A local convergence analysis of bilevel decomposition algorithms

The design of complex aerospace systems is a multidisciplinary design optimization (MDO) problem involving the interaction of multiple disciplines. However, because of the necessity of evaluating expensive black-box simulations, the enormous computational cost of solving MDO problems in aerospace systems has also become a problem in practice. To resolve this, metamodel-based design optimization techniques have been applied to MDO. With these methods, system models can be rapidly predicted using approximate metamodels to improve the optimization efficiency. This paper presents an overall survey of metamodel-based MDO for aerospace systems. From the perspective of aerospace system design, this paper introduces the fundamental methodology and technology of metamodel-based MDO, including aerospace system MDO problem formulation, metamodeling techniques, state-of-the-art metamodel-based multidisciplinary optimization strategies, and expensive black-box constraint-handling mechanisms. Moreover, various aerospace system examples are presented to illustrate the application of metamodel-based MDOs to practical engineering. The conclusions derived from this work are summarized in the final section of the paper. The survey results are expected to serve as guide and reference for designers involved in metamodel-based MDO in the field of aerospace engineering.

Multidisciplinary design optimization of long-range slender guided rockets considering aeroelasticity and subsidiary loads

Satellite constellation system design is a challenging and complicated multidisciplinary design optimization (MDO) problem involving a number of computation-intensive multidisciplinary analysis models. In this paper, the MDO problem of a constellation system consisting of small observation satellites is investigated to simultaneously achieve the preliminary design of constellation configuration and the satellite subsystems. The constellation is established based on Walker-? configuration considering the coverage performance. Coupled with the constellation configuration, several disciplines including payload, power, thermal control, and structure are taken into account for satellite subsystems design subject to various constraints (i.e., ground resolution, power usage, natural frequencies, etc.). Considering the mixed-integer and time-consuming behavior of satellite constellation system MDO problem, a novel sequential radial basis function (RBF) method using the support vector machine (SVM) for discrete-continuous mixed variables notated as SRBF-SVM-DC is proposed. In this method, a discrete-continuous variable sampling method is utilized to handle the discrete variables, i.e., the number of orbit planes and number of satellites, in the satellite constellation system MDO problem. RBF surrogates are constructed and gradually refined to represent the time-consuming simulations during optimization, which can efficiently lead the search to the optimum. Finally, the proposed SRBF-SVM-DC utilized to solve the satellite constellation system MDO problem is compared with a conventional integer coding based genetic algorithm (ICGA). The results show that SRBF-SVM-DC significantly decreases the system mass by about 28.63% subject to all the constraints, which greatly reduces the cost of the satellite constellation system. Moreover, the computational budget of SRBF-SVM-DC is saved by over 85% compared with ICGA, which demonstrates the effectiveness and practicality of the proposed surrogate assisted optimization approach for satellite constellation system design.

With the increasing design dimensionality, it is more difficult to solve multidisciplinary design optimization (MDO) problems. Many MDO decomposition strategies have been developed to reduce the dimensionality. Those strategies consider the design problem as a black-box function. However, practitioners usually have certain knowledge of their problem. In this paper, a method leveraging causal graph and qualitative analysis is developed to reduce the dimensionality of the MDO problem by systematically modeling and incorporating the knowledge about the design problem into optimization. Causal graph is created to show the input–output relationships between variables. A qualitative analysis algorithm using design structure matrix (DSM) is developed to automatically find the variables whose values can be determined without resorting to optimization. According to the impact of variables, an MDO problem is divided into two subproblems, the optimization problem with respect to the most important variables, and the other with variables of lower importance. The novel method is used to solve a power converter design problem and an aircraft concept design problem, and the results show that by incorporating knowledge in form of causal relationship, the optimization efficiency is significantly improved.

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.

Engineering systems have become quite complicated in recent days. The requirements of design are complex and it is hard to meet them by considering only one discipline. In this paper, we suggest a hybrid Multi-objective Genetic Algorithm (MOGA) for a Multidisciplinary Design Optimization (MDO) model in engineering design. In order to get Pareto optimal solutions, we use local search to enhance the efficiency and effectiveness. We solve a problem of airplane wings optimization using MOGA. After compared with other MDO algorithms, we find the result shows that the proposed algorithm’s performance has an advantage over previous work for these MDO problems. As the algorithm is evaluated based on practical data, we show that the approach can be used to solve MDO problems in the real world.

The programming language Java (recently referred to as the computer language of the Web) offers substantial possibilities for the type of complex engineering problems typically encountered in multidisciplinary design optimization (MDO) problems. In order to demonstrate the potential uses of Java for MDO problems, this paper presents the development of the Web Interface for complex engineering design (WICkED) software, which simulates the convergence of a decomposed complex system in a distributed computing environment and computes the sensitivity derivatives of the system with respect to the independent input variables using the GSE method or the finite difference method. In this application, one computer is designated as the server and sends out required inputs to a number of client subsystems over the Internet. A number of client computers can connect to the server and then receive the inputs necessary to calculate the solution to their model. As the code necessary to solve the model already exists at the client, only the inputs have to be sent over the network. When the client has solved the calculation, it returns the results to the server which processes the result to produce new inputs.

Additive manufacturing (AM) techniques are ideal for producing customized products due to their high design flexibility. Despite the previous studies on specific additive manufactured customized products such as biomedical implants and prostheses, the simultaneous optimization of components, materials, AM processes, and dimensions remains a challenge. Multidisciplinary design optimization (MDO) is a research area of solving complex design problems involving multiple disciplines which usually interact with each other. The objective of this research is to formulate and solve an MDO problem in the development of additive manufactured products customized for various customers in different market segments. Three disciplines, i.e. the customer preference modeling, AM production costing, and structural mechanics are incorporated in the MDO problem. The optimal selections of components, materials, AM processes, and dimensional parameters are searched with the objectives to maximize the functionality utility, match individual customers' personal performance requirements, and minimize the total cost. A multi-objective genetic algorithm with the proposed chromosome encoding pattern is applied to solve the MDO problem. A case study of designing customized trans-tibial prostheses with additive manufactured components is presented to illustrate the proposed MDO method. Clusters of multi-dimensional Pareto-optimal design solutions are obtained from the MDO, showing trade-offs among the objectives. Appropriate design decision can be chosen from the clusters based on the manufacturer's market strategy. Highlights An optimization problem for additive manufactured customized products is solved. Three disciplines are incorporated in multidisciplinary design optimization (MDO). The selection of component, material, additive manufacturing process and dimensions are optimized. A multiobjective genetic algorithm is applied to solve the MDO problem. Pareto-optimal solutions with different utilities and costs are obtained.

Multidisciplinary design optimization based on independent subspaces (MDOIS), which is a multidisciplinary design optimization (MDO) algorithm, has been recently proposed. Since MDOIS is relatively simple compared with other MDO algorithms, it is easy to apply MDOIS to practical engineering problems. In this research, an MDO problem is defined for the design of a belt-integrated seat (BIS) while considering crashworthiness. The crash model consists of an airbag, a BIS, an energy-absorbing steering system, and a safety belt. It is found that the current design problem has two disciplines - structural non-linear analysis and occupant analysis. The interdisciplinary relationship between the disciplines is identified. Interdisciplinary variables between the two disciplines are stiffness of the seat back frame and the belt load. The interdisciplinary relationship is addressed in the system analysis step in MDOIS. Prior to each independent subspace design, values of interdisciplinary variables at a given design point are determined in the system analysis step. The determined values are passed to corresponding subspaces, and the subspaces treat the received values of the interdisciplinary variables as constant parameters throughout the subspace design. For the present example, the belt load is passed to the structural analysis subspace and the stiffness of the seat back frame is passed to the occupant analysis subspace. Determined design variables in each subspace are passed to the system analysis step. In this way, the design process iterates until the convergence criterion is satisfied. As a result of the design, the weight of the BIS and the head injury criterion (HIC) of an occupant are reduced while the specified constraints are satisfied. Since the system analysis cannot be formulated in an explicit form in the current example, an optimization problem is formulated to solve the system analysis. The results from MDOIS are discussed.

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 discusses the performance of two discipline-oriented decompositions applied to the multidisciplinary design optimization (MDO) problem of a wing in transonic regime. A finite element model (FEM) serves to predict the wing structural deformations under cruise aerodynamic loads obtained from computational fluid dynamics (CFD) analyses. The aim of the design is to obtain a suitable wing structure and external shape that maximizes the cruise range under a lift constraint subjected to structural safety factor constraints for a given upwind gust load case. A methodology to quickly predict the performance of a decomposition method is presented. This methodology is applied to several possible formulations (single and bi-level decomposition) of the optimization problem. Based on the forecasted performance of the different decompositions, a bi-level FIO and a semi-decoupled decompositions are tested on the transonic wing design problem. The optimization results highlight the respective advantages of hierarchical decomposition and decoupling. With the hierarchical decomposition, the line search performed better because the structure is adapted for each external shape configuration. As for the semi-decoupled formulation, it was able to reduce the cost related to the resolution of the MDA. However, the bi-level FIO decomposition obtained a slightly better objective function than the semi-decoupled formulation. Also, the designs obtained by the optimizations are not representative of what is done in the industry because of a weakness in the objective function and because the load case is too conservative.

Large scale multidisciplinary design optimization (MDO) problems often involve massive computation over vast data sets; Regardless of the MDO problem solving methodology, advanced computing technologies and architectures are indispensable. The data parallelism inherent in some engineering problems makes massively parallel architectures a natural choice, but efficiently harnessing the power of massive parallelism requires sophisticated algorithms and techniques. This paper presents an effort to apply massively scalable distributed control and dynamic load balancing techniques to the reasonable design space identification phase of a variable complexity approach to the multidisciplinary design optimization of a high speed civil transport (HSCT). The scalability and performance of two dynamic load balancing techniques, random polling and global round robin with message combining, and two termination detection schemes, token passing and global task count, are studied. The extent to which such techniques are applicable to other MDO paradigms, and to the potential for parallel multidisciplinary design with current large-scale disciplinary codes, is of particular interest.

Effect of Variable Selection on Multidisciplinary Design Optimization: a Flight Vehicle Example

In order to reduce the cost of oceanographic exploration, a new underwater vehicle is designed to sail the required distance with the lowest energy consumed. Since the new underwater vehicle is a complicated multidisciplinary system, it is firstly decomposed into four smaller disciplines and then a multidisciplinary design optimization (MDO) problem is built based on these disciplines. The Multidisciplinary Feasible (MDF) architecture is adopted as the solution strategy to this optimization problem considering that it is easily implemented and a multidisciplinary feasible solution is always guaranteed throughout the optimization process. To solve this optimization problem efficiently, the coupled adjoint method is firstly introduced to improve the efficiency of gradient calculation and then a discipline-merging method is proposed to further enhance the computational efficiency. After this, the discipline-merging method is verified against the finite difference method in two aspects of solution accuracy and computational costs and the results show it is an effective and high efficient gradient calculation method. Finally, the multidisciplinary design optimization of the new underwater vehicle is performed efficiently under the MDF architecture with the discipline-merging method to calculate gradients.

The area of Multiparametric Optimization (MPO) solves problems that contain unknown problem data represented by parameters. The solutions map parameter values to optimal design and objective function values. In this paper, for the first time, MPO techniques are applied to improve and advance Multidisciplinary Design Optimization (MDO) to solve engineering problems with parameters. A multiparametric subgradient algorithm is proposed and applied to two MDO methods: Analytical Target Cascading (ATC) and Network Target Coordination (NTC). Numerical results on test problems show the proposed parametric ATC and NTC methods effectively solve parametric MDO problems and provide useful insights to designers. In addition, a novel Two-Stage ATC method is proposed to solve nonparametric MDO problems. In this new approach elements of the subproblems are treated as parameters and optimal design functions are constructed for each one. When the ATC loop is engaged, steps involving the lengthy optimization of subproblems are replaced with simple function evaluations.