Multidisciplinary Design Optimization Problems Research Articles

A surrogate based multistage-multilevel optimization procedure for multidisciplinary design optimization

Optimization procedure is one of the key techniques to address the computational and organizational complexities of multidisciplinary design optimization (MDO). Motivated by the idea of synthetically exploiting the advantage of multiple existing optimization procedures and meanwhile complying with the general process of satellite system design optimization in conceptual design phase, a multistage-multilevel MDO procedure is proposed in this paper by integrating multiple-discipline-feasible (MDF) and concurrent subspace optimization (CSSO), termed as MDF-CSSO. In the first stage, the approximation surrogates of high-fidelity disciplinary models are built by disciplinary specialists independently, based on which the single level optimization procedure MDF is used to quickly identify the promising region and roughly locate the optimum of the MDO problem. In the second stage, the disciplinary specialists are employed to further investigate and improve the baseline design obtained in the first stage with high-fidelity disciplinary models. CSSO is used to organize the concurrent disciplinary optimization and system coordination so as to allow disciplinary autonomy. To enhance the reliability and robustness of the design under uncertainties, the probabilistic version of MDF-CSSO (PMDF-CSSO) is developed to solve uncertainty-based optimization problems. The effectiveness of the proposed methods is verified with one MDO benchmark test and one practical satellite conceptual design optimization problem, followed by conclusion remarks and future research prospects.

Multidisciplinary Design Optimization for Complex Engineered Systems: Report From a National Science Foundation Workshop

Complex engineered systems are typically designed using a systems engineering framework that is showing its limitations. Multidisciplinary design optimization (MDO), which has evolved remarkably since its inception 25 years ago, offers alternatives to complement and enhance the systems engineering approach to help address the challenges inherent in the design of complex engineered systems. To gain insight into these challenges, a one-day workshop was organized that gathered 48 people from industry, academia, and government agencies. The goal was to examine MDO’s current and future role in designing complex engineered systems. This paper summarizes the views of five distinguished speakers on the “state of the research” and discussions from an industry panel comprised of representatives from Boeing, Caterpillar, Ford, NASA Glenn Research Center, and United Technologies Research Center on the “state of the practice.” Future research topics to advance MDO are also identified in five key areas: (1) modeling and the design space, (2) metrics, objectives, and requirements, (3) coupling in complex engineered systems, (4) dealing with uncertainty, and (5) people and workflow. Finally, five overarching themes are offered to advance MDO practice. First, MDO researchers need to engage disciplines outside of engineering and target opportunities outside of their traditional application areas. Second, MDO problem formulations must evolve to encompass a wider range of design criteria. Third, effective strategies are needed to put designers “back in the loop” during MDO. Fourth, the MDO community needs to do a better job of publicizing its successes to improve the “buy in” that is needed to advance MDO in academia, industry, and government agencies. Fifth, students and practitioners need to be better educated on systems design, optimization, and MDO methods and tools along with their benefits and drawbacks.

Hybrid Power/Energy Generation Through Multidisciplinary and Multilevel Design Optimization With Complementarity Constraints

The optimal design of hybrid power generation systems (HPGSs) can significantly improve the technical and economic performance of power supply. However, the discrete-time simulation with logical disjunctions involved in HPGS design usually leads to a nonsmooth optimization model, to which well-established techniques for smooth nonlinear optimization cannot be directly applied. This paper casts the HPGS design optimization problem as a multidisciplinary design optimization problem with complementarity constraints, a formulation that introduces a complementarity formulation of the nonsmooth logical disjunction, as well as a time horizon decomposition framework, to ensure a fast local solution. A numerical study of a stand-alone hybrid photovoltaic/wind power generation system is presented to demonstrate the effectiveness of the proposed approach.

Application of Multidisciplinary Design Optimization Based on Approximation Model to Marine Supercharger Turbo

In this paper, the multidisciplinary design optimization based on Approximation Model for supercharge turbo is studied. Temperature and pressure loads are transferred to the solid model by distance-weighted function, and structure deformation is transferred to aerodynamic model by mesh regenerated method in order to avoid mesh aberration. The Multidisciplinary analysis (MDA) model of supercharge turbo considering aerodynamic, heat transfer, strength and vibration is obtained on the basis of information transferring, which is solved by iterated three times. The Kriging Approximation Model which fits the sample space accurately is employed in the MDO process to reduce computational cost. Results show that performance of supercharge turbo is improvement on the MDO system based on Approximation Model, meanwhile the computational time of the optimization system is saved. Also, this method is suitable for other Multidisciplinary Design Optimization problems.

COLLABORATIVE OPTIMIZATION WITH DIMENSION REDUCTION

Collaborative optimization (CO) method is widely used in solving multidisciplinary design optimization (MDO) problems, yet its computation requirement has been an obstacle to the applications, leading to doubts about CO's convergence property. The feasible domain of CO problem is first examined and it is proven that feasible domain remains the same during the CO formulation. So is the same with extreme points. Then based on contemporary research conclusion that the system-level optimization problem suffers from inherent computational difficulties, it is further pointed out that the employment of meta-heuristic optimization methods in CO could eliminate these difficulties. To make CO more computational feasible, a new method collaborative optimization with dimension reduction (CODR) is proposed. It focused on optimization dimension reduction and lets local copy of common shared design variables equal system shared design variables directly. Thus, the number of dimensions that CODR could reduce equal the number of common shared design variables. Numerical experiment suggests that CODR reduces computations greatly without losing of optimization accuracy.

An Approach Based on Enhanced Collaborative Optimization and Kriging Approximation in Multidisciplinary Design Optimization

This paper concentrates on the computational challenge in multidisciplinary design optimization (MDO) and a comprehensive strategy combining enhanced collaborative optimization (ECO) and kriging approximation models is introduced. In this strategy, the computational and organizational advantages of original collaborative optimization (CO) are inherited by ECO, which can satisfy the strengthened consistency requirements. Kriging approximation models are constructed to replace high-fidelity simulation models in individual disciplines and reduce the expensive computational cost in practical MDO problems. The proposed methodology is demonstrated by solving the classical speed reducer design problem. The better results indicate that ECO using kriging approximation models can achieve a considerable reduction of computational expense while guaranteeing the accuracy of optimal solutions with efficient convergence.

A Nonhierarchical Formulation of Analytical Target Cascading

Analytical target cascading (ATC) is a method developed originally for translating system-level design targets to design specifications for the components that comprise the system. ATC has been shown to be useful for coordinating decomposition-based optimal system design. The traditional ATC formulation uses hierarchical problem decompositions, in which coordination is performed by communicating target and response values between parents and children. The hierarchical formulation may not be suitable for general multidisciplinary design optimization (MDO) problems. This paper presents a new ATC formulation that allows nonhierarchical target-response coupling between subproblems and introduces system-wide functions that depend on variables of two or more subproblems. Options to parallelize the subproblem optimizations are also provided, including a new bilevel coordination strategy that uses a master problem formulation. The new formulation increases the applicability of the ATC to both decomposition-based optimal system design and MDO. Moreover, it belongs to the class of augmented Lagrangian coordination methods, having thus convergence properties under standard convexity and continuity assumptions. A supersonic business jet design problem is used to demonstrate the flexibility and effectiveness of the presented formulation.

A Regularized Inexact Penalty Decomposition Algorithm for Multidisciplinary Design Optimization Problems With Complementarity Constraints

Economic and physical considerations often lead to equilibrium problems in multidisciplinary design optimization (MDO), which can be captured by MDO problems with complementarity constraints (MDO-CC)—a newly emerging class of problem. Due to the ill-posedness associated with the complementarity constraints, many existing MDO methods may have numerical difficulties solving this class of problem. In this paper, we propose a new decomposition algorithm for the MDO-CC based on the regularization technique and inexact penalty decomposition. The algorithm is presented such that existing proofs can be extended, under certain assumptions, to show that it converges to stationary points of the original problem and that it converges locally at a superlinear rate. Numerical computation with an engineering design example and several analytical example problems shows promising results with convergence to the all-in-one solution.

Multidisciplinary Design and Collaborative Optimization for Bus Body

A multidisciplinary design optimization(MDO),considering lightweight,stiffness,strength,vibration mode and rollover,is implemented to an integral bus body structure to improve its synthetic performance.To conquer the difficulty of high nonlinearity brought by rollover analysis and to resolve the time consuming problem of structural finite element(FE) analysis,multidisciplinary approximate models are built.The FE models of the bus body are established,in which the stiffness,strength and vibration mode analyses are finished by Msc.Nastran,and the rollover analysis is solved by the explicit dynamic software Ls-dyna.The disciplinary variables are selected on the basis of technological requirements and sensitivity analysis results.The experimental design is finished by using optimal Latin hypercube method.Then the multidisciplinary approximate models are constructed by using the response surface method.The MDO problem is solved by using collaborative optimization approach.The result shows that the optimal solution is better than the original design in aspects of lightweight,stiffness,strength,modal vibration,and safety of rollover.

Gradient-based Kriging approximate model and its application research to optimization design

In the process of multidisciplinary design optimization, there exits the calculation complexity problem due to frequently calling high fidelity system analysis models. The high fidelity system analysis models can be surrogated by approximate models. The sensitivity analysis and numerical noise filtering can be done easily by coupling approximate models to optimization. Approximate models can reduce the number of executions of the problem’s simulation code during optimization, so the solution efficiency of the multidisciplinary design optimization problem can be improved. Most optimization methods are based on gradient. The gradients of the objective and constrain functions are gained easily. The gradient-based Kriging (GBK) approximate model can be constructed by using system response value and its gradients. The gradients can greatly improve prediction precision of system response. The hybrid optimization method is constructed by coupling GBK approximate models to gradient-based optimization methods. An aircraft aerodynamics shape optimization design example indicates that the methods of this paper can achieve good feasibility and validity.

Robust Collaborative Optimization Method Based on Dual-response Surface

A novel method for robust collaborative design of complex products based on dual-response surface (DRS-RCO) is proposed to solve multidisciplinary design optimization (MDO) problems under uncertainty. Collaborative optimization (CO) which decomposes the whole system into a double-level nonlinear optimization problem is widely accepted as an efficient method to solve MDO problems. In order to improve the quality of complex product in design process, robust collaborative optimization (RCO) is developed to solve those problems under uncertain conditions. RCO does optimization on the linear sum of mean and standard deviation of objective function and gets an optimal solution with high robustness. Response surfaces method is an important way to do approximation in robust design. DRS-RCO is an improved RCO method in which dual-response surface replaces system uncertainty analysis module of CO. The dual-response surface is the approximate model of mean and standard deviation of objective function respectively. In DRS-RCO, All the information of subsystems is included in dual-response surfaces. As an additional item, the standard deviation of objective function is added to the subsystem optimization. This item guarantee both the mean and standard deviation of this subsystem is reaching the minima at the same time. Finally, a test problem with two coupled subsystems is conducted to verify the feasibility and effectiveness of DRS-RCO.

Bilevel Adaptive Weighted Sum Method for Multidisciplinary Multi-Objective Optimization

The primary goal of this research is to develop a framework for dealing with multi-objective, multidisciplinary optimization problems with a large number of variables. The proposed method is expected to provide a relatively uniformly spaced, widely distributed Pareto front. To achieve this end, a novel integration of the adaptive weighted sum method within a concurrent subspace optimization framework is presented. In the bilevel framework of concurrent subspace optimization, the adaptive weighted sum is used to make tradeoffs among multiple, conflicting objectives. To obtain better distributed solutions, two modifications are made. First, an additional equality constraint in suboptimization for each expected solution is relaxed because it causes slow convergence within the bilevel optimization framework. The probability of entrapment in local minima can also be reduced. Second, the mesh of the Pareto front patches is modified due to the low efficiency of the original scheme. The proposed method is demonstrated with three multidisciplinary design optimization problems: 1) a numerical multidisciplinary design optimization test problem with a convex Pareto front, available within the NASA multidisciplinary design optimization Test Suite; 2) a test problem with a nonconvex Pareto front, which is not easily solved; and 3) a conceptual design of a subsonic passenger aircraft, which consists of two objectives, four design variables, five coupling behavior variables, seven constraints in aerodynamics, and weight discipline. The primary results show that the proposed method is promising with regard to obtaining a uniformly spaced, widely distributed, and smooth Pareto front and is applicable in the design of large-scale, complex engineering systems such as aircraft.

Metamodel-based collaborative optimization framework

This paper focuses on the metamodel-based collaborative optimization (CO). The objective is to improve the computational efficiency of CO in order to handle multidisciplinary design optimization problems utilising high fidelity models. To address these issues, two levels of metamodel building techniques are proposed: metamodels in the disciplinary optimization are based on multi-fidelity modelling (the interaction of low and high fidelity models) and for the system level optimization a combination of a global metamodel based on the moving least squares method and trust region strategy is introduced. The proposed method is demonstrated on a continuous fiber-reinforced composite beam test problem. Results show that methods introduced in this paper provide an effective way of improving computational efficiency of CO based on high fidelity simulation models.

Multiobjective Collaborative Robust Optimization With Interval Uncertainty and Interdisciplinary Uncertainty Propagation

We present a new solution approach for multidisciplinary design optimization (MDO) problems that, for the first time in literature, has all of the following characteristics: Each discipline has multiple objectives and constraints with mixed continuous-discrete variables; uncertainty exists in parameters and as a result, uncertainty propagation exists within and across disciplines; probability distributions of uncertain parameters are not available but their interval of uncertainty is known; and disciplines can be fully (two-way) coupled. The proposed multiobjective collaborative robust optimization (McRO) approach uses a multiobjective genetic algorithm as an optimizer. McRO obtains solutions that are as best as possible in a multiobjective and multidisciplinary sense. Moreover, for McRO solutions, the variation of objective and/or constraint functions can be kept within an acceptable range. McRO includes a technique for interdisciplinary uncertainty propagation. The approach can be used for robust optimization of MDO problems with multiple objectives, or constraints, or both together at system and subsystem levels. Results from an application of McRO to a numerical and an engineering example are presented. It is concluded that McRO can solve fully coupled MDO problems with interval uncertainty and obtain solutions that are comparable to a single-disciplinary robust optimization approach.

Block-separable linking constraints in augmented Lagrangian coordination

Augmented Lagrangian coordination (ALC) is a provably convergent coordination method for multidisciplinary design optimization (MDO) that is able to treat both linking variables and linking functions (i.e. system-wide objectives and constraints). Contrary to quasi-separable problems with only linking variables, the presence of linking functions may hinder the parallel solution of subproblems and the use of the efficient alternating directions method of multipliers. We show that this unfortunate situation is not the case for MDO problems with block-separable linking constraints. We derive a centralized formulation of ALC for block-separable constraints, which does allow parallel solution of subproblems. Similarly, we derive a distributed coordination variant for which subproblems cannot be solved in parallel, but that still enables the use of the alternating direction method of multipliers. The approach can also be used for other existing MDO coordination strategies such that they can include block-separable linking constraints.

Nonlinear programming approaches in the multidisciplinary design optimization of a sailing yacht keel fin

AbstractWe consider the use of global optimization techniques for the minimization of a nonconvex function, arising from a Multidisciplinary Design Optimization problem of a ship. In ship design different disciplines must be often coupled, encompassing hydrodynamic and structural problems. This scenario requires the use of specific multidisciplinary formulations, where standard mathematical programming methods could be unsuitable. (© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)

Augmented Lagrangian coordination for distributed optimal design in MDO

AbstractQuite a number of coordination methods have been proposed for the distributed optimal design of large‐scale systems consisting of a number of interacting subsystems. Several coordination methods are known to have numerical convergence difficulties that can be explained theoretically. The methods for which convergence proofs are available have mostly been developed for so‐called quasi‐separable problems (i.e. problems with individual subsystems coupled only through a set of linking variables, not through constraints and/or objectives). In this paper, we present a new coordination approach for multidisciplinary design optimization problems with linking variables as well as coupling objectives and constraints. Two formulation variants are presented, offering a large degree of freedom in tailoring the coordination algorithm to the design problem at hand. The first, centralized variant introduces a master problem to coordinate coupling of the subsystems. The second, distributed variant coordinates coupling directly between subsystems. Our coordination approach employs an augmented Lagrangian penalty relaxation in combination with a block coordinate descent method. The proposed coordination algorithms can be shown to converge to Karush–Kuhn–Tucker points of the original problem by using the existing convergence results. We illustrate the flexibility of the proposed approach by showing that the analytical target cascading method of Kim et al. (J. Mech. Design‐ASME 2003; 125(3):475–480) and the augmented Lagrangian method for quasi‐separable problems of Tosserams et al. (Struct. Multidisciplinary Opt. 2007, to appear) are subclasses of the proposed formulations. Copyright © 2007 John Wiley & Sons, Ltd.

Extended Collaboration Pursuing Method for Solver Larger Multidisciplinary Design Optimization Problems

The collaboration pursuing method is a sampling-based multidisciplinary design optimization method, which does not rely on sensitivity analysis. It was found that the collaboration pursuing method is constrained by the effectiveness of sampling in a design space when solving larger multidisciplinary design optimization problems. Three new modules, that is, discrete sampling, new initialization process, and active design variable control, are developed in this work to extend the collaboration pursuing method's capability in dealing with larger multidisciplinary design optimization problems. Using the collaboration pursuing method with the new modules, called extended collaboration pursuing method, a conceptual aircraft design problem involving structures, aerodynamics, and propulsion is successfully solved. The extended collaboration pursuing method is a promising new multidisciplinary design optimization method to solve larger multidisciplinary design optimization problems with better accuracy and comparable efficiency, when compared with other multidisciplinary design optimization methods.

Boundary search and simplex decomposition method for MDO problems with a convex or star-like state parameter region

One major challenge in multidisciplinary design optimization (MDO) is the presence of couplings among state parameters, which demands an iterative and often expensive system analysis (SA) process for each function evaluation in optimization. This paper offers a new perspective and proposes a corresponding method for solving MDO problems. The proposed method, named the boundary search and simplex decomposition method (BSSDM), geometrically captures the relation among coupled state parameters with a feasible state parameter region. Given the feasible state parameter region, the SA can be avoided during the optimization of the system objective function. To identify the feasible state parameter region, a search strategy is developed to find boundary points of the region. In the boundary search process, a collaboration model (CM) is applied to maintain the feasibility of samples with respect to the SA. In search of the system optimum in the feasible region, a robust simplex decomposition algorithm is developed for convex and star-like feasible state parameter regions. The BSSDM is tested with two numerical cases, one of which is an MDO problem constrained by a convex state parameter region, and the other is a SA problem with a star-like state parameter region. All results are then validated, and the results show the promising capability of the proposed BSSDM.

Collaboration Pursuing Method for Multidisciplinary Design Optimization Problems

Multidisciplinary design optimization problems are dominated by couplings among subsystems formulated from different disciplines. Effective and efficient collaboration between subsystems is always desirable when solving multidisciplinary design optimization problems. This work proposes a new sampling-based methodology, named the collaboration pursuing method, for multidisciplinary design optimization problems. In the collaboration pursuing method, a new collaboration model, reflecting both physical and mathematical characteristics of couplings in multidisciplinary design optimization problems, is formulated to guide the search of feasible design solutions. The interdisciplinary consistency among coupled state parameters in multidisciplinary design optimization problems is reflected and maintained by the collaboration model. An adaptive sampling strategy is also developed to speed up the search of local optimal solutions. The new method is implemented using MATLAB® 6.0 and successfully applied to four test problems, including an engineering design application.