Search

This paper introduces an extension to the normal distribution through the polar method to capture bimodality and asymmetry, which are often observed characteristics of empirical data. The later two features are entirely controlled by a separate scalar parameter. Explicit expressions for the cumulative distribution function, the density function and the moments were derived. The stochastic representation of the distribution facilitates implementing Bayesian estimation via the Markov chain Monte Carlo methods. Some real-life data as well as simulated data are analyzed to illustrate the flexibility of the distribution for modeling asymmetric bimodality.

International audience; A general theoretical and numerical framework for the strength and mass optimisation of variable-stiffness composite laminates (VSCLs) is presented in this work. The optimisation is performed in the context of the first-level problem of the multi-scale two-level optimisation strategy (MS2LOS) for VSCLs. Both the failure load maximisation problem (subject to a constraint on the mass) and the mass minimisation one (with a constraint on the VSCL strength) are solved for two benchmark structures. The effect of the presence of a constraint on the maximum tow curvature is also investigated. The solutions are searched by means of a deterministic algorithm by considering different scenarios in terms of the VSCL macroscopic behaviour: the orthotropy type and shape, the direction of the main orthotropy axis and the thickness of the laminate are tailored either globally (uniform over the structure) or locally. The polar method is used to represent the point-wise elastic response of the VSCL at the macroscopic scale. The distributions of the polar parameters and of the thickness are described through B-spline entities: their properties are exploited in computing physical and geometrical response functions of the VSCL as well as their gradient. The VSCL strength at the macroscopic scale is assessed using a laminate-level failure criterion in the space of polar parameters. Numerical results show considerable improvements with respect to both quasi-homogeneous isotropic structures and an optimised VSCL solution taken from the literature obtained by using the design approach based on lamination parameters. These results confirm the effectiveness of the proposed strategy and the great potential of VSCLs.

International audience; This work deals with the multi-scale optimisation of composite structures by adopting a general global-local (GL) modelling strategy to assess the structure responses at different scales. The GL modelling approach is integrated into the multi-scale two-level optimisation strategy (MS2LOS) for composite structures. The resulting design strategy is, thus, called GL-MS2LOS and aims at proposing a very general formulation of the design problem, without introducing simplifying hypotheses on the laminate stack and by considering, as design variables, the full set of geometric and mechanical parameters defining the behaviour of the composite structure at each pertinent scale. By employing a GL modelling approach, most of the limitations of well-established design strategies, based on analytical or semi-empirical models, are overcome. The GL-MS2LOS makes use of the polar formalism to describe the anisotropy of the composite at the macroscopic scale (where it is modelled as an equivalent homogeneous anisotropic plate). In this work, deterministic algorithms are exploited during the solution search phase. The challenge, when dealing with such a design problem, is to develop a suitable formulation and dedicated operators, to link global and local models physical responses and their gradients. Closed-form expressions of structural responses gradients are rigorously derived by taking into account for the coupling effects when passing from global to local models. The effectiveness of the GL-MS2LOS is proven on a meaningful benchmark: the least-weight design of a cantilever wing subject to different design requirements. Constraints include maximum allowable displacements, maximum allowable strains, blending, manufacturability requirements and buckling factor.; This work deals with the multi-scale optimisation of composite structures by adopting a general global-local (GL) modelling strategy to assess the structure responses at different scales. The GL modelling approach is integrated into the multi-scale two-level optimisation strategy (MS2LOS) for composite structures. The resulting design strategy is, thus, called GL-MS2LOS and aims at proposing a very general formulation of the design problem, without introducing simplifying hypotheses on the laminate stack and by considering, as design variables, the full set of geometric and mechanical parameters defining the behaviour of the composite structure at each pertinent scale. By employing a GL modelling approach, most of the limitations of well-established design strategies, based on analytical or semi-empirical models, are overcome. The GL-MS2LOS makes use of the polar formalism to describe the anisotropy of the composite at the macroscopic scale (where it is modelled as an equivalent homogeneous anisotropic plate). In this work, deterministic algorithms are exploited during the solution search phase. The challenge, when dealing with such a design problem, is to develop a suitable formulation and dedicated operators, to link global and local models physical responses and their gradients. Closed-form expressions of structural responses gradients are rigorously derived by taking into account for the coupling effects when passing from global to local models. The effectiveness of the GL-MS2LOS is proven on a meaningful benchmark: the least-weight design of a cantilever wing subject to different design requirements. Constraints include maximum allowable displacements, maximum allowable strains, blending, manufacturability requirements and buckling factor.

A numerical study of the effect of uncertainties in ply angles and thicknesses on the flutter speed of a cantilevered composite plate wing was conducted in this paper. Reduction of the number of uncertain parameters was possible thanks to the use of the polar method, which also enabled a systematic analysis of the influence of material symmetries. From the polar domain of orthotropic laminates, several stacking sequences were reconstructed in order to propagate parametric uncertainties . Typical fabrication uncertainties on ply thicknesses and angles were considered in order to quantify their influence on the probabilistic aeroelastic response. The reduction of the set of stochastic parameters by the polar method enabled the use of a polynomial-chaos approach which was combined with machine-learning techniques in order to deal with the correlation and with discontinuities in the response surface. Results reveal that possible manufacturing tolerances cause significant deviations in the critical flutter speed from the nominal value, especially when mode switches occur. As these deviations surpass classical dimensioning margins, uncertainty quantification approaches can provide added safety.

This paper deals with the concept and application of general three-dimensional (3-D) type-2 fuzzy logic systems in the polar frame (G3DT2FLS). 1 1 Because many acronyms are used in this paper, they are summarized in Table I for the convenience of the reader. We focused on the automatic membership function (MF) Generator, general type-2 polar fuzzy membership, new geometric operators in polar frame, inference consisting of fuzzy 3-D polar rules and antecedents/consequents $\theta$ -slice and $\alpha$ -planes. The cubic smoothing spline is introduced to generate the upper and lower MFs according to the information theory. Three indices of compactness, smoothness, and entropy are employed for tuning of MFs. A measure of ultrafuzziness, min, max, equal, and reduced ultrafuzziness are suggested for the polar method. Additionally, the set theoretic operations of 3-D type-2 fuzzy sets in the polar frame are discussed. We also prove the join, meet, centroid, and type-reduction operations in the polar frame. Several rule sets are given to show the usefulness and complexity of the proposed method. The performance of the partial general 3D polar type-2 fuzzy logic system showed linear growth as the number of rules was increased. Computation time tests showed that the algorithm reduces the computation time by a maximum of 67% compared with discrete MF and is shown by an extreme 98% for the geometric procedure. These results indicate significant improvements in computation time for the spline interpolation over the existing methods.

The classical method (namely inclining experiment) has been used to estimate the vertical center of gravity (VCG or more often KG) of ships for many years. This method is based on the assumption that the metacenter is unchanged in the calculation of KG when the vessel is heeled. However, ships built today have knuckles, chines, dead-rise which may give rise to excessive change in the water-plane. The location of the metacenter is changed on these vessels when heeled. Therefore, determination of the vertical center of gravity may be somewhat erroneous. In this study, three different methods based on the assumption of unchanged metacenter have been examined. Employing the Graphical method, Polar method and Generalized method, the KG values of all vessels can be calculated without any dependence on the metacenter. The three methods mentioned in this study were studied and applied on ten different ships. Furthermore, the results from the classical method compared with those obtained from the recently developed methods. Based on this comparison, it is observed that the new methods developed have similar results to the classical method. Therefore, these methods may be a good alternative to the inclining experiment in the future. Moreover, uncertainty analyses have been performed for the results obtained from the classical method just to realize if there are any critical GM values within the margin of error.

Crofton formula has attracted much attention in differential geometry, which can be used to calculate the length of the curve. This article starts with the concept and definition of Crofton formula on the sphere, and gives out approximate calculation formula of a curve on spherical surface based on surf segmentation. The triangular grid of the geodesic dome and Marsaglia Polar Method are adopted to generate random points for an equal division of the sphere. This article takes the circular arc on the sphere as an example, through MATLAB simulation, the length was calculated by approximation of Crofton’s formula. This calculation method can be extended to approximate calculation of the length of any curve in space, which will have a profound impact on practical engineering applications.

The transition between regular reflection (RR) and Mach reflection (MR) of a Type V shock–shock interaction on a double-wedge geometry with non-equilibrium high-temperature gas effects is investigated theoretically and numerically. A modified shock polar method that involves thermochemical non-equilibrium processes is applied to calculate the theoretical critical angles of transition based on the detachment criterion and the von Neumann criterion. Two-dimensional inviscid numerical simulations are performed correspondingly to reveal the interactive wave patterns, the transition processes, and the critical transition angles. The theoretical and numerical results of the critical transition angles are compared, which shows evident disagreement, indicating that the transition mechanism between RR and MR of a Type V shock interaction is beyond the admissible scope of the classical theory. Numerical results show that the collisions of triple points of the Type V interaction cause the transition instead. Compared with the frozen counterpart, it is found that the high-temperature gas effects lead to a larger critical transition angle and a larger hysteresis interval.

Buckling strength is the most important design criterion for laminated composite structures subjected to compressive loading. The aim of the present work is to introduce an explicit procedure in designing laminated composite plates for maximum buckling load subjected to strength and stiffness constraints using the polar formalism, which relates stiffness to the critical buckling load in order to simplify the calculation and optimization of the buckling load. The polar representation of plane elasticity is effectively used in this work in order to introduce a unified optimization method. Design for maximum flexural stiffness as well as laminate effective elastic modulus was developed in order to achieve the optimal orientation of the layers giving the maximum buckling strength. Three optimization procedures were proposed. The first one is unconstrained optimization and the two other cases are constrained optimization subjected to the flexural stiffness and the elastic modulus constraints. The quasi-homogeneous laminates, which have been used as a special class of laminate stacks, were proposed as the general solution for the optimal stacking sequence of the layers.

A Polar Method based on parametric Cubic Spline technique (PM-CS) is presented for obtaining wave resonating quadruplets { K 1 , K 2 , K 3 , K 4 } in the calculation of the nonlinear source term of the wave model, with results for both deep and finite depths. This polar method is applied in the WRT method for the nonlinear term, and may be considered in addition to the Polar Method with Adaptive Stepping (PM-AS) by Van Vledder (2000) . The PM-CS is used to obtain an uniform distribution of points on the locus of K 2 , for input pair ( K 1 , K 3 ), similar to PM-AS. However, it differs from PM-AS in two ways: (i) no assumption has been made regarding the shape of the locus, and (ii) iterative procedure is avoided, except for locus points on the symmetry in finite depths. For a smaller set of points on the locus, the present PM-CS with variable spacing can be employed. This is illustrated through the convergence of the nonlinear source term by varying points on the locus and the transfer integral evaluated using the Gauss–Legendre quadrature method. Results presented for 1-D nonlinear source term using WRT method are shown comparable with existing WRT results of Resio and Perrie (1991) and Vledder and Hashimoto (2013) .

International audience; An optimization methodology to find concurrently material spatial distribution and material anisotropy repartition is proposed for orthotropic, linear and elastic two dimensional membrane structures. The shape of the structure is parameterized by a density variable that determines the presence or absence of material. The polar method is used to parameterize a general orthotropic material by its elasticity tensor invariants by change of frame. A global structural stiffness maximization problem written as a compliance minimization problem is treated and a volume constraint is applied. The compliance minimization can be put into a double minimization of complementary energy. An extension of the alternate directions algorithm is proposed to solve the double minimization problem. The algorithm iterates between local minimizations in each element of the structure and global minimizations. Thanks to the polar method, the local minimizations are solved explicitly providing analytical solutions. The global minimizations are performed with finite element calculations. The method is shown to be straightforward and efficient. Concurrent optimization of density and anisotropy distribution of a cantilever beam and a bridge are presented.

It is a well-known fact that the current method for calculating a ship's vertical centre of gravity ( V C G ) following inclining experiments is limited when considering magnitude of applied heel angle and accuracy achieved for certain hull-forms due to the assumption of unchanged metacentre position when the vessel is heeled. New methods for calculating the V C G have been proposed, notably the Generalised and the Graphical methods. This paper aims to test these methods on a range of vessels, as well as present and contrast a new method named, the Polar method. The test will establish the error potential for each method using a purely technical software-simulated inclining experiment. Using the established error potential, a corrected V C G is calculated from actual inclining V C G values, which have been evaluated against the loading conditions for each vessel to see if the stability margins have been compromised. The study confirms the Classical method's dependency on applied heel angle magnitude, the change in waterplane area and that it compromises safety in some cases. The other methods, especially the Generalised and the Polar, produce very accurate results for any floating position of the vessel, highlighting the need to tear down the wall-sided assumption implicit in the Classical method and replace it with the better and more flexible methods.

International audience; This paper focuses on the analytical formulation of a tensorial laminate-level failure criterion. The criterion is formulated and expressed in the framework of the first-order shear deformation theory (FSDT) in order to take into account the influence of the transverse shear stresses on the failure mechanisms. More precisely the most common polynomial ply-level failure criteria (expressed under a unified matrix formulation) are considered and reformulated at the laminate level. The proposed unified formulation relies on the utilisation of the polar formalism gener- alised to the FSDT framework. Through this approach all the considered criteria can be formulated in terms of tensor invariants. Furthermore, thanks to the polar representation, an important theoretical result is proven: the existence of a set of analytical relationships between the laminate strength and stiffness invariants.

Multistable shells have been recently proposed as an effective solution to design morphing structures . We describe a class of shallow shells which are bistable after one of their sides, initially curved, is clamped along a flat line. Supposing the shell being assembled as a composite laminate , we show how the anisotropy of the material can influence the multistable behaviour and the robustness of stable configurations. Specifically, we focus on orthotropic laminated shells using the polar method for a complete representation of the anisotropic elastic properties. Two experimental prototypes have been produced and tested to validate our analytical and numerical results.

This paper presents an optimization methodology in order to find simultaneously optimal density and anisotropy. The objective is to maximize the structure global stiffness measured by the compliance. The density and the elasticity tensor are defined as design variables. The numerical procedure is composed of finite element stress calculations and local minimization problems. Thanks to the polar method, these local minimization problems are solved analytically. The method proposed turns out to be straightforward. Two numerical results are illustrated in the 2D-case: optimal designs depending on anisotropy, and optimal design and anisotropy.

The Ziggurat Algorithm is a very fast rejection sampling method for generating PseudoRandom Numbers (PRNs) from statistical distributions. In the algorithm, rectangular sampling domains are layered on top of each other (resembling a ziggurat) to encapsulate the desired probability density function. Random values within these layers are sampled and then returned if they lie beneath the graph of the probability density function. Here, we present an implementation where ziggurat layers reside completely beneath the probability density function, thereby eliminating the need for any rejection test within the ziggurat layers. In the new algorithm, small overhanging segments of probability density remain to the right of each ziggurat layer, which can be efficiently sampled with triangularly-shaped sampling domains. Median runtimes of the new algorithm for exponential and normal variates is reduced to 58% and 53% respectively (collective range: 41–93%). An accessible C library, along with extensions into Python and MATLAB/Octave are provided.

In the present work, an innovative approach is developed and employed to investigate the buckling load of composite laminated plates using polar representation of the fourth-order flexural stiffness tensor. Based on the Rayleigh–Ritz method for calculating the buckling load of orthotropic uncoupled laminates, a new approach is proposed for calculating the critical buckling load as a function of fiber orientation and the number of layers. A new formula is also represented for measuring the possible numbers of buckling modes based on the length-to-width ratio of the plate. Moreover, the new methods make it easy to handle buckling problems, optimization and design of laminated plates subjected to in-plane loads. The efficiency and precision of this method are proved by numerical examples. The anisotropic parts of the buckling load are calculated, and their polar variations are investigated.

The paper presents technology and results of measurements of the steel construction of the skylight of the Main Hall of the Warsaw University of Technology. The new version of the automated measuring system has been used for measurements. This system is based on Leica TCRP1201+ total station and the TCcalc1200 software application, developed by the author, which operates on a laptop computer connected with the total station by the wire. Two test measurements were performed. Each of them consisted of cyclic measurement using the polar method, from one station; points located on the skylight construction, as well as control points located on concrete, bearing poles, were successively measured. Besides geometrical values (such as Hz, V angles and the slope distance D), the changes of temperature and atmospheric pressure, were also recorded. Processed results of measurements contained information concerning the behaviour of the skylight; asymmetry of horizontal displacements with respect to the X axis have been proved. Changes of parameters of the instrument telescope and changes of the instrument orientation were also stated; they were connected with changes of the temperature. The most important results of works have been presented in the form of diagrams.

Electronic tacheometers are currently the standard instruments used in geodetic work, including also geodetic engineering measurements. The main advantage connected with this equipment is among others high accuracy of the measurement and thus high accuracy of the final determinations represented for example by the points’ coordinates. One of many applications of the tacheometers is the measurement of crane rail axes. This measurement is based on polar method and it allows to get the spatial coordinates of points in 3D local system. The standard technology of measurement of crane rail axes and development of its calculations’ results is well-known and widely presented in the subject literature. At the same time new methods of observations results evaluation are developing. Some new proposals for the development of measurement results were already presented in (Kamiński, 2013). This paper is a generalisation of the paper quoted above. The authors developed the concept which was presented there by a proposal for determining rectification corrections for semi gantry crane rail axes. To carried out the task, the parametric method with conditions on parameters was used. Moreover the practical tests on simulated measurement results were conducted. The results obtained from alignment confirmed the theoretical assumptions. Despite the fact that analyses were carried out only on the simulated data, it is already possible to say that presented method for determination of rectification corrections for crane rail axes can be used for development of the observations from real measurement.

The paper presents surveying and processing procedures used at a reconstruction of a circumferential jacket of a 13-storey building, as well as possibilities of numeric and graphic presentation of measurement results. The suitability of specific used procedures depends mainly on the dimensions of the building and its space zoning. Usually, the space polar method is commonly used for surveying of mediating parameters. Analytic processing is considered to be the most suitable method of evaluation of measured point deviations from the vertical plane, or from the plane in general position, mainly because of its smartness, rapidity and universality.

Because of the characteristics associated with distribution network and scarcity of real time measurements, state estimation (SE) methods used in transmission system are not directly applicable to the distribution network. As more sensors are being employed in smart distribution networks, the amount of quasi real time data gathered by various automation systems is increasing. For customer side automations, advanced metering infrastructures (AMIs) are deployed by many utilities. AMI data can be utilized in distribution management system to establish more accurate load models. Acknowledging the uniqueness of distribution network characteristics and requirements for on-line applications, this paper presents a comparative study of three major weighted least square based distribution system state estimation methods, i.e. node voltage polar method, node voltage rectangular method, and branch current rectangular method. Performances are compared using indices that are relevant to the requirements for smart distribution grid applications, i.e. accuracy level of voltage and load estimates, numbers of iterations and execution time required under various measurement combinations and network parameters disparity.

The Ziggurat algorithm is an efficient way for building a Gaussian random number generator (GRNG), which is useful in many scientific and engineering applications. As the classic ziggurat-based GRNG includes nonlinear operations in judging the wedge and tail regions, which is complicated and resource consuming. An improved ziggurat algorithm is proposed by optimizing the accepting model with piecewise linear fitting, replacing the nonlinear operations with linear ones. The improved GRNG works with a 33% decrease of resource and has no significant performance loss. In addition, the uniform random number generator (URNG) in GRNG is implemented by metastability-based ring oscillators with XOR trees, which is the state-of-the-art technique for generating true random numbers in fully digital ways. FPGA results show that the proposed GRNG generates good normal distribution samples with a smooth power spectrum.

In this paper, we introduce the p-generalized polar methods for the simulation of the p-generalized Gaussian distribution. On the basis of geometric measure representations, the well-known Box–Muller method and the Marsaglia–Bray rejecting polar method for the simulation of the Gaussian distribution are generalized to simulate the p-generalized Gaussian distribution, which fits much more flexibly to data than the Gaussian distribution and has already been applied in various fields of modern sciences. To prove the correctness of the p-generalized polar methods, we give stochastic representations, and to demonstrate their adequacy, we perform a comparison of six simulation techniques w.r.t. the goodness of fit and the complexity. The competing methods include adapted general methods and another special method. Furthermore, we prove stochastic representations for all the adapted methods.

We consider here the existence of rari-constant anisotropic layers and show that actually there are two distinct classes of such materials, mutually exclusive. Also, we show that the correct condition for establishing that a material is of the rari-constant type is that the number of independent linear tensor invariants of the elastic tensors must reduce to one. We characterize these materials and show that they can be designed by using some basic rules of homogenization. Copyright © 2015 John Wiley & Sons, Ltd.

It is shown that the widely applied ziggurat algorithm for generating random values from monotone decreasing distributions and symmetric unimodal distributions can be extended to unimodal multidimensional distributions. A multidimensional version of the ziggurat that was implemented in very fast random number generators is presented.

In the present paper, a simple membrane model based on the wrinkle strain approach is revisited with the aim of examining how the material elastic constants affect the static response of anisotropic membranes when wrinkling is taken into account. Employing the polar method, we analyze the role played by the polar moduli, which enable expressing the elasticity matrix components of an anisotropic material in terms of its invariant quantities. With reference to orthotropic materials, we first address the issue of membrane susceptibility to wrinkling by investigating the influence of the three polar parameters characterizing the anisotropic part of the constitutive law. The stress and strain states at any given point in a wrinkled membrane are analyzed by searching for explicit expressions for the principal values of stress and wrinkle strain. Finally, a comparison between our results and those obtained by a numerical solution available in the literature is made in the basic case of a membrane subjected to a pure shear strain state.

The problem of formulating the design of laminated composite plates as a global optimization problem is discussed. In the paper, a general procedure, based upon the polar formalism for the use of tensor invariants as design variables is presented, along with a wide discussion of its peculiarities and of some numerical issues. An outlook about some open problems is also given, with a perspective on possible future issues and further developments.

A preprint version of a paper of 15 pages; In this paper it is shown that a laminate composed of identical anisotropic layers, like for instance composite plies, cannot reach all the possible elastic states that are allowed by the classical elastic bounds on anisotropic constants. The analysis is carried on and facilitated by the use of particular tensor invariants, the so-called polar parameters.

The manufacturing and assembly of a steel construction is a process requiring high demands for accuracy in terms of surveying works, in particular, a need for accuracy in checking measurements using engineering surveying methods. The required standard deviations are in millimeters, and observing them is not a trivial task. The check is usually performed by taking geodetic measurements using the polar method from multiple standpoints, where measurements from individual standpoints are transformed into a common system by means of points measured from more standpoints. In the event that checks are performed more frequently within the same space, e.g., in a production hall, a high accuracy micro-network may be measured in order to achieve easier, faster and more accurate measurements, and check measurements may be made using the free standpoint methods. A necessary precondition for this is the verified stability of the network point’s stabilization. Due to the required standard coordinate deviation of the check measurement at a value of 2 mm to 3 mm, depending on the purpose of the measurements, the requirement set for an accuracy analysis before the measurement is to determine the coordinates of the network points with a standard positional deviation maximally equaling 1.0 mm.

The strength optimisation of orthotropic materials is an open problem. The majority of the studies conducted on strength optimisation of anisotropic materials have been performed with the help of a phenomenological failure criterion that describe the conditions to obtain a limit state identified with the ultimate strength of the material. This paper is a study concerning the bidimensional linear case. Our approach concerns the formulation of a failure criterion through tensors invariants, in order to reduce the optimisation variables to only one: the orientation of material properties of the continuum. Finally, the analytical approach is introduced in order to minimise the failure index and results obtained with the use of three different failure criteria that are compared in order to understand the role of the three formulations with respect to the strength's behaviour. Copyright © 2012 John Wiley & Sons, Ltd.

This paper aims at promoting laminate designs with specific stiffness properties: quasi-homogeneous, quasi-isotropic quasi-homogeneous, and thermally stable laminates. For this, the paper first discusses the concept of design, reviews the classical laminated plate theory and the usual design rules for laminates, and introduces the polar method for plane anisotropy. Then it develops rules and methods to design the stiffness properties that are looked for. In conclusion, it suggests that the classical design rules are too restrictive and that innovative designs should be explored for a better use of composites.

This article deals with practical examples of the applications of modern tachymeters in the determination of the nonverticality of high industrial chimneys, as an alternative to the classic methods. The possibility of a reflectorless measuring mode to determine the point cloud at the chimney's circumference is described. Using these points, we are able to model the chimney as a mathematically correct body using the least squares adjustment method. Subsequently, the unknown nonverticality of the chimney is determined as the deviation of the primary axis of the modelled body from the vertical.

We transfer some known univariate Lyapunov-type inequalities to the multivariate setting of a shell via the polar method. Our results are better than some results of Anastassiou (2011) [6] for even-order partial differential equations. (c) 2014 Elsevier Inc. All rights reserved.

In the simple and widely used method of Box–Muller [G. Box and M. Muller, A note on the generation of random normal deviates, Ann. Math. Statist. 29 (1958), pp. 610–611], from a pair of uniform and independent random variables in (0,1), a pair of standard and independent normal variables is obtained. In this article, we present a very simple and elegant generalization of this method to obtain a pair of correlated standard normal variables with a given coefficient of correlation. This generalized method, which is computationally very easy, is interpreted in geometric terms, considering a translation of the uniform interval (0,1) and a rotation of a defined angle, both related to the coefficient of correlation. Some numerical results are simulated and statistically analysed, proving that the generalization is extremely simple and powerful.

The polar method is a minimal invariant representation in plane elasticity. A plane orthotropic elastic behaviour is expressed by five polar invariants related to the elastic symmetries. In this paper, considering the orthotropy orientation and the polar invariants as optimisation parameters, we discuss the problem of minimising the elastic energy for a given state of stress. The minimisation with respect to the orientation is solved in order to find the associated optimal elastic energy for given polar invariants. Then, this quantity is minimised with respect to the polar invariants which characterise the magnitude of the anisotropic components of the elastic stiffness tensor. Optimal uncoupled composite laminates corresponding to this optimum are presented for membrane and bending loadings.

In this paper it is shown that the well known special orthotropy of paper, explained by micromechanical models of complex bodies, actually corresponds to a special case of orthotropy of a plane elastic body. The existence of this special orthotropy was predicted by the author in a previous work.

This paper presents an adaptive PSO algorithm whose numerical parameters can be updated following a scheduled protocol respecting some known criteria of convergence in order to enhance the chances to reach the global optimum of a hard combinatorial optimization problem, such those encountered in global optimization problems of composite laminates. Some examples concerning hard design problems are provided, showing the effectiveness of the approach.

A general discussion of the optimal solutions to the problems of maximum stiffness, buckling load and modal frequencies for rectangular laminates is proposed. The study is based upon some invariant dimensionless material parameters. The conditions giving rise to the same objective function for the three cases are discussed and a new physical meaning of the objective as a stiffness modulus is proposed. The geometrical and material conditions giving rise to non-trivial optimal solutions are also discussed.

We present new Chebyshev–Gruss type inequalities on R N over spherical shells and balls by extending some basic univariate results of Pachpatte.

The derivation of the invariants for a tensor of the third order in a two dimensional space is presented in the paper. The mathematical technique to obtain the expression of the invariants is based upon the so-called polar method, proposed as early as 1979 by G. Verchery as an alternative method to represent a plane tensor by its invariants. The polar components of a third order planar tensor are also introduced. When considered in the framework of piezoelectricity, an energetic interpretation of these components is given.

The problem of finding laminates having null thermal and hygral expansion or curvature coefficients is addressed in the paper, as well as the case of laminates having an isotropic response to thermal or moisture variations. The main feature of this research is the fact that the problem is treated and solved in the most general case: no simplifying hypotheses are made about the type of the stacking sequence, and the design procedure can also take into account elastic properties of the laminate, such as orthotropy, uncoupling and so on. The authors state the problem as a nonlinear programming problem, where a unique objective function includes all the requirements to be satisfied by the solutions. The nonlinear optimisation problem is stated and solved in the framework of the polar-genetic approach: the formulation is a function of the laminate polar parameters, while a genetic algorithm is used as numerical solving technique. A large number of numerical results are shown in the paper.

We transfer here basic univariate Lyapunov inequalities to the multivariate setting of a shell by using the polar method.

This is a Corrigendum to a previous publication entitled: "An extension of the polar method to the First-order Shear Deformation Theory of laminates" Corrigendum to "An extension of the polar method to the First-order Shear Deformation Theory of laminates" [Compos. Struct. 127 (2015) 328-339]

International audience; In this paper, we deal with the problem of optimizing the anisotropy distribution of a laminated structure in order to maximize, simultaneously, its stiffness and strength. For these two objectives, two functionals are considered: the compliance as a measure of the plate stiffness and a laminate-level failure index as a measure of the strength. To solve this optimization problem we used a two-step hierarchical strategy: in the first step the aim is to find the best distribution of the stiffness and strength anisotropic tensors of an equivalent homogenized plate, while in the second one the objective is to find at least one laminate lay-up satisfying the optimal properties obtained as result of the first step. The polar formalism has been used to represent both the stiffness and strength tensors and allowed us to find analytical solutions to the local minimizations of the two functionals. A test case is finally presented to show the effectiveness of the proposed strategy.

In this paper the Verchery’s polar method is extended to the conceptual framework of the First-order Shear Deformation Theory (FSDT) of laminates. It will be proved that the number of independent tensor invariants characterising the laminate constitutive behaviour remains unchanged when passing from the context of the Classical Laminate Theory (CLT) to that of the FSDT. Moreover, it will also be shown that, depending on the considered formulation, the elastic symmetries of the laminate shear stiffness matrix depend upon those of membrane and bending stiffness matrices. As a consequence of these results a unified formulation for the problem of designing the laminate elastic symmetries in the context of the FSDT is proposed. The optimum solutions are found within the framework of the polar-genetic approach, since the objective function is written in terms of the laminate polar parameters, while a genetic algorithm is used as a numerical tool for the solution search. In order to support the theoretical results, and also to prove the effectiveness of the proposed approach, some novel and meaningful numerical examples are discussed in the paper.

This paper concerns laminates made of composite plies with piezoelectric actuators. The subject of the paper is the research of laminates having, in some directions, null expansion coefficients in extension or having the same expansion coefficient in each direction (isotropic behavior). The general equations for the problem are given in the theoretical framework of the so-called polar method for the representation of plane tensors. Some simple formulas are given for the case of expansion coefficients of a standard plate introduced in the paper. Some numerical examples close the paper.

A polar method for obtaining wave resonating quadruplets { K 1 , K 2 , K 3 , K 4 } in the computation of nonlinear wave–wave interaction source term of the wave model is presented with results for both deep and finite water depths. The method first determines the end radial points of the locus equation for K 2 , for each set of input wave vectors ( K 1 , K 3 ) on the symmetry. The locus of K 2 (and hence K 4 ) is then traced in the anti-clockwise direction starting with the maximum radial point on the line of symmetry. It is shown that when k 3 > k 1 , the number of points on the locus varies when the orientations of the input wave vectors are changed and reduces when the difference in the magnitude of the input wave vectors is increased. A significant advantage in this method is that the angular increment on the locus for K 2 can be kept constant.

We present efficient procedures for generating random exponential and normal deviates based on the acceptance-complement method [Kronmal, R.A. and Peterson, A.V., 1981, Journal of the American Statistical Association, 76, 446–451.]. We provide comparisons with the corresponding Ziggurat procedures proposed by Marsaglia and Tsang [Marsaglia, G. and Tsang, W.W., 1984, SIAM Journal on Scientific and Statistical Computing, 5, 349–359; Marsaglia, G. and Tsang, W.W., 2000, Journal of Statistical Software, 5(8), 1–7.]. The proposed procedures maintain good precision over the entire support of the respective densities and are very easy to set up and implement. The proposed exponential procedure compares favourably with the Ziggurat procedure in terms of speed, running up to 24% faster on some platform/compiler/uniform generator combinations tested.

The problem of designing the elastic properties of a laminate is considered. It is shown that a unique formulation for all the design problems with respect to elastic symmetries can be found using polar invariants of the stiffness tensors. In this way, the design of laminates having some general elastic properties is reduced to a classical optimisation problem: the search for the absolute minimum, whose value is 0, of a positive semi-definite form in the space of the polar invariants. A minimum characterisation of some important elastic properties is also given. Some numerical examples and a discussion of the results are also included in the paper.

This paper is a critical review of the so-called polar method, a mathematical technique based upon tensor invariants introduced as early as 1979 by Professor G. Verchery to treat anisotropic plane problems. The paper contains a brief presentation of the method and a summary of some results obtained by the author and co-workers in some problems mainly concerning elasticity and the elastic design of composite laminates.