Automatic Evaluation of Multidisciplinary Derivatives Using a Graph-Based Problem Formulation in OpenMDAO

Abstract

The optimization of multidisciplinary systems with respect to large numbers of design variables is best pursued using a gradient-based optimization together with a method that efficiently evaluates coupled derivatives, such as the coupled adjoint method. However, implementing such a method in a problem with more than a few disciplines is time consuming and error prone. To address this issue, we develop an automated procedure for assembling and solving the coupled derivative equations that takes into account the disciplinary couplings using the interdisciplinary dependency graph of the problem. The coupled derivatives can be computed completely analytically, if analytic derivatives are available for all disciplines; otherwise, the coupled derivatives are computed semi-analytically. The procedure determines the disciplinary analyses execution order, detects iterative cycles, and uses this information to converge the coupled analysis, and evaluate the coupled derivatives as efficiently as possible by exploiting sparsity. Sparsity can occur at two levels within a multidisciplinary problem: between disciplines, when certain analyses do not affect all outputs, and within a discipline when, the Jacobian of that discipline is sparse. The numerical procedures are implemented in NASA's OpenMDAO framework, providing a flexible API for declaring discipline-level derivatives that can handle sparsity within a discipline. The tool is demonstrated in two MDO problems: the design of a small satellite and its operation with the objective of maximizing downloaded data to a ground station, and the design of a horizontal-axis wind turbine with the objective of minimizing the cost of energy. In both cases, the method demonstrated improved efficiency by taking advantage of analytic gradients considering sparsity. This new capability in OpenMDAO greatly facilitates the implementation of system-level direct and adjoint coupled derivative evaluations, and is applicable for general problems.

Integrated design of downwind land‐based wind turbines using analytic gradients

“…The benefits of using analytic derivatives, as opposed to finite differencing, for this problem are highlighted in a recent publication . We found that despite care in deriving models with smooth output, global finite differencing was not good enough to converge many of the solutions.…”

“…The constitutive equations for a laminate sequence can be expressed as (5) * The study actually showed DLC 1.4 extreme direction change (EDC) as the condition producing the largest deflection; however, only a PI controller was used for speed regulation, and no controller logic was asserted for start-up and shutdowns. The controller response to ECD, had there been one, would have substantially reduced turbine loads and resulted in turbine shutdown in the event of an extreme direction change.…”

“…Furthermore, the accuracy of finite‐difference gradients is limited (generally to six significant figures or less), which can cause issues with convergence. Using a very similar setup to that described in this paper, we found that finite‐differencing led to inferior solutions and in many cases failed to converge at all .…”

Uniform Decomposition and Positive-Gradient Differential Evolution for Multi-Objective Design of Wind Turbine Blade

“…Various design variables and constraints are involved in describing the blade aerodynamic shape and structural layout, as well as meeting the geometric, load, stress and strain, vibration, and fatigue requirements of the system. In a typical design example [2], up to 32 design variables and 102 constraints are adopted in optimizing a wind turbine, resulting in an extremely challenging design process.…”

Aero structural optimization for sailplane wing in preliminary design

“…One other hand, OpenMDAO has ability to subdivide a problem into groups or components and link their inputs and outputs, made it easier to decompose large and complex problems. OpenMDAO has been used to solve MDO problems in satellite design [8], wind turbine design [9], aircraft design [10] and aircraft trajectory optimization [11].…”