Sequential Optimization And Reliability Assessment Research Articles

Probabilistic design and optimization of thermal protection system with variable thickness based on non-uniform aerodynamic heating

Thermal Protection System (TPS) is a critical component of space vehicles to protect them from damage by extreme aerothermal heating conditions. The uncertainty of the aerothermal heating conditions, including approaching velocities, atmospheric density, pressure, etc., significantly affects the aerothermal heating imposed on the TPS, vastly changing the initial design conditions. In this study, we developed a new uncertainty quantitative (UQ) analysis method to comprehensively consider all the uncertainty of TPS materials, structures, and aerothermal heating conditions. To address the challenges of a massive amount of input and determined optimization parameters, we first induced the response surface method and Monte Carlo algorithm to produce a large number of analysis samples for Probabilistic Design. Then, the key affecting factors, not all the input design parameters, were chosen for optimization with sequential optimization and reliability assessment (SORA) strategy by a sensitivity analysis. With such improvements, the optimization for UQ analysis becomes practical. We used this new UQ method to analyze the effects of uncertainty of the aerothermal heating conditions on the thermal protection layer. A variable thickness design is proposed to adapt the non-uniform aerothermal heating conditions for the TPS. The results show that significant light-weighting can be achieved without the reliability penalty. The optimized structure also reduces internal temperature gradients and facilitates a more uniform temperature field distribution for the TPS. This study provides a new UQ engineering design approach for the system with many designed parameters.

An Efficient Decoupled Reliability-Based Topology Optimization Method Based on a Performance Shift Strategy

AbstractIn this paper, a new efficient reliability-based topology optimization (RBTO) method is proposed for structures, in which the double-loop optimization is equivalently decoupled into a sequential process based on a performance shift strategy. First, a volume minimization RBTO formulation is built for structures considering displacement or compliance reliability constraints. Second, an efficient decoupling scheme is proposed to turn the double-loop RBTO into a series of deterministic topology optimization and reliability analysis. For the reliability analysis, the reliability probability is calculated based on the probability distribution function of the performance function, and the probability distribution function can be solved by the maximum entropy method based on the raw moments calculated by the multiplicative dimension reduction method. A performance shift strategy is then applied to build an equivalence between probabilistic constraint and deterministic constraint to formulate the deterministic topology optimization. Thirdly, the adjoint variable method is applied to obtain the sensitivity information for topological design variables, and a gradient-based optimization algorithm is used to update the design variables. Finally, four typical numerical examples are used to verify the advantage of the proposed method. Compared with the traditional sequential optimization and reliability assessment (SORA) method, the proposed RBTO method results in a design with a 5.4% smaller volume value to reach the same reliability index requirement for the cantilever beam, and the function calls are 354 times less than that of the SORA method.

Simulation-Based Reliability Design Optimization Method for Industrial Robot Structural Design

Robots are main elements in Industry 4.0. Research on the design optimization of robots has a great significance in manufacturing industries. There inevitably exist various uncertainties in robot design that have an important influence on the reliability of robots. At present, the design optimization of robots considering the uncertainties is mainly focused on joints design and trajectory optimization. However, for the structural design of robots, deterministic design optimization still plays a leading role. In this paper, a simulation-based reliability design optimization method is proposed to improve the reliability of robots’ structural design. In the proposed method, the Latin hypercube sampling (LHS), computer simulation, response surface method (RSM) and SORA (Sequential Optimization and Reliability Assessment) algorithm are integrated to complete the structural design of the robot. Firstly, samples of the uncertainty design variables were obtained by LHS, and then, the reliability performance constraint functions were firstly constructed through the RSM in which the joint simulation of MTLAB and ANSYS was adopted. Afterwards, the reliability design optimization model was established on the basis of the probabilistic reliability theory. At last, the SORA algorithm was employed to realize the optimization. The design optimization problems of the big arm and the small arm of a 6 Kg industrial robot were considered to verify the proposed method. The results showed that the weights of the big arm and the small arm were, respectively, reduced by 7.73% and 25.70% compared with those of the original design, and the design was more effective in ensuring the reliability requirements compared with the deterministic optimization. Moreover, the results also demonstrated that the proposed method has a better computational efficiency compared with the reliability design optimization of the double-loop method.

A decoupled double-loop method with the adaptive allowable limits for probabilistic performance-based design optimization

In this study, inspired by the idea of the sequential optimization and reliability assessment (SORA) method, a new decoupled double-loop method for probabilistic performance-based design optimization (PPBDO) has been proposed. The highlight of the proposed method is the independence of this method from the type of problem and the limit state function; in such a way that it can be used in non-linear problems with multiple probabilistic constraints with the implicit form of the limit state function. However, it requires less computational effort than conventional double-loop methods. In each cycle of this method, deterministic performance-based design optimization was performed and then the reliability of the optimal design was evaluated. If the results of the reliability assessment are not desirable, the allowable limit of the constraints in the deterministic optimization problem will be changed based on the results of the reliability assessment. This process repeats until the results of the reliability assessment become desirable. The metaheuristic enhanced vibrating particle system algorithm and the Monte Carlo simulation method were used for optimization process and reliability analysis, respectively. To evaluate the efficiency of the proposed method, the optimization problem was defined so as to reduce the structural weight while meeting the acceptance criteria for performance-based design of steel moment frames according to ASCE 41-13. The PPBDO of three- and nine-story steel moment frames were investigated and the results showed that the proposed method was able to reduce the volume of calculations and direct the search path toward a reliable space.

Systematic Reliability-Based Multidisciplinary Optimization by Parallel Adaptive Importance Candidate Region

Reliability-based design optimization (RBDO) has become a prevalent design for aeronautical and aerospace engineering. The main problem is that it is impractical in complex cases with multi-failure regions, especially in multi-objective optimization. The active learning method can obtain an adaptive size of samples to get a relatively acceptable accuracy. The problem of RBDO using the traditional active learning Kriging (ALK) method is that the design space is generally still and only one training point is selected, which is not reasonable based on the concept of importance sampling and parallel calculation. As a consequence, the accuracy improvement is limited. In this paper, we investigate the method of obtaining an optimal size of design and reliability to assess space in parallel, simultaneously. A strategy of parallel adaptive candidate (PAIC) region with ALK is proposed and a sequential optimization and reliability assessment (SORA) method is modified to efficiently improve the accuracy. Importance sampling is used as a demonstration for the modified SORA with more accuracy. The method is then verified using mathematical cases and a scooping system of an amphibious aircraft.

A new framework for reliability-based design optimization using metaheuristic algorithms

This paper presents a new framework for reliability-based design optimization (RBDO) using metaheuristic algorithms based on decoupled methods. The sequential optimization and reliability assessment (SORA) is utilized as a decoupled method in this study. In the present framework, unlike the previous RBDO based on decoupled methods, a metaheuristic is employed on both reliability assessment and the optimization parts. Here, first a new reliability assessment method based on the metaheuristics is introduced. Then, a new termination condition for the metaheuristic inspired by the termination condition of the gradient-based method is presented. In order to investigate the efficiency of the proposed framework, an enhanced shuffled shepherd optimization algorithm (ESSOA) is used as an optimization approach. The efficiency of the proposed framework is evaluated by four well-known RBDO benchmarks investigated in the previous studies using the gradient-based method. Also, the two new RBDO problems are introduced in this study. The results show that the proposed framework can have better performance than the gradient-based method in the RBDO and can easily be utilized in a wide range of RBDO problems.

An Extended SORA Method for Hybrid Reliability-Based Design Optimization

In many practical engineering problems, distributions of some random variables may not be precisely known or even only the variation ranges can be given, due to which the Reliability-Based Design Optimization (RBDO) methods cannot be applied directly. In this paper, a hybrid RBDO model is established to deal with hybrid uncertainties including random variables with interval distribution parameters and coexistence of random and interval variables. The hybrid uncertainties lead to an interval of reliability, thus giving rise to a triple-loop optimization problem, which burdens the computational effort significantly. Therefore, an extended Sequential Optimization and Reliability Assessment (SORA) method is proposed, where the worst-case point over the bounded domain of interval uncertainty is employed to replace reliability constraint with deterministic optimization. Therefore, the hybrid RBDO problem can be efficiently solved by a series of deterministic optimization procedures. The efficiency and accuracy of the proposed method are illustrated through several numerical examples.

Uncertainty quantification of quadratically nonlinear local damage size quantification using nonlinear ultrasonics

Backscattered and forward scattered harmonic waves from local damage modeled as quadratically nonlinear material in a 1D bar are used to solve a deterministic inverse problem to find the local damage size by Ghodake (2020). Due to highly nonlinear (|sin(x)/x|) nature of the proposed forward problem, the constrained nonlinear deterministic optimization problem was solved using non-gradient-based algorithms. The forward model is a function of input frequencies, linear wave velocity, and local damage size. In practice, slight uncertainty of these parameters gives a completely different solution. To address this problem, uncertainty quantification of the modified forward problem is carried out along with solving the inverse problem using reliability-based design optimization (RBDO) and Bayesian Inverse (BI) approaches. In RBDO studies, Quantile Monte Carlo (QMC) approach gives more reliable and accurate results in comparison with approaches such as reliability index approach (RIA), performance measure approach (PMA), sequential optimization and reliability assessment (SORA), and inverse first-order reliability method (FORM) integrated with optimization methods such as interior-point (IP) and sequential programming (SQP). Multiple models are solved using the BI approach. Support vector machines regression (SVR), polynomial chaos expansions (PCE), and Kriging surrogate models are used along with RBDO and BI to reduce computational time.

Reliability-based optimization of ship structure based on interest subdomain dynamic surrogate model

Objectives Aiming at the difficulty of ensuring the fitting accuracy and optimization efficiency of surrogate models due to the high nonlinearity in ship structure reliability-based optimization design, a reliability-based ship structure optimization method based on the interest subdomain dynamic surrogate model is proposed. Methods This method puts forward the concept of the interest subdomain based on the sequential optimization and reliability assessment (SORA) method, determines the range of interest subdomains and formulates adaptive spatial reduction rules based on information entropy function H, then proposes an adaptive spatial reduction sequential sampling strategy based on interest subdomains, thereby constructing a dynamic Kriging surrogate model that highly fits the subdomain of interest locally with as few sample points as possible, and embedding the surrogate model and multi-island genetic algorithm (MIGA) in the SORA method to undertake reliability-based optimization.This study proposes a probability constraint feasibility checking method to reduce unnecessary reliability assessment processes. A mathematical example is given to verify the reliability-based optimization method. Results The relative error between the optimal solution and theoretical solution is 0.066 8%, and the number of function calls is 40.6% less than those of the optimal method in the references, which proves the accuracy and efficiency of this method. Conclusions When the proposed method is applied to the reliability optimization design of a cabin structure, the total cabin mass is reduced by 0.511% compared with the references, and 94 fewer finite element calculations are required, proving the efficiency and applicability of this method.

Global Decoupling for Structural Reliability-Based Optimal Design Using Improved Differential Evolution and Chaos Control

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.

An active learning Kriging-assisted method for reliability-based design optimization under distributional probability-box model

Due to lack of sufficient data and information in engineering practice, it is often difficult to obtain precise probability distributions of some uncertain variables and parameters in reliability-based design optimization (RBDO). In this paper, distributional probability-box (p-box) model is employed to quantify these uncertain variables and parameters. To reduce the computational cost in RBDO associated with expensive and time-consuming constraints, an active learning Kriging-assisted method is proposed. In this method, the sequential optimization and reliability assessment (SORA) method is extended for RBDO under distributional p-box model. Kriging metamodels are constructed to make the replacement of actual constraints. To remove unnecessary computational expense on constructing Kriging metamodels, a screening criterion is built and employed for the judgment of active constraints in RBDO. Then, an active learning function is defined to find out update samples, which are adopted for sequentially refining Kriging metamodel of each active constraint by focusing on its limit-state surface (LSS) around the most probable target point (MPTP) at the solution of SORA. Several examples, including a welded beam problem and a piezoelectric energy harvester design, are provided to test the accuracy and efficiency of the proposed active learning Kriging-assisted method.

A new local update-based method for reliability-based design optimization

Reliability-based design optimization (RBDO) is often associated with computationally expensive computer simulations in practical engineering applications, in which both objective and constraint functions may be implicit and time-consuming to evaluate. To improve the efficiency of RBDO, metamodels are often employed to substitute the implicit and expensive functions. In this paper, a new local update-based method is proposed for RBDO under the approximation of both objective and constraint functions. In this method, Kriging metamodels are employed to replace both objective and constraint functions, and sequential optimization and reliability assessment (SORA) is adopted to perform RBDO. Meanwhile, a local update method is developed for the approximation of objective and constraint functions, termed as LUOC. In LUOC, two local update strategies are proposed. For the constraint functions, one local update strategy judges the active constraints and then sequentially refines their Kriging metamodels in the local regions around the most probable target points obtained by SORA. For the objective function, the other local update strategy based on expected improvement is developed to refine the Kriging metamodel in the local region around the SORA solution with considering the Kriging prediction variance. Three examples including a battery design application are presented to test the accuracy and efficiency of the proposed method for RBDO.

A Single-Loop Reliability-Based MDSDO Formulation for Combined Design and Control Optimization of Stochastic Dynamic Systems

Abstract Optimization of dynamic engineering systems requires an integrated approach that accounts for the coupling between embodiment design and control system design, simultaneously. Generally known as combined design and control optimization (co-design), these methods offer superior system’s performance and reduced costs. Despite the widespread use of co-design approaches in the literature, not much work has been done to address the issue of uncertainty in co-design problem formulations. This is problematic as all engineering models contain some level of uncertainty that might negatively affect the system’s performance, if overlooked. While in our previous study we developed a robust co-design approach, a more rigorous evaluation of probabilistic constraints is required to obtain the targeted reliability levels for probabilistic constraints. Therefore, we propose and implement a novel stochastic co-design approach based on the principles of reliability-based design optimization (RBDO) to explicitly account for uncertainties from design decision variables and problem parameters. In particular, a reliability-based, multidisciplinary dynamic system design optimization (RB-MDSDO) formulation is developed using the sequential optimization and reliability assessment (SORA) algorithm, such that the analysis-type dynamic equality constraints are satisfied at the mean values of random variables, as well as their most probable points (MPPs). The proposed approach is then implemented for two case studies, and the results were benchmarked through Monte Carlo simulation (MCS) to indicate the impact of including reliability measures in co-design formulations.

Reliability-Based Design Optimization Using Step Length Adjustment Algorithm and Sequential Optimization and Reliability Assessment Method

A step length adjustment (SLA) iterative algorithm was proposed for locating the minimum performance target point (MPTP) in the inverse reliability analysis. This paper elaborates SLA and two deliberately designed numerical examples are used to compare SLA with other algorithms appearing in recent literatures for locating MPTP. The results show that SLA is much more robust and efficient. Then SLA and sequential optimization and reliability assessment (SORA) are combined to solve reliability-based design optimization (RBDO) problems. In the reliability assessment of SORA, with the design obtained from the previous cycle, SLA is used to locate MPTP. Then in the deterministic optimization, the boundaries of violated constraints are shifted to the feasible direction according to the MPTP obtained in the reliability assessment. Several examples frequently cited in similar studies are used to compare SORA-SLA with other RBDO algorithms. The results indicate the effectiveness and robustness of SORA-SLA.

An enhanced nonlinear interval number programming method considering correlation of interval variables

Based on decoupling strategy, a novel efficient method is proposed to solve the nonlinear interval uncertainty optimization problem with correlated interval design variables or parameters. This method is applicable to cases where both objective function and constraints are nonlinear with uncertain parameters, and design variables and parameters can be correlated or independent. The uncertainty of design variables and interval parameters is expressed by a multidimensional parallelepiped model, with which the correlated variables and parameters can be converted into independent interval parameters, thus constituting the traditional interval optimization model for independent interval parameters. Based on the idea of sequential optimization and reliability assessment (SORA), the two-layer nested optimization involved in the above interval optimization model can be converted into a single loop problem which can be solved efficiently by the sequential deterministic optimization algorithm. Finally, three numerical examples are investigated to demonstrate the effectiveness of the proposed model.

Extending SORA method for reliability-based design optimization using probability and convex set mixed models

In many practical applications, probabilistic and bounded uncertainties often arise simultaneously, and these uncertainties can be described by using probability and convex set models. However, the computing cost becomes unacceptable when directly solving the reliability-based design optimization (RBDO) problem with these uncertainties involved. To address this issue, in this study, a sequential sampling strategy by extending classical sequential optimization and reliability assessment (SORA) method for RBDO is developed. The proposed strategy can successively select sample points to update the surrogate model at each step of the optimization process. New samples for reliability constraints are mainly chosen from the local region around the approximate minimum performance target point (MPTP) and worst-case point (WCP). Typical design examples, including one engineering application, are investigated to demonstrate the efficiency and accuracy of the proposed method.

Adaptive conjugate single-loop method for efficient reliability-based design and topology optimization

The single-loop approach (SLA) for reliability-based design optimization (RBDO) is one of the most efficient schemes for optimization problems with linear and weak nonlinear probabilistic constraints. However, it may produce unstable results or increase computational efforts when using the ordinary search direction to determine the optimum design in RBDO problems with highly nonlinear probabilistic constraints. The conjugate gradient (CG) is a promising sensitivity vector for locating the most probable point (MPP) of highly nonlinear concave performance functions. However, the MPP computation using the CG may require a high computational burden for convex constraints To overcome the drawbacks of the SLA the adaptive conjugate single-loop approach (AC-SLA) is proposed for RBDO problems with a large variety of nonlinear constraints. The sensitivity vector of the probabilistic constraints is adaptively computed using the CG vector with a dynamical conjugate scalar factor (DCF). The DCF is adjusted within the range from 0 to 2 using two adaptive coefficients, which are adapted based on the new and previous points. Moreover, the Lyapunov exponents are developed as a general tool for detecting the robustness of different MPP approximation algorithms. The method is also applied to solve a reliability-based topology optimization (RBTO) problem. The ability of the AC-SLA in six RBDO benchmark problems, one applicable to an RBDO aircraft engineering problem and one for RBTO problem, is compared in terms of both robustness and efficiency using the SLA, performance measure approach (PMA), reliability index approach (RIA), and sequential optimization and reliability assessment (SORA). The numerical experiments demonstrate that the convergence capability of the proposed AC-SLA is significantly superior to the PMA, RIA, and SORA. The computational effort of the AC-SLA is significantly reduced, with stable results.

An improved two-stage framework of evidence-based design optimization

In this paper, an improved two-stage framework is presented to handle the evidence-based design optimization (EBDO) problem under epistemic uncertainty. The improvements include two aspects: (1) in the first stage, the equal areas method is employed to transform evidence variables into random variables, which avoids the assumption that unknown evidence variables and parameters obey the normal distribution. Then, a reliability-based design optimization (RBDO) problem with random variables is defined and solved by the sequential optimization and reliability assessment (SORA) method; (2) in the second stage, an improved algorithm is presented, which can calculate the plausibility of constraint violation more efficiently by continuously recording the minimum and maximum values of limit-state functions. The computational accuracy and efficiency of the improved framework are tested by numerical and engineering examples.

A novel methodology of reliability-based multidisciplinary design optimization under hybrid interval and fuzzy uncertainties

Various uncertainties exist in engineering practice, which brings adverse effects to the reliability of complicated engineering systems. Considering the case that interval and fuzzy uncertainties exist simultaneously, a new reliability estimation model is proposed based on the level cut strategy and volume ratio theory. The new reliability model is better than the traditional one which is very conservative. Moreover, a sequential optimization and reliability assessment (SORA) approach for multidisciplinary systems under hybrid interval and fuzzy uncertainties is developed to decouple the reliability analysis from the deterministic multidisciplinary design optimization (MDO). In the framework of SORA, the deterministic MDO and reliability analysis are executed sequentially, thus the efficiency can be improved. For the multidisciplinary uncertainty analysis, the first order Taylor expansion method and the interval vertex method are formulated. The calculation of the safety possibility under the volume ratio theory and the calculation of the shifting distance are deduced. Both numerical and engineering examples are employed to demonstrate the validity of the proposed method.

Comparative study of methods for optimization of electromagnetic devices with uncertainty

PurposeThis paper compares six reliability-based design optimization (RBDO) approaches dealing with uncertainties for a simple mathematical model and a multidisciplinary optimization problem of a safety transformer to highlight the most effective.Design/methodology/approachThe RBDO and various approaches to calculate the probability of failure are is presented. They are compared in terms of precision and number of evaluations on mathematical and electromagnetic design problems.FindingsThe mathematical example shows that the six RBDO approaches have almost the same results except the approximate moment approach that is less accurate. The optimization of the safety transformer highlights that not all the methods can converge to the global solution. Performance measure approach, single-loop approach and sequential optimization and reliability assessment (SORA) method appear to be more stable. Considering both numerical examples, SORA is the most effective method among all RBDO approaches.Originality/valueThe comparison of six RBDO methods on the optimization problem of a safety transformer is achieved for the first time. The comparison in terms of precision and number of evaluations highlights the most effective ones.