MDO Problems Research Articles

A Sequential Approach for Robust Multidisciplinary Design Optimization Under Mixed Interval and Probabilistic Uncertainties

Uncertainties cannot be ignored in the design process of complex multidisciplinary systems. Robust multidisciplinary design optimization methods (RMDOs) can treat uncertainties as specified probabilistic distributions when enough statistical information is available while they assign intervals for nondeterministic variables since designers may not have enough information to obtain statistical distributions, especially in the early stage of design optimization processes. Both types of uncertainties are very likely to appear simultaneously. In order to obtain solutions to RMDO problems under mixed interval and probabilistic uncertainties, this work proposed a new sequential RMDO approach, mixed SR-MDO. First, the robust optimization (RO) problem in a single discipline under mixed uncertainties is formulated and solved. Then, following the SR-MDO framework from the previous work, MDO problems under mixed uncertainties are solved by handling probabilistic and interval uncertainties sequentially in decomposed subsystem problems. Interval uncertainties are handled by using the worst-case sensitivity analysis, and the influence of probabilistic uncertainties in objectives, constraints, as well as in discipline analysis models is characterized by corresponding mean and variance. The applied SR-MDO framework allows subsystems in its full autonomy RO and sequential RO stages to run independently in parallel. This makes mixed SR-MDO be efficient for independent disciplines to work simultaneously and be more time-saving. Computational complexity of the proposed approach mainly relates to the double-loop optimization process in the worst-case interval uncertainties analysis. Examples are presented to demonstrate the applicability and efficiency of the mixed SR-MDO approach.

Filter-based adaptive Kriging method for black-box optimization problems with expensive objective and constraints

To reduce the computational cost of solving engineering design optimization problems with both expensive objective and constraints, a novel filter-based adaptive Kriging method notated as FLT-AKM is proposed in this paper. In FLT-AKM, a probability of constrained improvement (PCI) criterion is developed based on the notion of filter to sequentially generate new samples for updating Kriging metamodels of objective and constraints. At each iteration, an infill sample point is allocated at the position where the PCI is maximized to achieve potential improvement in optimality and feasibility. And the Kriging metamodels are consecutively updated by the newly-added infill sample points, which leads the FLT-AKM search to rapidly converge to the global optimum. The performance of the proposed FLT-AKM method is tested on a number of numerical benchmark problems via comparing with several widely-used metamodel-based constrained optimization methods. The comparison results indicate that FLT-AKM generally outperforms the competitors in terms of global convergence and efficiency performance. Finally, FLT-AKM is successfully applied to an all-electric GEO satellite MDO problem. The optimization results show that FLT-AKM is able to find a better feasible design with fewer computational budgets compared with our previous study, which demonstrates the effectiveness and practicality of the proposed FLT-AKM method for solving real-world expensive black-box engineering design optimization problems.

Development of an MDO platform for aero-structural interactions

An interactive MDO platform for multivariate and high coupling aero-structural interactions is presented in this paper. Programs based on RSM and GA are utilized as optimization solver, which is connected with CAD/CAE software by a friendly GUI. To generate and modify CAD models in the analysis and optimization process, CATIA Automation Technique is adopted to create a CAD model generator. During the analysis process, high-fidelity models in both disciplines are used. For aircraft wing MDO problems, optimal wing shape and structural configuration can be obtained using this platform in the preliminary design phase. While the current platform is aiming at solving Aero-Structural problems in the preliminary design phase of UAV components, it also provides a reference solution for other related MDO problems. Two cases are provided to verify the validity and practicability of this platform. Besides the design ideas and techniques, this paper also demonstrates its further development capabilities.

Sub-space approximations for MDO problems with disparate disciplinary variable dependence

An approach to solving multidisciplinary design optimisation problems using approximations built in sub-spaces of the design variable space is proposed. Each approximation is built in the sub-space significant to the corresponding discipline while the optimisation problem is solved in the full design variable space. Since the approximations are built in a space of reduced dimensionality, the computational budget associated with building them can be reduced without compromising their quality. The method requires the designer to make assumptions on which design variables are significant to each discipline. If such assumptions are deficient, the resulting approximations suffer from errors that are not possible to reduce by additional sampling. Therefore a recovery mechanism is proposed that updates the values of the insignificant variables at the end of each iteration to align with the current best point. The method is implemented within a trust region based optimisation framework and demonstrated on a multidisciplinary optimisation of a thin-walled beam section subject to stiffness and impact requirements.

A modified multidisciplinary feasible formulation for MDO using integrated coupled approximate models

This paper is concerned with the modification of multidisciplinary feasible formulation for MDO problems using the integrated coupled approximate models. A drawback of conventional MDFs is the numerical difficulty in decomposing the design variables and deriving the coupled equations of state. To overcome such a drawback of conventional methods, the coupling in analysis and design is resolved by approximating the state variables in each discipline by the response surface method and by modifying the optimization formulation using the corresponding integrated coupled approximate models. The validity, reliability and effectiveness of the proposed method are illustrated and verified through two optimization problems, a mathematical MDF problem and the multidisciplinary optimum design of suspension unit of wheeled armored vehicle.

Interaction prediction optimization in multidisciplinary design optimization problems.

The distributed strategy of Collaborative Optimization (CO) is suitable for large-scale engineering systems. However, it is hard for CO to converge when there is a high level coupled dimension. Furthermore, the discipline objectives cannot be considered in each discipline optimization problem. In this paper, one large-scale systems control strategy, the interaction prediction method (IPM), is introduced to enhance CO. IPM is utilized for controlling subsystems and coordinating the produce process in large-scale systems originally. We combine the strategy of IPM with CO and propose the Interaction Prediction Optimization (IPO) method to solve MDO problems. As a hierarchical strategy, there are a system level and a subsystem level in IPO. The interaction design variables (including shared design variables and linking design variables) are operated at the system level and assigned to the subsystem level as design parameters. Each discipline objective is considered and optimized at the subsystem level simultaneously. The values of design variables are transported between system level and subsystem level. The compatibility constraints are replaced with the enhanced compatibility constraints to reduce the dimension of design variables in compatibility constraints. Two examples are presented to show the potential application of IPO for MDO.

Improving collaborative optimization for MDO problems with multi-objective subsystems

This paper investigates collaborative optimization (CO) for multidisciplinary design optimization problems with multi-objective subsystems. A multi-stage optimization strategy is presented in the system level to enhance the performance of CO that may be weakened by the initial point in some cases. Resolving the multi-objective problem in subsystems with the preference-based algorithm requires setting the preference values for all the objective functions. Therefore, we transform the incompatibility function into a disciplinary constraint aiming to avoid providing the preference value, and propose two transformation methods, namely, closest distance and relaxed distance. At the end of the paper, the results of two engineering examples demonstrate the performance of the multi-stage optimization strategy, and the effectiveness of the transformation for the incompatibility function.

Optimization Method Based on a Mix Strategy

To solve problems that exist in optimal design such as falling into local optimal solution easily and low efficiency in collaborative optimization, a new mix strategy optimization method combined design of experiments (DOE) with gradient optimization (GO) was proposed. In order to reduce the effect on the result of optimization made by the designers decision, DOE for preliminary analysis of the function model was used, and the optimal values obtained in DOE stage was taken as the initial values of design variables in GO stage in the new optimization method. The reducer MDO problem was taken as a example to confirm the global degree, efficiency, and accuracy of the method. The results show the optimization method could not only avoid falling into local solution, but also have an obvious superiority in treating the complex collaborative optimization problems.

Systematic design space exploration and rearrangement of the MDO problem by using probabilistic methodology

This study proposes a probabilistic approach for systematic design space exploration and rearrangement. The stochastic quantities and qualities of design space are explored using a reliability index, and the design space is rearranged to a higher feasible region by using Chebyshev inequality. Four test cases composed of algebraic functions are used to investigate the validity of the proposed method. Re- sults show that the converged design space can include the feasible region outside the initial design space to the rearranged design space. Moreover, the results show that the proposed method is applicable to the aircraft wing design optimization problem, which considers collaborative optimization with three subsystems, namely, aerodynamics, structure, and performance. Wing design optimizations are also performed separately for the initial and converged design spaces. The results indicate that the range obtained in the converged design space improved compared with that in the initial design space. In conclusion, the proposed design space rearrangement method was shown to have the capability to search for feasible regions that are excluded in the initial design phase.

Multidisciplinary Design Optimization of Multi-Stage Launch Vehicle using Flight Phases Decomposition

Optimal design of launch vehicles is a complex multidisciplinary problem. The traditional way to optimize such vehicles is to decompose the problem into the different disciplines and to combine specific solvers and a global optimizer. This paper presents a new MDO method to optimize the configuration of an expendable launch vehicle and describes the application of this method to the design optimization of a small liquid fueled multi-stage launch vehicle. The objective is the payload mass maximization. The proposed bi-level method splits up the original MDO problem into the flight phases and optimizes concurrently the different stages of the launch vehicle. This transversal decomposition allows to reduce the search domain at system-level because the optimization problem at this level is analog to a coordination one. Two formulations of the method are presented and the results are compared to the ones obtained by a standard MDO method.

Sequential optimization and reliability assessment for multidisciplinary design optimization under aleatory and epistemic uncertainties

In engineering design, to achieve high reliability and safety in complex and coupled systems (e.g., Multidisciplinary Systems), Reliability Based Multidisciplinary Design Optimization (RBMDO) has been received increasing attention. If there are sufficient data of uncertainties to construct the probability distribution of each input variable, the RBMDO can efficiently deal with the problem. However there are both Aleatory Uncertainty (AU) and Epistemic Uncertainty (EU) in most Multidisciplinary Systems (MS). In this situation, the results of the RBMDO will be unreliable or risky because there are insufficient data to precisely construct the probability distribution about EU due to time, money, etc. This paper proposes formulations of Mixed Variables (random and fuzzy variables) Multidisciplinary Design Optimization (MVMDO) and a method of MVMDO within the framework of Sequential Optimization and Reliability Assessment (MVMDO-SORA). The MVMDO overcomes difficulties caused by insufficient information for uncertainty. The proposed method enables designers to solve MDO problems in the presence of both AU and EU. Besides, the proposed method can efficiently reduce the computational demand. Examples are used to demonstrate the proposed formulations and the efficiency of MVMDO-SORA.

공통설계변수를 고려한 독립적하부시스템에 의한 다분야통합최적설계

Multidisciplinary design optimization based on independent subspaces (MDOIS) is a simple and practical method that can be applied to the practical engineering MDO problems. However, the current version of MDOIS does not handle the common design variables. A new version of MDOIS is proposed and named as MDOIS/2006. It is a two-level MDO method while the original MDOIS is a single-level method. At first, system analysis is performed to solve the coupling in the analysis. If the termination criteria are not satisfied, each discipline solves its own design problem. Each discipline in the lower level solves the problem with common design variables while they are constrained by equality constraints. In the upper level, the common design variables of related disciplines are determined by using the optimum sensitivity of the objective function. To validate MDOIS/2006, mathematical problem and NASA test bed problem are solved. The results are compared with those from other MDO methods. Finally, MDOIS/2006 is applied to flow patterner design and shows that it can be successfully applied to the practical engineering MDO problem.