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The concept of fuzzy random variable combines two sources of uncertainty: randomness and fuzziness, whereas the model of ordered fuzzy numbers provides a representation of inaccurate quantitative data, and is an alternative to the standard fuzzy numbers model proposed by Zadeh. This paper develops the model of ordered fuzzy numbers by defining the concept of fuzzy random variables for these numbers, called further ordered fuzzy random variables. Thanks to the well-defined arithmetic of ordered fuzzy numbers (existence of neutral and opposite elements) and the introduced ordered fuzzy random variables; it becomes possible to construct fully fuzzy stochastic time series models such as e.g., the autoregressive model or the GARCH model in the form of classical equations, which can be estimated using the least-squares or the maximum likelihood method. Furthermore, the concept of normality of ordered fuzzy random variables and the method to generate pseudo-random ordered fuzzy variables with normal distribution are introduced.

Various technological challenges are essentially material problems in our times. New functional and functional graded nanomaterials are constructed of components with predefined properties. The design of nanostructures with predefined mechanical properties was considered in this paper. This study applies the evolutionary algorithm (EA) to the optimization problem in the design of nanomaterials. The optimal design combined EA with molecular dynamics to identify the size of the void for the prescribed elastic properties in monolayer 2D MoS2 nanostructures. The numerical results show that the proposed EA and the use of optimization method allowed accurately obtaining nanostructures with predefined mechanical material properties by introducing elliptical voids in the 2D MoS2 nanosheets.

The variation of elastic constants stiffness coefficients with respect to different percentage ratios of defects in monolayer molybdenum disulfide (MLMoS2) is reported for a particular set of atomistic nanostructural characteristics. The common method suggested is to use conventional defects such as single vacancy or di vacancy, and the recent studies use stone-walled multiple defects for highlighting the differences in the mechanical and electronic properties of 2D materials. Modeling the size influence of monolayer MoS2 by generating defects which are randomly distributed for a different percentage from 0% to 25% is considered in the paper. In this work, the geometry of the monolayer MoS2 defects modeled as randomized over the domain are taken into account. For simulation, the molecular static method is adopted and study the effect of elastic stiffness parameters of the 2D MoS2 material. Our findings reveals that the expansion of defects concentration leads to a decrease in the elastic properties, the sheer decrease in the elastic properties is found at 25%. We also study the diffusion of Molybdenum (Mo) in Sulphur (S) layers of atoms within MoS2 with Mo antisite defects. The elastic constants dwindle in the case of antisite defects too, but when compared to pure defects, the reduction was to a smaller extent in monolayer MoS2. Nevertheless, the Mo diffusion in sulfur gets to be more and more isotropic with the increase in the defect concentrations and elastic stiffness decreases with antisite defects concentration up to 25%. The distribution of antisite defects plays a vital role in modulating Mo diffusion in sulfur. These results will be helpful and give insights in the design of 2D materials.

This chapter is devoted to the single and multiobjective optimization problems. The chapter contains formulation of diverse mechanical and thermo-mechanical problems as well as practical applications of optimal design for the problems considered. The shape and topology optimization for various types of problems is considered. Several objectives for optimization problems are proposed, formulated and implemented. The bio-inspired algorithms and hybrid algorithms coupled with FEM or BEM are used in numerical examples. The formulation and solutions of sample problems for linear elastic, nonlinear elastoplastic, composites and structures with crack are described in detail. Optimization problems with more than one criterion are presented in the context of optimization for coupled fields problems.

The chapter presents various methods that can be qualified as intelligent computing ones. Different bio-inspired methods and techniques in the form of evolutionary algorithms (EAs), artificial immune systems (ANNs), particle swarm optimizers (PSOs) and artificial immune systems (AISs) are described. Moreover, information granularity attitude is introduced to model some uncertainties in material properties, geometry or boundary conditions. Granular computing techniques using interval numbers, fuzzy numbers and random variables are presented. Combinations of EAs and granular computing techniques in the form of fuzzy and stochastic EAS are proposed. Various hybrid computational intelligence algorithms combining different, intelligent or conventional techniques (e.g. gradient optimization methods) are described. A brief comparison of the effectiveness of selected bio-inspired optimization methods (PSO, EA and AIS) for the chosen test functions is also included.

This chapter contains a description of computational models of structures. The finite-element method (FEM) is presented for linear elastic structures. An alternative computational technique based on the boundary element method (BEM) is also described. The advantages and disadvantages of both the methods are identified. The coupling of the finite-element method with the boundary element method is also considered. The advantages of the coupled approach are depicted.

This chapter is devoted to the application of the bio-inspired methods in solving various inverse problems for mechanical systems. Several identification problems are considered. Various problems such as identification of boundary conditions for thermo-elastic and cracked structures, defects in elastic, thermo-elastic and cracked structures, and of material properties are formulated, numerically implemented and solved. Appropriate identification functionals for specific problems are formulated and implemented. Real object models were simulated using finite-element method (FEM) or boundary element method (BEM). The problems are solved for both ideal, deterministic values of measurements in sensors and noisy data. Evolutionary algorithms (EAs) and artificial immune systems (AISs) are used to solve the identification problems. Additionally, identification methods and procedures are supported by artificial neural networks (ANNs) and neuro-fuzzy inference system (NFISs). The effectiveness of the proposed method is demonstrated in several numerical examples of identification.

Two potentially new, 2D- graphene -like materials have been generated by the two stage searching strategy combining molecular and ab initio approach. The two candidates obtained from the evolutionary based algorithm and molecular calculations were then in depth analysed using first-principles Density Functional Theory from the mechanical, structural, phonon and electronic properties point of view. Both proposed polymorphs of graphene ( oP8 -P2mm) are mechanically and dynamically stable and can be metallic-like.

Potentially new, single-atom thick semiconducting 2D-graphene-like material, called Anisotropic-cyclicgraphene, have been generated by the two stage searching strategy linking molecular and ab initio approach. The candidate derived from the evolutionary based algorithm and molecular simulations was then profoundly analysed using first-principles density functional theory from the structural, mechanical, phonon, and electronic properties point of view. The proposed polymorph of graphene (rP16-P1m1) is mechanically, dynamically, and thermally stable and can be semiconducting with a direct band gap of 0.829 eV., Comment: 15 pages, 14 figures

The flat, two dimensional materials play important role in the research and industrial applications in the last 15 years. The new materials with flat atomic structures are discovered almost every month. The focus of the paper is on the discrete modeling of the single layer molybdenum disulphide based material (SLMoS2). Two methods, based on the molecular statics and molecular dynamics of estimation of material properties and numerical simulations at the nanolevel are described and discussed.

The article describes the application of a Hybrid Parallel Evolutionary Algorithm (HPEA) to optimal searching for new, stable atomic arrangements of two-dimensional graphene-like carbon lattices. The proposed approach combines the parallel evolutionary algorithm and the conjugated-gradient optimization technique. The main goal of the optimization is to find stable arrangements of carbon atoms under certain imposed conditions (e.g. density, shape and size of the unit cell). The fitness function is formulated as the total potential energy of an atomic system. The optimized structure is considered as a discrete atomic model and interactions between atoms are modeled using the AIREBO potential, especially developed for carbon and hydrocarbon materials. The parallel approach used in computations allows significant reduction of computation time. Validation of the obtained results and examples of the models of the new 2D materials obtained using the described algorithm are presented, along with their mechanical properties.

Instytuty Polskiej Akademii Nauk mają wśród polskich instytucji naukowych czołowy potencjał merytoryczny. W instytutach PAN w latach 2013-2016 powstało 19,7% prac afiliowanych w polskich instytucjach i umieszczonych w najbardziej prestiżowych pismach naukowych danych dziedzin (górne 10% z list pism danej dziedziny nauki uszeregowanych według rosnącego współczynnika wpływu, IF). Jest to najlepszy wynik wśród polskich instytucji akademickich. Kadra instytutów PAN zajmuje się nie tylko badaniami, ale też dydaktyką; 1607 osób (stan na 31 grudnia 2016) było na stacjonarnych studiach doktoranckich w Instytutach PAN. Stopień umiędzynarodowienia studiów w PAN (8%) jest największy wśród polskich uczelni. W związku z tym uzasadnione wydaje się powołanie Uniwersytetu PAN (UPAN), który mógłby stać się pierwszą w Polsce uczelnią badawczą. Planuje się, że docelowo kształciłoby się na nim 2,5 tys. osób, w przeważającej części studentek i studentów III stopnia. Taka liczba pozwoli na zagwarantowanie indywidualnej merytorycznej opieki każdemu studentowi, a także na wprowadzenie i przetestowanie nowatorskich programów dydaktycznych, właściwych erze Internetu i e-learningu, prowadzenia studiów online poprzez courser, studiów inter- i crossdyscyplinarnych. UPAN ma realne szanse stać się wizytówką polskiej nauki i szkolnictwa wyższego, gdyż otwarte, międzynarodowe wieloletnie programy konkursowe na pozycje: wizytujących profesorów, stażystów podoktorskich i doktorantów mogą podnieść umiędzynarodowienie zarówno kadry, jak i studentów UPAN do poziomu właściwego najlepszym uczelniom świata. Z uwagi na wielkość naszego budżetu nauki i szkolnictwa wyższego osiągnięcie takiego umiędzynarodowienia jest finansowo realne w najbliższym czasie tylko dla uczelni tak małej jak UPAN. Opisane działania będą projakościowe także dla samych instytutów PAN i istotnie podniosą ich poziom merytoryczny. Po kilku latach UPAN ma realne szanse na uplasowanie się w międzynarodowych rankingach na bardzo dobrych pozycjach, w pierwszej dwusetce, a nawet w pierwszej setce najlepszych światowych uczelni. Należy podkreślić, że warunkiem tego jest zapewnienie finansowania badań w instytutach PAN na co najmniej takim jak obecnie poziomie, uelastycznienie ich sieci, wytworzenie mechanizmów synergii pomiędzy instytutami (wspólny cel – UPAN) oraz wsparcie tego projektu długoletnim programem umiędzynarodowienia kadry i studentów.

A volatility forecasting comparative study between the most popular original GARCH model and the same model defined based on concepts of Ordered Fuzzy Numbers and Ordered Fuzzy Candlsticks is presented. These approaches offer a suitable tool to handle both imprecision of measurements and uncertainty associated with financial data. Therefore, they are particularly useful for volatility forecasting, since the volatility is unobservable and a proxy for it is used (realised volatility). In presented study, based on intra-daily data of the Warsaw Stock Exchange Top 20 Index (WIG 20), one showed that based on the adjusted-R squared and several prediction measurements, the fuzzy approach does perform better than the original GARCH model and forecasts more precisely in both the in-sample and out-of-sample predictions.

The purpose of this chapter is to present how Ordered Fuzzy Numbers (OFNs) can be used with financial high-frequency time series. Considering this approach the financial data are modeled using OFNs called further ordered fuzzy candlesticks. Their use allows modeling uncertainty associated with financial data and maintaining more information about price movement at an assumed time interval than compared to commonly used price charts (e.g., Japanese Candlestick chart). Furthermore, in a simple way, it is possible to include the information about volume and the bid-ask spread. Thanks to the well-defined arithmetic of OFNs, one can be used in technical analysis or to construct models of fuzzy time series in the form of classical equations. Examples of an ordered fuzzy moving average indicator and ordered fuzzy autoregressive process are presented.

The paper deals with the two-scale approach to the identification of material constants in composite materials. Structures made of unidirectionally fibre-reinforced composites are examined. Composite constituents’ elastic constants in a micro scale are identified on the basis of measurements performed in a macro scale. Numerical homogenization methods using a representative volume element are employed. Static (displacements in sensor points) and dynamic (eigenfrequencies) data are considered as measurements. Ideal and disturbed measurements are taken into account. Computational intelligence methods in the form of evolutionary algorithms and artificial immune systems are used to perform the identification procedure. Finite element method is used to solve the boundary-value problem for composites in both scales. Numerical examples presenting the effectiveness of the proposed approach are attached. Statistical data are considered to compare the efficiency of the identification procedure for both algorithms and different measurement data.

The goal of the paper is to present an experimental evaluation of fuzzy time series models which are based on ordered fuzzy numbers to predict financial time series. Considering this approach the financial data is modeled using Ordered Fuzzy Numbers (OFNs) called further by Ordered Fuzzy Candlesticks (OFCs). The use of them allows modeling uncertainty associated with financial data and maintaining more information about price movement at assumed time interval than comparing to commonly used price charts (e.g. Japanese Candlestick chart). Thanks to well-defined arithmetic of OFN, one can construct models of fuzzy time series, such as an Ordered Fuzzy Autoregressive Process (OFAR), where all input values are OFC, while the coefficients and output values are arbitrary OFN; in the form of classical equations, without using rule-based systems. In an empirical study ordered fuzzy autoregressive models are applied to modeling and predict price movement of futures contracts on Warsaw Stock Exchange Top 20 Index.

The paper deals with an application of the evolutionary algorithm (EA) and the particle swarm optimizer (PSO), to the optimization of shape, topology and material properties of spatial structures. Structures considered in this work are analyzed by the finite element method (FEM). Optimization criteria that are taken into account concern minimize mass and/or compliance. Numerical examples demonstrate that the method based on soft computing is a very effective technique for solving computer aided optimal design.

The paper deals with an application of a hybrid artificial immune system (HAIS) to the identification problems. The HAIS is applied to identify complex impedances of room walls. This approach is based on the mechanism discovered in biological immune systems. The numerical example demonstrates that the method based on immune computation is an effective technique for solving computer aided in identification problem.

The paper is devoted to the optimization of energy of carbon based atomic structure with use of the memetic algorithm. The graphene like atoms structure is coded into floating point genes and underwent evolutionary changes. The global optimization algorithm is supported by local gradient based improvement of chromosomes. The optimization problem is solved with the use of Intel PHI (Intel Many Integrated Core Architecture – Intel MIC). The example of optimization and speedup measurement for parallel optimization are given in the paper.

The paper is devoted to the application of the swarm methods and the finite element method to optimization of the stiffeners location in the 2-D structures (plane stress, bending plates and shells). The structures are optimized for the stress and displacement criteria. The numerical examples demonstrate that the method based on the swarm computation is an effective technique for solving the computer aided optimal design. The additional comparisons of the effectiveness of the particle swarm optimizer (PSO) and evolutionary algorithms (EA) are presented.

In present paper an improved multi-objective evolutionary algorithm is used for Pareto optimization of selected coupled problems. Coupling of mechanical, electrical and thermal fields is considered. Boundary-value problems of the thermo-elasticity, piezoelectricity and electro-thermo-elasticity are solved by means of finite element method (FEM). Ansys Multiphysics and MSC.Mentat/Marc software are used to solve considered coupled problems. Suitable interfaces between optimization tool and the FEM software are created. Different types of functionals are formulated on the basis of results obtained from the coupled field analysis. Functionals depending on the area or volume of the structure are also proposed. Parametric curves NURBS are used to model some optimized structures. Numerical examples for exemplary three-objective optimization are presented in the paper.

This article deals with an application of the artificial immune system (AIS) to the identification problem of piezoelectric structures analysed by the boundary element method (BEM). The AIS is applied to identify material properties of piezoelectrics. The AIS is a computational adaptive system inspired by the principles, processes and mechanisms of biological immune systems. The algorithms typically use the characteristics of immune systems, such as learning and memory to simulate and solve a problem in a computational manner. The main advantage of the AIS, contrary to gradient methods of optimization, is the fact that it does not need any information about the gradient of fitness function.

During the last decade artificial immune systems have drawn much of the researchers' attention. All the work that has been done allowed to develop many interesting algorithms which come in useful when solving engineering problems such as data mining and analysis, anomaly detection and many others. Being constantly developed and improved, the algorithms based on immune metaphors have some limitations, though. In this paper we elaborate on the concept of a novel artificial immune algorithm by considering the possibility of combining the clonal selection principle and the well known K-means algorithm. This novel approach and a new way of performing suppression (based on the usefulness of the evolving lymphocytes) in clonal selection result in a very effective and stable immune algorithm for both unsupervised and supervised learning. Further improvements to the cluster analysis by means of the proposed algorithm, immune K-means, are introduced. Different methods for clusters construction are compared, together with multi-point cluster validity index and a novel strategy based on minimal spanning tree (mst) and a analysis of the midpoints of the edges of the (mst). Interesting and useful improvements of the proposed approach by means of negative selection algorithms are proposed and discussed.

An application of evolutionary algorithms and the finite-element method to the topology optimization of 2D structures (plane stress, bending plates, and shells) and 3D structures is described. The basis of the topological evolutionary optimization is the direct control of the density material distribution (or thickness for 2D structures) by the evolutionary algorithm. The structures are optimized for stress, mass, and compliance criteria. The numerical examples demonstrate that this method is an effective technique for solving problems in computer-aided optimal design. †This is an extended and enhanced version of work presented at the mini−symposium on Evolutionary Algorithms: Recent Applications in Engineering and Science organized by Dr William Annicchiarico at the 7th World Congress on Computational Mechanics, Los Angeles, July 2006.

The paper presents a new algorithm for feature selection and classification. The algorithm is based on an immune metaphor, and combines both negative and clonal selection mechanisms characteristic for B- and T-lymphocytes. The main goal of the algorithm is to select the best subset of features for classification. Two level evolution is used in the proposed system for detectors creation and feature selection. Subpopulations of evolving detectors (T-lymphocytes) are able to discover subsets of features well suited for classification. The subpopulations cooperate during evolution by means of a novel suppression mechanism which is compared to the traditional suppression mechanism. The proposed suppression method proved to be superior to the traditional suppression in both recognition performance and its ability to select the proper number of subpopulations dynamically. Some results in the task of ECG signals classification are presented. The results for binary and real coded T-lymphocytes are compared and discussed.

In the paper an application of the particle swarm optimizer (PSO) and artificial immune system (AIS) to optimization problems is presented. Reinfored structures considered in this work are dynamically loaded and analyzed by the coupled boundary and finite element method (BEM/FEM). The metod is applied to optimize location of stiffeners in plates using criteria depended on displacements. The main advantage of the particle swarm optimizer, contrary to gradient methods of optimization, is the fact that it does not need any information about the gradient of fitness function. A comparison of the PSO, artificial immune system and evolutionary algorithm (EA) is also shown and it proves the efficiency of the former over other artificial intelligence methods of optimization. The coupled BEM/FEM, which is used to analyse structures, is very accu-rate in analysis and attractive in optimization tasks. It is because of problem dimensionality reduction in comparison with more frequently used domain methods, like for instance the FEM. Numerical examples demonstrate that the combination of the PSO with the BEM/FEM is an effective technique for solving computer aided optimal design problems, both with respect to accuracy and computational resources.

The paper deals with an application of the artificial immune system (AIS) to the optimization of shape, topology and material properties of 2-D and 3-D structures. Structures considered in this work are analyzed by the finite element method (FEM). Optimization criteria that are taken into account concern minimize mass of the structure. Numerical examples demonstrate that the method based on soft computing is a very effective technique for solving computer aided optimal design.

The aim of the paper is to present the application of the distributed evolutionary algorithms to selected optimization and defect identification problems. The coupling of evolutionary algorithms with the finite element method and the boundary element method creates a computational intelligence technique that is very suitable in computer aided optimal design. Several numerical examples for shape, topology optimization and identification are presented for elastic, thermoelastic and elastoplastic structures.

The paper is devoted to application of evolutionary algorithms and the boundary element method to shape optimization of structures for various thermomechanical criteria, inverse problems of finding an optimal distribution of temperature on the boundary and identification of unknown boundary. Design variables are specified by Bezier curves. Several numerical examples of evolutionary computation are presented.

The paper is devoted to applications of two bio-inspired methods: artificial immune systems and particle swarm optimizers to selected shape and topology optimization problems of structures. It contains numerical examples and comparisons of immune and swarm approaches with evolutionary optimization.