Collaborative Reliability Analysis under the Environment of Multidisciplinary Design Optimization

In order to improve the computational efficiency of reliability analysis for MDO, a hierarchical hybrid reliability analysis strategy is proposed. This strategy is composed of two layers, the sequential optimization and reliability assessment (SORA) method is adopted in the upper level to decouple the reliability analysis from MDO, which guarantee that the reliability analysis and deterministic MDO can be executed in a serial way. In the lower layer, the collaborative optimization (CO) strategy is integrated both in deterministic MDO and reliability analysis. Furthermore, based on the traditional theory of most probable point (MPP), the reliability analysis combined with the CO strategy is conducted concurrently. And also, the deterministic MDO is implemented in parallel with the CO strategy. Finally, the proposed strategy is validated with an illustrative example and the results are compared with the existing reliability analysis methods in MDO, showing that the proposed strategy is efficient and competent for evaluating design feasibility in MDO environment with uncertainties.

The optimal design of multidisciplinary systems under uncertainty involves three types of iterative analyses- multidisciplinary analyses, reliability analyses, and optimization. If these processes are carried out in a nested manner in reliability based multidisciplinary design optimization (RBMDO), the computational expense can be enormous. Computationally efficient simultaneous analysis and design (SAND) or All-AtOnce methods have been developed earlier for deterministic MDO. Single loop RBDO methods have also been developed to integrate reliability analysis and optimization in a single loop. This paper develops a novel formulation of RBMDO problems with both system and component level reliability constraints. The formulation allows for solution with a single design optimization loop by using the KKT optimality conditions of the first-order reliability method (FORM) as constraints in the design optimization along with disciplinary consistency constraints. Alternative solution processes which remove the FORM MPP’s from the decision space and many constraints from the problem are developed. The alternative solution methods are more efficient and more robust than using nonlinear programming to solve the single loop formulation. A numerical example is solved with the two proposed solution algorithms. I. Introduction In the design of multidisciplinary systems it is often necessary to make decisions under uncertainty. This requires that multidisciplinary systems modeling be integrated with uncertainty analysis and optimization. A common example of such an engineering design application would be the design of an aircraft wing under uncertain aerodynamic loading conditions, with multiple structural components with uncertain capacities, such that the entire structural system achieves a prescribed level of reliability. The problem has coupled physics, in that the pressure distribution is dependent upon the deflection, and the deflection is dependent on the pressure distribution. Such designs have been performed using multidisciplinary optimization (MDO). Other examples where MDO may be applied include electronics packaging, in which the disciplines of heat transfer and circuit analysis are coupled, the design of new types of aircraft and integrating them into an existing fleet of aircraft, and even the management of interconnected homeland infrastructures. While MDO has matured with the development of different types of algorithms for various types of problems and design organizations, there has been little work to incorporate reliability analysis into MDO. However, system and component reliabilities are often a major concern in the design of multidisciplinary systems. MDO can be computationally expensive due to the need to achieve consistency across multiple disciplinary analysis codes; each disciplinary analysis code may be computationally expensive to run. Further, the optimal design of multidisciplinary systems under uncertainty involves three types of iterative procedures- multidisciplinary analyses, reliability analyses, and optimization. The integration of these three procedures is referred to in this paper as reliability based multidisciplinary design optimization (RBMDO). If these processes are carried out simultaneously, the computational expense can be enormous. Computational savings can often be achieved with simultaneous analysis and design (SAND) or All-At-Once methods, which allow for the solution of the multidisciplinary analyses and the optimization in one single loop. Single loop RBDO methods allow for the integration of reliability analysis and optimization in a single loop. Applying SAND and single-loop RBDO together may allow for large computational savings when applied to problems in RBMDO. This paper develops a novel formulation of RBMDO problems with both system and component level reliability constraints based on the SAND method for MDO and by use of the KKT conditions of the first order reliability method (FORM). The resulting formulation may be solved as a standard nonlinear programming problem. While the single loop formulation presented in this paper is a solution of the RBMDO problem, robustness and efficiency are improved through two improved algorithms which can remove the MPP’s from the decision space, thereby significantly reducing the optimizer’s workload.

Collaborative optimization (CO), a decomposition-based multidisciplinary design optimization (MDO) method, is generally used in aeronautical systems. However, the subsystems optimizations of CO method are nested into system optimization of it in the execution process, numerous subsystem analyses are required to achieve consistency for entire system. For this problem, a nonnested collaborative optimization (NNCO) method is developed, in which system and subsystem optimizations are solved sequentially and repeatedly, and system consistancy is coordinated by a loop process together with the update of penalty parameters. Through test results of common analytical problems, which are generally used in the experiments of CO method, it's shown that the NNCO method requires two third fewer calls of the subsystem analysis than the CO method at least. Further more, CO and NNCO method are used to solve a satellite MDO problem, 0.05 and 8.05 hours are consumed for these two methods respectively. The efficiency of NNCO method in practical engineering design problem is proved.

This article focuses on the efficient multi-objective particle swarm optimization algorithm to solve multidisciplinary design optimization problems. The objective is to extend the formulation of collaborative optimization which has been widely used to solve single-objective optimization problems. To examine the proposed structure, racecar design problem is taken as an example of application for three objective functions. In addition, a fuzzy decision maker is applied to select the best solution along the pareto front based on the defined criteria. The results are compared to the traditional optimization, and collaborative optimization formulations that do not use multi-objective particle swarm optimization. It is shown that the integration of multi-objective particle swarm optimization into collaborative optimization provides an efficient framework for design and analysis of hierarchical multidisciplinary design optimization problems.

This paper presents a new improved collaborative optimization (CO) model that provides solution capabilities for multiobjective multidisciplinary optimization problems. Reasons that cause computational difficulties in CO algorithm are firstly analyzed. Then a new system level objective function is advised to minimize relative value between the collaborative objective function and single disciplinary objective function. And it eliminates the effect of dimensions and magnitude orders among objectives. A new subsystem level objective function is developed that includes the disciplinary objective function and the consistency constraint. A new CO framework which is more suitable for multilevel distributed design is advised. In this CO framework, the system level optimizer does not only independently invoke the subdisciplinary analysis tools, but also invoke its subdiscipline optimizer. The improved CO model proposed in this work is demonstrated with two examples. The results of examples show the improved CO is not only feasible, reliable and efficient, but also well suitable to solve the multiobjective optimization problems in multidisciplinary design environment.

In this work, the reliability analysis is employed to take into account the uncertainties in a structure. Reliability analysis is a tool to compute the probability of failure corresponding to a given failure mode. In this study, one of the most commonly used reliability analysis method namely first order reliability method is used to calculate the probability of failure. Since finding the most probable point (MPP) or design point is a constrained optimization problem, in contrast to all the previous studies based on the penalty function method or the preference of the feasible solutions technique, in this study one of the latest versions of the differential evolution metaheuristic algorithm named improved (μ+λ)-constrained differential evolution (ICDE) based on the multi-objective constraint-handling technique is utilized. The ICDE is very easy to implement because there is no need to the time-consuming task of fine tuning of the penalty parameters. Several test problems are used to verify the accuracy and efficiency of the ICDE. The statistical comparisons revealed that the performance of ICDE is better than or comparable with the other considered methods. Also, it shows acceptable convergence rate in the process of finding the design point. According to the results and easier implementation of ICDE, it can be expected that the proposed method would become a robust alternative to the penalty function based methods for the reliability assessment problems in the future works.

With a great capability of dealing with epistemic uncertainty, evidence theory has been utilised to conduct reliability analysis for engineering systems recently. Unfortunately, the discontinuous nature of uncertainty quantification using evidence theory incurs a huge computational cost. This paper proposes an efficient method to improve the computational efficiency of evidence theory for reliability analysis. In this method, evidence variables are transformed into random variables. Support vector regression is used to construct the approximation model of the limit-state function. The most probable point (MPP) of the approximate reliability problem with only random variables is searched out. Based on the MPP, the most probable focal element (MPFE) of the original problem with evidence variables is identified. According to the MPFE and the monotonicity of the limit-state function, contributions of some focal elements to belief and plausibility in evidence theory can be judged directly. Hence, the number of focal elements involved in the calculation of extreme values of the limit-state function is reduced. Four numerical examples are utilised to test the performance of the proposed method. Results indicate that the proposed method can reduce the computational cost on reliability analysis under epistemic uncertainty while ensuring the high accuracy of reliability analysis results.

A hybrid self-adaptive conjugate first order reliability method for robust structural reliability analysis

Design engineers and decision-makers across various fields are constantly working to make optimal design decisions for multidisciplinary engineering systems in an effort to improve performance and reduce costs. The multiple disciplines that decision-makers are forced to consider can range from different physical components of a system, to competing physical phenomena influencing a component (e.g. flow forces and structural strength), to completely separate models of interest to a system (e.g. engineering performance and lifecycle cost). The common element that all these decision-making scenarios share is the presence of couplings between the considered disciplines (or subsystems). How the values for these coupling parameters are determined within a decision-making or optimization framework is the subject of countless research efforts. At present the multidisciplinary design optimization (MDO) community has settled on a few proven techniques such Collaborative Optimization (CO) and Analytic Target Cascading (ATC). However, current MDO techniques have issues that limit their effectiveness in solving various MDO problems. Many of these strategies require close coordination between subsystem optimization solvers, require significant effort by decision-makers to pose their problems in a suitable format, and/or can have large computational efficiency problems due to the fact that they involve solving nested optimization problems. In an effort to alleviate some of these issues and make MDO easier to implement and more computationally efficient, a new sequential MDO algorithm called Cooperative Design Optimization (CDO) is proposed. The CDO approach functions through a series of subsystem optimizations using a successively smaller cooperation space. The cooperation space is analogous to the design space of a traditional optimization problem, but includes only the coupling parameters that are a factor in multiple subsystems. A single iteration of the approach can be thought of as a negotiation between all of the subsystems regarding the boundaries of the cooperation space. To facilitate this cooperation, each subsystem optimization problem is restructured as a multi-objective optimization problem so that a Pareto set of optimal solutions, or agreeable design alternatives, are produced. As a result, after all subsystem optimizations are accomplished, each subsystem’s obtained Pareto optimal solution sets are compared with respect to the coupling parameters and new bounds in the cooperation space are determined for the next iteration. This process allows each subsystem to be optimized completely independently within the boundaries of the agreed upon cooperation space while coordination is achieved through an analysis of obtained optimal solutions for each subsystem. Iterations, or negotiations, are repeated until an acceptable solution or set of solutions is obtained for all included subsystems through this cooperative process.

Model-Based Systems Engineering (MBSE) supports the system-level design of complex products effectively. Currently, system design and optimization for complex products are two distinct processes that must be executed using different software or platforms, involving intricate data conversion processes. Applying multidisciplinary optimization to validate system optimization often necessitates remodeling the optimization objects, which is time-consuming, labor-intensive, and highly error-prone. A critical activity in systems engineering is identifying the optimal design solution for the entire system. Multidisciplinary Design Optimization (MDO) and reliability analysis are essential methods for achieving this. This paper proposes a SysML-based multidisciplinary reliability design optimization modeling method. First, by analyzing the definitions and mathematical models of multidisciplinary reliability design optimization, the SysML extension mechanism is employed to represent the optimization model based on SysML. Next, model transformation techniques are used to convert the SysML optimization model generated in the first stage into an XML description model readable by optimization solvers. Finally, the proposed method’s effectiveness is validated through an engineering case study of an in-vehicle environmental control integration system. The results demonstrate that this method fully utilizes SysML to express MDO problems, enhancing the efficiency of design optimization for complex systems. Engineers and system designers working on complex, multidisciplinary projects can particularly benefit from these advancements, as they simplify the integration of design and optimization processes, facilitating more reliable and efficient product development.

ABSTRACTThis work deals with multi-objective optimization for the suspension parameters of railcars and proposes an optimization strategy combining the collaborative optimization method and the metamodel method to optimize the suspension parameters of railcars. In this strategy, the system-level optimization and subsystem-level optimization problems are efficiently calculated by the collaborative optimization method, and the metamodel replaces the actual model in the subsystem-level to improve the convergence and robustness of the optimization algorithm. Finally, an optimal combination of suspension parameters is determined by the effective implementation of the proposed optimization method, and compared with the original design in terms of performance and robustness. Here, the lateral running stability and vertical running smoothness of railcars were increased by 10.96% and 10.1%, respectively, and the derailment coefficient and the reduction ratio of wheel weight of railcars were increased by 6.14% and 6.36%, respectively. The optimization results show that the dynamic performance of railcars is improved remarkably with the robust collaborative optimization of the suspension parameters, and the reliability and effectiveness of the proposed robust collaborative optimization method for the suspension parameters' optimization of the railcars have also been verified.Abbreviation: SBD: simulation-based design; DOF: degree of freedom; RCO: robust collaborative optimization

Uncertainties in multidisciplinary design optimization (MDO) have a significant influence on the whole design process of the engineering system. If uncertainty in the engineering system is not considered, optimization results may be unreliable. On the basis of the existing methods for uncertainty analysis, a new method for MDO under uncertainty, i.e. First-Order Second-Moment based collaborative optimization (FOSM-CO), is proposed, and its particular computational process is described in detail. The FOSM-CO is an optimization approach that utilizes reliability index to evaluate the probability that the random variables are in the safe region and instead of the original constraints with the probability. The computation results of the example indicate that FOSM-CO is an effective reliability design method for MDO under uncertainty.

Multidisciplinary design optimization (MDO) is expected to play a major role in the competitive transportation industries of tomorrow, i.e., in the design of aircraft and spacecraft, of high speed trains, boats, and automobiles. All of these vehicles require maximum performance at minimum weight to keep fuel consumption low and conserve resources. Here, MDO can deliver mathematically based design tools to create systems with optimum performance subject to the constraints of disciplines such as structures, aerodynamics, controls, etc. Although some applications of MDO are beginning to surface, the key to a widespread use of this technology lies in the improvement of its efficiency. This aspect is investigated here for the MDO subset of structural optimization, i.e., for the weight minimization of a given structure under size, strength, and displacement constraints. Specifically, finite element based multilevel optimization of structures (here, statically indeterminate trusses and beams for proof of concept) is performed. In the system level optimization, the design variables are the coefficients of assumed displacement functions, and the load unbalance resulting from the solution of the stiffness equations is minimized. Constraints are placed on the deflection amplitudes and the weight of the structure. In the subsystems level optimizations, the weight of each element is minimized under the action of stress constraints, with the cross sectional dimensions as design variables. This approach is expected to prove very efficient, especially for complex structures, since the design task is broken down into a large number of small and efficiently handled subtasks, each with only a small number of variables. This partitioning will also allow for the use of parallel computing, first, by sending the system and subsystems level computations to two different processors, ultimately, by performing all subsystems level optimizations in a massively parallel manner on separate processors. It is expected that the subsystems level optimizations can be further improved through the use of controlled growth, a method which reduces an optimization to a more efficient analysis with only a slight degradation in accuracy. The efficiency of all proposed techniques is being evaluated relative to the performance of the standard single level optimization approach where the complete structure is weight minimized under the action of all given constraints by one processor and to the performance of simultaneous analysis and design which combines analysis and optimization into a single step. It is expected that the present approach can be expanded to include additional structural constraints (buckling, free and forced vibration, etc.) or other disciplines (passive and active controls, aerodynamics, etc.) for true MDO.

First and second order approximate reliability analysis methods using evidence theory

In this paper, an efficient method for the reliability analysis of systems with nonlinear limit states is described. It combines optimization-based and simulation-based approaches and is particularly applicable for problems with highly nonlinear and implicit limit state functions, which are difficult to solve by conventional reliability methods. The proposed method consists of two major parts. In the first part, an optimization-based method is used to search for the most probable point (MPP) on the limit state. This is achieved by using adaptive response surface approximations. In the second part, a multi-modal adaptive importance sampling method is proposed using the MPP information from the first part as the starting point. The proposed method is applied to the reliability estimation of a vehicle body-door subsystem with respect to one of the important quality issues -- the door closing effort. The superiority of the proposed method, in terms of efficiency and accuracy, is demonstrated with a numerical example of highly nonlinear limit state problem, as well as an automotive door closing effort application. A generalized framework for reliability estimation is also proposed for problems with large numbers of random variables and complicated limit states.