval Runge-Kutta methods are given. Section 4 deals with all possible forms of numbers which are exactly represented in floating-point arithmetic. In Section 5 some approximations of the widths of interval-solutions are discussed. Section 6 concludes the paper.

On the numerical solution of singular integral equations using sandikidze's approximaton

Abstract: In this work we present a direct method for an approximation solution of a singular integral equation (S.I.E) on a piecewise smooth integration path. Many studies devoted to the numerical procedure are developed for solving (S.I.E) over a finite interval, especially [-1,1]. Cauchy type singular integral equations are often encountered in problems of mathematical physics when solving problems in elasticity theory, aerodynamics, electrodynamics and other branches of sciences and technology. Also we note that, the solution of a large class of boundary-value problems in mathematical physics can be reduced to singular integral equations (S.I.E) of the form

Molecular dynamics study of short and medium range order in modified bgo glasses

Abstract: In the paper we present our recent Molecular Dynamics (MD) simulations of the structure of bismuth-germanate (BGO) glasses of x(pBi2 (1!p)Bi2O3)(1!x)GeO2 composition, where x denotes the content of the bismuth oxide in unmodified glasses, and p - the fraction of neutral bismuth that can appear in the surface modification process (e.g. annealing in hydrogen atmosphere). We consider glasses of compositions x = 0.1, 0.2, 0.3 and reduction degrees p = 0, 0.25, 0.5, 0.75, 1. The simulation results are analysed in detail and compared with the structural data provided by other authors.

On the link between dna sequencing and graph theory

Abstract: The methods cited in this paper solve the combinatorial part of DNA sequencing by hybridization, basing on known approaches from graph theory. It is assumed here, that the length of oligonucleotides used in the hybridization experiment is constant within a library.

The structure of porous and spontaneously densified amorphous PbSiO3 : a molecular dynamics study

Abstract: A b s t r a c t : In the paper we propose and test a \"gel-drying\" method of obtaining porous oxide glasses in Molecular Dynamics (MD) simulations. The simulation is started with low (screened) values of ionic charges. Then, the charges are gradually increased (to mimic the gradual elimination of a polar solvent) up to ful l ionic charges (a completely dry gel). This computational trick is applied to produce a porous PbSiO 3 system. The structure of the resulting low-density samples is analysed in detail. Then, the porous structures are submitted to spontaneous densification, and the structure of the obtained dense bulk glasses is analysed. Finally, the structures of bulk glass obtained via spontaneous densification (density ρ = 8250 kg/m ) and bulk glass of the same density obtained via isotropic compression are compared.

An approximation method for simulating temperature dependence of poisson's ratios of self-expanding auxegens

Abstract: A combined molecular dynamics and molecular mechanics method has been developed for estimating Poisson's ratios of certain types of molecular auxetics at various temperatures. The temperature dependence of the auxeticity of a special class of molecular auxetics, namely, a self-expanding supramolecular network of auxegens containing alternating phenyl and acetylene links, is studied with use of this approximation method. The simulation results show that as temperature increases from 0 to 300 K, the auxeticity of the resulting superlattice or van der Waals network of auxegens decreases from the initial self-expandability to two negatively small Poisson ratios on the xoy plane.

Theoretical and experimental 1h and 13c nmr spectra of 3-hydroxypyridine, 3-methoxypyridine, and n-ethyl-3-oxypyridinium betaine*

Abstract: H and C NMR chemical shifts for neutral (3-hydroxypyridine and 3-methoxypyridine) and zwitterionic (N-ethyl-3-oxypyridinium betaine and 3-pyridone) molecules were calculated by GIAO/B3LYP/6-31G(d,p) and IGLO/deMon/NMR approaches. Linear correlations between the calculated and experimental H and C NMR chemical shifts for 3-hydroxypyridine, 3-methoxypyridine, and N-ethyl-3-oxypyridinium betaine suggest that the 3-hydroxy tautomer is dominant in DMSO-d6. The lack of such a correlation for 3-pyridone indicates an absence of this species in DMSO-d6 solution.

Elastic properties of inverse power fluids

Abstract: We investigated the consequences of variable potential softness on elastic properties, using the repulsive inverse power potential. With this potential the softness can be changed continuously from very soft to extremely steep or hard. An explicit formula for the equation of state is derived and discussed. It is shown how this formula can be exploited to determine the infinite frequency elastic properties of the inverse power fluid. Explicit formulae for the elastic constants, the high-frequency elastic moduli, the longitudinaland transverse-wave velocities and Poisson's ratio are obtained. Their behaviour in the steeply repulsive limit is discussed. It is demonstrated that the softness directly determines the Poisson's ratio, and it is shown that in order to decrease the value of the Poisson's ratio a harder potential interaction must be applied.

Surface response optimisation of auxetic homogenised cellular plates using genetic programming

Abstract: This paper describes an approach based on Genetic Programming to perform the metamodelling of cellular structure properties with in-plane auxetic behaviour. Common procedures to design microstructure topologies with complex shape is to use analytical and/or Finite Element (FE) models and quantify the variability of their homogenised mechanical properties versus internal cell parameters. For the FE case, the large number of computations involved can rule out many approaches due to the expense of carrying out many runs. One way of circumnavigating this problem is to replace the true system by an approximate surrogate/replacement model, which is fast-running compared to the original. In traditional approaches using response surfaces a simple least-squares multinomial model is often adopted. The object of this paper is to extend the class of possible models considerably by carrying out a general symbolic regression using a Genetic Programming approach. The approach is demonstrated on the optimisation of the unit cell of centresymmetric auxetic cellular solids composing a simply supported plate for maximum central deflection under transverse uniform pressure. 1. I N T R O D U C T I O N Engineering computation has played a larger role compare to a few decades ago. With the advances of computer, various engineering computational programs have been developed, tailored for different aspects of engineering. Finite Element Analysis (FEA) alongside with others has become the essential tools to master. Though the computers have made the life of engineers easier, the new challenges of engineering design nowadays are constantly pushed to solve more complicated problems. This will lead to the issue when the designs are very dependent to the computational power. While the design optimisations are taking too long to perform, engineers and scientists choose for some solutions that create a replacement/surrogate model of the simulation or experimental data. *Corresponding author: e-mail: t.lew@shef.ac.uk, phone: (+44) 0114 222 77 21, fax: (+44) 0114 222 7890. 170 T. L. Lew et al. This replacement model will be used for the optimisation process, instead of re-generating the data from the simulation/experiments upon each optimisation step. The creation of the replacement/surrogate model is also known as metamodelling. Traditional metamodelling techniques usually involve using a multivariate response surface model. The research in this field reached its mature state some time ago, but there are some known problems in using these approaches. For example, difficulties with the polynomial basis function, problems in trying to fit a highly nonlinear model using response surface functions, explosion of coefficients and low extrapolation-abilities. It also requires some prior knowledge of the data before choosing the right functions to fit [1]. Overview of the state of art metamodelling techniques can be found in [2]. Recent findings in metamodelling has developed into techniques that are inspired by the natural evolution, which created the most successful and remarkable designs known by mankind. Most engineer and scientists prefer to use an algorithm that is extensively proven for its reusability and robustness, rather spending time, efforts and money customising the data with the chosen functions. This technique, with today 's phrase, it is called evolutionary algorithms. Evolutionary algorithms include a few subclasses, such as evolutionary computation, genetic algorithms, genetic programming etc. Evolutionary algorithms were proven to have successful results in all types of automated design system [3]. In this work, an example of simple analytical optimisation is presented with the use of genetic programming as a metamodelling technique. The genetic programming code was developed using Java programming language; it is used to fit the structural properties of the auxetic honeycomb, which generated f rom [4]. The optimisation routine was taken from [5], and is performed in Matlab using Sequential Quadratic Programming (SQP) method. A simply supported plate made by a cellular structure composed by centresymmetric auxetic grid provides the test case considered. Auxetic (i.e., negative Poisson's ratio) materials and structures have recently attracted much attention in the research community for their unusual characteristics and the fact that auxetic behaviour is often accompanied by interesting performance of other multiphysics characteristics. Extended references and descriptions on auxetic systems can be found in [6-9]. Centresymmetric honeycomb structures have been between the first examples of system exhibiting in-plane negative Poisson 's ratios. The conventional hexagonal unit cell of a regular honeycomb can be made re-entrant, with an internal negative angle, and exhibits lateral expansion when pulled along one direction [4, 10]. Re-entrant cell honeycombs are highly anisotropic, but features higher Voigt bounds for transverse shear modulus compared their conventional counterpart [11], directional band-gaps behaviour in flexural wave propagation cases [11], and strong dielectric anisotropicity that can be used together with the mechanical one to design electromagnetic compatibility characteristics in microwave absorbers [12, 13]. Surface Response Optimisation of Auxetic Homogenised Cellular Plates The creation of the initial random population is a blind search in the problem domain, whereby the birth of each individual is achieved by randomly generating a root node, and its subsequent branches. The root node must be chosen f rom the function set, as illustrated in Fig. 1, a root node ' + ' has arity 2, with each argument being represented by a connection to a subsequent node. It is then randomly combined with other nodes f rom either the terminal or Fig. 1. Creation of individual of initial population There is however one rule to obey: each function must be applicable to any values returned by other functions and any values carried by the terminal nodes. (2) (1) where F and T represent function sets and terminal sets respectively; M and N are the number of functions and terminals included in the GP. Each function f i takes in a specific number of arguments the number also known as the funct ion ' s arity, while the terminals have null arity 2 . G E N E T I C P R O G R A M M I N G T H E O R Y As mentioned above. Genetic programming is a subclass of evolutionary computation techniques, however unlike other evolutionary techniques that evolve numerical values, it evolves functions as a solution for a given problem. Genetic Programming (GP) shares the same concepts as the well-known Genetic Algorithm (GA) but increases the complexity by allowing the structure of the solution to undergo adaptation. The structure is typically a hierarchical computer program or mathematical function of dynamically varying size and shape. GP starts with an initial population of randomly generated individuals, which consists of function and terminal nodes appropriate to the problem domain. The appropriateness of the functions are less strict than the traditional metamodel , as it is jus t to decide if the function used would be arithmetical, logical etc., one can even use some user-defined functions that are suitable for the problem domain. Therefore, depending on the problem domain, the individuals may be real, complex, vector, symbolic, multiple valued etc. Each individual is built f rom repeatedly combining all possible functions and terminals: 171 172 T. L. Lew et al. function set. If a terminal node was chosen, it will stop branching out; else the growth will continue. The individual created from Fig. 1 represents an expression of (x · y) + z. There are a few growth strategies, namely full method, grow method and ramped half-andhalf [14] The full method of generating the initial population involves creating individuals that grow up 'til the maximum allowable level. The grow method involves creating individuals that are variably shaped, whereby the depth can be of any level less than or equal to the maximum allowable level. The ramped half-and-half is a mixture of both. The grow method is adopted in this work. To prevent the individuals from growing infinitely, a limit was enforced on the depth of the tree structure and on the total number of nodes a tree can have, after which only terminal nodes are chosen. Some other bloat (i. e. rapid increase of individual tree size) control methods can refer to [15, 16]. The driving force of GP, as for GA, is by computing the fitness of the individuals before 'mating' them to produce subsequent generations. Each individual in the population is measured in terms of its performance merit, known as the fitness measure. There are several ways of assigning a fitness measure based on different problem domains. [14] The fitness measure enables the best individuals to be chosen to inherit across the generations, and eliminates the unfit ones. The fitness of an individual in this work is assigned by the inverse of its percentage Mean Square Error (MSE):

Fast summation of double infinite modal series involved in analysis of shielded microstrip circuits

Abstract: The paper addresses some aspects connected with computational methods involved in analysis of shielded microstrip circuits in the frame of the IE-MoM approach. The paper is focused on a method for efficient evaluation of double infinite modal series, which arise in the analysis of vertical metallizations embedded in a waveguide or cavity filled with a multilayer medium. Generally, the modal series converge very slowly, when treated in its original form, and from practical point of view it makes the IE-MoM approach inefficient. The rate of convergence of the modal series can be significantly increased by means of a transformation of the double infinite series into a fast converging single series. The transformation exploits the contour integral and the residue theorem method in conjunction with the well known Kummer's transformation. The proposed method proved to be very efficient since it enables radical savings in computational time and this feature makes the method a good candidate for practical purposes, especially for electromagnetic CAD tools.

Advanced methods of heart rate signals processing and their usefulness in diagnosis support ii. univariate statistical techniques

Abstract: (Rec. 29 February 2004) Abstract: A tentative cardiological database was established using virtual instrumentation described in the first part of presented paper. Some additional not heart rate variability parameters were added. Three selected univariate statistical techniques were used for illustration diagnosis support techniques in discrimination between healthy and coronary heart disease people. Comparison of nonparametric MannWhitney test, receiver operating characteristic ROC analysis and univariate logistic regression results was performed. In all used methods long term heart rate variability indices were most useful in prediction of patient's status. The correctness of classification was between 55 to 79 percent with ROC technique and 68 to 78 percent with logistic regression. However high number of false negative FN cases excludes univariate techniques as reliable screening test.