Extreme events, such as rogue waves, earthquakes, and stock market crashes, occur spontaneously in many dynamical systems. Because of their usually adverse consequences, quantification, prediction, and mitigation of extreme events are highly desirable. Here, we review several aspects of extreme events in phenomena described by high-dimensional, chaotic dynamical systems. We especially focus on two pressing aspects of the problem: (i) mechanisms underlying the formation of extreme events and (ii) real-time prediction of extreme events. For each aspect, we explore methods relying on models, data, or both. We discuss the strengths and limitations of each approach as well as possible future research directions.

The purpose of Part 1 of this paper is to provide a review of recent results from 1991 through 2003 in the area of theoretical aspects of statistical and equivalent linearization in the analysis of structural and mechanical nonlinear stochastic dynamic systems. First, a discussion about misunderstandings appearing in the literature in derivation of linearization coefficients for mean-square linearization criterion is presented. In Secs. 3–6 new theoretical results, including new types of criteria, nonlinearities, and excitations in the context of linearization methods, are reviewed. In particular, moment criteria called energy criteria, linearization criteria in the space of power spectral density functions and probability density functions are discussed. A survey of a wide class of so-called nonlinearization techniques, including equivalent quadratization and equivalent cubicization methods, is given in Sec. 7. New linearization techniques for nonlinear stochastic systems with parametric Gaussian excitations and external non-Gaussian excitations are discussed in Secs. 8 and 9, respectively. In the last sections, four surveys of papers where stochastic linearization is used as a mathematical tool in other theoretical approaches, namely, models of dynamic systems with hysteresis, finite element method, and control of nonlinear stochastic systems and linearization with sensitivity analysis, are given. A discussion of the accuracy analysis of linearization techniques and some general conclusions close this paper. There are 217 references cited in this revised article.

The article presents a review of the theory of reliability and its use for design code making procedures based on the probabilistic approach. Principles of structural safety in design code making are considered. To do so, the basic principles of the limit state methods in the form of partial factors and ways of their development are analyzed. The problem of manufactured structures or specimen reliability evaluation is discussed according to the control test results. Classification of loads and actions, which were adopted during the code making, is presented. Descriptions of the load combinations are considered. Evaluation of the risk in civil engineering is observed and estimations of the acceptable and optimal risk are discussed. Development of reliability based design principles is also presented. Based on review of preceding work, general requirements for probabilistic codes are described and the critical observations of the modern methods of the analysis of the failure probability are presented. This review article cites 125 references.

Major aspects of design and operation of pneumatically agitated bioreactors are reviewed. The focus is on considerations that are relevant to industrial practice. Airlift bioreactors are emphasized. The treatment covers hydraulics, hydrodynamics, gas-liquid and solid-liquid mass transfer, heat transfer, mixing, and suspension. Newtonian and non-Newtonian systems are discussed. Applications in microbial fermentations, animal and plant cell culture, biotransformations with immobilized enzymes, and treatment of wastewater are outlined. Comparisons with more conventional bioreactor technologies are made. Design features for sterile processing in airlift systems are detailed. The evidence for superior performance of airlift bioreactors is overwhelming. Excellent productivities have been demonstrated with yeasts, bacteria, and filamentous fungi. Processes that produce highly viscous broths, including several biopolymer producing fermentations, have been proven in airlift devices. Similarly, many hybridoma cultures and plant cell suspensions have given good results. As a general rule, volumetric productivity of airlift bioreactors equals or betters that of conventional stirred tanks. Typically, this level of performance is achieved at substantially lower power input than in stirred vessels. Furthermore, the probability of mechanical failure and likelihood of loss of sterility are lower with airlift bioreactors. In wastewater treatment, too, airlift devices have far outperformed conventional systems. Airlift bioreactors accept higher BOD loadings, produce less sludge, and the degradation rate is faster; performance improves with increasing scale of operation. This review article includes 328 references.

A numerical procedure of evaluating the exceedance probabilities of MDOF-systems under non-stationary random excitation is presented. The method is based on a newly developed Controlled Monte Carlo simulation procedure applicable to dynamical systems. It uses “Double and Clump” to assess the low probability domain and employs further intermediate thresholds to increase the efficiency of MCS for estimating first passage probabilities. Applied to a hysteretic type of MDOF-system, the method shows good results when compared with direct MCS.

The present work studies two deductions used to determine the local probability of fracture and whose final expressions coincide only in the case of a Weibull specific risk of fracture function of two parameters. As a result thereof, the inconsistency of one of the two deductions is shown and it is explained how to differentiate the two approaches experimentally. Three-point bending of a rectangular and a round beam is studied by resorting to the integral equations method in order to find the Weibull specific risk function.

Grain sorting is a process observed in fluvial, coastal, and estuarian environments whereby the selective transport of different fractions of a sediment mixture gives rise to a non-uniform spatial distribution of the grain size probability density function. The formation of some fluvial bedforms (bedloadsheets and sand ridges) is shown to be generated or dominantly controlled by grain sorting. However, most bedforms (river dunes, free and forced fluvial bars, coastal ripples) are moderately affected by the sorting mechanism which is invariably found to lead to a damping effect on bedform growth. Recent investigations of the above phenomena are reviewed and new methodologic aspects arising in treating the instability of flow and bed topography in the presence of mixtures are pointed out.

Crack propagation in heterogeneous media is of primary interest for engineering purposes, in order to predict the overall toughness and the probability of fracture from data on the microstructure. Probabilistic models for mode I crack propagation in two dimensions are presented. They are developed for brittle elastic materials with a random distribution of fracture energy. These models enable us to calculate in a closed form the probability of fracture involving crack nucleation and propagation that differ from the usual fracture statistics models based on the weakest link model. The use of the Griffith’s crack arrest criterion is applied to random function models for the distribution of the fracture energy and for various loading conditions resulting in stable or unstable crack propagation. From the models are deduced some statistical size effects.

In the last decade, progress in brittle material technology (CC-composites, ceramics, ceramic composites, etc) made these so attractive for structural applications that one has to put up with their tendency to fail unpredictably, provided that one can predict with confidence probabilities of failure . The present paper addresses prediction of life-time scatter for brittle materials. In the paper, we model a commonly observed mode of slow crack growth in brittle materials, namely a Markovian stochastic pattern of a microscopic random jump, followed by a random waiting time, followed by a random jump, and so on. The waiting times are related, on physical grounds, to random energy barriers at the arrest points, whereas random magnitudes of the jumps are treated within the existing framework of Crack Diffusion Theory. This leads to a description of crack growth as a random process whose transition probability density satisfies a hyperbolic PDE. Relation to probabilistic life-time prediction is discussed and a simple illustrative example considered.

The principles of Fracture-Statistics Mechanics are presented using two functional equations, namely one for the cumulative probability of fracture, and another for the local probability of fracture. These two functional equations are independent and they become compatible only when the volume considered is very small. The determination of the specific-risk function can be made by means of integral equations, without having to specify the analytical expression for this function. This two principles give two principles of uncertainty. Some applications to seismology are given where it is shown that the possibilities of predicting the instant of occurrence and the magnitude of an earthquake are null. Only the probability of occurrence of an earthquake of a given magnitude in a given place can be known. The instant of occurrence, the magnitude and the location are aleatory.

The role of homoclinic effects in solution of a reconstruction problem of system attractors and model equations from experimental observable in the presence of external noise is investigated numerically. It is shown that the possibility of reconstruction essentially depends on character of origin system homoclinic trajectories and noise intensity. If the homoclinic structure belongs to the attractor, then the reconstruction results in restoration origin system attractors. A small noise influence causes in this case a small perturbation of attractors probability measure and practically disappears due to filtering properties of the reconstruction algorithm. The homoclinic structure does not belong to the attractor, then in the absence of noise the probability measure concentrates at the attractor, the structure of which is not defined by the homoclinics. The noise perturbation induces new regimes. Then the attractor structure essentially depends on the homoclinics structure and noise level. In this case the model system attractor of which reproduces “invisible” homoclinic structure, is obtained as a result of reconstruction.

To address the issues of aging of energy production and distribution systems, and those associated with durability and reliability of new systems, improvements in the methods for service life prediction are needed to adequately account for the contributions of material and environmental variables. A mechanistically based probability approach is proposed. The efficacy of this approach and the importance of materials considerations are discussed. Support of research, utilizing this approach to address specific failure mechanisms, is recommended.

In this paper a micromechanical approach to damage growth in graph-representable microstructures is presented. Damage is defined as an elastic-inelastic transition in the grain boundaries and is represented in terms of a binary or ternary random field Z on the graph. A method based on the percolation theory brings out the size effects in scatter of strength, and the fractal character of damage geometry, and thus provides a basis for a multifractal model of a range of damage phenomena. The Markov property of field Z leads to a description of Z in terms of Gibbs probability measures and establishes a link between the entropy of disorder of Z and the physical entropy of damage in the ensemble of material specimens. Derivation of stochastic constitutive laws is outlined using the formalism of free energy and the dissipation function extended to random media.

Three types of aleatory variables can be considered here as determinants of the probabilistic strength of a structure, namely, material properties, boundary conditions, and external agents. The probability of fracture or yielding of some material subjected to external forces verifies F˜ ( V 1 + V 2 , σ) = F˜ ( V 1 , σ) F˜ ( V 2 , σ), where F = 1 – F˜ is the cumulative probability of fracture or yielding and σ is a constant stress on the material. In this manner there can be treated general frameworks or truss structures, soil problems, pressure vessels in general, tunnels in the rock, fatigue, and finally seisms caused by sudden earth-crust fractures.

This review examines the field of structural analysis where finite element methods (FEMs) are used in a probabilistic setting. The finite element method is widely used, and its application in the field of structural analysis is universally accepted as an efficient numerical solution method. The analysis of structures, whether subjected to random or deterministic external loads, has been developed mainly under the assumption that the structure’s parameters are deterministic quantities. For a significant number of circumstances, this assumption is not valid, and the probabilistic aspects of the structure need to be taken into account. We present a review of this emerging field: stochastic finite element methods. The terminology denotes the application of finite element methods with a probabilistic context. This broad definition includes two classes of methods: (i) first- and second-order second moment methods, and (ii) reliability methods. This paper addresses only the first category, leaving the second to specialists in that area. The contribution of this review is to illustrate the similarities and differences of the various methods falling in the first category. Also excluded from this review are simulation methods such as Monte Carlo and response surface, and methods that use FEM to solve deterministic equations (Fokker–Planck) governing probability densities. The essential conclusion is that the second moment methods are mathematically identical to the second order (except for the Neumann expansion). The essential distinction that can be made regarding stochastic FEM is the nature of the structure: It can be deterministic or random. By random structure is meant one with parameters that have associated uncertainties, and thus which must be modeled in a random form. Although the randomness in the structure can be of three types, random variable, random process in space, and random process in time, discussion will be limited to the first two categories. While keeping the emphasis on finite element methods, other techniques involving finite differences, which are useful in the study of multi-degree-of-freedom systems, are briefly mentioned. The present review covers only developments that are derived from the engineering literature, thus implying near-term applicability.

For offshore structures the fatigue limit state is governing the structural dimensions of several members and joint connections. Safety against fatigue failure is achieved through a combination of design requirements and performance of in-service inspections with repair of detected fatigue cracks. A review of uncertainties involved in fatigue life predictions by fracture mechanics is presented with particular reference to steel structures. Sources of uncertainties considered are: environmental conditions, hydrodynamic loading, global structural analysis, local stress calculation at fatigue sensitive points, and fatigue crack growth modeling by fracture mechanics. A probabilistic model using the fracture mechanics in probabilistic form is presented. This model accounts for uncertainties in loading, initial and critical defect sizes, material parameters, and in the uncertainty related to computation of the stress intensity factor. Failure probabilities are computed by first-order reliability methods and sensitivity factors are determined. Model updating based on in-service inspection results is formulated. Uncertainties with respect to detecting a crack and to correctly sizing a crack are included. Experience on application of the analysis method is presented.