A reliability-based multidisciplinary design optimization procedure based on combined probability and evidence theory

To address the reliability-based multidisciplinary design optimization (RBMDO) problem under mixed aleatory and epistemic uncertainties, an RBMDO procedure is proposed in this paper based on combined probability and evidence theory. The existing deterministic multistage-multilevel multidisciplinary design optimization (MDO) procedure MDF-CSSO, which combines the multiple discipline feasible (MDF) procedure and the concurrent subspace optimization (CSSO) procedure to mimic the general conceptual design process, is used as the basic framework. In the first stage, the surrogate based MDF is used to quickly identify the promising reliable regions. In the second stage, the surrogate based CSSO is used to organize the disciplinary optimization and system coordination, which allows the disciplinary specialists to investigate and optimize the design with the corresponding high-fidelity models independently and concurrently. In these two stages, the reliability-based optimization both in the system level and the disciplinary level are computationally expensive as it entails nested optimization and uncertainty analysis. To alleviate the computational burden, the sequential optimization and mixed uncertainty analysis (SOMUA) method is used to decompose the traditional double-level reliability-based optimization problem into separate deterministic optimization and mixed uncertainty analysis sub-problems, which are solved sequentially and iteratively until convergence is achieved. By integrating SOMUA into MDF-CSSO, the Mixed Uncertainty based RBMDO procedure MUMDF-CSSO is developed. The effectiveness of the proposed procedure is testified with one simple numerical example and one MDO benchmark test problem, followed by some conclusion remarks.     

Sequential optimization and reliability assessment for multidisciplinary systems design

With higher reliability and safety requirements, reliability-based design has been increasingly applied in multidisciplinary design optimization (MDO). A direct integration of reliability-based design and MDO may present tremendous implementation and numerical difficulties. In this work, a methodology of sequential optimization and reliability assessment for MDO is proposed to improve the efficiency of reliability-based MDO. The central idea is to decouple the reliability analysis from MDO with sequential cycles of reliability analysis and deterministic MDO. The reliability analysis is based on the first-order reliability method (FORM). In the proposed method, the reliability analysis and the deterministic MDO use two MDO strategies, the multidisciplinary feasible approach and the individual disciplinary feasible approach. The effectiveness of the proposed method is illustrated with two example problems.     

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.     

Reliability based multidisciplinary design optimization of cooling turbine blade considering uncertainty data statistics

Considering the coupling among aerodynamic, heat transfer and strength, a reliability based multidisciplinary design optimization method for cooling turbine blade is introduced. Multidisciplinary analysis of cooling turbine blade is carried out by sequential conjugated heat transfer analysis and strength analysis with temperature and pressure interpolation. Uncertainty data including the blade wall, rib thickness, elasticity Modulus and rotation speed is collected. Data statistics display the probability models of uncertainty data follow three-parameter Weibull distribution. The thickness of blade wall, thickness and height of ribs are chosen as design variables. Kriging surrogate model is introduced to reduce time-consuming multidisciplinary reliability analysis in RBMDO loop. The reliability based multidisciplinary design optimization of a cooling turbine blade is carried out. Optimization results shows that the RBMDO method proposed in this work improves the performance of cooling turbine blade availably.

     

A surrogate based multistage-multilevel optimization procedure for multidisciplinary design optimization

Optimization procedure is one of the key techniques to address the computational and organizational complexities of multidisciplinary design optimization (MDO). Motivated by the idea of synthetically exploiting the advantage of multiple existing optimization procedures and meanwhile complying with the general process of satellite system design optimization in conceptual design phase, a multistage-multilevel MDO procedure is proposed in this paper by integrating multiple-discipline-feasible (MDF) and concurrent subspace optimization (CSSO), termed as MDF-CSSO. In the first stage, the approximation surrogates of high-fidelity disciplinary models are built by disciplinary specialists independently, based on which the single level optimization procedure MDF is used to quickly identify the promising region and roughly locate the optimum of the MDO problem. In the second stage, the disciplinary specialists are employed to further investigate and improve the baseline design obtained in the first stage with high-fidelity disciplinary models. CSSO is used to organize the concurrent disciplinary optimization and system coordination so as to allow disciplinary autonomy. To enhance the reliability and robustness of the design under uncertainties, the probabilistic version of MDF-CSSO (PMDF-CSSO) is developed to solve uncertainty-based optimization problems. The effectiveness of the proposed methods is verified with one MDO benchmark test and one practical satellite conceptual design optimization problem, followed by conclusion remarks and future research prospects.     

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.     

Continuum topology optimization with non-probabilistic reliability constraints based on multi-ellipsoid convex model

Using a quantified measure for non-probab ilistic reliability based on the multi-ellipsoid convex model, the topology optimization of continuum structures in presence of uncertain-but-bounded parameters is investigated. The problem is formulated as a double-loop optimization one. The inner loop handles evaluation of the non-probabilistic reliability index, and the outer loop treats the optimum material distribution using the results from the inner loop for checking feasibility of the reliability constraints. For circumventing the numerical difficulties arising from its nested nature, the topology optimization problem with reliability constraints is reformulated into an equivalent one with constraints on the concerned performance. In this context, the adjoint variable schemes for sensitivity analysis with respect to uncertain variables as well as design variables are discussed. The structural optimization problem is then solved by a gradient-based algorithm using the obtained sensitivity. In the present formulation, the uncertain-but bounded uncertain variations of material properties, geometrical dimensions and loading conditions can be realistically accounted for. Numerical investigations illustrate the applicability and the validity of the present problem statement as well as the proposed numerical techniques. The computational results also reveal that non-probabilistic reliability-based topology optimization may yield more reasonable material layouts than conventional deterministic approaches. The proposed method can be regarded as an attractive supplement to the stochastic reliability-based topology optimization.     

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.     

Efficient strategies for reliability-based design optimization of variable stiffness composite structures

This study investigates efficient design optimization frameworks for composite structures with uncertainties related to material properties and loading. The integration of two decoupled reliability-based design optimization methodologies with a decoupled discrete material optimization is proposed to determine material and fiber orientation for three-dimensional composite structures. First, a deterministic and decoupled discrete material optimization is used for baseline comparison. The objective is to minimize the cost of composite structures with the design variables comprising of the piecewise patch orientations and material properties of the fiber reinforced composites. The reliability-based design optimization includes a hybrid method, and also the sequential optimization and reliability assessment method. In the sequential optimization and reliability assessment method, the inverse reliability analysis is evaluated using a stochastic response surface method and a first order reliability approach. Comparing the methods based on the optimal material and fiber orientations, the uncertainties in loads and material properties lead to different optimal layouts compared to the deterministic solutions. The numerical results also reveal that the hybrid method applied in reliability based designs results in negligible additional computational cost.     

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.     

Sequential approach to reliability analysis of multidisciplinary analysis systems

The goal of this study is to present an efficient strategy for reliability analysis of multidisciplinary analysis systems. Existing methods have performed the reliability analysis using nonlinear optimization techniques. This is mainly due to the fact that they directly apply multidisciplinary design optimization (MDO) frameworks to the reliability analysis formulation. Accordingly, the reliability analysis and the multidisciplinary analysis (MDA) are tightly coupled in a single optimizer, which hampers the use of recursive and function-approximation-based reliability analysis methods such as the first-order reliability method (FORM). In order to implement an efficient reliability analysis method for multidisciplinary analysis systems, we propose a new strategy named sequential approach to reliability analysis for multidisciplinary analysis systems (SARAM). In this approach, the reliability analysis and MDA are decomposed and arranged in a sequential manner, making a recursive loop. The key features are as follows. First, by the nature of the recursive loop, it can utilize the efficient advanced first-order reliability method (AFORM). It is known that AFORM converges fast in many cases and requires only the value and the gradient of the limit-state function. Second, the decomposed architecture makes it possible to execute concurrent subsystem analyses for both the reliability analysis and MDA. The concurrent subsystem analyses are conducted by using the global sensitivity equation (GSE). The efficiency of the SARAM method was verified using two illustrative examples taken from the literatures. Compared with existing methods, it showed the least number of subsystem analyses over the other methods while maintaining accuracy.     

A robust study of reliability-based optimization methods under eigen-frequency

Nowadays, the search in reliability-based design optimization is becoming an important engineering design activity. Traditionally for these problems, the objective function is to minimize a cost function while satisfying the reliability constraints. The reliability constraints are usually formulated as constraints on the probability of failure. This paper focuses on the study of a particular problem with the failure mode on vibration of structure. The difficulty in evaluating reliability constraints comes from the fact that modern reliability analysis methods are themselves formulated as an optimization problem. Solving such nested optimization problems is extremely expensive for large-scale multidisciplinary systems which are likewise computationally intensive. With this in mind research, we propose in this paper a new method to treat reliability-based optimization methods under frequencies constraint. The goal of this development has resolved just one problem of optimization and reduced the cost of computation. Aircraft wing design typically involves multiple disciplines such as aerodynamics and structure; this numerical example demonstrated the different advantages of the proposed method.     

A sequential approximate programming strategy for reliability-based structural optimization

Although reliability-based structural optimization (RBSO) is recognized as a rational structural design philosophy that is more advantageous to deterministic optimization, most common RBSO is based on straightforward two-level approach connecting algorithms of reliability calculation and that of design optimization. This is achieved usually with an outer loop for optimization of design variables and an inner loop for reliability analysis. A number of algorithms have been proposed to reduce the computational cost of such optimizations, such as performance measure approach, semi-infinite programming, and mono-level approach. Herein the sequential approximate programming approach, which is well known in structural optimization, is extended as an efficient methodology to solve RBSO problems. In this approach, the optimum design is obtained by solving a sequence of sub-programming problems that usually consist of an approximate objective function subjected to a set of approximate constraint functions. In each sub-programming, rather than direct Taylor expansion of reliability constraints, a new formulation is introduced for approximate reliability constraints at the current design point and its linearization. The approximate reliability index and its sensitivity are obtained from a recurrence formula based on the optimality conditions for the most probable failure point (MPP). It is shown that the approximate MPP, a key component of RBSO problems, is concurrently improved during each sub-programming solution step. Through analytical models and comparative studies over complex examples, it is illustrated that our approach is efficient and that a linearized reliability index is a good approximation of the accurate reliability index. These unique features and the concurrent convergence of design optimization and reliability calculation are demonstrated with several numerical examples.     

Reliability-based design optimization using a family of methods of moving asymptotes

In this study, an effective method for reliability-based design optimization is proposed enhancing sequential optimization and reliability assessment (SORA) method by a family of methods of moving asymptotes (MMA) approximations. In SORA, reliability estimation and deterministic optimization are performed sequentially. And the sensitivity and function value of probabilistic constraint at the most probable point (MPP) are obtained in the process of finding reliability information. In this study, a family of MMA approximations are constructed by utilizing the sensitivity and function value of the probabilistic constraint at the MPP. So, no additional evaluation of the probabilistic constraint is required in constructing MMA approximations. Moreover, no additional evaluation of the probabilistic constraint is required in the deterministic optimization of SORA by using a family of MMA approximations. The efficiency and accuracy of the proposed method were verified through numerical examples.     

Novel numerical methods for reliability analysis and optimization in engineering fuzzy heat conduction problem

Engineering fuzzy heat conduction problem with subjective uncertainties in input parameters constitutes a significant challenge. Based on fuzzy and interval theory, this paper presents novel numerical methods to efficiently identify the effect of fuzzy uncertainty on the system reliability analysis and optimization design. Firstly using the interval ranking strategy, the interval safety possibility in the transition state can be precisely quantified, and the eventual fuzzy safety possibility is calculated by integral operation. Then a fuzzy reliability-based optimization model is established with considerable computational cost caused by the two-layer nested loop. In order to improve the computational efficiency, a subinterval perturbation method based on the first-order Taylor series is presented to replace the inner loop. Comparing numerical results with traditional reliability model, two numerical examples are provided to evidence the superiority of proposed model and method for fuzzy reliability analysis and optimization in practical engineering.     

基于Isight的CSSO算法框架

概述多学科优化设计(Multidisciplinary Design Optimization,MDO)在复杂产品设计中的必要性,阐述CSSO算法的基本内容,以齿轮减速器为例,利用Isight搭建CSSO计算框架并对每一步进行分析和说明.将CSSO计算结果与传统优化结果进行对比,结果表明CSSO算法比传统算法的计算效率高、准确性好.     

Reliability-based design optimization of aeroelastic structures

Aeroelastic phenomena are most often either ignored or roughly approximated when uncertainties are considered in the design optimization process of structures subject to aerodynamic loading, affecting the quality of the optimization results. Therefore, a design methodology is proposed that combines reliability-based design optimization and high-fidelity aeroelastic simulations for the analysis and design of aeroelastic structures. To account for uncertainties in design and operating conditions, a first-order reliability method (FORM) is employed to approximate the system reliability. To limit model uncertainties while accounting for the effects of given uncertainties, a high-fidelity nonlinear aeroelastic simulation method is used. The structure is modelled by a finite element method, and the aerodynamic loads are predicted by a finite volume discretization of a nonlinear Euler flow. The usefulness of the employed reliability analysis in both describing the effects of uncertainties on a particular design and as a design tool in the optimization process is illustrated. Though computationally more expensive than a deterministic optimum, due to the necessity of solving additional optimization problems for reliability analysis within each step of the broader design optimization procedure, a reliability-based optimum is shown to be an improved design. Conventional deterministic aeroelastic tailoring, which exploits the aeroelastic nature of the structure to enhance performance, is shown to often produce designs that are sensitive to variations in system or operational parameters.     

Reliability-based multi-scale design optimization of composite frames considering structural compliance and manufacturing constraints

Structural and Multidisciplinary Optimization - The paper proposes an efficient methodology for concurrent reliability-based multi-scale design optimization (RBMDO) of composite frames to minimize...     

An approximate sequential optimization and reliability assessment method for reliability-based design optimization

Sequential optimization and reliability assessment (SORA) is one of the most popular decoupled approaches to solve reliability-based design optimization (RBDO) problem because of its efficiency and robustness. In SORA, the double loop structure is decoupled through a serial of cycles of deterministic optimization and reliability assessment. In each cycle, the deterministic optimization and reliability assessment are performed sequentially and the boundaries of violated constraints are shifted to the feasible direction according to the reliability information obtained in the previous cycle. In this paper, based on the concept of SORA, approximate most probable target point (MPTP) and approximate probabilistic performance measure (PPM) are adopted in reliability assessment. In each cycle, the approximate MPTP needs to be reserved, which will be used to obtain new approximate MPTP in the next cycle. There is no need to evaluate the performance function in the deterministic optimization since the approximate PPM and its sensitivity are used to formulate the linear Taylor expansion of the constraint function. One example is used to illustrate that the approximate MPTP will approach the accurate MPTP with the iteration. The design variables and the approximate MPTP converge simultaneously. Numerical results of several examples indicate the proposed method is robust and more efficient than SORA and other common RBDO methods.