Deterministic design and a priori parameters are used in traditional optimization approaches. The material characteristics of solid wood are not deterministic in reality. Hence, realistic optimization and simulation methods need to take the uncertainties of parameters into account. The uncertainty characteristics of wood are mainly originated in natural variation. In addition to this, incertitudes from lack of knowledge are inherent. Accordingly, the aleatoric approach of randomness can be expanded to a polymorphic uncertainty model. Fuzzy probability based randomness is used in this work. Therefore, the epistemic approach of fuzziness is taken into account. The distribution functions of random variables are parametrized by fuzzy variables. So coupling of both, aleatoric and epistemic uncertainties, is involved.Interactions of fuzzy variables and crosscorrelations of random variables are considered among and within the parameters. Crosscorrelated random fields are used to represent spatial variation of material parameters. The autocovariance structures are modeled structurally dependent on the tree trunk axes. FEM results are applied as basic solutions of a loaded timber structure. A local orthotropic material formulation with respect to specifically located tree trunk axes is used. The optimal positions of the tree trunk axes for each wooden log are examined as design parameters. Polymorphic uncertainty is used to describe a priori parameters. The developed methods for uncertainty analysis are embedded in an automated and parallelized optimization processing. An analysis of a two-tier glulam beam, according to a purlin of a timber roof construction, is shown as numerical example for the optimization framework.

A Discrete Direct (DD) model calibration and uncertainty propagation approach is explained and demonstrated on a 4-parameter Johnson-Cook (JC) strain-rate dependent material strength model for an Aluminum alloy. The methodology's performance is characterized in many trials involving four random realizations of strain-rate dependent material-test data curves per trial, drawn from a large synthetic population. The JC model is calibrated to particular combinations of the data curves to obtain calibration parameter sets which are then propagated to "Can Crush" structural model predictions to produce samples of predicted response variability. These are processed with appropriate sparse-sample uncertainty quantification (UQ) methods to estimate various statistics of response with an appropriate level of conservatism. This is tested on 16 output quantities (von Mises stresses and equivalent plastic strains) and it is shown that important statistics of the true variabilities of the 16 quantities are bounded with a high success rate that is reasonably predictable and controllable. The DD approach has several advantages over other calibration-UQ approaches like Bayesian inference for capturing and utilizing the information obtained from typically small number of replicate experiments in model calibration situations-especially when sparse replicate functional data are involved like force-displacement curves from material tests. The DD methodology is straightforward and efficient for calibration and propagation problems involving aleatory and epistemic uncertainties in calibration experiments, models, and procedures.

Multiplying the capacity of communication links by using the multiple-input multiple-output mechanism has become an essential part of various wireless standards. In this paper, we focus on the bit error rate in such systems and consider its optimization under parameter uncertainty from a formal point of view (including methods with result verification). The theoretical results are demonstrated using a close-to-life application.

Dry powder inhalers, used as a means for pulmonary drug delivery, typically contain a combination of active pharmaceutical ingredients (API) and significantly larger carrier particles. The micro-sized drug particles - which have a strong propensity to aggregate and poor aerosolization performance - are mixed with significantly large carrier particles that cannot penetrate the mouth-throat region to deagglomerate and entrain the smaller API particles in the inhaled airflow. Therefore, a DPI's performance depends on the carrier-API combination particles' entrainment and the time and thoroughness of the individual API particles' deagglomeration from the carrier particles. Since DPI particle transport is significantly affected by particle-particle interactions, particle sizes and shapes present significant challenges to CFD modelers to model regional lung deposition from a DPI. We employed the Particle-In-Cell method for studying the transport/deposition and the agglomeration and deagglomeration for DPI carrier and API particles in the present work. The proposed development will leverage CFD-PIC and sensitivity analysis capabilities from the Department of Energy laboratories: Multiphase Flow Interface Flow Exchange and Dakota UQ software. A data-driven framework is used to obtain the reliable low order statics of the particle's residence time in the inhaler. The framework is further used to study the effect of drug particle density, carrier particle density and size, fluidizing agent density and velocity, and some numerical parameters on the particles' residence time in the inhaler.

This paper studies the propagation of uncertainties on serviceability assessment of footbridges in unrestricted traffic condition based on a non-deterministic approach. Multi-pedestrian loading is modeled as a stationary Gaussian random process through the Equivalent Spectral Model [1] which yields analytical expressions of the spectral moments of the footbridge dynamic response. The uncertain pedestrian-induced loading parameters and structural dynamic properties are modeled as interval variables. An approximate analytical procedure, based on the Improved Interval Analysis [2], is introduced as an efficient alternative to classical optimization in order to propagate interval uncertainties. The presented procedure allows us to derive closed-form expressions of the bounds of the spectral moments of the response, as well as of the expected value and cumulative distribution function of the maximum footbridge acceleration. Two strategies are proposed to assess footbridges' serviceability. The first one leads to the definition of a range of comfort classes. The second strategy enables us to estimate an interval of probability of reaching at least a suitable comfort level.

The Bhattacharyya distance has been developed as a comprehensive uncertainty quantification metric by capturing multiple uncertainty sources from both numerical predictions and experimental measurements. This work pursues a further investigation of the performance of the Bhattacharyya distance in different methodologies for stochastic model updating. The first procedure is the Bayesian model updating where the Bhattacharyya distance is utilized to define an approximate likelihood function and the transitional Markov chain Monte Carlo algorithm is employed to obtain the posterior distribution of the parameters. In the second model updating procedure, the Bhattacharyya distance is utilized to construct the objective function of an optimization problem. The objective function is defined as the Bhattacharyya distance between the samples of numerical prediction and the samples of the target data. The comparison study is performed on a four degree-of-freedoms mass-spring system. A challenging task is raised in this example by assigning different distributions to the parameters with imprecise distribution coefficients. This requires the stochastic updating procedure to calibrate not the parameters themselves, but their distribution properties. The performance of the Bhattacharyya distance in both Bayesian updating and optimization-based updating procedures are presented and compared. The results demonstrate the Bhattacharyya distance as a comprehensive and universal uncertainty quantification metric in stochastic model updating.

In this article, a method is proposed to conduct a global sensitivity analysis of epistemic uncertainty on both system input and system structure, which is very common in early stage of system development, using Dempster-Shafer theory (DST). In system reliability assessment, the input corresponds to component reliability and system structure is given by system reliability function, cut sets, or truth table. A method to propagate real-number mass function through set-valued mappings is introduced and applied on system reliability calculation. Secondly, we propose a method to model uncertain system with multiple possible structures and how to obtain the mass function of system level reliability. Finally, we propose an indicator for global sensibility analysis. Our method is illustrated, and its efficacy is proved by numerical application on two case studies.

Reliability analysis evaluates the failure probability of structures considering random variables of a system. Existing methods such as First-order reliability (FORM) and Second-order reliability (SORM) are difficult to predict the failure probability of implicit functions in mechanical structures. Monte Carlo Simulation (MCS) can predict the failure probability with a high accuracy but it is time-consuming. Agent-based methods such as the Kriging model have the approved performance to predict the failure probability in both efficiency and accuracy. An active method is proposed in this paper to improve the efficiency of predicting the probability of failures by combining the Kriging model and MCS using a new learning function and its stopping condition. A representative selection strategy is developed based on spectral clustering to decide sample points in the design of experiments (DoE). The new learning function integrates uncertainty and similarity of predicted Kriging values to search the next best sample point for updating the initial DoE. The learning process is terminated based on the stopping condition for a given accuracy of predicting probability of failures. Four case studies are conducted to validate the proposed method. Results show that the proposed method can predict the probability of failures with the improved accuracy and reduced time.

The treatment of uncertainty using extra-probabilistic approaches, like intervals or p-boxes, allows for a clear separation between epistemic uncertainty and randomness in the results of risk assessments. This can take the form of an interval of failure probabilities; the interval width W being an indicator of "what is unknown". In some situations, W is too large to be informative. To overcome this problem, we propose to reverse the usual chain of treatment by starting with the targeted value of W that is acceptable to support the decision-making, and to quantify the necessary reduction in the input p-boxes that allows achieving it. In this view, we assess the feasibility of this procedure using two case studies (risk of dike failure, and risk of rupture of a frame structure subjected to lateral loads). By making the link with the estimation of excursion sets (i.e. the set of points where a function takes values below some prescribed threshold), we propose to alleviate the computational burden of the procedure by relying on the combination of Gaussian Process metamodels and sequential design of computer experiments. The considered test cases show that the estimates can be achieved with only a few tens of calls to the computationally intensive algorithm for mixed aleatory/epistemic uncertainty propagation.

In this study, we use possibility distribution as a basis for parameter uncertainty quantification in one-dimensional consolidation problems. A Possibility distribution is the one-point coverage function of a random set and viewed as containing both partial ignorance and uncertainty. Vagueness and scarcity of information needed for characterizing the coefficient of consolidation in clay can be handled using possibility distributions. Possibility distributions can be constructed from existing data, or based on transformation of probability distributions. An attempt is made to set a systematic approach for estimating uncertainty propagation during the consolidation process. The measure of uncertainty is based on Klir's definition (1995). We make comparisons with results obtained from other approaches (probabilistic…) and discuss the importance of using possibility distributions in this type of problems.

The well-known SPAR-H methodology is widely used for analysis of human reliability in complex technological systems. It allows assessing the human error probability taking into account eight important groups of performance shaping factors. Application of this methodology to practical problems traditionally involves assumptions which are difficult to verify under the conditions of uncertainty. In particular, it introduces only two possible values of the nominal human error probabilities (for diagnosis and for actions) which do not cover the whole spectrum of the tasks within operator's activity. In addition, although the traditional methodology considers the probabilities of human errors as the random variables, it operates only on a single predefined type of distribution for these variables and does not deal with the real situations in which the type of distribution remains uncertain. The paper proposes modification to the classical approach to enable more adequate modelling of real situations with the lack of available information. The authors suggest usage of the interval-valued probability technique and of the expert judgement on the maximum probability density for actual probabilities of human errors. Such methodology allows obtaining generic results that are valid for the entire set of possible distributions (not only for one of them). The modified methodology gives possibility to derive final assessments of human reliability in interval form indicating "the best case" and "the worst case". A few numerical examples illustrate the main stages of the suggested procedure.

This paper presents a novel methodology to solve an inverse uncertainty quantification problem where only the variation of the system response is provided by a small set of experimental data. Furthermore, the method is extended for cases where the uncertainty of the response quantities is given by an incomplete set of statistical moments. For both cases, the uncertainty on the output space is represented by a minimum volume enclosing ellipsoid (MVEE). The actual inverse uncertainty quantification is conducted by identifying also a hyper-ellipsoid for the input parameters which has an image on the output space that matches the MVEE as close as possible. Hence, the newly introduced approach is a contribution to the field of non-probabilistic uncertainty quantification methods. Compared to literature, the new approach has often superior accuracy and especially an improved efficiency for high-dimensional problems. The method is validated first by an analytical test case and subsequently applied to a jet engine performance model, where this type of inverse uncertainty quantification has to be solved to allow for a consistent and integrated solution procedure. In both cases, the results are compared with an inverse method where the variability on the input side is quantified by a multi-dimensional interval. It can be shown that the hyper-ellipsoid approach is superior with respect to the computation time in high-dimensional problems encountered not only in jet engine design.

In a probabilistic design approach for cylindrical shells Gaussian random fields are used to simulate geometric imperfections. The shape of imperfections depends, among others, on the autocorrelation properties of the random field. Underlying uncertainties like a small sample size or imprecise measurements make it practically impossible to define a crisp correlation function. For a more realistic description of the imprecise correlation structure, the classical probabilistic approach is extended to a fuzzy stochastic approach. More exactly, the polymorphic uncertainty approach is used taking into account natural variability and incompleteness. Consequently, geometric imperfections are represented as fuzzy probability based random fields. Therefore, the required correlation parameters are described as polymorphic uncertain parameters. The quantification of uncertainties is demonstrated on real data. Furthermore, the polynomial chaos surrogate model is used for the alpha-level optimization in the fuzzy analysis. The sensitivity indices as a by-product of the surrogate model show the influence of the input parameters on the statistical parameters of the critical buckling load factor. The main purpose of this paper is to show how the presented methods can support the design process of cylindrical shells.

Quantifying effects of system-wide uncertainties (i.e., affecting structural, piezoelectric, and/or electrical components) in the analysis and design of piezoelectric vibration energy harvesters have recently been emphasized. The present investigation proposes first a general methodology to model these uncertainties within a finite element model of the harvester obtained from an existing finite element software. Needed from this software are the matrices relating to the structural properties (mass, stiffness), the piezoelectric capacitance matrix as well as the structural-piezoelectric coupling terms of the mean harvester. The thermal analogy linking piezoelectric and temperature effects is also extended to permit the use of finite element software that do not have piezoelectric elements but include thermal effects on structures. The approach is applied to a beam energy harvester. Both weak and strong coupling configurations are considered, and various scenarios of load resistance tuning are discussed, i.e., based on the mean model, for each harvester sample, or based on the entire set of harvesters. The uncertainty is shown to have significant effects in all cases even at a relatively low level, and these effects are dominated by the uncertainty on the structure versus the one on the piezoelectric component. The strongly coupled configuration is shown to be better as it is less sensitive to the uncertainty and its variability in power output can be significantly reduced by the adaptive optimization, and the harvested power can even be boosted if the target excitation frequency falls into the power saturation band of the system.

A framework that allows the use of well-known dynamic estimators in piezoelectric harvesters (PEHs) (i.e., deterministic performance estimators) and that accounts for the random error associated with the mathematical model and the uncertainties of model parameters is presented here. This framework may be employed for Posterior Robust Stochastic analysis, such as when a harvester can be tested or is already installed and the experimental data are available. In particular, the framework detailed here is introduced to update the electromechanical properties of PEHs using Bayesian techniques. The updated electromechanical properties are identified by adopting a Transitional Markov Chain Monte Carlo. A well-known device with a nonlinear constitutive relationship is employed for experiments in this study, and the results demonstrated the capability of the proposed framework to update nonlinear electromechanical properties. The importance of including model parameter uncertainties to generate robust predictive tools is also supported by the results. Therefore, this framework constitutes a powerful tool for the robust design and prediction of PEH performance.

An energy harvesting dynamic vibration absorber (EHDVA) is studied to suppress undesirable vibrations in a host structure as well as to harvest electrical energy from vibrations using piezoelectric transduction. This work studies the feasibility of using vibration absorber for harvesting energy under random excitation and in presence of parametric uncertainties. A two degrees-of-freedom model is considered in the analytical formulation for the host along with the absorber. A separate equation is used for energy generation from piezoelectric material. Two studies are reported here: (i) with random excitation where the base input is considered to be Gaussian and (ii) parametric uncertainty is considered with harmonic excitation. Under random base excitation, the analytical results show that, with the proper selection of parameters, harvested electrical energy can be increased along with the reduction in vibration of the host structure. Graphs are reported showing tradeoff between harvested energy and vibration control. Whereas, Monte Carlo simulations are carried out to analyze the system with parametric uncertainty. This showed that the mean harvested power decreases with an increase in uncertainties in the natural frequency as well as damping ratio. In addition, optimal electrical parameters for obtaining maximum power for the case of uncertain parameters are also reported in this study.

The paper discusses applications of the domain-independent method of algebraic inequalities, for reducing uncertainty and risk. Algebraic inequalities have been used for revealing the intrinsic reliability of competing systems and ranking the systems in terms of reliability in the absence of knowledge related to the reliabilities of their components. An algebraic inequality has also been used to establish the principle of the well-ordered parallel-series systems which, in turn, has been applied to maximize the reliability of common parallel-series systems. The paper introduces linking an abstract inequality to a real process by a meaningful interpretation of the variables entering the inequality and its left- and right-hand parts. The meaningful interpretation of a simple algebraic inequality led to a counterintuitive result. If two varieties of items are present in a large batch, the probability of selecting randomly two items of different variety is smaller than the probability of selecting randomly two items of the same variety.

Reliability assessment of linear discretized structures with interval parameters subjected to stationary Gaussian multicorrelated random excitation is addressed. The interval reliability function for the extreme value stress process is evaluated under the Poisson assumption of independent up-crossing of a critical threshold. Within the interval framework, the range of stress-related quantities may be significantly overestimated as a consequence of the so-called dependency phenomenon, which arises due to the inability of the classical interval analysis to treat multiple occurrences of the same interval variables as dependent ones. To limit undesirable conservatism in the context of interval reliability analysis, a novel sensitivity-based procedure relying on a combination of the interval rational series expansion and the improved interval analysis via extra unitary interval is proposed. This procedure allows us to detect suitable combinations of the endpoints of the uncertain parameters which yield accurate estimates of the lower bound and upper bound of the interval reliability function for the extreme value stress process. Furthermore, sensitivity analysis enables to identify the most influential parameters on structural reliability. A numerical application is presented to demonstrate the accuracy and efficiency of the proposed method as well as its usefulness in view of decision-making in engineering practice.