Appendix I. Optimal allocations (2006-2012)

Multidisciplinary Design Optimization (MDO) approaches are tasked with optimizing a com-plex system that requires evaluating the interactions of multiple disciplines or equivalently multiple subsystems. To explain some of the difficulties in MDO, Figure 1-2 presents a design structure matrix (DSM) which demonstrates feed-forward and feedback between dis-ciplines in a multidisciplinary system. All disdis-ciplines are placed on a diagonal, feed-forward is represented as connections above the diagonal, and feedback is indicated by connections below the diagonal. The feedback from Discipline 2 to Discipline 1 means that there must be some iterative solution between Disciplines 1 and 2 to find the resulting output that Discipline 3 requires, and this iterative solution must be done for each set of design vari-ables tested during optimization. This can be further complicated when both Disciplines 1 and 2 have their own constraints that must be satisfied for their individual feasibility. The seminal publication [34], Problem Formulation for Multidisciplinary Optimization, defined multidisciplinary feasibility as when (1) all single disciplines are individual feasible and (2)

the input to each discipline corresponds to the output of the other via the interdisciplinary mapping. With respect to Figure 1-2, this means that all the constraints within Disciplines 1, 2 and 3 are satisfied, and that when an output of one discipline is an input to another discipline, the value output and the value input must be equal. This corresponds to the iteration between Discipline 1 and Discipline 2 having been solved. Solving for all of the interacting variables between disciplines such that the system is multidisciplinary feasible is a called a multidisciplinary analysis (MDA).

Figure 1-2: Block diagram for a sys-tem with feed-forward from discipline 1 to 2 and 3, and feedback from disci-pline 2 to 1. The system objective is only a function of discipline 3.

The remainder of this section presents common MDO formulations. The discussion highlights the degree to which the formulations enable multifidelity methods, are parallelizable, and whether or not the methods have a mathematical proof of convergence to an optimal system design. In terms of paralleliza-tion of the system optimizaparalleliza-tion there are three ways in which an optimization method can be parallelized, (i) by evaluating the function in parallel, e.g., eval-uating each required function evaluation for a finite difference calculation at the same time, (ii) perform-ing the linear algebra and optimization calculations in parallel, and (iii) decomposing the optimization into smaller subproblems that can each be performed

in parallel [102]. This thesis will only consider parallelizability of the optimization itself and not the linear algebra, i.e., only techniques (i) and (iii) are considered.

1.3.1 Fundamental MDO Methods

Another major contribution from the paper Problem Formulation for Multidisciplinary Op-timization is to define three fundamental multidisciplinary opOp-timization approaches which allow for the use of nonlinear programming techniques to solve system optimization prob-lems. The first formulation is called the Multidisciplinary Feasible (MDF) formulation, and

in this formulation evaluating the system objective function value for any given set of de-sign variables requires a full MDA. This formulation can be computationally intensive as calculations of the objective function used for sensitivity information also require solving a complete MDA. The second approach is called the All-At-Once (AAO) formulation, and with this approach no feasibility is ensured until convergence. All of the analysis variables and coupling variables are treated as additional design variables, and all of the analyses and disciplinary couplings are treated as constraints. In this formulation the optimization routine is required to solve the nonlinear systems of constraints in order to find a feasible solution; however, the full MDA is not required in a large portion of the design space when feasibility is impossible. The final approach presented is the Individual Discipline Feasible (IDF) formulation. In the IDF approach individual discipline feasibility is ensured; however, the multidisciplinary feasibility is left to the optimizer as a constraint. The IDF approach is flexible and numerous formulations are posed [34].

Using the MDF approach, multifidelity optimization and parallelization are only possible at the system level. Surrogate models of the system performance can be created from full MDAs at many designs. In addition, each MDA may be carried out simultaneously, but the iterative solution of finding feasible designs is only parallelizable if a system exhibits atypical subsystem couplings. Convergence of an MDF optimization only relies on the capabilities of the iterative technique used in the MDA and on the nonlinear programming (NLP) solver used in the optimization, therefore this technique can typically be mathematically guaranteed to find an optimal system given that the assumptions of the iterative solution method and NLP solver are met. The essential drawback of this method is when each subsystem contains a high-fidelity simulation; in this case, the iterative solve in the MDA can be prohibitively expensive and preclude the use of the MDF formulation.

The IDF approach offers a potential benefit in that the subsystems only communicate through the level optimization problem. This enables parallelizing the overall system-level optimization problem into separately finding feasible subsystem designs, and also allows the possibility to evaluate the full system performance in parallel. There is also the potential to use multifidelity methods for the system-level and subsystem-level analyses. The

conver-gence of this technique only relies on converconver-gence properties of the NLP solver used for the optimization, so in many cases the approach can be proven to converge.

The AAO approach, also known as Simultaneous Analysis and Design (SAND), has the issue that all discipline analyses need to be converted to residual form, which is atypical of customary disciplinary analyses. However, it is theoretically possible to evaluate all dis-ciplines and the system objective in parallel and to use multifidelity methods to solve the equality constrained optimization problem. The convergence of this method is only depen-dent on the convergence properties of the NLP solver. In fact, Alexandrov et al. developed provably convergent trust-region methods to solve the AAO formulation as a multilevel prob-lem [4]. Their algorithm separates the constraints into blocks and finds solutions that are in the null space of the constraint block Jacobians, and then updates the trust-region according to a merit function that includes constraint violations. The advantage of the AAO formula-tion is that the problem can be solved without using a constraint following algorithm which given the large number of constraints in an MDO problem, should prove fruitful.

1.3.2 Open Issues in Multidisciplinary Optimization

Multidisciplinary system design optimization is a challenge due to the nonlinear couplings between subsystems and the expense of estimating the performance of subsystems. These challenges suggest that a parallelizable framework for MDO is a necessity, and that rigorous multifidelity optimization techniques should be possible within that framework. Moreover, it is common that the sensitivity of a subsystem’s performance to its design variables will be available for some but not for all subsystems. Therefore MDO frameworks should also allow for gradient-free optimization, but must be able to exploit gradient information when it is available in order to rapidly find good system designs.

When considering the single-level MDO frameworks, MDF, IDF, and AAO, the under-lying NLP solver can provide a guarantee of finding an optimal design. As NLP solvers with a convergence guarantee are available for both gradient-free and gradient-exploiting optimization problems, this guarantee applies in both cases. However, the ability to mix gradient-free and gradient-exploiting handling for individual disciplines and the

paralleliz-ability of these basic frameworks is limited. The multi-level MDO methods, Collaborative Optimization [18, 60], Bilevel Integrated System Synthesis [107–109], Concurrent Subspace Optimization [106, 119], and Analytical Target Cascading [78, 79, 113], are more paralleliz-able, but fast convergence or even just convergence for the case of a non-convex optimization problem is not guaranteed. One benefit of these multilevel methods is that the formulations based on response surfaces offer a chance to use either gradient-free or gradient-exploiting handling for the subsystems as appropriate. For example, BLISS [108] requires global sen-sitivity information for each subproblem optimization, but BLISS2000 [109] approximates this sensitivity information with quadratic response surfaces. Therefore novel MDO methods that enable a mixture of gradient-free and gradient-exploiting optimization methods and are parallelizable are a clear need. Multifidelity optimization techniques offer a possible solution to this challenge. Surrogate-based multifidelity methods typically create smooth surrogate models for each discipline’s performance with known bounds for the error in surrogate-based sensitivity information. This surrogate sensitivity information can exploit high-fidelity gra-dients when available, but does not require that information. In addition, these surrogate models can be used to optimize the system and likely will save a significant number of discipline- and system-level performance evaluations.