Multilevel, multidisciplinary optimization methods are important for designing complex systems, wiht many componeents and and many disciplines involved in a distributed design environment. We study rigorous methods to partition and coordinate the decomposed problem solution, and to examine solution properties under conditions of uncertainty. Applications include hybrid powertrains, vehicle design, electronic devices, product development, architecture.

Keywords: multidisciplinary optimization (MDO), analytical target cascading (ATC), reliability-based design optimization (RBDO), robust optimal design, optimal partitioning and coordination, uncertainty, validation, verification and testing

Optimal Partitioning and Coordination Decisions in Complex System Design Optimization
by James Allison, PhD Candidate in Mechanical Engineering

Many engineering systems are too complicated to design as a single integrated entity, and require decomposition-based design. Several complex system design optimization methods have been developed to solve system design problems that have been partitioned into smaller and easier to solve subproblems. These methods employ a coordination algorithm that guides repeated solution of subproblems toward a consistent and optimal system design.

Before decomposition-based design optimization can be applied the system designer must first decide how to divide a system into subproblems (i.e., partition), and decide on a strategy for coordinating the subproblems. The decisions strongly influence the success of the system design process. Optimal partitioning decisions that minimize interactions between subproblems have been studied, and optimal subproblem sequencing, one aspect of coordination, has also been investigated. The relationship between partitioning and coordination decisions, however, has not yet been studied.

This work has shown that partitioning and coordination decisions are in fact coupled, and should be made together. All partitioning and coordination options for three example systems were enumerated, and the coordination problem size (CS) and subproblem size (SS) for each option was calculated. Both of these metrics need to be minimized to reduce computational expense of the design problem. The plots below show the tradeoff between CS and SS for the example systems, and demonstrate that the simultaneous decision approach (P||C) identified all Pareto-optimal solutions, while the non-simultaneous approaches did not. This is evidence that partitioning and coordination decisions are coupled.

   

A simultaneous decision approach has been applied to several example systems, including automotive electric water pump design, generalized truss design, and integrated electric vehicle design. In the latter case study three vehicle systems (powertrain, chassis, and structure) are considered, along with interactions between these systems.

Downloadable materials: ODE poster

Sequential Linear Programming Coordination for Analytical Target Cascading under Uncertainty
by Jeongwoo Han, PhD Candidate in Mechanical Engineering

Engineers make decisions under various forms of uncertainty. Moreover, many products today are large and complex systems whose design requires multidisciplinary analyses involving significant interactions. Inclusion of uncertain quantities in these interactions can strongly couple subsystems to each other. Use of decomposition strategies in Multi-Disciplinary Optimization (MDO) of such system may be the only available solution approach. Uncertainty propagation of random variables, however, can be expensive for highly nonlinear functions. Moreover, matching the entire distributions of random linking variables is practically impossible in MDO, including probabilistic analytical target cascading (PATC). This work focuses on Analytical Target Cascading (ATC), an optimization method for multilevel hierarchical systems and develops efficient algorithms for multilevel system design under uncertainty and reliability target cascading using sequential linearization and suspension strategy.

Sequential linearization can be effective for probabilistic ATC because SLP-based probabilistic ATC requires only means and variances to represent the random variables that can be efficiently estimated and matched. Suspension strategy takes advantage of the properties of weakly-coupled elements and can reduce the number of function evaluations by 15 ~ 20%. The accuracy and effectiveness of the proposed SLP-based PATC strategy is demonstrated with several numerical examples. The solution strategy will be applied for hybrid vehicle design problems, including battery and fuel cell design models and an enterprise decision model.

Downloadable materials: ODE poster

Optimization of an HEV System with Reduced Order Models
by Michael J. Alexander, MS Candidate in Mechanical Engineering

My research currently involves optimizing dynamical systems that occur in hierarchical design problems. In some cases, these problems require function-valued quantities that describe the behavior of the subsystems to be communicated. When decomposition-based optimization is used to solve these systems and partitions are made across function-valued communication lines, these function-valued inputs become design variables during optimization. Reduced-dimension representations of these function-valued quantities approximating the original representations are required to enable efficient optimization. This can be accomplished through reduced order models (ROMs), which are an efficient and accurate means to model function-valued inputs or outputs of dynamical systems. The ROM algorithms are mathematically-based methods for dimension reduction that are effective at managing representation error.

In my case study, the goal is to optimize a hybrid electric vehicle (HEV) system using ROM representations for engine maximum torque curves and power loss maps. The optimization technique to be used is analytical target cascading (ATC), decomposed into a bi-level (vehicle system and engine subsystems) problem. The key ROMs to be investigated are RBF neural networks, proper orthogonal decomposition (PODs) and hybrid PODs with image warping transformations. Finally, the approximations of the ROMs to the original engine curves and maps will be quantified using the Accuracy and Validity Algorithm for Simulation (AVASIM).

Downloadable materials: ODE poster