Reliability-based Design Optimization Algorithm Research Articles

Reliability-based design optimization of the spiral water jacket for motorized spindle

Spiral water jackets are essential refrigeration units in motorized spindles. The design of spiral water jackets directly affects the machining accuracy and reliability of motorized spindles. In this regard, a reliability-based design optimization (RBDO) method is presented to obtain the optimal configuration of the spiral water jacket. The proposed RBDO algorithm aims to minimize the volume of the spiral water jacket and power loss of the cooling equipment while ensuring machining reliability. The geometric correlation and machining reliability are selected as constraints. The Kriging surrogate model is established to approximate the time-consuming finite element model for improving computational efficiency. An efficient metaheuristic algorithm is introduced to perform optimal exploration for the proposed RBDO problem. The risk speed of the motorized spindle is investigated by the experiment and finite element analysis. Subsequently, the effects of various cooling configurations on thermal error are investigated thoroughly. Eventually, the proposed RBDO is conducted on a motorized spindle. Through probabilistic evaluation, machining reliability is guaranteed by the proposed RBDO compared to deterministic design optimization. This paper provides a new design paradigm for cooling configurations in motorized spindles from the perspective of machining reliability.

Efficient decoupling-assisted evolutionary/metaheuristic framework for expensive reliability-based design optimization problems

Reliability-based design optimization (RBDO) algorithm is to minimize the objective under the probabilistic factors. While gradient-based and classical evolutionary RBDO algorithms provide promising performance on simple optimization problems, they are likely to perform poorly on challenging problems, including the multimodal functions, discrete design spaces, non-differential problems, etc. This paper proposes a unified framework to improve the performance of existing RBDO algorithms for complex RBDO problems. Our framework is based on three new strategies: generalized decoupling evolutionary and metaheuristic RBDO framework, particle’s memory saving strategy, and adaptive fractional-order equilibrium optimizer algorithm. The proposed algorithm is characterized by a decoupling strategy to enable the parallel operation of the inner reliability computation and outer deterministic optimization, a particle’s memory saving strategy to provide effective guidance from the previous iteration, and the adaptive fractional-order equilibrium optimizer algorithm to enhance the search efficiency and global convergence capacity. To evaluate the performance of the proposed algorithm, a wide range of experiments are conducted on different types of use cases. The experimental results demonstrate that our algorithm provides superior performance over other comparative algorithms.

Efficient method using Whale Optimization Algorithm for reliability-based design optimization of labyrinth spillway

Spillways are essential parts of dams, in which the main task of these structures is to allow the passing of excess water and floods from the upstream to the downstream. In this regards, the main goal of this paper is proposing a novel framework for the probabilistic design of labyrinth spillway structures using a new developed reliability-based design optimization (RBDO) approach. In this RBDO approach, the total volume of the spillway is considered as the objective function of the optimization problem under uncertainties, while the labyrinth spillway parameters are considered as the design variables. Hereafter, to solve the formulated RBDO problem of the labyrinth spillway design, a new proposed model that consist of coupling the Monte Carlo Simulation (MCS) with a hybrid Artificial Neural Network (ANN) based Whale Optimization Algorithm (WOA) model is developed. The hybrid ANN-WOA is utilized to approximate the labyrinth spillway response in order to reduce the computational cost during the RBDO analysis. The proposed MCS-ANN-WOA model was implemented on the Ute dam labyrinth spillway at Logan, New Mexico (USA). The obtained results showed that the proposed RBDO model performance is more accurate and robust compared to the deterministic optimization (DO) approaches for an optimal design of the labyrinth spillway shape with the consideration of the safety levels.

A Broyden–Fletcher–Goldfarb–Shanno algorithm for reliability-based design optimization

Classic mean value methods of performance measure approach in reliability-based design optimization may become computationally challenging due to their instability and inefficiency as the degree of nonlinearity grows in problems. In this paper, sequential quadratic programming method associated with the Broyden–Fletcher–Goldfarb–Shanno algorithm is introduced to take information about the curvature of the performance function into account and determine the search direction of the minimum performance target point within the performance measure approach. A new mean value method of performance measure approach is developed in accordance with the search direction defined by a technique akin to the non-monotone line search of globalization strategy in the unconstrained optimization problem. The minimum performance target point, on the one hand, is rapidly located in the local domain along the search direction by the steepest descend method, and on the other hand, globally explored along a direction from the origin with exponentially decreasing step sizes. The behavior of the proposed method is illustrated and compared with some other existing algorithms through two reliability examples with highly-nonlinear performance functions and two representative reliability-based design optimization problems in engineering practice. The results show that the proposed method is robust, accurate and efficient with relatively less iterations in contrast with the other representative algorithms of performance measure approach in the literature.

Efficient System Reliability-Based Design Optimization Study for Replaced Hip Prosthesis Using New Optimized Anisotropic Bone Formulations.

An efficient reliability algorithm is developed to transfer the system reliability problem to a single-component reliability problem, considering the uncertainty of loading cases and the material properties. The main difficulty is that femoral bone densities change after hip arthroplasty and, thus, the mechanical properties of the distinctive bone tissues and, therefore, the corresponding elasticity modulus and yield stress values change. Therefore, taking these changes into account during the hip prosthesis design process is strongly needed. As the bone possesses anisotropic behaviors, as the material properties in both radial and tangential directions in long bone (femur, tibia) are almost similar, the bone anisotropy is represented in this study by transversal isotropy. Two optimized formulations for yield stress against the elasticity modulus relationship are first developed and then integrated into an efficient reliability algorithm. Thus, a coupling between reliability and optimization, so-called reliability-based design optimization (RBDO), is introduced in order to control the reliability level. The proposed RBDO algorithm using optimum safety factors (OSF) takes into account the material uncertainties and leads to new stem dimensions. An in-depth numerical analysis on a cementless hip prosthesis is implemented to demonstrate the appropriateness of the proposed algorithm with the consideration of many different loading cases. The results show that the studied model can be effectively used when compared to previous works, which concerns the changes in both geometry and material properties.

Reliability-Based Design Optimization of Complex Problems With Multiple Design Points via Narrowed Search Region

Abstract The expensive computational cost is always a major concern for reliability-based design optimization (RBDO) of complex problems. The performance of RBDO can be lowered by the inaccuracy of reliability analysis (RA) which is caused by multiple local optimums and multiple design points in highly non-linear space. In order to reduce the computational burden and guarantee the accuracy of RA (and thus to improve the RBDO performance), a global RBDO algorithm by adopting an improved constraint boundary sampling (GRBDO-ICBS) method is proposed. Specifically, the GRBDO-ICBS method first narrows the concerned search region by using a Kriging-based global search. The accuracies of the design points are verified by the expected risk function (ERF), and the corresponding inaccurate design points are added into training samples to update Kriging. Then a multi-start gradient-based sequential RBDO is carried out, which tries to find out all multiple design points in the concerned search region. The performance of GRBDO-ICBS is demonstrated by four examples. All results have shown that the proposed method can achieve similar accuracy as Monte Carlo simulation (MCS)-based RBDO but with a much lower computational cost.

Reliability-Based Design Optimization Using Step Length Adjustment Algorithm and Sequential Optimization and Reliability Assessment Method

A step length adjustment (SLA) iterative algorithm was proposed for locating the minimum performance target point (MPTP) in the inverse reliability analysis. This paper elaborates SLA and two deliberately designed numerical examples are used to compare SLA with other algorithms appearing in recent literatures for locating MPTP. The results show that SLA is much more robust and efficient. Then SLA and sequential optimization and reliability assessment (SORA) are combined to solve reliability-based design optimization (RBDO) problems. In the reliability assessment of SORA, with the design obtained from the previous cycle, SLA is used to locate MPTP. Then in the deterministic optimization, the boundaries of violated constraints are shifted to the feasible direction according to the MPTP obtained in the reliability assessment. Several examples frequently cited in similar studies are used to compare SORA-SLA with other RBDO algorithms. The results indicate the effectiveness and robustness of SORA-SLA.

A performance measure approach for risk optimization

In recent years, several approaches have been proposed for solving reliability-based design optimization (RBDO), where the probability of failure is a design constraint. The same cannot be said about risk optimization (RO), where probabilities of failure are part of the objective function. In this work, we propose a performance measure approach (PMA) for RO problems. We first demonstrate that RO problems can be solved as a sequence of RBDO sub-problems. The main idea is to take target reliability indexes (i.e., probabilities of failure) as design variables. This allows the use of existing RBDO algorithms to solve RO problems. The idea also extends the literature concerning RBDO to the context of RO. Here, we solve the resulting RBDO sub-problems using the PMA. Sensitivity expressions required by the proposed approach are also presented. The proposed approach is compared to an algorithm that employs the first-order reliability method (FORM) for evaluation of the probabilities of failure. The numerical examples show that the proposed approach is efficient and more stable than direct employment of FORM. This behavior has also been observed in the context of RBDO, and was the main reason for the development of PMA. Consequently, the proposed approach can be seen as an extension of PMA approaches to RO, which result in more stable optimization algorithms.

Reliability-Based Design Optimization on Qualitative Objective With Limited Information

Reliability-based design optimization (RBDO) algorithms typically assume a designer's prior knowledge of the objective function along with its explicit mathematical formula and the probability distributions of random design variables. These assumptions may not be valid in many industrial cases where there is limited information on variable variability and the objective function is subjective without mathematical formula. A new methodology is developed in this research to model and solve problems with qualitative objective functions and limited information about random variables. Causal graphs and design structure matrix are used to capture designer's qualitative knowledge of the effects of design variables on the objective. Maximum entropy theory and Monte Carlo simulation are used to model random variables' variability and derive reliability constraint functions. A new optimization problem based on a meta-objective function and transformed deterministic constraints is formulated, which leads close to the optimum of the original mathematical RBDO problem. The developed algorithm is tested and validated with the Golinski speed reducer design case. The results show that the algorithm finds a near-optimal reliable design with less initial information and less computation effort as compared to other RBDO algorithms that assume full knowledge of the problem.

An improved binary bat flexible sampling algorithm for reliability-based design optimization of truss structures with discrete-continuous variables

PurposeReliability-based design optimizations (RBDOs) of engineering structures involve complex non-linear/non-differentiable performance functions, including both continuous and discrete variables. The gradient-based RBDO algorithms are less than satisfactory for these cases. The simulation-based approaches could also be computationally inefficient, especially when the double-loop strategy is used. This paper aims to present a pseudo-double loop flexible RBDO, which is efficient for solving problems, including both discrete/continuous variables.Design/methodology/approachThe method is based on the hybrid improved binary bat algorithm (BBA) and weighed simulation method (WSM). According to this method, each BBA’s movement generates proper candidate solutions, and subsequently, WSM evaluates the reliability levels for design candidates to conduct swarm in a low-cost safe-region.FindingsThe accuracy of the proposed enhanced BBA and also the hybrid WSM-BBA are examined for ten benchmark deterministic optimizations and also four RBDO problems of truss structures, respectively. The solved examples reveal computational efficiency and superiority of the method to conventional RBDO approaches for solving complex problems including discrete variables.Originality/valueUnlike other RBDO approaches, the proposed method is such organized that only one simulation run suffices during the optimization process. The flexibility future of the proposed RBDO framework enables a designer to present multi-level design solutions for different arrangements of the problem by using the results of the only one simulation for WSM, which is very helpful to decrease computational burden of the RBDO. In addition, a new suitable transfer function that enhanced convergence rate and search ability of the original BBA is introduced.

Enhanced single-loop method for efficient reliability-based design optimization with complex constraints

Reliability-based design optimization (RBDO) has been widely implemented for engineering design optimization when considering the uncertainty. The single loop approaches (SLA) are highly efficient but is prone to converge with inappropriate results for highly nonlinear probabilistic constraints. In this paper, a novel RBDO algorithm is proposed based on single loop approach and the enhanced chaos control method, named as enhanced single-loop method (ESM). The performance of SLA is enhanced using an adaptive inverse reliability method with limited number of iterations. The adaptive step size is computed based on a merit function which is computed using the results of the new and previous iterations. The iterations of the probabilistic constraints of RBDO models are manually controlled in the range from 1 to 10 in ESM. The efficiency and accuracy of the ESM are compared through four nonlinear RBDO problems with complex constraints, including a nonlinear mathematical problem, two engineering problems and a practical complex stiffened panel example with complex buckling constraint for aircraft design. Results illustrate that the proposed ESM is more efficient and robust than the performance measure approach and reliability index approach for RBDO problems.

The relevance of reliability-based topology optimization in early design stages of aircraft structures

This paper aims to compare the structural designs derived from Deterministic Topology Optimization (DTO) and Reliability-Based Topology Optimization (RBTO). The target is to prove that including uncertain data in preliminary stages of the design process such as topology optimization can help to reduce the weight of the component to be conceived. To compare both approaches the same Reliability-Based Design Optimization (RBDO) sizing problem is formulated over the bar structures emerged from the DTO and RBTO results. The RBTO is performed through the RBDO algorithm Sequential Optimization and Reliability Assessment (SORA) based on its robustness, accuracy and decoupled nature, which allows to use an external optimization software when dealing with the topology optimization problem. Three application examples, including two 2D bar structures and a 3D simplified aircraft rear fuselage are studied. The results show that the designs based on RBTO are lighter than the designs based on DTO when both structures are compared in a subsequent reliability-based size optimization including the same probabilistic information.

Reliability based design optimization for a rock tunnel support system with multiple failure modes using response surface method

Reliability-based design optimization (RBDO) is applied to a rock tunnel support system, considering the ground-support interaction and the parametric uncertainties. Both reliability index approach (RIA) and performance measure approach (PMA) are employed as the RBDO algorithms. The first-order reliability method (FORM) and the response surface method (RSM) are used to perform the reliability analysis. The shotcrete thickness and its installation position are considered as two design variables to be optimized in an illustrative example of a circular rock tunnel. The influence of the two design variables on the reliability results is first investigated with respect to three failure modes, namely, the support capacity criterion, the tunnel convergence criterion and the rockbolt length criterion. Then, the proposed RBDO procedure is used to seek for the design variables of minimum shotcrete thickness and the optimal installation position for the rock tunnel example. The computational efficiency and stability between RIA and PMA are compared. The influences of the rock mass quality, the variation of the shotcrete thickness and the target reliability requirement on the optimized results are also investigated and discussed.

A Hybrid Method for Truss Mass Minimization considering Uncertainties

In real-world structural problems, a number of factors may cause geometric imperfections, load variability, or even uncertainties in material properties. Therefore, a deterministic optimization procedure may fail to account such uncertainties present in the actual system leading to optimum designs that are not reliable; the designed system may show excessive safety or sometimes not sufficient reliability to carry applied load due to uncertainties. In this paper, we introduce a hybrid reliability-based design optimization (RBDO) algorithm based on the genetic operations of Genetic Algorithm, the position and velocity update of the Particle Swarm Algorithm (for global exploration), and the sequential quadratic programming, for local search. The First-Order Reliability Method is used to account uncertainty in design and parameter variables and to evaluate the associated reliability. The hybrid method is analyzed based on RBDO benchmark examples that range from simple to complex truss parametric sizing optimizations with stress, displacements, and frequency deterministic and probabilistic constraints. The proposed final problem, which cannot be handled by single loop RBDO algorithms, highlights the importance of the proposed approach in cases where the discrete design variables are also random variables.

An efficient method of solving design optimization problems with arbitrary random distributions in gradient-based design spaces

Reliability-Based Design Optimization (RBDO) algorithms, such as Reliability Index Approach (RIA) and Performance Measure Approach (PMA), have been developed to solve engineering optimization problems under design uncertainties. In some existing methods, the random design space is transformed to standard normal design space and the reliability assessment, such as reliability index from RIA or performance measure from PMA, is estimated in order to evaluate the failure probability. When the random variable is arbitrarily distributed and cannot be properly fitted to any known form of probability density function, the existing RBDO methods cannot perform reliability analysis in the original design space. This paper proposes a novel Ensemble of Gradient-based Transformed Reliability Analyses (EGTRA) to evaluate the failure probability of any arbitrarily distributed random variables in the original design space. The arbitrary distribution of the random variable is approximated by a merger of multiple Gaussian kernel functions in a single-variate coordinate that is directed toward the gradient of the constraint function. The failure probability is then estimated using the ensemble of each kernel reliability analysis. This paper further derives a linearly approximated probabilistic constraint at the design point with allowable reliability level in the original design space using the aforementioned fundamentals and techniques. Numerical examples with generated random distributions show that existing RBDO algorithms can improperly approximate the uncertainties as Gaussian distributions and provide solutions with poor assessments of reliabilities. On the other hand, the numerical results show EGTRA is capable of efficiently solving the RBDO problems with arbitrarily distributed uncertainties.

Reliability-based performance optimization of TMD for vibration control of structures with uncertainty in parameters and excitation

Recent development of system identification using Bayesian models or stochastic filtering provides probabilistic descriptions (i.e., probability density function or statistical parameters like mean and variance) of the identified model parameters (e.g., mass, stiffness, and damping). Optimal design of passive controllers for these systems whose parameters are uncertain has remained an open problem. With this in view, the present study aims to develop numerical solution scheme for the optimal design of tuned mass damper (TMD) operating in uncertain environment. Deterministic design of TMD in these cases suffers detuning as the system parameters are random. Thus, a reliability-based design optimization (RBDO) scheme is presented in this paper for better performance of the TMD when exposed to uncertainties. To solve the RBDO problem, response surface methodology is used along with the moving least squares technique. Dual response surfaces are used for separate handling of optimization and reliability analysis. First response surface performs optimization of the design variables of TMD, while the second response surfaces are used for the estimation of the statistical properties like mean and variance to satisfy the constrained conditions. Numerical analysis is presented to show the effectiveness of the proposed algorithm for RBDO of single degree of freedom-TMD system as a proof of concept. The proposed meta-model-based algorithm can be applied for the optimal design of controller for large structures where conventional technique may face difficulty to handle both optimization and uncertainty quantification simultaneously. Copyright © 2016 John Wiley & Sons, Ltd.

An incremental shifting vector approach for reliability-based design optimization

This paper proposes a decoupling algorithm for reliability-based design optimization (RBDO) with high performance in terms of efficiency and convergence, which provides an effective tool for reliability design of many complex structures. The algorithm proceeds by performing a shifting vector calculation and then solving a deterministic design optimization in each step, and eventually converges to the optimal solution. An incremental shifting strategy is proposed to ensure stable convergence in the iteration process. In each step, the shifting vector preserves the information from the previous step, and only an adjustment is made for it through a shifting vector increment. A computation method is given for the shifting vector increment, avoiding solving an optimization problem during the reliability analysis and thus greatly reducing the computational cost of the iteration process. Six numerical examples and two engineering applications are presented to validate the effectiveness of the method proposed in this paper.

A single loop reliability-based design optimization using EPM and MPP-based PSO

A reliability-based design optimization (RBDO) incorporates a probabilistic analysis with an optimization technique to find a best design within a reliable design space. However, the computational cost of an RBDO task is often expensive compared to a deterministic optimization, which is mainly due to the reliability analysis performed inside the optimization loop. Theoretically, the reliability of a given design point can be obtained through a multidimensional integration. Integration with multiple variables over the safety domain is, unfortunately, formidable in most cases. Monte- Carlo simulation (MCS) is often used to solve this difficulty. However, the inherit statistic uncertainty associated with MCS sometimes causes an unstable RBDO solution. To avoid this unstable solution, this study transforms a multi-variable constraint into a single variable constraint using an exponential function with a polynomial coefficient (EPM). The adaptive Gauss-Kronrod quadrature is used to compute the constraint reliability. The calculated reliability and its derivative are incorporated with an optimizer such as sequential quadratic programming (SQP) or most probable point particle swarm optimization (MPP-based PSO) to conduct the RBDO task. To ensure the design accuracy, the stability of the RBDO algorithm with respect to the initial point is investigated through several numerical examples.

An evolutionary based Bayesian design optimization approach under incomplete information

Design optimization in the absence of complete information about uncertain quantities has been recently gaining consideration, as expensive repetitive computation tasks are becoming tractable due to the invention of faster and parallel computers. This work uses Bayesian inference to quantify design reliability when only sample measurements of the uncertain quantities are available. A generalized Bayesian reliability based design optimization algorithm has been proposed and implemented for numerical as well as engineering design problems. The approach uses an evolutionary algorithm (EA) to obtain a trade-off front between design objectives and reliability. The Bayesian approach provides a well-defined link between the amount of available information and the reliability through a confidence measure, and the EA acts as an efficient optimizer for a discrete and multi-dimensional objective space. Additionally, a GPU-based parallelization study shows computational speed-up of close to 100 times in a simulated scenario wherein the constraint qualification checks may be time consuming and could render a sequential implementation that can be impractical for large sample sets. These results show promise for the use of a parallel implementation of EAs in handling design optimization problems under uncertainties.

Reliability-Based Multidisciplinary Design Optimization Using Probabilistic Gradient-Based Transformation Method

Recently, solving the complex design optimization problems with design uncertainties has become an important but very challenging task in the communities of reliability-based design optimization (RBDO) and multidisciplinary design optimization (MDO). The MDO algorithms decompose the complex design problem into the hierarchical or nonhierarchical optimization structure and distribute the workloads to each discipline (or subproblem) in the decomposed structure. The coordination of the local responses is crucial for the success of finding the optimal design point. The problem complexity increases dramatically when the existence of the design uncertainties is not negligible. The RBDO algorithms perform the reliability analyses to evaluate the probabilities that the random variables violate the constraints. However, the required reliability analyses build up the degree of complexity. In this paper, the gradient-based transformation method (GTM) is utilized to reduce the complexity of the MDO problems by transforming the design space to multiple single-variate monotonic coordinates along the directions of the constraint gradients. The subsystem responses are found using the monotonicity principles (MP) and then coordinated for the new design points based on two general principles. To consider the design uncertainties, the probabilistic gradient-based transformation method (PGTM) is proposed to adapt the first-order probabilistic constraints from three different RBDO algorithms, including the chance constrained programming (CCP), reliability index approach (RIA), and performance measure approach (PMA), to the framework of the GTM. PGTM is efficient because only the sensitivity analyses and the reliability analyses require function evaluations (FE). The optimization processes of monotonicity analyses and the coordination procedures are free of function evaluations. Several mathematical and engineering examples show the PGTM is capable of finding the optimal solutions with desirable reliability levels.