The early design stage of mechanical structures is often characterized by unknown or only partially known boundary conditions and environmental influences. Particularly, in the case of safety-relevant components, such as the crumple zone structure of a car, those uncertainties must be appropriately quantified and accounted for in the design process. For this purpose, possibility theory provides a suitable tool for the modeling of incomplete information and uncertainty propagation. However, the numerical propagation of uncertainty described by possibility theory is accompanied by high computational costs. The necessarily repeated model evaluations render the uncertainty analysis challenging to be realized if a model is complex and of large scale. Oftentimes, simplified and idealized models are used for the uncertainty analysis to speed up the simulation while accepting a loss of accuracy. The proposed multifidelity scheme for possibilistic uncertainty analysis, instead, takes advantage of the low costs of an inaccurate low-fidelity model and the accuracy of an expensive high-fidelity model. For this purpose, the functional dependency between the high- and low-fidelity model is exploited and captured in a possibilistic way. This results in a significant speedup for the uncertainty analysis while ensuring accuracy by using only a low number of expensive high-fidelity model evaluations. The proposed approach is applied to an automotive car crash scenario in order to emphasize its versatility and applicability.

We consider biotransport in tumors with uncertain heterogeneous material properties. Specifically, we focus on the elliptic partial differential equation (PDE) modeling the pressure field inside the tumor. The permeability field is modeled as a log-Gaussian random field with a prespecified covariance function. We numerically explore dimension reduction of the input parameter and model output. Specifically, truncated Karhunen–Loève (KL) expansions are used to decompose the log-permeability field, as well as the resulting random pressure field. We find that although very high-dimensional representations are needed to accurately represent the permeability field, especially in presence of small correlation lengths, the pressure field is not sensitive to high-order KL terms of the input parameter. Moreover, we find that the pressure field itself can be represented accurately using a KL expansion with a small number of terms. These observations are used to guide a reduced-order modeling approach to accelerate computational studies of biotransport in tumors.

The consideration of uncertainty is especially important for the design of complex systems. Because of high complexity, the total system is normally divided into subsystems, which are treated in a hierarchical and ideally independent manner. In recent publications, e.g., (Zimmermann, M., and von Hoessle, J. E., 2013, “Computing Solution Spaces for Robust Design,” Int. J. Numer. Methods Eng., 94(3), pp. 290–307; Fender, J., Duddeck, F., and Zimmermann, M., 2017, “Direct Computation of Solution Spaces,” Struct. Multidiscip. Optim., 55(5), pp. 1787–1796), a decoupling strategy is realized via first the identification of the complete solution space (solutions not violating any design constraints) and second via derivation of a subset, a so-called box-shaped solution space, which allows for decoupling and therefore independent development of subsystems. By analyzing types of uncertainties occurring in early design stages, it becomes clear that especially lack-of-knowledge uncertainty dominates. Often, there is missing knowledge about overall manufacturing tolerances like limitations in production or subsystems are not even completely defined. Furthermore, flexibility is required to handle new requirements and shifting preferences concerning single subsystems arising later in the development. Hence, a set-based approach using intervals for design variables (i.e., interaction quantities between subsystems and the total system) is useful. Because in the published approaches, no uncertainty consideration was taken into account for the computation of these intervals, they can possibly have inappropriate size, i.e., being too narrow. The work presented here proposes to include these uncertainties related to design variables. This allows now to consider lack-of-knowledge uncertainty specific for early phase developments in the framework of complex systems design. An example taken from a standard crash load case (frontal impact against a rigid wall) illustrates the proposed methodology.

Accident modeling is a vital step, which helps in designing preventive measures to avoid future accidents, and thus, to enhance process safety. Bayesian networks (BN) are widely used in accident modeling due to its capability to represent accident scenarios from their causes to likely consequences. However, to assess likelihood of an accident using the BN, it requires exact basic event probabilities, which are often obtained from expert opinions. Such subjective opinions are often inconsistent and sometimes conflicting and/or incomplete. In this work, evidence theory has been coupled with BN to address inconsistency, conflict and incompleteness in the expert opinions. It combines the acquired knowledge from various subjective sources, thereby rendering accuracy in probability estimation. Another source of uncertainty in BN is model uncertainty. To represent multiple interactions of a cause–effect relationship Noisy-OR and leaky Noisy-AND gates are explored in the study. Conventional logic gates, i.e., OR/AND gates can only provide a linear interaction of cause–effect relationship hence introduces uncertainty in the assessment. The proposed methodology provides an impression how dynamic risk assessment could be conducted when the sufficient information about a process system is unavailable. To illustrate the execution of a proposed methodology, a tank equipped with a basic process control system has been used as an example. A real-life case study has also been used to validate the proposed model and compare its results with those using a deterministic approach.

Semi-active actuators have been used in engineering systems for vibration control purposes. For instance, magnetorheological (MR) dampers are applied in support of vehicle seats and smart suspensions of bridges and buildings. Parametric and nonparametric approaches were developed to model MR actuators, in which the former presents well-established and representative models. In this context, this work aims at comparing the so-called Bingham, modified Bouc-Wen (BW), and hysteretic models dedicated to MR actuators. Typical inverse problems were solved to minimize the difference between the forces determined by using these models and experimental data. The obtained results demonstrated that the hysteretic model is better adapted to represent the considered MR actuator, presenting lower computational cost and easy implementation. Additionally, uncertainty and sensitivity analyses based on the interval approach were applied on the updated MR models aiming to determine the working envelopes associated with the most important parameters of the models.

In this contribution, the optimization of systems under uncertainty is considered. The possibilistic evaluation of the fuzzy-valued constraints and the adoption of a multicriteria decision making technique for the fuzzy-valued objective function enable a meaningful solution to general fuzzy-valued optimization problems. The presented approach is universally applicable, which is demonstrated by reformulating and solving the linear quadratic regulator problem for fuzzy-valued system matrices and initial conditions.

With the increasing size and complexity of modern infrastructure networks rises the challenge of devising efficient and accurate methods for the reliability analysis of these systems. Special care must be taken in order to include any possible interdependencies between networks and to properly treat all uncertainties. This work presents a new approach for the reliability analysis of complex interconnected networks through Monte Carlo simulation and survival signature. Application of the survival signature is key in overcoming limitations imposed by classical analysis techniques and facilitating the inclusion of competing failure modes. The (inter)dependencies are modeled using vine copulas while the uncertainties are handled by applying probability boxes and imprecise copulas. The proposed method is tested on a complex scenario based on the IEEE reliability test system, proving its effectiveness and highlighting the ability to model complicated scenarios subject to a variety of dependent failure mechanisms.

In this work, we study the effect of uncertainties modeling and the choice of objective function on the results of optimization design problems in deterministic and probabilistic contexts. Uncertainties modeling are studied in two cases identified in the literature. The results show how the choice of two different objective functions, which lead to the same results in deterministic case, may lead to opposite results in probabilistic case. Also, the results show how the uncertainties modeling type can affect the antagonism between mean and standard deviation in the reliability-based robust design optimization (RBRDO) problems. Three mechanical applications chosen from the literature are used to illustrate these cases.

Risk analyses require proper consideration and quantification of the interaction between humans, organisation and technology in high-hazard industries. Quantitative Human Reliability Analysis approaches require the estimation of human error probabilities, often obtained from human performance data on different tasks in specific contexts (also known as performance shaping factors). Data on human errors are often collected from simulated scenarios, near-misses report systems, and experts with operational knowledge. However, these techniques usually miss the realistic context where human errors occur. The present research proposes a realistic and innovative approach for estimating human error probabilities using data from major accident investigation reports. The approach is based on Bayesian Networks used to model the relationship between performance shaping factors and human errors. The proposed methodology allows minimising the expert judgement of human error probabilities, by using a strategy that is able to accommodate the possibility of having no information to represent some conditional dependencies within some variables. Therefore, the approach increases the transparency about the uncertainties of the human error probability estimations. The approach also allows identifying the most influential performance shaping factors, supporting assessors to recommend improvements or extra controls in risk assessments. Formal verification and validation processes are also presented.

Robust design strategies continue to be relevant during concept-stage complex system design to minimize the impact of uncertainty in system performance due to uncontrollable external failure events. Historical system failures such as the 2003 North American blackout and the 2011 Arizona-Southern California Outages show that decision making, during a cascading failure, can significantly contribute to a failure's magnitude. In this paper, a scalable, model-based design approach is presented to optimize the quantity and location of decision-making agents in a complex system, to minimize performance loss variability after a cascading failure, regardless of where the fault originated in the system. The result is a computational model that enables designers to explore concept-stage design trade-offs based on individual risk attitudes for system performance and performance variability, after a failure. The IEEE RTS-96 power system test case is used to evaluate this method, and the results reveal key topological locations vulnerable to cascading failures, that should not be associated with critical operations. This work illustrates the importance of considering decision-making when evaluating system level trade-offs, supporting robust design.

This paper focuses on the computational homogenization of graphene sheet-reinforced composites with randomly dispersed inclusions and uncertainty in the constituent materials. Material uncertainty of the matrix and of the graphene inclusions are considered separately and their relative effect on the homogenized properties is assessed. The uncertainty in the inclusion material is due to structural defects of the graphene lattice and is taken into account using random variables for each component of the elasticity matrix. Moreover, Monte Carlo simulation is used to extract the statistical characteristics of the homogenized properties of the composite material. The results lead to useful conclusions regarding the effect of material and geometrical uncertainty on the macroscopic properties of graphene sheet-reinforced composites.

The main aim of this work is to study a significance of structural microdefects and their uncertainty in structural steel on its elastoplastic large deformations subjected to tensile test with the use of the generalized stochastic perturbation method. Elastoplastic behavior of the macroscopically homogeneous material is defined by the Gurson–Tvergaard–Needleman (GTN) constitutive model, where Young's modulus and this model constants q 1 and q 2 are consecutively randomized according to the Gauss probability distribution. The stochastic finite element method (SFEM) analysis has been carried out in the system abaqus for the problem of necking under tension to compute the first four probabilistic moments and coefficients of displacements, deformations, and stresses. The tenth-order perturbation scheme has been implemented via statistically optimized least-squares method (LSM) determination of the structural nodal polynomial response functions. A comparison with Monte Carlo simulation (MCS) as well as the semi-analytical integral technique based on the same polynomial bases confirms applicability of the method proposed for the input uncertainty not larger than 0.10. Further numerical experiments with this constitutive law including stochastic nucleation and/or coalescence would be necessary to better understand deformations and stresses of stochastic porous plastic materials. This model may find its applications in various stress states of the plastic materials with voids as well as in numerical simulations of the composite materials with imperfect interphases, for instance, where some parameters exhibit initial Gaussian statistical scattering.

This work uses a new method of determining a parameterization, resampling, and dimension search of an uncertainty model that can be used for efficient engineering models in control design. An algorithm using the Cayley–Menger determinant as a measure of the dimension test geometry (volume/area/length) of the parametric data points is presented to search for a reduced number of dimensions that can be used to represent the parameters of a model that captures the uncertainty in a dynamic system (uncertainty model). A genetic algorithm (GA) is utilized to solve the nonconvex problem of finding the coefficients of a parameterization of the uncertainty model. A resampling approach for the uncertainty model is also presented. The methods presented here are demonstrated on an electrohydraulic valve control system problem. This demonstration includes consideration of the dimensional search, data resampling, and parameterizing of an uncertainty class determined from test data for 30 replications of an electrohydraulic flow control valve which were experimentally modeled in the lab. The suggested resampling method and the parameterization of the uncertainty are used to analyze the robust stability of a control system for the class of valves using both frequency domain h-infinity methods and analysis of closed-loop poles for the resampled uncertainty model.

The uncertain free vibration analysis of engineering structures with the consideration of nonstochastic spatially dependent uncertain parameters is investigated. A recently proposed concept of interval field is implemented to model the intrinsic spatial dependency of the uncertain-but-bounded system parameters. By employing the appropriate discretization scheme, evaluations of natural frequencies for engineering structures involving interval fields can be executed within the framework of the finite element method. Furthermore, a robust, yet efficient, computational strategy is proposed such that the extreme bounds of natural frequencies of the structure involving interval fields can be rigorously captured by performing two independent eigen-analyses. Within the proposed computational analysis framework, the traditional interval arithmetic is not employed so that the undesirable effect of the interval overestimation can be completely eliminated. Consequently, both sharpness and physical feasibility of the results can be guaranteed to a certain extent for any discretized interval field. The plausibility of the adopted interval field model, as well as the feasibility of the proposed computational scheme, is clearly demonstrated by investigating both academic-sized and practically motivated engineering structures.

This paper addresses the sensitivity analysis of time-dependent computer models. Often, in practice, we partition the inputs into a subset of inputs relevant to the application studied, and a complement subset of nuisance inputs that are not of interest. We propose sensitivity measures for the relevant inputs of such dynamic computer models. The subset of nuisance inputs is used to create replication-type information to help quantify the uncertainty of sensitivity measures (or indices) for the relevant inputs. The method is first demonstrated on an analytical example. Then we use the proposed method in an application about the safety of restraint systems in light tactical vehicles. The method indicates that chest deflection curves are more sensitive to the addition of pretensioners and load limiters than to the type of seatbelt.

Accurate state-of-charge (SOC) estimation of lithium-ion battery packs is technically challenging because of the cell-to-cell variability due to the manufacturing tolerance. In addition, there is no unanimous definition of the pack SOC since each cell has its own SOC and the pack can be configured in different ways. This study first adopts a suitable pack SOC definition among existing ones, then proposes uncertainty modeling and propagation analysis for pack SOC estimation considering the cell-to-cell variability, and finally conducts the SOC estimation for one serially connected battery pack using the one state hysteresis model with the extended Kalman filter (EKF). The results reveal that pack SOC variability is inevitable due to the cell-to-cell variability and accurate pack SOC estimation is challenging considering both the cell-level SOC accuracy and the pack-level estimation efficiency.

Uncertainties cannot be ignored in the design process of complex multidisciplinary systems. Robust multidisciplinary design optimization methods (RMDOs) can treat uncertainties as specified probabilistic distributions when enough statistical information is available while they assign intervals for nondeterministic variables since designers may not have enough information to obtain statistical distributions, especially in the early stage of design optimization processes. Both types of uncertainties are very likely to appear simultaneously. In order to obtain solutions to RMDO problems under mixed interval and probabilistic uncertainties, this work proposed a new sequential RMDO approach, mixed SR-MDO. First, the robust optimization (RO) problem in a single discipline under mixed uncertainties is formulated and solved. Then, following the SR-MDO framework from the previous work, MDO problems under mixed uncertainties are solved by handling probabilistic and interval uncertainties sequentially in decomposed subsystem problems. Interval uncertainties are handled by using the worst-case sensitivity analysis, and the influence of probabilistic uncertainties in objectives, constraints, as well as in discipline analysis models is characterized by corresponding mean and variance. The applied SR-MDO framework allows subsystems in its full autonomy RO and sequential RO stages to run independently in parallel. This makes mixed SR-MDO be efficient for independent disciplines to work simultaneously and be more time-saving. Computational complexity of the proposed approach mainly relates to the double-loop optimization process in the worst-case interval uncertainties analysis. Examples are presented to demonstrate the applicability and efficiency of the mixed SR-MDO approach.

This paper presents a reliability analysis method for automated vehicles equipped with adaptive cruise control (ACC) and autonomous emergency braking (AEB) systems to avoid collision with an obstacle in front of the vehicle. The proposed approach consists of two main elements, namely uncertainty modeling of traffic conditions and model-based reliability analysis. In the uncertainty modeling step, a recently developed Gaussian mixture copula (GMC) method is employed to accurately represent the uncertainty in the road traffic conditions using the real-world data, and to capture the complicated correlations between different variables. Based on the uncertainty modeling of traffic conditions, an adaptive Kriging surrogate modeling method with an active learning function is then used to efficiently and accurately evaluate the collision-avoidance reliability of an automated vehicle. The application of the proposed method to the Department of Transportation Safety Pilot Model Deployment database and an in-house built Advanced Driver Assist Systems with ACC and AEB controllers demonstrate the effectiveness of the proposed method in evaluating the collision-avoidance reliability.