Reliability-based design optimization of composite stiffened panels in post-buckling regime

This paper focuses on Deterministic and Reliability Based Design Optimization (DO and RBDO) of composite stiffened panels considering post-buckling regime and progressive failure analysis. The ultimate load that a post-buckled panel can hold is to be maximised by changing the stacking sequence of both skin and stringers composite layups. The RBDO problem looks for a design that collapses beyond the shortening of failure obtained in the DO phase with a target reliability while considering uncertainty in the elastic properties of the composite material. The RBDO algorithm proposed is decoupled and hence separates the Reliability Analysis (RA) from the deterministic optimization. The main code to drive both the DO and RBDO approaches is written in MATLAB and employs Genetic Algorithms (GA) to solve the DO loops because discrete design variables and highly nonlinear response functions are expected. The code is linked with Abaqus to perform parallel explicit nonlinear finite element analyses in order to obtain the structural responses at each generation. The RA is solved through an inverse Most Probable failure Point (MPP) search algorithm that benefits from a Polynomial Chaos Expansion with Latin Hypercube Sampling (PCE-LHS) metamodel when the structural responses are required. The results led to small reductions in the maximum load that the panels can bear but otherwise assure that they will collapse beyond the shortening of failure imposed with a high reliability.

     

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.     

Optimum values of structural safety factors for a predefined reliability level with extension to multiple limit states

In the field of deterministic structural optimization, the designer reduces the structural cost without taking into account uncertainties concerning materials, geometry and loading. This way, the resulting optimum solution may represent a lower level of reliability and thus a higher risk of failure. It is the objective of reliability-based design optimization (RBDO) to design structures that should be both economic and reliable. The coupling between mechanical modeling, reliability analyses and optimization methods leads to very high computational costs and weak convergence stability. Since the traditional RBDO solution is achieved by alternating between reliability and optimization iterations, the structural designers performing deterministic optimization do not consider the RBDO model as a practical tool for the design of real structures. Fortunately, a hybrid method based on simultaneous solution of the reliability and the optimization problem, has successfully reduced the computational time problem. The hybrid method allows us to satisfy a required reliability level, but the vector of variables here contains both deterministic and random variables. The hybrid RBDO problem is thus more complex than that of deterministic design. The major difficulty lies in the evaluation of the structural reliability, which is carried out by a special optimization procedure. In this paper a new methodology is presented with the aim of finding a global solution to RBDO problems without additional computing cost for the reliability evaluation. The safety factor formulation for a single limit state case has been used to efficiently reduce the computational time . This technique is fundamentally based on a study of the sensitivity of the limit state function with respect to the design variables. In order to demonstrate analytically the efficiency of this methodology, the optimality condition is then used. The efficiency of this technique is also extended to multiple limit state cases. Two numerical examples are presented at the end of the paper to demonstrate the applicability of the new methodology.     

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.     

Convergence control of single loop approach for reliability-based design optimization

For solution of reliability-based design optimization (RBDO) problems, single loop approach (SLA) shows high efficiency. Thus SLA is extensively used in RBDO. However, the iteration solution procedure by SLA is often oscillatory and non-convergent for RBDO with nonlinear performance function. This prevents the application of SLA to engineering design problems. In this paper, the chaotic single loop approach (CLSA) is proposed to achieve the convergence control of original iterative algorithm in SLA. The modification involves automated selection of the chaos control factor by solving a novel one-dimensional optimization model. Additionally, a new oscillation-checking method is constructed to detect the oscillation of iterative process of design variables. The computational capability of CLSA is demonstrated through five benchmark examples and one stiffened shell application. The comparison of numerical results indicates that CSLA is more efficient than the double loop approach and the decoupled approach. CSLA also solves the RBDO problems with highly nonlinear performance function and non-normal random variables stably.     

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.     

Single loop single vector approach using the conjugate gradient in reliability based design optimization

Conventional reliability-based design optimization (RBDO) approaches require high computing costs. Among the existing RBDO methods, the single loop single vector approach (SLSV) converts the RBDO problem into a single loop deterministic optimization. Hence, it can efficiently reduce the design cost compared to other methods. However, this method has a weakness in that instability or inaccuracy in convergence can be increased according to the problem characteristics. It often happens when the performance function is highly nonlinear or concave. In this study, a novel method is proposed to overcome the problems. It is an SLSV method using the conjugate gradient that is calculated with the gradient directions at the most probable points (MPP) of the previous cycles. Mathematical examples and structural applications are solved to verify the proposed method. The numerical performances of the proposed method are compared with other RBDO methods such as the RIA, PMA, SORA and SLSV approaches. It is shown that the SLSV method using the conjugate gradient (SLSVCG) is not greatly influenced by problem characteristics and the convergence capability is quite superior. Also, the computational cost of the proposed method is significantly reduced and an excellent solution satisfying the specified reliability is obtained.     

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.     

Inverse analysis method using MPP-based dimension reduction for reliability-based design optimization of nonlinear and multi-dimensional systems

There are two commonly used analytical reliability analysis methods: linear approximation - first-order reliability method (FORM), and quadratic approximation - second-order reliability method (SORM), of the performance function. The reliability analysis using FORM could be acceptable in accuracy for mildly nonlinear performance functions, whereas the reliability analysis using SORM may be necessary for accuracy of nonlinear and multi-dimensional performance functions. Even though the reliability analysis using SORM may be accurate, it is not as much used for probability of failure calculation since SORM requires the second-order sensitivities. Moreover, the SORM-based inverse reliability analysis is rather difficult to develop.This paper proposes an inverse reliability analysis method that can be used to obtain accurate probability of failure calculation without requiring the second-order sensitivities for reliability-based design optimization (RBDO) of nonlinear and multi-dimensional systems. For the inverse reliability analysis, the most probable point (MPP)-based dimension reduction method (DRM) is developed. Since the FORM-based reliability index (β) is inaccurate for the MPP search of the nonlinear performance function, a three-step computational procedure is proposed to improve accuracy of the inverse reliability analysis: probability of failure calculation using constraint shift, reliability index update, and MPP update. Using the three steps, a new DRM-based MPP is obtained, which estimates the probability of failure of the performance function more accurately than FORM and more efficiently than SORM. The DRM-based MPP is then used for the next design iteration of RBDO to obtain an accurate optimum design even for nonlinear and/or multi-dimensional system. Since the DRM-based RBDO requires more function evaluations, the enriched performance measure approach (PMA+) with new tolerances for constraint activeness and reduced rotation matrix is used to reduce the number of function evaluations.     

An optimal shifting vector approach for efficient probabilistic design

The application of reliability-based design optimization (RBDO) is hindered by the unbearable computational cost in the structure reliability evaluating process. This study proposes an optimal shifting vector (OSV) approach to enhance the efficiency of RBDO. In OSV, the idea of using an optimal shifting vector in the decoupled method and the notation of conducting reliability analysis in the super-sphere design space are proposed. The shifted limit state function, instead of the specific performance function, is used to identify the inverse most probable point (IMPP) and derive the optimal shifting vector for accelerating the optimization process. The super-sphere design space is applied to reduce the number of constraints and design variables for the novel reliability analysis model. OSV is very efficient for highly nonlinear problems, especially when the contour lines of the performance functions vary widely. The computation capability of the proposed method is demonstrated and compared to existing RBDO methods using four mathematical and engineering examples. The comparison results show that the proposed OSV approach is very efficient.     

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.     

A decoupling approach for evidence-theory-based reliability design optimization

For the problem of evidence-theory-based reliability design optimization (EBDO), this paper presents a decoupling approach which provides an effective tool for the reliability design of some complex structures with epistemic uncertainty. The approach converts the original nested optimization into a sequential iterative process including design optimization and reliability analysis. In each iteration step, through the uniformity algorithm, the original EBDO is firstly transformed to a conventional reliability-based design optimization (RBDO) and an optimal solution is obtained by solving it. At the solution, the first-order approximate reliability analysis method (FARM) is then used to perform the evidence-theory-based reliability analysis for each constraint. In addition, the RBDO solving and the evidence-theory-based reliability analysis are carried out alternately until reaching the convergence. Finally, two numerical examples and a practical engineering application show the effectiveness of the proposed method.     

Reliability-based design optimization using SORM and SQP

In this work a second order approach for reliability-based design optimization (RBDO) with mixtures of uncorrelated non-Gaussian variables is derived by applying second order reliability methods (SORM) and sequential quadratic programming (SQP). The derivation is performed by introducing intermediate variables defined by the incremental iso-probabilistic transformation at the most probable point (MPP). By using these variables in the Taylor expansions of the constraints, a corresponding general first order reliability method (FORM) based quadratic programming (QP) problem is formulated and solved in the standard normal space. The MPP is found in the physical space in the metric of Hasofer-Lind by using a Newton algorithm, where the efficiency of the Newton method is obtained by introducing an inexact Jacobian and a line-search of Armijo type. The FORM-based SQP approach is then corrected by applying four SORM approaches: Breitung, Hohenbichler, Tvedt and a recent suggested formula. The proposed SORM-based SQP approach for RBDO is accurate, efficient and robust. This is demonstrated by solving several established benchmarks, with values on the target of reliability that are considerable higher than what is commonly used, for mixtures of five different distributions (normal, lognormal, Gumbel, gamma and Weibull). Established benchmarks are also generalized in order to study problems with large number of variables and several constraints. For instance, it is shown that the proposed approach efficiently solves a problem with 300 variables and 240 constraints within less than 20 CPU minutes on a laptop. Finally, a most well-know deterministic benchmark of a welded beam is treated as a RBDO problem using the proposed SORM-based SQP approach.     

A direct decoupling approach for efficient reliability-based design optimization

This paper develops an efficient methodology to perform reliability-based design optimization (RBDO) by decoupling the optimization and reliability analysis iterations that are nested in traditional formulations. This is achieved by approximating the reliability constraints based on the reliability analysis results. The proposed approach does not use inverse first-order reliability analysis as other existing decoupled approaches, but uses direct reliability analysis. This strategy allows a modular approach and the use of more accurate methods, including Monte-Carlo-simulation (MCS)-based methods for highly nonlinear reliability constraints where first-order reliability approximation may not be accurate. The use of simulation-based methods also enables system-level reliability estimates to be included in the RBDO formulation. The efficiency of the proposed RBDO approach is further improved by identifying the potentially active reliability constraints at the beginning of each reliability analysis. A vehicle side impact problem is used to examine the proposed method, and the results show the usefulness of the proposed method.     

Single-loop system reliability-based topology optimization considering statistical dependence between limit-states

This paper presents a single-loop algorithm for system reliability-based topology optimization (SRBTO) that can account for statistical dependence between multiple limit-states, and its applications to computationally demanding topology optimization (TO) problems. A single-loop reliability-based design optimization (RBDO) algorithm replaces the inner-loop iterations to evaluate probabilistic constraints by a non-iterative approximation. The proposed single-loop SRBTO algorithm accounts for the statistical dependence between the limit-states by using the matrix-based system reliability (MSR) method to compute the system failure probability and its parameter sensitivities. The SRBTO/MSR approach is applicable to general system events including series, parallel, cut-set and link-set systems and provides the gradients of the system failure probability to facilitate gradient-based optimization. In most RBTO applications, probabilistic constraints are evaluated by use of the first-order reliability method for efficiency. In order to improve the accuracy of the reliability calculations for RBDO or RBTO problems with high nonlinearity, we introduce a new single-loop RBDO scheme utilizing the second-order reliability method and implement it to the proposed SRBTO algorithm. Moreover, in order to overcome challenges in applying the proposed algorithm to computationally demanding topology optimization problems, we utilize the multiresolution topology optimization (MTOP) method, which achieves computational efficiency in topology optimization by assigning different levels of resolutions to three meshes representing finite element analysis, design variables and material density distribution respectively. The paper provides numerical examples of two- and three-dimensional topology optimization problems to demonstrate the proposed SRBTO algorithm and its applications. The optimal topologies from deterministic, component and system RBTOs are compared with one another to investigate the impact of optimization schemes on final topologies. Monte Carlo simulations are also performed to verify the accuracy of the failure probabilities computed by the proposed approach.     

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.     

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.     

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.     

A hybrid sufficient performance measure approach to improve robustness and efficiency of reliability-based design optimization

The stable convergence and efficiency of reliability-based design optimization (RBDO) using performance measure approach (PMA) are the major issue to develop the reliability methods based on modified chaos control (MCC), hybrid chaos control (HCC) and finite-step length adjustment (FSL). However, these methods may be inefficient for RBDO problems with convex and concave probabilistic constraints. In this paper, an adaptive modified chaos control (AMC) is proposed to provide the robust and efficient results in RBDO. The proposed AMC is adjusted using dynamical chaos control factor, which is extracted using sufficient descent condition for PMA. Using sufficient criterion, the proposed AMC is adaptively combined with advanced mean value (AMV) to improve the performance of PMA, named as hybrid adaptive modified chaos control (HAMC). Considering the robustness and efficiency, the proposed HAMC is compared with several existing reliability methods by three nonlinear structural/mathematical performance functions and two RBDO problems. The results indicate that the proposed HAMC with sufficient descent condition provides superior convergences in terms of both robustness and efficiency, compared to existing PMA methods using AMV, MCC, HCC and FSL.

     

Efficient reliability-based design optimization using a hybrid space with application to finite element analysis

The design of high technology structures aims to define the best compromise between cost and safety. The Reliability-Based Design Optimization (RBDO) allows us to design structures which satisfy economical and safety requirements. However, in practical applications, the coupling between the mechanical modelling, the reliability analyses and the optimization methods leads to very high computational time and weak convergence stability. Traditionally, the solution of the RBDO model is achieved by alternating reliability and optimization iterations. This approach leads to low numerical efficiency, which is disadvantageous for engineering applications on real structures. In order to avoid this difficulty, we propose herein a very efficient method based on the simultaneous solution of the reliability and optimization problems. The procedure leads to parallel convergence for both problems in a Hybrid Design Space (HDS). The efficiency of the proposed methodology is demonstrated on the design of a steel hook, where the RBDO is combined with Finite Element Analysis (FEA).