# Research

## Background: large-scale design optimization

Design optimization is the application of numerical optimization to the solution of engineering design problems. The LSDO Lab focuses on large-scale design optimization problems: those involving hundreds or more design variables and computational models with many heterogeneous parts (multiple engineering disciplines, types of numerical models, programming languages, etc.).

*General optimization problem statement*

Large-scale problems are solved with gradient-based optimization algorithms and advanced derivative computation methods. Efficiently and accurately computing derivatives of complex models is the most difficult challenge for LSDO. In the past decade, Prof. Hwang contributed to a series of advances [1-3] that have made this feasible, centered on his discovery of a unified derivatives equation (UDE) using the inverse function theorem [4]. The UDE lies at the heart of NASA's OpenMDAO software framework [5]. OpenMDAO is a modeling environment that computes derivatives semi-automatically by working with a modular implementation of complex models. OpenMDAO has enabled rapid growth recently in LSDO. See below for a presentation on recent advances in LSDO.

- Martins, J. R. R. A., and Hwang, J. T., Review and unification of methods for computing derivatives of multidisciplinary computational models,
*AIAA journal*, Vol. 51, No. 11, 2013, pp. 2582--2599. [bibtex] [doi] - Gray, J. S., Hearn, T. A., Moore, K. T., Hwang, J., Martins, J., and Ning, A., Automatic evaluation of multidisciplinary derivatives using a graph-based problem formulation in OpenMDAO,
*15th AIAA/ISSMO Multidisciplinary Analysis and Optimization Conference*. (AIAA 2014-2042) [bibtex] [doi] - Hwang, J. T., and Martins, J., Parallel allocation-mission optimization of a 128-route network,
*16th AIAA/ISSMO Multidisciplinary Analysis and Optimization Conference*. (AIAA 2015-2321) [bibtex] [doi] - Hwang, J. T., and Martins, J. R. R. A., A Computational Architecture for Coupling Heterogeneous Numerical Models and Computing Coupled Derivatives,
*ACM Trans. Math. Softw.*, Vol. 44, No. 4, 2018, pp. 37:1--37:39. [bibtex] [doi] - Gray, J. S., Hwang, J. T., Martins, J. R. R. A., Moore, K. T., and Naylor, B. A., OpenMDAO: An open-source framework for multidisciplinary design, analysis, and optimization,
*Structural and Multidisciplinary Optimization*, Vol. 59, No. 4, 2019, pp. 1075--1104. [bibtex] [doi]

*Right: This design structure matrix visualizes data flow (off-diagonal blocks) between components (diagonal blocks) in an LSDO algorithm. It shows an example of a model with multiple, coupled disciplines.*

## A new architecture for LSDO

*(Sponsor: National Science Foundation, CMMI)*

*(Sponsor: National Science Foundation, CMMI)*

In large-scale design optimization, discipline outputs, *y*, are often defined using implicit functions of the form, *R(x,y)=0*. An important choice to be made is whether to treat these quantities computed by discipline models as state variables (reduced-space method) or design variables (full-space method), which are visualized below. In the reduced-space method, the variables are computed by solvers that are part of the model. In the full-space method, the optimizer is responsible for computing the state variables. The reduced-space method results in a smaller, easier-to-solve optimization problem, while the full-space method has more inexpensive model evaluations. This project investigates a new hybrid method [1] enabled by a new algorithm that combines both formulations, which has the potential to achieve the best of both worlds: the efficiency of the full-space method and the robustness of the reduced-space method.

- Joshy, A. J., and Hwang, J. T., A new architecture for large-scale system design optimization,
*AIAA Journal (Accepted subject to revisions)*, 2020. [bibtex]

### New hybrid method

### Existing reduced-space method

### Existing full-space method

## LSDO of urban air mobility aircraft concepts

*(Sponsor: Hyundai Motor Company)*

*(Sponsor: Hyundai Motor Company)*

Urban air mobility (UAM) has the potential to transformatively impact the lives of Americans by enabling point-to-point, on-demand air transportation. UAM is the vision of facilitating convenient air travel in cities using electric aircraft carrying 1-4 passengers by leveraging recent advances in electric propulsion, autonomy, and connectivity, among other technologies. UAM will utilize electric vertical takeoff & landing (eVTOL) aircraft, and billions of dollars have been invested in the development of eVTOL concepts.

Large-scale design optimization is an invaluable tool in the eVTOL concept design problem because of the rapidly changing requirements in UAM, the tight integration of propulsion and airframe, and the wide range of concepts being considered for UAM. This project incorporates physics-based models for airframe aerodynamics, rotor aerodynamics, motor, battery, weights, stability, and performance. Below is a visualization of the LSDO of the wing design, propulsion system sizing, and rotor layout in Uber's eCRM-002 concept.

## LSDO of CubeSat swarms

*(Sponsor: National Science Foundation, AGS)*

*(Sponsor: National Science Foundation, AGS)*

The Virtual Super-resolution Optics using Reconfigurable Swarms (VISORS) mission will launch two to three CubeSats into low-Earth orbit in 2023. These CubeSats will maintain tight formation to create a virtual telescope that will be used to observe the Sun at a resolution that is an order-of-magnitude higher than what is currently possible. This project involves 10 institutions led by the University of Illinois, and is in collaboration with NASA Goddard Space Center.

In this project, we use large-scale design optimization to minimize propellant use and maximize the amount of data downloaded while satisfying the alignment constraints during the observation window in the orbit. The scope of the model includes orbit dynamics, attitude dynamics, propulsion, and communication modeling, and we simulate a full 90-minute orbit. Below is a notional visualization of an optimization problem involving the VISORS CubeSats.