An efficient strategy for reliability-based multidisciplinary design optimization using BLISS

This paper presents an efficient reliability-based multidisciplinary design optimization (RBMDO) strategy. The conventional RBMDO has tri-level loops: the first level is an optimization in the deterministic space, the second one is a reliability analysis in the probabilistic space, and the third one is the multidisciplinary analysis. Since it is computationally inefficient when high-fidelity simulation methods are involved, an efficient strategy is proposed. The strategy [named probabilistic bi-level integrated system synthesis (ProBLISS)] utilizes a single-level reliability-based design optimization (RBDO) approach, in which the reliability analysis and optimization are conducted in a sequential manner by approximating limit state functions. The single-level RBDO is associated with the BLISS formulation to solve RBMDO problems. Since both the single-level RBDO and BLISS are mainly driven by approximate models, the accuracy of models can be a critical issue for convergence. The convergence of the strategy is guaranteed by employing the trust region–sequential quadratic programming framework, which validates approximation models in the trust region radius. Two multidisciplinary problems are tested to verify the strategy. ProBLISS significantly reduces the computational cost and shows stable convergence while maintaining accuracy.     

A cell evolution method for reliability-based design optimization

The reliability-based design optimization (RBDO) presents to be a systematic and powerful approach for process designs under uncertainties. The traditional double-loop methods for solving RBDO problems can be computationally inefficient because the inner reliability analysis loop has to be iteratively performed for each probabilistic constraint. To solve RBDOs in an alternative and more effective way, Deb et al. [1] proposed recently the use of evolutionary algorithms with an incorporated fastPMA. Since the imbedded fastPMA needs the gradient calculations and the initial guesses of the most probable points (MPPs), their proposed algorithm would encounter difficulties in dealing with non-differentiable constraints and the effectiveness could be degraded significantly as the initial guesses are far from the true MPPs. In this paper, a novel population-based evolutionary algorithm, named cell evolution method, is proposed to improve the computational efficiency and effectiveness of solving the RBDO problems. By using the proposed cell evolution method, a family of test cells is generated based on the target reliability index and with these reliability test cells the determination of the MPPs for probabilistic constraints becomes a simple parallel calculation task, without the needs of gradient calculations and any initial guesses. Having determined the MPPs, a modified real-coded genetic algorithm is applied to evolve these cells into a final one that satisfies all the constraints and has the best objective function value for the RBDO. Especially, the nucleus of the final cell contains the reliable solution to the RBDO problem. Illustrative examples are provided to demonstrate the effectiveness and applicability of the proposed cell evolution method in solving RBDOs. Simulation results reveal that the proposed cell evolution method outperforms comparative methods in both the computational efficiency and solution accuracy, especially for multi-modal RBDO problems.     

Component and system reliability-based topology optimization using a single-loop method

We perform reliability-based topology optimization by combining reliability analysis and material distribution topology design methods to design linear elastic structures subject to random inputs, such as random loadings. Both component reliability and system reliability are considered. In component reliability, we satisfy numerous probabilistic constraints which quantify the failure of different events. In system reliability, we satisfy a single probabilistic constraint which encompasses the component events. We adopt the first-order reliability method to approximate the component reliabilities and the inclusion-exclusion rule to approximate the system reliability. To solve the probabilistic optimization problem, we use a variant of the single loop method, which eliminates the need for an inner reliability analysis loop. The proposed method is amenable to implementation with existing deterministic topology optimization software, and hence suitable for practical applications. Designs obtained from component and system reliability-based topology optimization are compared to those obtained from traditional deterministic topology optimization and validated via Monte Carlo simulation.     

An enhanced simulation-based design method coupled with meta-heuristic search algorithm for accurate reliability-based design optimization

The enhanced weighted simulation-based design method in conjunction with particle swarm optimization (PSO) is developed as a pseudo double-loop algorithm for accurate reliability-based design optimization (RBDO). According to this hybrid method, generated samples of weighed simulation method (WSM) are considered as initial population of the PSO. The proposed population is then employed to evaluate the safety level of each PSO swarm (design candidates) during movement. Using this strategy, there is no required to conduct new sampling for reliability assessment of design candidates (PSO swarms). Employing PSO as the search engine of RBDO and WSM as the reliability analyzer provide more accurate results with few samples and also increase the application range of traditional WSM. Besides, a shift strategy is also introduced to increase the capability of the WSM to investigate general RBDO problems including both deterministic and random design variables. Several examples are investigated to demonstrate the accuracy and robustness of the method. Results demonstrate the computational efficiency and superiority of the proposed method for practical engineering problems with highly nonlinear and implicit probabilistic constrains.     

Extension of optimum safety factor method to nonlinear reliability-based design optimization

In the reliability-based design optimization (RBDO) model, the mean values of uncertain system variables are usually applied as design variables, and the cost is optimized subject to prescribed probabilistic constraints as defined by a nonlinear mathematical programming problem. Therefore, a RBDO solution that reduces the structural weight in uncritical regions does not only provide an improved design but also a higher level of confidence in the design. In this paper, we present recent developments for the RBDO model relative to two points of view: reliability and optimization. Next, we develop several distributions for the hybrid method and the optimum safety factor methods (linear and nonlinear RBDO). Finally, we demonstrate the efficiency of our safety factor approach extended to nonlinear RBDO with application to a tri-material structure.     

Performance assessment of a multi-objective parametric optimization algorithm with application to a multi-physical engineering system

The original problem of reliability-based design optimization (RBDO) is mathematically a nested two-level structure that is computationally time consuming for real engineering problems. In order to overcome the computational difficulties, many formulations have been proposed in the literature. These include SORA (sequential optimization and reliability assessment) that decouples the nested problems. SLA (single loop approach) further improves efficiency in that reliability analysis becomes an integrated part of the optimization problem. However, even SLA method can become computationally challenging for real engineering problems involving many reliability constraints. This paper presents an enhanced version of SLA where the first phase is based on approximation at nominal design point. After convergence of first iterative phase is reached the process transitions to a second phase where approximations of reliability constraints are carried out at their respective minimum performance target point (MPTP). The solution is implemented in Altair OptiStruct, where adaptive approximation and constraint screening strategies are utilized in the RBDO process. Examples show that the proposed two-phase approach leads to reduction in finite element analyses while preserving equal solution quality.     

Response surface single loop reliability-based design optimization with higher-order reliability assessment

Reliability-based design optimization (RBDO) aims at determination of the optimal design in the presence of uncertainty. The available Single-Loop approaches for RBDO are based on the First-Order Reliability Method (FORM) for the computation of the probability of failure, along with different approximations in order to avoid the expensive inner loop aiming at finding the Most Probable Point (MPP). However, the use of FORM in RBDO may not lead to sufficient accuracy depending on the degree of nonlinearity of the limit-state function. This is demonstrated for an extensively studied reliability-based design for vehicle crashworthiness problem solved in this paper, where all RBDO methods based on FORM strongly violates the probabilistic constraints. The Response Surface Single Loop (RSSL) method for RBDO is proposed based on the higher order probability computation for quadratic models previously presented by the authors. The RSSL-method bypasses the concept of an MPP and has high accuracy and efficiency. The method can solve problems with both constant and varying standard deviation of design variables and is particularly well suited for typical industrial applications where general quadratic response surface models can be used. If the quadratic response surface models of the deterministic constraints are valid in the whole region of interest, the method becomes a true single loop method with accuracy higher than traditional SORM. In other cases, quadratic response surface models are fitted to the deterministic constraints around the deterministic solution and the RBDO problem is solved using the proposed single loop method.     

A new efficient decoupled reliability-based design optimization method with quantiles

Structural and Multidisciplinary Optimization - The problems of reliability-based design optimization (RBDO) can generally be solved by double-loop methods, single-loop methods or decoupled...     

Reliability-based topology optimization of double layer grids using a two-stage optimization method

This paper presents a two-stage optimization method for reliability-based topology optimization (RBTO) of double layer grids that considers uncertainties in applied loads. The optimization is performed using a two-stage optimization by employing the method of moving asymptotes (MMA) and ant colony optimization (ACO), which is called MMA-ACO method. For implementation of MMA-ACO, the structural stiffness is maximized using MMA, first. Then, the results of MMA are used to enhance ACO through the following four modifications: (I) finding the structural importance rate of elements or joints and using this to achieve a better topology, (II) determining the number of compressive and tensile element types, (III) changing the lower limit of available cross-sectional areas for the elements of each group and (IV) modifying the generation of random stable structures. In reliability analysis, multiple criteria i.e. stiffness and eigenvalue are considered where the probability of failure in each mode is calculated by Monte Carlo simulation (MCS). To reduce the computational time, the eigenvalues are evaluated using the third order approximation (TOA). Through numerical examples, reliability-based topology designs of typical double layer grids are obtained by ACO and MMA-ACO methods. The numerical results reveal the computational advantages and effectiveness of the proposed MMA-ACO method for the RBTO of double layer grids. Also, the importance of considering uncertainties is then demonstrated by comparing the results obtained by those of other failure probabilities.     

Reliability-based topology optimization

The objective of this work is to integrate reliability analysis into topology optimization problems. The new model, in which we introduce reliability constraints into a deterministic topology optimization formulation, is called Reliability-Based Topology Optimization (RBTO). Several applications show the importance of this integration. The application of the RBTO model gives a different topology relative to deterministic topology optimization. We also find that the RBTO model yields structures that are more reliable than those produced by deterministic topology optimization (for the same weight).     

Reliability-based topology optimization using a standard response surface method for three-dimensional structures

In this paper, a reliability-based topology optimization (RBTO) for 3-D structures was performed using bi-directional evolutionary structural optimization (BESO) and the standard response surface method (SRSM). In order to get a stable optimal topology, the most recently-developed filter scheme was implemented with BESO, and SRSM was used to generate an approximate limit state function. These results were compared with the recently announced results of RBTO for 2-D structures, and the differences between the results for the 3-D and 2-D structures were examined. A cantilever beam and an MBB beam were selected as the numerical examples. The comparison showed that the optimal topologies of deterministic topology optimization (DTO) and RBTO for the 2-D and 3-D MBB beams, respectively, are very different. Specifically, the two-support member on the left hand side comes into being along the width for the 3-D case, but not for the 2-D case. This shows that RBTO for 3-D structures should be performed as part of the design process.     

Relaxed performance measure approach for reliability-based design optimization

The efficiency and robustness of reliability analysis methods are important factors to evaluate the probabilistic constraints in reliability-based design optimization (RBDO). In this paper, a relaxed mean value (RMV) approach is proposed in order to evaluate probabilistic constraints including convex and concave functions in RBDO using the performance measure approach (PMA). A relaxed factor is adaptively determined in the range from 0 to 2 using an inequality criterion to improve the efficiency and robustness of the inverse first-order reliability methods. The performance of the proposed RMV is compared with six existing reliability methods, including the advanced mean value (AMV), conjugate mean value (CMV), hybrid mean value (HMV), chaos control (CC), modified chaos control (MCC), and conjugate gradient analysis (CGA) methods, through four nonlinear concave and convex performance functions and three RBDO problems. The results demonstrate that the proposed RMV is more robust than the AMV, CMV, and HMV for highly concave problems, and slightly more efficient than the CC, MCC, and CGA methods. Furthermore, the proposed relaxed mean value guarantees robust and efficient convergence for RBDO problems with highly nonlinear performance functions.     

Reliability-based topology optimization of continuum structures subject to local stress constraints

Topology optimization of continuum structures is a challenging problem to solve, when stress constraints are considered for every finite element in the mesh. Difficulties are compounding in the reliability-based formulation, since a probabilistic problem needs to be solved for each stress constraint. This paper proposes a methodology to solve reliability-based topology optimization problems of continuum domains with stress constraints and uncertainties in magnitude of applied loads considering the whole set of local stress constrains, without using aggregation techniques. Probabilistic constraints are handled via a first-order approach, where the principle of superposition is used to alleviate the computational burden associated with inner optimization problems. Augmented Lagrangian method is used to solve the outer problem, where all stress constraints are included in the augmented Lagrangian function; hence sensitivity analysis may be performed only for the augmented Lagrangian function, instead of for each stress constraint. Two example problems are addressed, for which crisp black and white topologies are obtained. The proposed methodology is shown to be accurate by checking reliability indices of final topologies with Monte Carlo Simulation.     

Extensions of design potential concept for reliability-based design optimization to nonsmooth and extreme cases

The reliability-based design optimization (RBDO) can be described by the design potential concept in a unified system space, where the probabilistic constraint is identified by the design potential surface of the reliability target that is obtained analytically from the first-order reliability method (FORM). This paper extends the design potential concept to treat nonsmooth probabilistic constraints and extreme case design in RBDO. In addition, refinement of the design potential surface, which yields better optimum design, can be obtained using more accurate second-order reliability method (SORM). By integrating performance probability analysis into the iterative design optimization process, the design potential concept leads to a very effective design potential method (DPM) for robust system parameter design. It can also be applied effectively to extreme case design (ECD) by directly representing a probabilistic constraint in terms of the system performance function. Received July 25, 2000     

Reliability based design optimization of wire bonding in power microelectronic devices

This paper presents a numerical investigation of the probabilistic approach in estimating the reliability of wire bonding, and develops a reliability-based design optimization Methodology (RBDO) for microelectronic device structures. The objective of the RBDO method is to design structures which should be both economical and reliable where the solution reduces the structural weight in uncritical regions. It does not only provide an improved design, but also a higher level of confidence in the design. The Finite element simulation model intends to analyze the sequence of the failure events in power microelectronic devices. This numerical model is used to estimate the probability of failure of power module regarding the wire bonding connection. However, due to time-consuming of the multiphysics finite element simulation, a response surface method is used to approximate the response output of the limit state, in this way the reliability analysis is performed directly to the response surface by using the First and the Second Order Reliability Methods FORM/SORM. Subsequently the reliability analysis is integrated in the optimization process to improve the performance and reliability of structural design of wire bonding. The sequential RBDO algorithm is used to solve this problem and to find the best structural designs which realize the best compromise between cost and safety.     

Reliability-based design optimization for problems with interval distribution parameters

Traditional reliability-based design optimization (RBDO) generally describes uncertain variables using random distributions, while some crucial distribution parameters in practical engineering problems can only be given intervals rather than precise values due to the limited information. Then, an important probability-interval hybrid reliability problem emerged. For uncertain problems in which interval variables are included in probability distribution functions of the random parameters, this paper establishes a hybrid reliability optimization design model and the corresponding efficient decoupling algorithm, which aims to provide an effective computational tool for reliability design of many complex structures. The reliability of an inner constraint is an interval since the interval distribution parameters are involved; this paper thus establishes the probability constraint using the lower bound of the reliability degree which ensures a safety design of the structure. An approximate reliability analysis method is given to avoid the time-consuming multivariable optimization of the inner hybrid reliability analysis. By using an incremental shifting vector (ISV) technique, the nested optimization problem involved in RBDO is converted into an efficient sequential iterative process of the deterministic design optimization and the hybrid reliability analysis. Three numerical examples are presented to verify the proposed method, which include one simple problem with explicit expression and two complex practical applications.     

Application of reliability-based optimization to earth-moving machine: hydraulic cylinder components design process

Reliability-based design optimization (RBDO) incorporates probabilistic analysis into optimization process so that an optimum design has a great chance of staying in the feasible design space when the inevitable variability in design variables/parameters is considered. One of the biggest drawbacks of applying RBDO to practical problem is its high computational cost that is often impractical to industries. In search of the most suitable RBDO method for industrial applications, we first evaluated several existing RBDO approaches in details such as the double-loop RBDO, the sequential optimization and reliability assessment, and the response surface method. Then, based on industry needs, a platform incorporating/integrating the existing algorithm of optimization and reliability analysis is built for a practical RBDO problem. Effectiveness of the proposed RBDO approach is demonstrated using a simple cantilever beam problem and a more complicated industry problem.     

Experience with approximate reliability-based optimization methods II: an exhaust system problem

Traditional reliability-based design optimization (RBDO) requires a double-loop iteration process. The inner optimization loop is to find the reliability and the outer is the regular optimization loop to optimize the RBDO problem with reliability objectives or constraints. It is known that the computation can be prohibitive when the associated function evaluation is expensive. This situation is even worse when a large number of reliability constraints are present. As a result, many approximate RBDO methods, which convert the double loop to a single loop, have been developed. In this research, an engineering problem with a large number of constraints (144) is designed to test RBDO methods based on the first-order reliability method (FORM), including single- and double-loop methods. In addition to the number of constraints, this problem possesses many local minimums. Some original authors of the RBDO methods are also asked to solve the same problem. The results and the efficiencies for different methods are published and discussed.     

Reliability-based design optimization by adaptive-sparse polynomial dimensional decomposition

This paper puts forward two new methods for reliability-based design optimization (RBDO) of complex engineering systems. The methods involve an adaptive-sparse polynomial dimensional decomposition (AS-PDD) of a high-dimensional stochastic response for reliability analysis, a novel integration of AS-PDD and score functions for calculating the sensitivities of the failure probability with respect to design variables, and standard gradient-based optimization algorithms, encompassing a multi-point, single-step design process. The two methods, depending on how the failure probability and its design sensitivities are evaluated, exploit two distinct combinations built on AS-PDD: the AS-PDD-SPA method, entailing the saddlepoint approximation (SPA) and score functions; and the AS-PDD-MCS method, utilizing the embedded Monte Carlo simulation (MCS) of the AS-PDD approximation and score functions. In both methods, the failure probability and its design sensitivities are determined concurrently from a single stochastic simulation or analysis. When applied in collaboration with the multi-point, single-step framework, the proposed methods afford the ability of solving industrial-scale design problems. Numerical results stemming from mathematical functions or elementary engineering problems indicate that the new methods provide more computationally efficient design solutions than existing methods. Furthermore, shape design of a 79-dimensional jet engine bracket was performed, demonstrating the power of the AS-PDD-MCS method developed to tackle practical RBDO problems.     

An inverse-measure-based unilevel architecture for reliability-based design optimization

Reliability-based design optimization (RBDO) is a methodology for finding optimized designs that are characterized with a low probability of failure. Primarily, RBDO consists of optimizing a merit function while satisfying reliability constraints. The reliability constraints are constraints on the probability of failure corresponding to each of the failure modes of the system or a single constraint on the system probability of failure. The probability of failure is usually estimated by performing a reliability analysis. During the last few years, a variety of different formulations have been developed for RBDO. Traditionally, these have been formulated as a double-loop (nested) optimization problem. The upper level optimization loop generally involves optimizing a merit function subject to reliability constraints, and the lower level optimization loop(s) compute(s) the probabilities of failure corresponding to the failure mode(s) that govern(s) the system failure. This formulation is, by nature, computationally intensive. Researchers have provided sequential strategies to address this issue, where the deterministic optimization and reliability analysis are decoupled, and the process is performed iteratively until convergence is achieved. These methods, though attractive in terms of obtaining a workable reliable design at considerably reduced computational costs, often lead to premature convergence and therefore yield spurious optimal designs. In this paper, a novel unilevel formulation for RBDO is developed. In the proposed formulation, the lower level optimization (evaluation of reliability constraints in the double-loop formulation) is replaced by its corresponding first-order Karush–Kuhn–Tucker (KKT) necessary optimality conditions at the upper level optimization. Such a replacement is computationally equivalent to solving the original nested optimization if the lower level optimization problem is solved by numerically satisfying the KKT conditions (which is typically the case). It is shown through the use of test problems that the proposed formulation is numerically robust (stable) and computationally efficient compared to the existing approaches for RBDO.