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

Reliability-based design optimization and uncertainty quantification for optimal conditions of composite structures with non-linear behavior

In this paper, an efficient technique is developed to predict failure probability of three failure modes(case rupture, fracture and bolt breakage) related to solid rocket motor case due to the inner pressure during the mission flight. The overall procedure consists of the steps: 1) design parameters affecting the case failure are identified and their uncertainties are modelled by probability distribution, 2) combustion analysis in the interior of the case is carried out to obtain maximum expected operating pressure(MEOP), 3) stress and other structural performances are evaluated by finite element analysis(FEA), and 4) failure probabilities are calculated for the above mentioned failure modes. Axi-symmetric assumption for FEA is employed for simplification while contact between bolted joint is accounted for. Efficient procedure is developed to evaluate failure probability which consists of finding first an Most Probable Failure Point(MPP) using First-Order Reliability Method(FORM), next making a response surface model around the MPP using Latin Hypercube Sampling(LHS), and finally calculating failure probability by employing Importance Sampling.

In this research, a reliability based topology optimization (RBTO) for structural design methodology using the Hybrid Cellular Automata (HCA) method is proposed. More speciflcally, a decoupled reliability based design optimization (RBDO) approach is utilized, so that the topology optimization is separate from the reliability analysis. In this paper, a maximum allowable displacement failure mode is considered. In this methodology, starting from a continuum design space of uniform material distribution and initial uncertain variable values, a deterministic topology optimization is followed by a reliability assessment of the resulting structure to determine the most probable point of failure (MPP) for the current structure. The MPP is determined with respect to the maximum allowable de∞ection of the structure when loaded. This is generally a computationally expensive process using traditional techniques due to the large number of design variables associated with topology optimization problem. However, combining the e‐cient methods of the non-gradient HCA algorithm with the decoupled approach for RBDO aims to reduce this burden. The topology optimization was without constraint in previous applications of the HCA method. To accommodate RTBO, a mechanism for a global constraint for maximum allowable displacement is developed. This paper details the methodology for the six-sigma design of structures using topology optimization.

복합재료가 신재생 에너지 산업 관련 구조물 및 해양 구조물에서 좀 더 신뢰도 있는 주 하중 부재로 사용되기 위하여 복합재료평판의 확률론적 비선형 초기 파단 하중과 원공과 곡률이 있는 복합재료판의 확률론적 비선형 좌굴 하중이 평가되었다. 주어진 설계 추출점에서의 확정론적 유한요소해석 결과를 바탕으로 반응면기법을 이용하여 한계상태면을 확률변수로 이루어진 2차 다항식으로 근사하였다. 또한, MPFP 근처에서 좀 더 정확하게 한계상태면을 근사하기 위하여 반복적 선형보간법이 적용되었다. 파괴확률을 평가하기 위하여 근사된 한계상태면 상에서 향상된 일계이차모멘트법과 몬테카를로법이 수행되었다. 마지막으로 파단에 영향을 주는 주요한 확률변수를 파악하기 위하여 변환된 확률변수에 대한 신뢰도지수의 감도를 계산하였다. Probabilistic nonlinear first ply failure loads of flat composite panels and nonlinear buckling loads of curved composite panels with cutouts are estimated to provide the more reliable main load carrying structure in the renewable energy industry and offshore structures. The response surface method approximates limit state surface to a second order polynomial form of random variables with the results of deterministic finite element analyses at given sampling design points. Furthermore, the iterative linear interpolation scheme is used to obtain a more accurate approximation of the limit state surface near the most probable failure point (MPFP). The advanced first order second moment method and the Monte Carlo method are performed on an approximated limit state surface to evaluate the probability of failure. Finally, the sensitivity of the reliability index with respect to transformed random variables is investigated to figure out the main random variables that have an effect on failures.

<div class="htmlview paragraph">This paper presents an enriched performance measure approach (PMA+) for reliability-based design optimization (RBDO) to substantially improve computational efficiency when applied to large-scale applications. Three aspects of PMA+ are presented: as a way to launch RBDO at a deterministic optimum design, as an efficient probabilistic feasibility check, and as a fast reliability analysis under the condition of design closeness. It is found that deterministic design optimization helps improve numerical efficiency by reducing some RBDO iterations. Unlike deterministic design optimization, a significant computational burden is imposed on the feasibility check of constraints in the RBDO process due to the costs of a reliability analysis. Such difficulties can be effectively resolved by using a mean value (MV) first-order method with an allowable accuracy for the purpose of feasibility identification, and by carrying out the refined reliability analysis using the hybrid mean value (HMV) first-order method for <i>ε</i>-active and violate constraints in the RBDO process. In addition, the fast reliability analysis method is proposed by reusing some of the information obtained at the previous RBDO iteration to efficiently evaluate probabilistic constraints at the current design iteration under the condition of design closeness. Other RBDO methods have recently been developed to enhance numerical efficiency of RBDO while maintaining numerical accuracy and stability. Thus, the PMA+ is compared to existing RBDO methods from a numerical efficiency and stability point of view. For a numerical understanding of the RBDO process, several numerical examples are provided, including a large-scale multi-crash application.</div>

The design of reinforced concrete (RC) structures involves several kinds of uncertainties, which are usually considered through the partial safety factors prescribed in the codes of practice. The traditional design optimization of RC structures uses deterministic information of the problem. The partial safety factors are used to consider loading fluctuation and the variability of material properties. The use of these safety factors in deterministic optimization usually leads to over-designing structures, as these safety factors are calibrated for a large class of structures. In the deterministic optimization procedure the reliability cannot be controlled. For this reason, the Reliability-Based Design Optimization (RBDO) is devoted to design economical and reliable structures. However, the RBDO problem involves the evaluation of probabilistic constraints performed by the reliability analysis. The most common RBDO formulations are based on a nested optimization problem, with an outer loop for the design optimization and an inner loop for the reliability analysis. Therefore, an expensive computation effort is required to solve the RBDO problem. In this paper, a new RBDO method of RC structures is proposed. The RBDO problem is decomposed to several cycles of deterministic design optimization (DDO) based on new safety factors called Optimal System Safety Factors (OSSF). At each cycle of the DDO, the system reliability analysis is performed to verify the reliability of the optimal design, the OSSF are computed by a probabilistic method on the basis of the previous system reliability analysis, then the updated OSSF are provided for the next cycle of the DDO. The application to the design of an RC structure shows the interest and the efficiency of the proposed method.

With the rapid advancement of computing technology, mathematical modeling and computer simulations have become common and standard methods for design of complex systems. To exercise these models intelligently and eliminate the burden of manual iteration, manipulating inputs and reviewing outputs, optimization strategies are applied in a simulation-based design environment. In reality, the knowledge that engineers have about the design problem is imperfect and incomplete, uncertainty always exists in simulation-based design. The conventional deterministic optimization techniques do not consider the impact of such uncertainties. If the system responses driving the design decisions are very sensitive to these uncertainties, serious problems might confront the designer. Performance of the actual system may differ greatly from expected optimum performance. Again, if the design solution is near one or more constraint boundaries, then even slight uncertainties or changes in the operating environment could produce failed, unsafe designs, and/or result in substantial performance degradation. In recent years, the concept of reliability and robust design has become very popular. Probabilistic design analysis and statistical modeling allow the identification of designs that qualify as not only feasible, but as consistently feasible in the face of uncertainty. Reliability based topology optimization (RBTO) extends the reliability notion to the area of structural topology optimization. Although reliability based design optimization (RBDO) is a well established research area, very few work has been presented in the area of RBTO. In this study, we give an overview of the various works done in the field of RBTO. Ever since Bendsoe and Kikuchi introduced the topology optimization using a homogenization method, many topology optimization methods have been developed for the both linear and nonlinear structures. Traditional topology optimization methods drive the topology of a structure to an optimum design based on the constraint on mass. Kharmanda et al. introduced reliability constraints in deterministic topology optimization problem. They proposed a heuristic strategy that aims to reduce mass while improving the reliability level of the structure without greatly increasing its weight. But the limit state function used by them was not based on failure criteria for the structure. Their formulation considered uncertainty with respect to ∗Design Automation Laboratory †Advisor geometrical dimension and applied load only. Also their reliability analysis seems to be independent of the boundary and loading condition, so their results showed similar values for the uncertain variables for different structures. Tovar et al. have developed the Hybrid Cellular Automaton (HCA) method for structural synthesis of continuum material where the state of each cell is defined by both density and strain energy. In Agarwal, a decoupled RBDO approach is employed such that the topology optimization is separate from the reliability analysis. Patel et al. showed the use of RBTO using the gradient free Hybrid Cellular Automata (HCA) method. Their formulation incorporates uncertainty with respect to material property also. They considered limit state function based on failure modes on the output displacements. This study aims at giving a comparative study on these methods. It points out the advantages and disadvantages of the methods, and their significance in the field of RBTO. As large numbers of design variables are associated with continuum topology optimization problems, RBTO methods are inherently computationally expensive because of additional required system analysis associate with RBDO. Based on the observations, this study presents an efficient method for incorporating reliability analysis in topology optimization.

An enriched performance measure approach is presented for reliability-based design optimization to substantially improve computational efficiency when applied to large-scale applications. In the enriched performance measure approach, four improvements are made over the original performance measure approach: as a way to launch reliability-based design optimization at a deterministic optimum design, as a new enhanced hybrid-mean value method, as an efficient probabilistic feasibility check, and as a fast reliability analysis under the condition of design closeness. It is found that deterministic design optimization helps improve numerical efficiency by reducing some reliability-based design optimization iterations. In reliability-based design optimization, a computational burden on the feasibility check of constraints can be significantly reduced by using a mean value first-order method and by carrying out the refined reliability analysis using the enhanced hybrid-mean value method for e-active and violated constraints. The enhanced hybrid-mean value method is developed to handle nonlinear and/or nonmonotonic constraints in reliability analysis. The fast reliability analysis method is proposed to efficiently evaluate probabilistic constraints under the condition of design closeness. Moreover, two numerical examples are provided to compare the enriched performance measure approach to existing reliability-based design optimization methods from a numerical efficiency and stability point of view.

Reliability based design optimization is a methodology for nding optimized designs that are characterized with a low probability of failure. Primarily, reliability based design optimization 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 dierent formulations have been developed for reliability based design optimization. Traditionally, these have been formulated as a doubleloop (nested) optimization problems. The upper level optimization loop generally involves optimizing a merit function subject to reliability constraints and the lower level optimization loop(s) compute the probabilities of failure corresponding to the failure mode(s) that govern 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 reliability based design optimization is developed. In the proposed formulation, the lower level optimization (evaluation of reliability constraints in the double-loop formulation) is replaced by its corresponding rst order Karush-Kuhn-Tucker (KKT) necessary optimality conditions at the upper level optimization. Such a replacement is equivalent to solving the original nested optimization if the constraint qualication conditions are satised. It is shown through the use of test problems that the proposed formulation is numerically robust (stable) and computationally ecient compared to the existing approaches for reliability based design optimization.

In this paper, a surrogate‐based iterative importance sampling (IS) method is proposed to efficiently evaluate failure probability (FP) of engineering structures. The surrogate is constructed by the training points centered at the most probable failure point (MPP), which is estimated by an approximate algorithm without any additional evaluations of the real limit state function (LSF), and the IS method using the estimated MPP is further obtained; and an iterative procedure combined with the surrogate and the IS method is finally proposed to improve the accuracy of the estimated result for the FP. Three numerical examples are employed to demonstrate the accuracy of the proposed method, and further, an inside flap of an aircraft is employed to demonstrate the practical application of the proposed method in engineering problems. The results show that the reasonable estimated result of the FP within the range of 10−3‐10−5 can be evaluated by the proposed method with only hundreds of the evaluations of the real LSF.

This paper presents a new method for performing reliability-based design optimization (RBDO) of structures based on sequential optimization and reliability assessment (SORA). SORA is an effective method for solving RBDO that separates uncertainty analysis from optimization loops for reducing computational cost. However, SORA has some limitations, such as an inability to deal with problems involving discrete design variables or discontinuity in a domain, the dependency of solutions on the starting point of numerical solutions, and the lack of guarantee that global optimum solutions will be found. In this paper, a global decoupling method is proposed to tackle these limitations and improve the performance of SORA. This method links SORA enhanced with modified chaos control (ESORA) to an improved differential evolution (IDE) to perform RBDO. To deal with RBDO problems with discrete design variables, a rounding method is integrated into IDE. IDE also utilizes an adaptive selection scheme in a mutation step and an elitist strategy in the selection phase. To improve the efficiency of the method for RBDO problems with highly nonlinear performance functions and nonnormal random variables, a modified chaos control is employed to assess reliability constraints. Five numerical examples are considered to investigate the strength of the proposed method, illustrating its appropriate efficiency and accuracy. The proposed method is also extendable to more complex problems such as system reliability-based structural optimization and RBDO of nonlinear structures.

This study performed reliability-based robust design optimization (RBRDO) of stator and rotor shape of the IPM type electric motor to reduce the torque ripple considering uncertainties. A metamodel based on the design of experiments results, which was calculated by using 2-D electromagnetic finite element analysis, was used to calculate manufacturing uncertainties and material property distribution. As a result, the mean torque ripple of the optimum electric motor design was reduced by 56.4 % and the robustness was increased by 61.8 % compared with the initial design. The RBRDO design was compared with the designs of deterministic design optimization and reliability-based design optimization, and it was shown that, by using the RBRDO method, the optimum IPMSM can be designed to satisfying not only the minimum torque ripple but also the robustness and the reliability of output torque and torque ripple.

Use of approximations for calculating the structural failure probability under uncertainties is a necessary aspect. But efficient use of suitable approximations at different stages of reliability prediction and design makes this valuable tool practical for many large scale structures. The objective of this paper is to address the modern approximation concepts and their utility in structural reliability estimation. Emphasis is placed on building the higher order approximations using intervening variables to closely represent the nonlinear limit state functions. The most probable failure point (MPP) search can be carried either in X-space or U-space for a wide variety of random variable distributions. Several test problems with highly nonlinear performance functions are used to demonstrate the accuracy of the proposed method.

<p>The primary objective of this study is to develop new computational methods for robust design optimization (RDO) and reliability-based design optimization (RBDO) of high-dimensional, complex engineering systems. Four major research directions, all anchored in polynomial dimensional decomposition (PDD), have been defined to meet the objective. They involve: (1) development of new sensitivity analysis methods for RDO and RBDO; (2) development of novel optimization methods for solving RDO problems; (3) development of novel optimization methods for solving RBDO problems; and (4) development of a novel scheme and formulation to solve stochastic design optimization problems with both distributional and structural design parameters.</p> <p>The major achievements are as follows. Firstly, three new computational methods were developed for calculating design sensitivities of statistical moments and reliability of high-dimensional complex systems subject to random inputs. The first method represents a novel integration of PDD of a multivariate stochastic response function and score functions, leading to analytical expressions of design sensitivities of the first two moments. The second and third methods, relevant to probability distribution or reliability analysis, exploit two distinct combinations built on PDD: the PDD-SPA method, entailing the saddlepoint approximation (SPA) and score functions; and the PDD-MCS method, utilizing the embedded Monte Carlo simulation (MCS) of the PDD approximation and score functions. For all three methods developed, both the statistical moments or failure probabilities and their design sensitivities are both determined concurrently from a single stochastic analysis or simulation. Secondly, four new methods were developed for RDO of complex engineering systems. The methods involve PDD of a high-dimensional stochastic response for statistical moment analysis, a novel integration of PDD and score functions for calculating the second-moment sensitivities with respect to the design variables, and standard gradient-based optimization algorithms. The methods, depending on how statistical moment and sensitivity analyses are dovetailed with an optimization algorithm, encompass direct, single-step, sequential, and multi-point single-step design processes. Thirdly, two new methods were developed for 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, resulting in 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 SPA and score functions; and the AS-PDD-MCS method, utilizing the embedded MCS of the AS-PDD approximation and score functions. In addition, a new method, named as the augmented PDD method, was developed for RDO and RBDO subject to mixed design variables, comprising both distributional and structural design variables. The method comprises a new augmented PDD of a high-dimensional stochastic response for statistical moment and reliability analyses; an integration of the augmented PDD, score functions, and finite-difference approximation for calculating the sensitivities of the first two moments and the failure probability with respect to distributional and structural design variables; and standard gradient-based optimization algorithms, leading to a multi-point, single-step design process.</p> <p>The innovative formulations of statistical moment and reliability analysis, design sensitivity analysis, and optimization algorithms have achieved not only highly accurate but also computationally efficient design solutions. Therefore, these new methods are capable of performing industrial-scale design optimization with numerous design variables.</p>

A multiobjective, reliability-based design optimization (RBDO) is performed considering manufacturing uncertainties in order to facilitate the performance and reliability of a centrifugal compressor. During the RBDO procedure, Monte Carlo Simulation (MCS) is utilized to calculate failure probability. Latin hypercube design (LHD) is adopted to sample and form simulation experiment scheme. Besides, response surface model (RSM) is constructed to reduce the computational cost of RBDO. Results show that in deterministic optimization (DO), the reliability of mass flow rate is only 0.7843 while in RBDO the reliability of mass flow rate reaches 0.9976. Meanwhile, the value of efficiency and total pressure ratio descend 0.00197 and 0.00822 respectively, but both of them still satisfy the reliability requirements. Accordingly, a more reliable design scheme of the compressor is obtained.