Reliability-based Design Optimization Procedure Research Articles

Sliding stability analyses of a rock slope using deterministic, semi-probabilistic and probabilistic methods

Stability analyses of geotechnical structures in rock are traditionally performed using deterministic methods. In Europe, Eurocode7, introduced in the beginning of the 21st century, adopts limit state design and semi-probabilistic methods, using partial factors for the design of geotechnical structures. Meanwhile, reliability-based design, using probabilistic methods, is becoming more common in practical cases. The paper considers an intentionally simple case study—the analysis of a slope in a rock mass with one discontinuity, considered in a discrete way, forming a rock block to be stabilised by anchors—to compare the results obtained with the different methods. The objective is to calculate the force applied by the anchors so that the ultimate limit states of sliding of the rock block is verified. Deterministic-based design optimization considering both the traditional global safety factor approach and the partial factor approach following the Eurocode 7 are first applied. A reliability-based design optimization procedure—which takes geometrical and mechanical properties of the discontinuities as random variables—is then used, and the results are compared to the former ones. A discussion is presented concerning the consistency of the obtained results.

Reliability-based design optimization for vertical vibrations of a modified electric vehicle using fourth-moment polynomial standard transformation method

This article presents a new reliability-based design optimization procedure for the vertical vibration issues raised by a modified electric vehicle using fourth-moment polynomial standard transformation method. First, the fourth-moment polynomial standard transformation method with polynomial chaos expansion is used to obtain the reliability index of uncertain constraints in the reliability-based design optimization which is highly precise and saves computing time compared with other common methods. Next, the half-car model with nonlinear suspension parameters for the modified electric vehicle is investigated, and the response surface methodology is adopted to approximate the complex and time-consuming vertical vibration calculation to the polynomial expressions, and the approximation is validated for reliability-based design optimization results within permissible error level. Then, reliability-based design optimization results under both deterministic and uncertain load parameters are shown and analyzed. Unlike the traditional vertical vibration optimization that only considers one or several sets of load parameters, which lacks versatility, this article presents the reliability-based design optimization with uncertain load parameters which is more suitable for engineering. The results show that the proposed reliability-based design optimization procedure is an effective and efficient way to solve vertical vibration optimization problems for the modified electric vehicle, and the optimization statistics, including the maximum probability interval, can provide references for other suspension dynamical optimization.

Reliability-based design optimization of composite battery box based on modified particle swarm optimization algorithm

The application of carbon fiber reinforced polymer (CFRP) material introduces great challenges to the optimization design process, such as complex non-linear material behavior, the inherent uncertainty of design variables and multilevel characteristics of the structure. This paper aims at developing a reliability-based design optimization (RBDO) method to solve the CFRP battery box lightweight design problem considering both meso- and macro-scopic parameters. The method has three kernel parts: the uncertainty quantification and propagation part, the finite element analysis part and the optimization part. In the first part, the internal geometry variability of plain woven CFRP was obtained by X-ray micro-CT images. Representative Volume Element (RVE) models are established to predict the elastic and strength properties of the studied composites, and the constitutive model of material was adapted in stiffness and strength analysis of the battery box structure in the second part. Then a RBDO procedure considering design variables across two scales is developed using a modified particle swarm optimization and surrogate modeling techniques. The structure of the CFRP battery box achieved by the proposed multiscale optimization procedure realizes a weight loss of 22.14%, and the performance demands are satisfied with high reliability, which further reveals the advantages of using this methodology.

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

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

Reliability-based design optimization of reinforced concrete structures including soil–structure interaction using a discrete gravitational search algorithm and a proposed metamodel

A new discrete gravitational search algorithm (DGSA) and a metamodelling framework are introduced for reliability-based design optimization (RBDO) of reinforced concrete structures. The RBDO of structures with soil–structure interaction (SSI) effects is investigated in accordance with performance-based design. The proposed DGSA is based on the standard gravitational search algorithm (GSA) to optimize the structural cost under deterministic and probabilistic constraints. The Monte-Carlo simulation (MCS) method is considered as the most reliable method for estimating the probabilities of reliability. In order to reduce the computational time of MCS, the proposed metamodelling framework is employed to predict the responses of the SSI system in the RBDO procedure. The metamodel consists of a weighted least squares support vector machine (WLS-SVM) and a wavelet kernel function, which is called WWLS-SVM. Numerical results demonstrate the efficiency and computational advantages of DGSA and the proposed metamodel for RBDO of reinforced concrete structures.

Unified reliability and design optimization for earthquake engineering

In this paper, a methodology is put forward that builds upon the emerging focus on structural damage in earthquake engineering. A versatile “unified” limit-state formulation is employed to complement the traditional focus on collapse limit-states. The methodology underscores the advantages of formulating explicit probabilistic models for all predictions; including ground motion, structural response, damage, and ensuing losses. Next, by formulating limit-state functions in terms of total cost, including construction and damage costs, a novel reliability-based design optimization procedure is offered. Notably, the design optimization is carried out in the tail of the total cost distribution. Consequently, the unlikely but potentially devastating losses are explicitly addressed. This is significant for many real decision makers. This also addresses the criticism of the traditional design optimization that it focuses solely on expected (mean) cost. The proposed approach is shown to be akin to a risk averse decision strategy. Numerical examples including a six-storey structure with random structural properties and random ground motion are presented to illustrate the methodology.

Reliability-Based Design of a Slat-Track Fatigue LifeUsing Mesh Morphing Technology

Although the aerospace production process is much better controlled than in other industries, it remains true that very small manufacturing tolerances exist in the geometrical parameters such as flange thickness and hole diameters. In the current design process, the effect of this manufacturing variability on the structural durability and safety cannot be accurately assessed and is hence compensated for by applying safety factors. This is not an ideal situation, because it may lead to slightly overdesigned structures. A much more promising approach is to include probabilistic models of design variables into the mechanical simulation process. With a new methodology based on reliability analysis, engineers can obtain a better understanding of the actual effect of the manufacturing tolerances and of variability in material properties. Based on the analysis results, the robustness and reliability of the design can be assessed and improved if needed. In this paper, the aforementioned probabilistic approach is demonstrated on a slat-track structure. Measurements of different geometrical properties were collected during the manufacturing process and their variability was characterized probabilistically with statistical models. Then a reliability analysis was carried out using mesh morphing technology and fatigue life predictions with an industrial-sized finite element model of the slat track to assess the reliability of the structure in terms of fatigue life. The outcome of the analysis consists of a probabilistic model of the structural performance (e.g., fatigue life for the slat track), given the variability in the geometrical parameters. Then a reliability-based design optimization procedure was carried out to improve the design of the slat track while maintaining the same reliability of the nominal design.

신뢰성을 고려한 효율적인 공력 형상 최적 설계에 대한 연구

일반적인 신뢰성 최적 설계는 결정론적 최적 설계에 비해 매우 많은 계산비용이 필요하므로 공력 형상 최적화와 같은 큰 문제에 직접 적용하기는 매우 어렵다. 본 연구에서는 이러한 한계를 극복하기 위하여 이점 근사화 기법과 adjoint 민감도 해석 기법을 결합한 효율적인 신뢰성 설계 과정을 제안한다. 이 방법은 계산비용은 결정론적 방법과 거의 동일하지만 계산 결과에 있어서는 기존의 신뢰성 설계 기법과 유사한 결과를 얻을 수 있다. 이를 이용하여 3차원 공력 형상 최적 설계를 매우 효율적으로 수행할 수 있었다. The conventional reliability based design optimization(RBDO) methods require high computational cost compared with the deterministic design optimization(DO) methods, therefore it is hard to apply directly to large-scaled problems such as an aerodynamic shape design optimization. In this study, to overcome this computational limitation the efficient RBDO procedure with the two-point approximation(TPA) and adjoint sensitivity analysis is proposed, that the computational requirement is nearly the same as DO and the reliability accuracy is good compared with that of RBDO. Using this, the 3-D aerodynamic shape design optimization is performed very efficiently.

온도영향을 고려한 연수기 압력탱크의 신뢰성 최적설계

Deterministic optimum designs that are obtained without consideration of uncertainties could lead to unrealiable designs, Such deterministic engineering optimization tends to promote the structural system with less reliability redundancy than obtained with conventional design procedures using the factor of safety, Consequently, deterministic optimized structures will usually have higher failure probabilities than unoptimized structures. This paper proposes the reliability based design optimization technique for a pressure tank considering temperature effect. This paper presents an efficient and stable reliability based design optimization method by using the advanced first order second moment method, which evaluates a probabilistic constraint far more accuracy. In addition, the response surface method is utilized to approximate the performance functions describing the system characteristics in the reliability based design optimization procedure.

Reliability based design optimization using response surfaces in application to multidisciplinary systems

Traditionally, reliability based design optimization (RBDO) is formulated as a nested optimization problem. For these problems the objective is to minimize a cost function while satisfying the reliability constraints. The reliability constraints are usually formulated as constraints on the probability of failure corresponding to each of the failure modes or a single constraint on the system probability of failure. The probability of failure is usually estimated by performing a reliability analysis. 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. In this research, a framework for performing reliability based multidisciplinary design optimization using approximations is developed. Response surface approximations (RSA) of the limit state functions are used to estimate the probability of failure. An outer loop is incorporated to ensure that the approximate RBDO converges to the actual most probable point of failure. The framework is compared with the exact RBDO procedure. In the proposed methodology, RSAs are employed to significantly reduce the computational expense associated with traditional RBDO. The proposed approach is implemented in application to multidisciplinary test problems, and the computational savings and benefits are discussed.