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Guest Editorial

ASME J. Risk Uncertainty Part B. 2017;3(3):030301-030301-1. doi:10.1115/1.4036707.
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The fractional calculus, namely, the calculus of integrals and derivatives of real or complex order, has captured considerable popularity and importance in engineering applications during the last decades. The considerable number of scientific publications concerning fractional calculus and its applications in various and widespread fields of science and engineering demonstrates the importance of this mathematical tool. The growing interest in this field is due to the fact that the generalization of differential and integral operators provides some new tools to model, simulate, represent, and solve different kinds of engineering problems. Some examples are viscoelasticity, diffusive transport, electrical networks, probability and statistics, control theory of dynamical systems, chemical physics, optics and signal processing, and so on. Moreover, recently some important applications have regarded various engineering problems in stochastic mechanics. In fact, it has been proved that the fractional calculus is a valuable mathematical tool for the characterization of stochastic processes and random variables, for the simulation of external loads on structures, for the modeling of the internal damping of the medium, for the solution of the Fokker–Planck equation, etc. Basically, the fractional operators are commonly used as purely mathematical tools to solve stochastic differential equations and/or to describe and model mechanical behavior of real structures. In the first case, some contributions regard the solution of Langevin equations, the use of Mellin transform for the evaluation of complex-order spectral moments, the description of random variable and random processes, etc. In the second case, several applications of Caputo and Riemann–Liouville operators have showed the capabilities of these operators in the description of the mechanical behavior of structures and the modeling of real material constitutive laws. In particular, they are able to model nonlocality, viscoelasticity and fading memory effect, multiscale structures, Brownian motions and anomalous diffusion, etc.

Commentary by Dr. Valentin Fuster

SPECIAL SECTION PAPERS

ASME J. Risk Uncertainty Part B. 2017;3(3):030901-030901-6. doi:10.1115/1.4036700.

In the last decades, the research community has shown an increasing interest in the engineering applications of fractional calculus, which allows to accurately characterize the static and dynamic behavior of many complex mechanical systems, e.g., the nonlocal or nonviscous constitutive law. In particular, fractional calculus has gained considerable importance in the random vibration analysis of engineering structures provided with viscoelastic damping. In this case, the evaluation of the dynamic response in the frequency domain presents significant advantages, once a probabilistic characterization of the input is provided. On the other hand, closed-form expressions for the response statistics of dynamical fractional systems are not available even for the simplest cases. Taking advantage of the residue theorem, in this paper the exact expressions of the spectral moments of integer and complex orders (i.e., fractional spectral moments of linear fractional oscillators driven by acceleration time histories obtained as samples of stationary Gaussian white noise processes are determined.

Commentary by Dr. Valentin Fuster
ASME J. Risk Uncertainty Part B. 2017;3(3):030902-030902-5. doi:10.1115/1.4036806.

In this paper, we yield with a nonlocal elastic rod problem, widely studied in the last decades. The main purpose of the paper is to investigate the effects of the statistic variability of the fractional operator order s on the displacements u of the rod. The rod is supposed to be subjected to external distributed forces, and the displacement field u is obtained by means of numerical procedure. The attention is particularly focused on the parameter s, which influences the response in a nonlinear fashion. The effects of the uncertainty of s on the response at different locations of the rod are investigated by the Monte Carlo simulations. The results obtained highlight the importance of s in the probabilistic feature of the response. In particular, it is found that for a small coefficient of variation of s, the probability density function of the response has a unique well-identifiable mode. On the other hand, for a high coefficient of variation of s, the probability density function of the response decreases monotonically. Finally, the coefficient of variation and, to a small extent, the mean of the response tend to increase as the coefficient of variation of s increases.

Commentary by Dr. Valentin Fuster
ASME J. Risk Uncertainty Part B. 2017;3(3):030903-030903-5. doi:10.1115/1.4036701.

The paper deals with the stochastic dynamics of a vibroimpact single-degree-of-freedom system under a Gaussian white noise. The system is assumed to have a hard type impact against a one-sided motionless barrier, located at the system's equilibrium. The system is endowed with a fractional derivative element. An analytical expression for the system's mean squared response amplitude is presented and compared with the results of numerical simulations.

Commentary by Dr. Valentin Fuster
ASME J. Risk Uncertainty Part B. 2017;3(3):030904-030904-7. doi:10.1115/1.4036702.

Recently, a displacement-based nonlocal bar model has been developed. The model is based on the assumption that nonlocal forces can be modeled as viscoelastic (VE) long-range interactions mutually exerted by nonadjacent bar segments due to their relative motion; the classical local stress resultants are also present in the model. A finite element (FE) formulation with closed-form expressions of the elastic and viscoelastic matrices has also been obtained. Specifically, Caputo's fractional derivative has been used in order to model viscoelastic long-range interaction. The static and quasi-static response has been already investigated. This work investigates the stochastic response of the nonlocal fractional viscoelastic bar introduced in previous papers, discretized with the finite element method (FEM), forced by a Gaussian white noise. Since the bar is forced by a Gaussian white noise, dynamical effects cannot be neglected. The system of coupled fractional differential equations ruling the bar motion can be decoupled only by means of the fractional order state variable expansion. It is shown that following this approach Monte Carlo simulation can be performed very efficiently. For simplicity, here the work is limited to the axial response, but can be easily extended to transverse motion.

Commentary by Dr. Valentin Fuster
ASME J. Risk Uncertainty Part B. 2017;3(3):030905-030905-7. doi:10.1115/1.4036703.

In this paper, the probabilistic response of nonlinear systems driven by alpha-stable Lévy white noises is considered. The path integral solution is adopted for determining the evolution of the probability density function of nonlinear oscillators. Specifically, based on the properties of alpha-stable random variables and processes, the path integral solution is extended to deal with Lévy white noises input with any value of the stability index alpha. It is shown that at the limit when the time increments tend to zero, the Einstein–Smoluchowsky equation, governing the evolution of the response probability density function, is fully restored. Application to linear and nonlinear systems under different values of alpha is reported. Comparisons with pertinent Monte Carlo simulation data and analytical solutions (when available) demonstrate the accuracy of the results.

Commentary by Dr. Valentin Fuster
ASME J. Risk Uncertainty Part B. 2017;3(3):030906-030906-7. doi:10.1115/1.4036704.

It is well known that mechanical parameters of polymeric materials, e.g., epoxy resin, are strongly influenced by the temperature. On the other hand, in many applications, the temperature is not known exactly during the design process and this introduces uncertainties in the prevision of the behavior also when the stresses are deterministic. For this reason, in this paper, the mechanical behavior of an epoxy resin is characterized by means of a fractional viscoelastic model at different temperatures; then, a simple method to characterize the response of the fractional viscoelastic material at different temperatures modeled as a random variable with assigned probability density function (PDF) subjected to deterministic stresses is presented. It is found that the first- and second-order statistical moments of the response can be easily evaluated only by the knowledge of the PDF of the temperature and the behavior of the parameters with the temperature. Comparison with Monte Carlo simulations is also performed in order to assess the accuracy and the reliability of the method.

Commentary by Dr. Valentin Fuster
ASME J. Risk Uncertainty Part B. 2017;3(3):030907-030907-6. doi:10.1115/1.4036705.

In this paper, interval fractional derivatives are presented. We consider uncertainty in both the order and the argument of the fractional operator. The approach proposed takes advantage of the property of Fourier and Laplace transforms with respect to the translation operator, in order to first define integral transform of interval functions. Subsequently, the main interval fractional integrals and derivatives, such as the Riemann–Liouville, Caputo, and Riesz, are defined based on their properties with respect to integral transforms. Moreover, uncertain-but-bounded linear fractional dynamical systems, relevant in modeling fractional viscoelasticity, excited by zero-mean stationary Gaussian forces are considered. Within the interval analysis framework, either exact or approximate bounds of the variance of the stationary response are proposed, in case of interval stiffness or interval fractional damping, respectively.

Commentary by Dr. Valentin Fuster
ASME J. Risk Uncertainty Part B. 2017;3(3):030908-030908-8. doi:10.1115/1.4036706.

This paper presents a nonlinear rubber spring model for the primary suspension of the railway vehicle, which can effectively describe the amplitude dependency and the frequency dependency of the rubber spring, by taking the elastic force, the fractional derivative viscous force, and nonlinear friction force into account. An improved two-dimensional vehicle–track coupled system is developed based on the nonlinear rubber spring model of the primary suspension. Nonlinear Hertz theory is used to couple the vehicle and track subsystems. The railway vehicle subsystem is regarded as a multibody system with ten degrees-of-freedom, and the track subsystem is treated as finite Euler–Bernoulli beams supported on a discrete–elastic foundation. Mechanical characteristic of the rubber spring due to harmonic excitations is analyzed to clarify the stiffness and damping dependencies on the excitation frequency and the displacement amplitude. Dynamic responses of the vehicle–track coupled dynamics system induced by the welded joint irregularity and random track irregularity have been performed to illustrate the difference between the Kelvin–Voigt model and the proposed model in the time and frequency domain.

Commentary by Dr. Valentin Fuster

Research Papers

ASME J. Risk Uncertainty Part B. 2017;3(3):031001-031001-12. doi:10.1115/1.4035399.

Corrosion in ship structures is influenced by a variety of factors that are varying in time and space. Existing corrosion models used in practice only partially address the spatial variability of the corrosion process. Typical estimations of corrosion model parameters are based on averaging measurements for one ship type over structural elements from different ships and operational conditions. Most models do not explicitly predict the variability and correlation of the corrosion process among multiple locations in the structure. This correlation is of relevance when determining the necessary inspection coverage, and it can influence the reliability of the ship structure. In this paper, we develop a probabilistic spatiotemporal corrosion model based on a hierarchical approach, which represents the spatial variability and correlation of the corrosion process. The model includes as hierarchical levels vessel–compartment–frame–structural element–plate element. At all levels, variables representing common influencing factors (e.g., coating life) are introduced. Moreover, at the lowest level, which is the one of the plate element, the corrosion process can be modeled as a spatial random field. For illustrative purposes, the model is trained through Bayesian analysis with measurement data from a group of tankers. In this application, the spatial dependence among corrosion processes in different parts of the ships is identified and quantified using the proposed hierarchical model. Finally, how this spatial dependence can be exploited when making inference on the future condition of the ships is demonstrated.

Topics: Corrosion , Vessels , Ships
Commentary by Dr. Valentin Fuster
ASME J. Risk Uncertainty Part B. 2017;3(3):031002-031002-9. doi:10.1115/1.4035439.
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Optimization for crashworthiness is of vast importance in automobile industry. Recent advancement in computational prowess has enabled researchers and design engineers to address vehicle crashworthiness, resulting in reduction of cost and time for new product development. However, a deterministic optimum design often resides at the boundary of failure domain, leaving little or no room for modeling imperfections, parameter uncertainties, and/or human error. In this study, an operational model-based robust design optimization (RDO) scheme has been developed for designing crashworthiness of vehicle against side impact. Within this framework, differential evolution algorithm (DEA) has been coupled with polynomial correlated function expansion (PCFE). An adaptive framework for determining the optimum basis order in PCFE has also been presented. It is argued that the coupled DEA–PCFE is more efficient and accurate, as compared to conventional techniques. For RDO of vehicle against side impact, minimization of the weight and lower rib deflection of the vehicle are considered to be the primary design objectives. Case studies by providing various emphases on the two objectives have also been performed. For all the cases, DEA–PCFE is found to yield highly accurate results.

Commentary by Dr. Valentin Fuster
ASME J. Risk Uncertainty Part B. 2017;3(3):031003-031003-8. doi:10.1115/1.4035704.

Industry has been implementing condition monitoring (CM) for turbines to minimize losses and to improve productivity. Deficient conditions can be identified before losses occur by monitoring the equipment parameters. For any loss scenario, the effectiveness of monitoring depends on the stage of the loss scenario when the deficient condition is detected. A scenario-based semi-empirical methodology was developed to assess various types of condition monitoring techniques, by considering their effect on the risk associated with mechanical breakdown of steam turbines in the forest products (FPs) industry. A list of typical turbine loss scenarios was first generated by reviewing loss data and leveraging expert domain knowledge. Subsequently, condition monitoring techniques that can mitigate the risk associated with each loss scenario were identified. For each loss scenario, an event tree analysis (ETA) was used to quantitatively assess the variations in the outcomes due to condition monitoring, and resultant changes in the risk associated with turbine mechanical breakdown. An application was developed following the methodology to evaluate the effect of condition monitoring on turbine risk mitigation.

Commentary by Dr. Valentin Fuster
Select Articles from Part A: Civil Engineering

Technical Papers

ASCE-ASME Journal of Risk and Uncertainty in Engineering Systems, Part A: Civil Engineering. 2015;3(2):. doi:10.1061/AJRUA6.0000841.
Abstract 

Abstract  In stochastic analysis of engineering systems, the task of generating samples according to a target probability distribution involving some performance function of the system response often arises. This paper introduces an adaptive method for rejection sampling that uses adaptive kernel sampling densities (AKSD) as proposal densities for the rejection sampling algorithm in an iterative approach. The AKSD formulation relies on having available (1) a small number of samples from the target density, as well as (2) evaluations of the system performance function over some other sample set. This information is then used to establish the adaptive features of the stochastic sampling involving (1) an explicit optimization of the kernel characteristics for reduction of the computational burden, and so maximizing sampling efficiency, and (2) selection of the exact model parameters to target so that potential problems when forming proposal densities for high-dimensional vectors are avoided. Beyond this theoretical formulation of the adaptive stochastic sampling, its implementation within the context of Subset Simulation (SS) is also demonstrated, with the AKSD method utilized for generating independent conditional failure samples. Additionally, a modified rejection sampling algorithm is proposed for using AKSD in SS that can significantly reduce the required number of simulations of the system model response.

Topics:
Density , Simulation , Approximation , Sampling methods
ASCE-ASME Journal of Risk and Uncertainty in Engineering Systems, Part A: Civil Engineering. 2016;3(2):. doi:10.1061/AJRUA6.0000881.
Abstract 

Abstract  Axle loading spectrum inputs obtained from existing weigh-in-motion (WIM) stations are one of the key data elements required in the pavement mechanistic-empirical (ME) design. Because of limited number of WIM stations within a state agency, it is critical to implement clustering approaches to identifying similar traffic patterns and developing cluster average Level 2 inputs for a particular pavement design. Even though several states have applied clustering methods for this purpose, they rely solely on hierarchical-based method. Many other types of clustering techniques based on different induction principles are available but have not been tested. In this paper, four types of clustering methods, including agglomerative hierarchical, partitional K-means, model-based, and fuzzy c-means algorithms, are implemented to cluster traffic attributes for pavement ME design using data sets from 39 WIM sites in Michigan. Two case studies, one flexible pavement and one rigid pavement, are conducted. The impacts of using various clustering methods for preparation of Level 2 traffic inputs on pavement performance are examined. Cosine similarity analyses reveal that the four clustering methodologies generate highly comparable traffic inputs and predicted pavement performance as compared to the Level 1 results. However, the equivalent single axle loads (ESALs) from the four clustering methods can result in 12.7 mm (0.5 in.) of difference of designed surface layer thickness. The hierarchical method consistently has the lowest cosine similarity values, the fuzzy-based method has the highest similarity, while the other two clustering methods generally outperform the hierarchical method if Level 1 site-specific results are set as the benchmark. This study raises the awareness that more research is desired to select the most appropriate clustering approach for the development of Level 2 traffic inputs based on existing WIM data sets for pavement ME design.

Topics:
Design , Traffic , Trucks , Pavement
ASCE-ASME Journal of Risk and Uncertainty in Engineering Systems, Part A: Civil Engineering. 2016;3(2):. doi:10.1061/AJRUA6.0000889.
Abstract 

Abstract  Data mining is a discovery procedure to explore and visualize useful but less-than-obvious information or patterns in large collections of data. Given the amount and varying parameter types in a large data set such as that of the National Bridge Inventory (NBI), using traditional clustering techniques for discovery is impractical. As a consequence, the authors have applied a novel data discovery tool, called Two-step cluster analysis, to visualize associations between concrete bridge deck design parameters and bridge deck condition ratings. Two-step cluster analysis is a powerful knowledge discovery tool that can handle categorical and interval data simultaneously and is capable of reducing dimensions for large data sets. The analysis, of a total of 9,809 concrete highway bridge decks in the Northeast climatic region, found that bridges with cast-in-place decks that have a bituminous wearing surface, a preformed fabric membrane, and epoxy-coated reinforcement protection are strongly associated with the high condition ratings for bridge decks regardless of the average daily truck traffic (ADTT). Conversely, results show that bridges with cast-in-place bridge decks that have a bituminous wearing surface but have neither a deck membrane nor deck reinforcement protection are strongly associated with low condition ratings for bridge decks regardless of the ADTT. It was concluded that Two-step cluster analysis is a useful tool for bridge owners and agencies to visualize general trends in their concrete bridge deck condition data and to support them in their decision-making processes to effectively allocate limited funds for maintenance, repair, and design of bridge decks.

Topics:
Bridges (Structures) , Concretes , Data mining
ASCE-ASME Journal of Risk and Uncertainty in Engineering Systems, Part A: Civil Engineering. 2015;3(2):. doi:10.1061/AJRUA6.0000859.
Abstract 

Abstract  As traditionally infrastructure-centric industries such as the railways deploy ever more complex information systems, data interoperability becomes a challenge that must be overcome in order to facilitate effective decision making and management. In this paper, the authors propose a system based on semantic data modeling techniques to allow integration of information from diverse and heterogeneous sources. The results of work, which aimed to demonstrate how semantic data models can be used in the rail industry, are presented. These include a novel domain ontology for the railways, and a proof-of-concept real time passenger information system based on semantic web technologies. Methods and patterns for creating such ontologies and real world systems are discussed, and ontology-based techniques for integrating data with legacy information systems are shown.

Topics:
Ontologies , Data fusion , Rails
ASCE-ASME Journal of Risk and Uncertainty in Engineering Systems, Part A: Civil Engineering. 2016;3(2):. doi:10.1061/AJRUA6.0000868.
Abstract 

Abstract  A maintenance problem can be regarded as an optimization task, where the solution is a trade-off between the costs associated with inspection and repair activities and the benefits related to the faultless operation of the infrastructure. The optimization aims at minimizing the total cost while tuning some parameters, such as the number, time, and quality of inspections. Due to the unavoidable uncertainties, the expected cost of maintenance and failure can only be estimated by assessing the reliability of the system. The problem is, therefore, formulated as a time-variant reliability-based optimization, where both objective and constraint functions require the assessment of reliability with time. This paper proposes an efficient general numerical technique to solve this problem by means of just one single reliability analysis, while explicitly taking the diverse forms of uncertainty into account. The technique is generally applicable to any problem where the ageing or damage propagation process is known by means of input-output relationships, which apply to a great number of the cases. This technique exploits a Monte Carlo strategy derived from the concept of forced simulation, which significantly increases the efficiency of computing the optimal solution. The efficiency and accuracy of the proposed approach is shown by means of an example involving a fatigue-prone weld in a bridge girder.

Topics:
Inspection , Maintenance , Simulation , Design

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