Reliability-based Optimization Problems Research Articles

Reliability-Based Design Optimization of the PEMFC Flow Field with Consideration of Statistical Uncertainty of Design Variables

Recently, with the fourth industrial revolution, the research cases that search for optimal design points based on neural networks or machine learning have rapidly increased. In addition, research on optimization is continuously reported in the field of fuel cell research using hydrogen as fuel. However, in the case of optimization research, it often requires a large amount of training data, which means that it is more suitable for numerical research such as CFD simulation rather than time-consuming research such as actual experiments. As is well known, the design range of fuel cell flow channels is extremely small, ranging from hundreds of microns to several millimeters, which means the small tolerance could cause fatal performance loss. In this study, the general optimization study was further improved in terms of reliability by considering stochastic tolerances that may occur in actual industry. The optimization problem was defined to maximize stack power, which is employed as objective function, under the constraints such as pressure drop and current density standard deviation; the performance of the optimal point through general optimization was about 3.252 kW/L. In the reliability-based optimization problem, the boundary condition for tolerance was set to 0.1 mm and tolerance was assumed to occur along a normal distribution. The optimal point to secure 99% reliability for the given constraints was 2.918 kW/L, showing significantly lower performance than the general optimal point.

Reliability-Based Design Optimization Applied to a Rotor Supported by Hydrodynamic Bearings

Rotating machines are an important part of industrial equipment. It is essential to improve their performance while reducing the manufacturing, operating, and maintenance costs. Ensuring their reliability is also crucial because a machine breakdown can result in significant costs and potential environmental and safety damage. Reliability-based optimization is an approach that aims to find an optimal and robust design that guarantees a machine’s reliability. In this study, we focused on optimizing the shaft diameter and oil temperature of a rotor supported by hydrodynamic bearings. We considered the materials’ elastic moduli, density, and bearing clearance as uncertain parameters. Our goal was to ensure 99% reliability regarding both the vibration amplitude and stability threshold. To model the machine, we used the finite element method and represented the bearings using stiffness and damping coefficients, considering the linear short bearing model. Due to the complexity of the model, we employed surrogate models to solve the reliability-based optimization problem. Our results showed that the optimization problem could be solved successfully using Kriging, polynomial chaos expansion, and polynomial chaos Kriging.

Travel time reliability-based optimization problem for CAVs dedicated lanes

This paper proposed an optimization problem that determines the deployment pattern of dedicated lanes to connected autonomous vehicles (CAVs) considering the stochastic traffic demand and the stochastic traffic capacity. The difference between CAVs and regular human-piloted vehicles (RHVs) is driving behavior. The driving behavior of CAVs is expected to be more standardized than that of RHVs. Therefore, we assume that when the penetration ratio of CAVs increases in the lane flow, the mean lane capacity will increase, and the lane capacity variance will decrease. The mean and the variance of lane travel time decrease when the penetration ratio increases. Following this assumption, the difference in the stochastic properties between CAVs and RHVs is considered in a traffic assignment model. The traffic assignment model is formulated as a variational inequality problem. The network design problem with equilibrium constraints was solved by a simulated annealing algorithm in a test network.

Fully decoupled reliability-based optimization of linear structures subject to Gaussian dynamic loading considering discrete design variables

Reliability-based optimization (RBO) offers the possibility of finding an optimal design for a system according to a prescribed criterion while explicitly taking into account the effects of uncertainty. However, due to the necessity of solving simultaneously a reliability problem nested in an optimization procedure, the corresponding computational cost is usually high, impeding the applicability of the methods. This computational cost is even further enlarged when one or several design variables must belong to a discrete set, due to the requirement of resorting to integer programming optimization algorithms. To alleviate this issue, this contribution proposes a fully decoupled approach for a specific class of problems, namely minimization of the failure probability of a linear system subjected to an uncertain dynamic load of the Gaussian type, under the additional constraint that the design variables are integer-valued. Specifically, by using the operator norm framework, as developed by the authors in previous work, this paper shows that by reducing the RBO problem with discrete design variables to the solution of a single deterministic optimization problem followed by a single reliability analysis, a large gain in numerical efficiency can be obtained without compromising the accuracy of the resulting optimal design. The application and capabilities of the proposed approach are illustrated by means of three examples.

Fully decoupled reliability-based design optimization of structural systems subject to uncertain loads

Reliability-based optimization (RBO) offers the possibility of finding the best design for a system according to a prescribed criterion while explicitly taking into account the effects of uncertainty. Although the importance and usefulness of RBO is undisputed, it is rarely applied to practical problems, as the associated numerical efforts are usually extremely large due to the necessity of solving simultaneously a reliability problem nested in an optimization procedure. To alleviate this issue, this contribution proposes an approach for solving a particular class of problems in RBO: the minimization of the failure probability of a linear system subjected to an uncertain load. The main characteristic of this approach is that it is fully decoupled. That is, the solution of the RBO problem is reduced to the solution of a single deterministic optimization problem followed by a single reliability analysis. Such an approach implies a complete change of paradigm with respect to the more classical double-loop and sequential methods for RBO. The theoretical basis for the proposed approach lies in the application of the operator norm theorem. The application and capabilities of the proposed approach are illustrated by means of three examples.

A Two-Level Kriging-Based Approach with Active Learning for Solving Time-Variant Risk Optimization Problems

Several methods have been proposed in the literature to solve reliability-based optimization problems, where failure probabilities are design constraints. However, few methods address the problem of life-cycle cost or risk optimization, where failure probabilities are part of the objective function. Moreover, few papers in the literature address time-variant reliability problems in life-cycle cost or risk optimization formulations; in particular, because most often computationally expensive Monte Carlo simulation is required. This paper proposes a numerical framework for solving general risk optimization problems involving time-variant reliability analysis. To alleviate the computational burden of Monte Carlo simulation, two adaptive coupled surrogate models are used: the first one to approximate the objective function, and the second one to approximate the quasi-static limit state function. An iterative procedure is implemented for choosing additional support points to increase the accuracy of the surrogate models. Three application problems are used to illustrate the proposed approach. Two examples involve random load and random resistance degradation processes. The third problem is related to load-path dependent failures. This subject had not yet been addressed in the context of risk-based optimization. It is shown herein that accurate solutions are obtained, with extremely limited numbers of objective function and limit state functions calls.

Simplified algorithm for reliability sensitivity analysis of structures: A spreadsheet implementation.

An important segment of the reliability-based optimization problems is to get access to the sensitivity derivatives. However, since the failure probability is not a closed-form function of the input variables, the derivatives are not explicitly computable and rather require a full reliability analysis which is computationally expensive. In this paper, a step-by-step algorithm has been presented to calculate the derivatives of the probability of failure and safety index with respect to the input parameters based on the advanced first-order second-moment (AFOSM) reliability method. The proposed algorithm is then implemented in a spreadsheet using Visual Basic for Application (VBA) programming language. Two geotechnical and structural examples are then presented to examine the program and describe the modeling procedure. The robustness of the proposed method is examined using a Gaussian random perturbation. The capability of the proposed method in the calculation of the sensitivity derivatives of the model uncertainty is explained in a separate section. Finally, the proposed model has been compared to the forward finite difference (FFD) method and the results are validated.

Seismic risk sensitivity of structures equipped with anti-seismic devices with uncertain properties

Damping and isolation devices are often employed to control and enhance the seismic performance of structural systems. However, the effectiveness of these devices in mitigating the seismic risk may be significantly affected by manufacturing tolerances, and systems equipped with devices whose properties deviate from the nominal ones may exhibit a performance very different than expected.The paper analyzes this problem by proposing a general framework for investigating the sensitivity of the seismic risk of structural systems with respect to system properties varying in a prescribed range. The proposed framework is based on the solution of a reliability-based optimization (RBO) problem, aimed to search for the worst combination of the uncertain anti-seismic device parameters, within the allowed range of variation, that maximizes the seismic demand hazard. A hybrid probabilistic approach is employed to speed up the reliability analyses required for evaluating the objective function at each iteration of the RBO process. This approach combines a conditional method for estimating the seismic demand at a given intensity level, with a simulation approach for representing the seismic hazard.The proposed method is applied to evaluate the influence of the variability of the properties of linear and nonlinear fluid viscous dampers on the seismic risk of a low-rise steel building. The study results show that the various response parameters considered are differently affected by the damper properties and unveil the capability of the proposed approach to evaluate the potentially worst conditions that jeopardize the system reliability.

Smooth size design for the natural frequencies of curved Timoshenko beams using isogeometric analysis

Curved thick beams with smoothly variable cross-section size are very common in practical engineering problems. Designing the variable size based on the traditional finite element methods often leads to non-smooth solutions. To guarantee the smoothness of size distribution, an isogeometric analysis (IGA)-based design approach is proposed in this work to optimize the cross-section size of curved Timoshenko beams for natural frequencies. Due to the geometric exactness and high-order continuity of IGA, there is high accuracy of natural frequency prediction and sensitivity analysis for design optimization. It is found that small numbers of design variables lead to parameterization-dependent solutions, while large numbers of design variables induce design fluctuation. To avoid the undesirable design fluctuation, a stability transformation method-based K-S aggregation constraint scheme is proposed to regularize the size distribution and also achieve the stable convergence of optimal solutions. Multiple design studies including the deterministic and reliability-based optimization problems are performed to demonstrate the applicability and effectiveness of the proposed approach.

Nonprobabilistic Reliable LQR Design Method for Active Vibration Control of Structures with Uncertainties

The linear quadratic regulator (LQR), which allows a natural consideration of the constraints on state variables and control effort, is widely used in modern control theory for the active control of structural vibration. However, uncertainties are unavoidable in the design of active control systems. Thus, in the design process of an LQR controller, it is important to deal with these uncertainties, which are always bounded and have a great influence on the final performance of the closed-loop system. In this paper, the nonprobabilistic reliability concept is extended to the selection of weighting matrices for determining the LQR controller. First, a new uncertainty propagation analysis method for an active control system is presented. To adapt to the special problem of controller design, a novel nonprobabilistic reliability index is proposed. The reliability and safety margin of the closed-loop system can be continuously characterized by this modified nonprobabilistic reliability index. The weighing matrices used to design the LQR controller via the Riccati equation are finally determined by solving a reliability-based optimization problem. The proposed method preserves the excellent characteristics of LQR control theory and provides a quantitative method for selecting the weight matrices for LQR controller design in consideration of the inherent uncertainties. Two numerical examples are presented to demonstrate the effectiveness and feasibility of the proposed method. The results from these examples reveal that the uncertainties in active control system design can be addressed from a reliability perspective.

Reliability-based trajectory optimization using nonintrusive polynomial chaos for Mars entry mission

This paper presents the reliability-based sequential optimization (RBSO) method to settle the trajectory optimization problem with parametric uncertainties in entry dynamics for Mars entry mission. First, the deterministic entry trajectory optimization model is reviewed, and then the reliability-based optimization model is formulated. In addition, the modified sequential optimization method, in which the nonintrusive polynomial chaos expansion (PCE) method and the most probable point (MPP) searching method are employed, is proposed to solve the reliability-based optimization problem efficiently. The nonintrusive PCE method contributes to the transformation between the stochastic optimization (SO) and the deterministic optimization (DO) and to the approximation of trajectory solution efficiently. The MPP method, which is used for assessing the reliability of constraints satisfaction only up to the necessary level, is employed to further improve the computational efficiency. The cycle including SO, reliability assessment and constraints update is repeated in the RBSO until the reliability requirements of constraints satisfaction are satisfied. Finally, the RBSO is compared with the traditional DO and the traditional sequential optimization based on Monte Carlo (MC) simulation in a specific Mars entry mission to demonstrate the effectiveness and the efficiency of the proposed method.

A certified Branch & Bound approach for reliability-based optimization problems

Reliability-based optimization problems are optimization problems considering a constraint that measures reliability of the modelled system: the probability of a safety event with respect to controllable decision variables and uncertain random variables. Most solving approaches use approximate techniques for evaluating this reliability constraint. As a consequence, the reliability of the computed optimal decision is not guaranteed. In this paper, we investigate an interval-based Branch & Bound for solving globally reliability-based optimization problems with numerical guarantee. It combines an interval Branch & Bound framework with a certified reliability analysis technique. This technique considers the reliability constraint and induced safety region modelled within Probabilistic Continuous Constraint Programming paradigm. The certified reliability analysis is numerically handled by an interval quadrature algorithm. In addition, a new interval quadrature function for two random variables, based on linear models of the safety region is described. Two implementations of the Branch & Bound, which differ on how the certified reliability analysis is handled throughout the optimization process, are presented. A numerical study of these two variants shows the relevance of the interval linear model-based quadrature function.

A derivative-free trust-region algorithm for reliability-based optimization

In this note, we present a derivative-free trust-region (TR) algorithm for reliability based optimization (RBO) problems. The proposed algorithm consists of solving a set of subproblems, in which simple surrogate models of the reliability constraints are constructed and used in solving the subproblems. Taking advantage of the special structure of the RBO problems, we employ a sample reweighting method to evaluate the failure probabilities, which constructs the surrogate for the reliability constraints by performing only a single full reliability evaluation in each iteration. With numerical experiments, we illustrate that the proposed algorithm is competitive against existing methods.

Efficient Approach for the Reliability-Based Design of Linear Damping Devices for Seismic Protection of Buildings

This paper presents an efficient reliability-based methodology for the seismic design of viscous/viscoelastic dissipative devices in independent and/or coupled buildings. The proposed methodology is consistent with modern performance-based earthquake engineering frameworks and explicitly considers the uncertainties affecting the seismic input and the model parameters, as well as the correlation between multiple limit states. The proposed methodology casts the problem of the dampers’ design for a target performance objective in the form of a reliability-based optimization problem with a probabilistic constraint. The general approach proposed in this study is specialized to stochastic seismic excitations and performance levels for which the structural behavior can be assumed as linear elastic. Under these conditions, the optimization problem is solved efficiently by taking advantage of existing analytical techniques for estimating the system reliability. This analytical design solution is an approximation of the optimal design and can be used as a hot-start point for simulation-based techniques, which can be employed to find the optimal design solution. An efficient correction formula is proposed to obtain an improved design solution that is generally sufficiently close for engineering purposes to the optimal design solution obtained from significantly more-computationally-expensive simulation-based techniques. The proposed design methodology is illustrated and validated by considering two steel buildings modeled as linear elastic multiple-degree-of-freedom systems for different linear damper properties and collocation, for both independent and coupled configurations.

Stochastic structural and reliability based optimization of tuned mass damper

The purpose of the current work is to present and discuss a technique for optimizing the parameters of a vibration absorber in the presence of uncertain bounded structural parameters. The technique used in the optimization is an interval extension based on a Taylor expansion of the objective function. The technique permits the transformation of the problem, initially non-deterministic, into two independents deterministic sub-problems. Two optimization strategies are considered: the Stochastic Structural Optimization (SSO) and the Reliability Based Optimization (RBO). It has been demonstrated through two different structures that the technique is valid for the SSO problem, even for high levels of uncertainties and it is less suitable for the RBO problem, especially when considering high levels of uncertainties.

Reliability Based Optimization with Metaheuristic Algorithms and Latin Hypercube Sampling Based Surrogate Models

Reliability based optimization (RBO) is one of the most appropriate methods for structural design under uncertainties. It searches for the best compromise between cost and safety while considering system uncertainties by incorporating reliability measures within the optimization. Despite the advantages of RBO, its application to practical engineering problem is still quite challenging. In this paper, we propose an effective method to decouple the loops of reliability assessment analysis and optimization by creating surrogate models. The Latin Hypercube sampling approach is applied to a structural finite element model to obtain an effective database for building surrogate models. In order to avoid premature convergence of the optimization process, the RBO problem is solved with metaheuristic methods such as genetic algorithm and simulated annealing. The relative efficiency of surrogate models and their relationship with metaheuristic search engine are discussed in the article.

Reliability-Based Structural Optimization of Wave Energy Converters

More and more wave energy converter (WEC) concepts are reaching prototypelevel. Once the prototype level is reached, the next step in order to further decrease thelevelized cost of energy (LCOE) is optimizing the overall system with a focus on structuraland maintenance (inspection) costs, as well as on the harvested power from the waves.The target of a fully-developed WEC technology is not maximizing its power output,but minimizing the resulting LCOE. This paper presents a methodology to optimize thestructural design of WECs based on a reliability-based optimization problem and the intentto maximize the investor’s benefits by maximizing the difference between income (e.g., fromselling electricity) and the expected expenses (e.g., structural building costs or failure costs).Furthermore, different development levels, like prototype or commercial devices, may havedifferent main objectives and will be located at different locations, as well as receive varioussubsidies. These points should be accounted for when performing structural optimizationsof WECs. An illustrative example on the gravity-based foundation of the Wavestar deviceis performed showing how structural design can be optimized taking target reliability levelsand different structural failure modes due to extreme loads into account.

Development of a GA-based method for reliability-based optimization of structures with discrete and continuous design variables using OpenSees and Tcl

This paper develops a method based on genetic algorithm (GA) for reliability-based optimization (RBO) of structures through a software application. This method employs GA as an optimization technique in which reliability constraints of RBO problem are evaluated using finite element reliability analysis modules offered by OpenSees. Since Tcl is a programmable and interpreted language, the proposed method is implemented using Tcl scripting language together with OpenSees software framework. Developing the RBO method through this software application leads to the important advantage of utilizing new advanced structural reliability methods implemented in OpenSees without compiling the source code. In the RBO problem, the cost of design is defined as an objective function and a number of design requirements are defined as reliability constraints. This method is also able to deal with RBO problems with discrete and continuous design variables. Six numerical examples are provided to investigate the strengths of the method through the software application. Results illustrate the advantages of this application in RBO of structures. Since OpenSees is used for finite element reliability analysis, this application is readily extendable to nonlinear structures.

Efficient approach for reliability-based optimization based on weighted importance sampling approach

An efficient methodology is presented to perform the reliability-based optimization (RBO). It is based on an efficient weighted approach for constructing an approximation of the failure probability as an explicit function of the design variables which is referred to as the ‘failure probability function (FPF)’. It expresses the FPF as a weighted sum of sample values obtained in the simulation-based reliability analysis. The required computational effort for decoupling in each iteration is just single reliability analysis. After the approximation of the FPF is established, the target RBO problem can be decoupled into a deterministic one. Meanwhile, the proposed weighted approach is combined with a decoupling approach and a sequential approximate optimization framework. Engineering examples are given to demonstrate the efficiency and accuracy of the presented methodology.

An efficient reliability method for probabilistic H-infinity robust control of uncertain linear dynamic systems

Achieving balance between robustness and performance is always a fundamental challenge we are faced with in control design of systems in the presence of uncertainties. H∞ robust control theory has been extensively used to deal with the problems of robust stabilization and disturbance attenuation of uncertain systems. However, in traditional H∞ robust control, the most frequently used method for dealing with uncertainties is adopting the assumption of norm-bounded perturbations of arbitrary structure about a nominal plant and the design techniques are based on the deterministic worst-case scenarios that may never occur in a particular control system and thus often led to over-conservative results. In this paper we focus on developing a reliability method for probabilistic H∞ robust control of linear uncertain systems by describing the uncertain parameters as random variables. A new efficient reliability method for robust control of dynamic systems with probabilistic parametric uncertainties is presented systematically. Robust control design is carried out by solving a reliability-based optimization problem where the disturbance attenuation and control cost are minimized under the condition that reliability requirement is satisfied. One of the advantages of the presented reliability method is that it can be used directly for robust control design of parametric uncertain systems. Another advantage of the presented method is that it makes it possible to consider the factors such as system performance, control cost, and reliability simultaneously in an integrated framework and provides an essential basis for the coordination and tradeoffs between these important factors in control design of uncertain systems. Compared to traditional H∞ robust control, the presented method is less-conservative and more reasonable for dealing with probabilistic uncertainties. The presented formulations are within the framework of linear matrix inequality and thus can be carried out conveniently. Two numerical examples are investigated to demonstrate the effectiveness and feasibility of the presented method.