We present Ozone, a Python library for solving ordinary differential equations (ODEs) within gradient-based optimization algorithms. Ozone makes available the entire family of explicit and implicit Runge–Kutta methods and implements several solution approaches including parallel-in-time approaches. A unique feature is the ability to perform sensitivity analysis for any combination of method and solution approaches. Ozone is implemented as a library in the Computational System Design Language (CSDL), enabling the automatic calculation of the derivatives of the optimization objective and constraints that are outputs of the larger system within which the Ozone model is embedded. Ozone allows researchers to easily incorporate ODEs in multidisciplinary optimization and direct-transcript optimal control models. This paper describes the components of the software implementation of Ozone and demonstrates its features through four illustrative applications.

A major bottleneck in multidisciplinary design optimization (MDO) is its high computational cost, with large-scale problems often requiring hours or days to complete. To address this issue, engineers turn to low-fidelity models or data-driven surrogate models such as neural networks. However, while these approaches significantly reduce runtime, they often sacrifice accuracy, which can impact the quality of optimization results. Physics-informed neural networks (PINNs) offer a promising middle ground by embedding physical governing equations directly into the training process, enabling both quick evaluation and high accuracy. Despite their potential, traditional PINNs often converge to non-physical solutions, limiting their use in real-world applications. Thus, many applications of PINNs in existing literature are limited to academic problems. In this paper, we demonstrate accurate flow field predictions on a three-dimensional domain around a complex aircraft geometry using PINNs, achieving a nondimensional PDE residual of less than 5 x 10(-3) and an average surface tangency error of less than 2 degrees. We avoid common failure modes associated with PINNs by leveraging inexpensive approximate data to help guide the network to a physical solution, qualitatively matching the reference solution. We also improve convergence speed compared to standard neural networks by encoding geometric information in the neural network architecture. We believe this demonstration on a practical application takes a step towards the use of PINN-based models in MDO workflows.

The recently introduced graph-accelerated non-intrusive polynomial chaos (NIPC) method has shown effectiveness in solving a broad range of uncertainty quantification (UQ) problems with multidisciplinary systems. It uses integration-based NIPC to solve the UQ problem and generates the quadrature rule in a desired tensor structure, so that the model evaluations can be efficiently accelerated through the computational graph transformation method, Accelerated Model evaluations on Tensor grids using Computational graph transformations (AMTC). This method is efficient when the model's computational graph possesses a certain type of sparsity which is commonly the case in multidisciplinary problems. However, it faces limitations in high-dimensional cases due to the curse of dimensionality. To broaden its applicability in high-dimensional UQ problems, we propose AS-AMTC, which integrates the AMTC approach with the active subspace (AS) method, a widely-used dimension reduction technique. In developing this new method, we have also developed AS-NIPC, linking integration-based NIPC with the AS method for solving high-dimensional UQ problems. AS-NIPC incorporates rigorous approaches to generate orthogonal polynomial basis functions for lower-dimensional active variables and efficient quadrature rules to estimate their coefficients. The AS-AMTC method extends AS-NIPC by generating a quadrature rule with a desired tensor structure. This allows the AMTC method to exploit the computational graph sparsity, leading to efficient model evaluations. In an 81-dimensional UQ problem derived from an air-taxi trajectory optimization scenario, AS-NIPC demonstrates a 30% decrease in relative error compared to the existing methods, while ASAMTC achieves an 80% reduction.

Large-scale gradient-based Multidisciplinary Design Optimization (MDO) can aid in the exploration of high-dimensional design spaces for novel air vehicle concepts, thereby leading to more efficient and economic designs. This paper builds on past works where we demonstrated large-scale physics-based MDO capabilities and applied these to NASA's lift-plus-cruise electric air taxi concept. We extend this comprehensive mid-fidelity system-level optimization problem with high(er)-fidelity subsystem-level optimizations of various aircraft systems and mission phases, with the aim of further enhancing the accuracy and scope of the aforementioned system-level optimization problem: 1) Power minimization during the transition mission phase; 2) Electrical powertrain topology optimization; 3) Cell chemistry modeling and thermomechanical battery pack topology optimization; 4) Shell-based coupled aero-elastic wing structure optimization. An Analytical Target Cascading-like distributed MDO architecture allows us to couple the system- and subsystem-level optimizations in order to arrive at a consistent and feasible design. We find that the system-level design is most heavily impacted by the reduced battery pack-level energy densities that stem from power peaks in the mission profile. These power peaks seem to result from the inefficient lift rotor blade designs that are needed to satisfy noise constraints. We draw conclusions based on trends in our results and give recommendations for future MDO studies of electrical air taxi vehicle concepts.

Multidisciplinary design optimization (MDO) models are built by assembling sub-models that represent varying disciplines and design conditions. Consequently, such models are often computationally expensive, requiring their evaluation in a parallel computing environment to decrease solution time. MDO modeling frameworks, which are software tools to allow construction of MDO models, generally require manual user input to specify where to parallelize their code. However, this process can be tedious if the model is large or complex. In this paper, we propose a method heavily inspired by scheduling theory that aims to fully automate parallelization of MDO models. We use static task scheduling to partition mathematical operations in the model to different processors prior to model execution. Additionally, our approach supports adjoint-based derivative computation to enable the use of gradient-based optimization algorithms. Our results show a significant performance improvement when applying our method to several MDO applications. We believe our approach enables MDO modelers to solve computationally expensive problems in a more efficient manner.

Aircraft conceptual design is a high-dimensional optimization problem, involving up to hundreds of continuous design variables and constraints. The design of novel aircraft concepts such as electric vertical takeoff and landing (eVTOL) vehicles can benefit from the systematic exploration of the design space through the use of gradient-based optimization. In this paper, we demonstrate the application of large-scale multidisciplinary design optimization (MDO) to NASA's lift-plus-cruise electric air taxi concept, using low and mid-fidelity physics-based simulations to model the aircraft. We improve the modeling fidelity from our previous work on the same air taxi concept through refined rotor-aerodynamic and aeroacoustic models as well as in-the-loop structural analysis. In addition, we expand the scope of the analysis to include several steady design conditions, a quasi-steady transition maneuver, structural sizing conditions, as well as one-engine-inoperative (OEI) scenarios, amounting to 18 design conditions. To perform the analysis and optimization, we use a newly developed software library called the Comprehensive Aircraft high-Dimensional Design Environment (CADDEE), which facilitates the integration of all discipline models. We solve a gross weight minimization problem with over 300 design variables and over 200 constraints, spanning all modeled disciplines. Results show a decrease in gross weight of 10% after a total optimization time of about 17 hours. These results demonstrate the effectiveness of applying of large-scale MDO to the aircraft conceptual design problem.

Methods such as non-intrusive polynomial chaos (NIPC), and stochastic collocation are frequently used for uncertainty propagation problems. Particularly for low-dimensional problems, these methods often use a tensor-product grid for sampling the space of uncertain inputs. A limitation of this approach is that it encounters a significant challenge: the number of sample points grows exponentially with the increase of uncertain inputs. Current strategies to mitigate computational costs abandon the tensor structure of sampling points, with the aim of reducing their overall count. Contrastingly, our investigation reveals that preserving the tensor structure of sample points can offer distinct advantages in specific scenarios. Notably, by manipulating the computational graph of the targeted model, it is feasible to avoid redundant evaluations at the operation level to significantly reduce the model evaluation cost on tensor-grid inputs. This paper presents a pioneering method: Accelerated Model Evaluations on Tensor grids using Computational graph transformations (AMTC). The core premise of AMTC lies in the strategic modification of the computational graph of the target model to algorithmically remove the repeated evaluations on the operation level. We implemented the AMTC method within the compiler of a new modeling language called the Computational System Design Language (CSDL). We demonstrate the effectiveness of AMTC by using it with the full-grid NIPC method to solve four low-dimensional UQ problems involving an analytical piston model, a multidisciplinary unmanned aerial vehicle design model, a multi-point air taxi mission analysis model, and a single-disciplinary rotor model, respectively. For three of the four test problems, AMTC reduces the model evaluation cost by between 50% and 90%, making the full-grid NIPC the most efficacious method to use among the UQ methods implemented.

The univariate dimension reduction (UDR) method stands as an economical solution for addressing the challenges of high-dimensional uncertainty quantification (UQ). UDR's fundamental strategy is to approximate the original function using univariate functions so that the UQ cost only scales linearly with the dimension of the problem. Nonetheless, UDR's effectiveness can diminish when uncertain inputs have high variance, particularly when assessing the output's second and higher-order statistical moments. This paper proposes a new method, gradient-enhanced univariate dimension reduction (GUDR), that enhances the accuracy of UDR by incorporating univariate gradient function terms into the UDR approximation function. Theoretical results indicate that the GUDR approximation is expected to be one order more accurate than UDR in approximating the original function and it is expected to generate more accurate results in computing the output's second and higher-order statistical moments. Our proposed method uses a computational graph transformation strategy to efficiently evaluate the GUDR approximation function on tensor-grid quadrature inputs, and the tensor-grid input-output data can be easily used to compute any risk measure of the output. With an efficient automatic differentiation method to compute the gradients, our methods preserve UDR's linear cost-scaling with problem dimension. Numerical results show that the GUDR is more accurate than UDR in estimating the standard deviation of the output and has comparable performance with the method of moments using 3rd-order Taylor series expansion.

The adjoint method provides an efficient way to compute sensitivities for system models with a large number of inputs. However, implementing the adjoint method requires significant effort that limits its use. The effort is exacerbated in large-scale multidisciplinary design optimization. We propose the adoption of a three-stage compiler as the method for constructing computational models for large-scale multidisciplinary design optimization to enable accurate and efficient adjoint sensitivity analysis. We develop a new modeling language called the Computational System Design Language that provides an appropriate input to the compiler front end that works well with multidisciplinary models. This paper describes the three-stage compiler methodology and the Computational System Design Language. The proposed solution uses a graph representation of the numerical model to automatically generate a computational model that computes adjoint-based sensitivities for use within an optimization framework. For two engineering models, this approach reduces the amount of user code by a factor of approximately two compared to their original implementations, without a measurable increase in computation time. This paper also includes a best-case complexity analysis that is built into the compiler implementation to allow users to estimate the memory required to evaluate a computational model and its derivatives, which is independent of the compiler back end that ultimately generates the computational model. Future compiler implementations are expected to approach the theoretical best-case memory cost and improve run time performance for both model evaluation and derivative computation.

The univariate dimension reduction (UDR) method stands as a way to estimate the statistical moments of the output that is effective in a large class of uncertainty quantification (UQ) problems. UDR's fundamental strategy is to approximate the original function using univariate functions so that the UQ cost only scales linearly with the dimension of the problem. Nonetheless, UDR's effectiveness can diminish when uncertain inputs have high variance, particularly when assessing the output's second and higher-order statistical moments. This paper proposes a new method, gradient-enhanced univariate dimension reduction (GUDR), that enhances the accuracy of UDR by incorporating univariate gradient function terms into the UDR approximation function. Theoretical results indicate that the GUDR approximation is expected to be one order more accurate than UDR in approximating the original function, and it is expected to generate more accurate results in computing the output's second and higher-order statistical moments. Our proposed method uses a computational graph transformation strategy to efficiently evaluate the GUDR approximation function on tensor-grid quadrature inputs, and use the tensor-grid input-output data to compute the statistical moments of the output. With an efficient automatic differentiation method to compute the gradients, our method preserves UDR's linear scaling of computation time with problem dimension. Numerical results show that the GUDR is more accurate than UDR in estimating the standard deviation of the output and has a performance comparable to the method of moments using a third-order Taylor series expansion.

This paper presents a framework under development for enabling large-scale multi-fidelity modeling and optimization of electric vertical takeoff and landing concepts (eVTOL). The key features of the framework are a geometry-centric approach to multidisciplinary design optimization (MDO), a modular functional-form representation of the disciplines involved in aircraft design, and fully automated derivative computations thereby allowing efficient gradient-based optimization. The framework is first presented in a general manner agnostic to the vehicle concept or the physics-based analyses used. The key disciplines involved in the design of eVTOL aircraft and the couplings between the disciplines are described. The complex multidisciplinary nature of the design is emphasized. The framework is then applied to the design of the NASA lift-plus-cruise concept. Low-fidelity solvers of aerodynamics, propulsion, structural estimation, acoustics, powertrain, and battery are coupled together. The optimization considers 108 design variables and 16 constraints. It is shown that MDO considering geometric variables results in a design with lower gross mass than when the geometric variables are not considered. The optimization turnaround time of 30 minutes on a standard workstation demonstrates the capabilities of fully-coupled large-scale MDO using gradient-based optimization. The framework is under development and an open-source version will be released in the near-future.

Large-scale multidisciplinary design optimization (MDO) is the solution of numerical optimization problems involving dozens or more design variables and multidisciplinary models. It can provide a systematic approach for dealing with high-dimensional design problems where the engineer’s intuition is of limited value. One of the critical barriers is the complexity of implementation for adjoint sensitivity analysis which is needed for efficient large-scale MDO. Recently, a new modeling language, called CSDL, was developed that enables the construction of a computational graph for the model. The graph can be used to implement automatic adjoint sensitivity analysis, but the proposed compiler for CSDL incurs significant computational overhead as the cost of the automation. This paper describes a method that eliminates this overhead in the CSDL compiler by using a code generation technique that utilizes graph traversal algorithms. Together, with CSDL, this method bears similarities to reverse-mode algorithmic differentiation but has notable differences such as the application of the implicit function theorem and the use of hybrid dense-sparse linear algebra. The proposed method shows an order of magnitude improvement in computation time compared to the previous compiler. The results indicate that this method is an effective method for sensitivity analysis of large-scale MDO models that achieves both efficiency and automation.

Electric Vertical Takeoff and Landing (eVTOL) aircraft experience complex, unsteady aerodynamic interactions between rotors, wings, and fuselage that can make design difficult. We introduce a new framework for predicting aerostructural interactions. Specifically, we demonstrate the coupling of a finite element solver with Reissner-Mindlin shell theory for computing deflections and a viscous vortex particle for capturing wakes. We perform convergence studies of the aerodynamics and the coupled aerostructural model. Finally, we share some preliminary results of the dynamic aeroelastic response of Uber's eCRM-002 main wing, and share some qualitative observations.

The optimization of large-scale and multidisciplinary engineering systems is now prevalent in many fields, exemplified in areas such as aircraft, satellite, and wind turbine design. The rise of advanced modeling frameworks has expanded the scope of large-scale optimization techniques across various research domains. However, novel applications have emerged that pose efficiency-related computational challenges to the existing methods employed in large-scale, gradient-based optimization. The authors previously proposed a new paradigm for accelerating the optimization of large-scale and complex-engineered systems, laying the groundwork for a new approach. The new paradigm is based on a hybrid optimization architecture called SURF which stands for strong unification of reduced-space and full-space. SURF has the potential to expedite optimization of models with state variables that are iteratively computed by solving nonlinear systems. This paper extends the existing paradigm by providing new theoretical results that unify the reduced-space and full-space algorithms in a practical optimization setting that considers line searches and quasi-Newton methods. We also present a practical, SQP-based SURF algorithm that can be applied to general, inequality-constrained problems. The new algorithm also includes an adaptive hybrid selection strategy for robust convergence and faster solutions. We test the new algorithm on a low-fidelity motor optimization problem and a wind farm layout optimization problem to validate the optimization results, and to demonstrate its computational benefits. In one of the problems, SURF was able to speed up the traditional optimization by approximately 25 percent. In the other problem, SURF was able to converge to a better optimal solution.

View Video Presentation: https://doi.org/10.2514/6.2022-3997.vid Uncertainty quantification (UQ) methods such as integration-based non-intrusive polynomial chaos (NIPC) and stochastic collocation often use a tensor-product grid to sample the space of random inputs. In these situations, the number of quadrature points increases exponentially with the number of uncertain inputs. If we view the model as a computational graph with only elementary functions and operations, the current framework evaluates each operation node in the graph for all of the quadrature points. However, each node in the model only has distinct values on the quadrature points from the uncertain inputs on which the operation's output depends. This means that the current framework could create many unnecessary repeated evaluations on certain operation nodes. In this paper, we propose a new method that uses a computational graph representation of the model to algorithmically remove these unnecessary repeated evaluations so that the evaluation cost on full-grid quadrature points can be significantly reduced. This method is implemented in the Computational System Design Language, an algebraic modeling language, to achieve the cost reduction on UQ problems in a general way. The process of detecting and achieving savings from the model structure is abstracted by the modeling language and its back-end. This method is applied to a multi-point aircraft mission analysis problem with two uncertain inputs. Our method reduces the computing time of the integration-based NIPC method by an order of magnitude. For a wide range of forward UQ problems, we expect this method to significantly improve time scaling with the number of the uncertain inputs.

Electric air taxis are envisioned to enable a new form of aviation in dense, urban locations. One of the feasibility barriers is the limited range due to the much lower energy densities that batteries have compared to fuel. A promising research direction to overcome this bottleneck is the use of multi-functional battery packs that dissipate heat but also carry structural loads. Here, we propose a scalable approach with automated derivative computation to design a battery pack of an eVTOL aircraft considering the power profile in takeoff, which is presumed to be the sizing condition. The problem uses a multi-fidelity submodel consisting of an equivalent circuit model to represent battery cells and a high-fidelity dynamic thermoelastic submodel of the battery pack. First, a steady-state topology optimization problem is solved based on the maximum heat output of the battery cells, computed from the aircraft power profile. Second, a dynamic topology optimization problem is solved based on the transient portion of the transition from liftoff to forward flight. Compared to the steady-state topology optimization, a 9.94% reduction in time-averaged compliance is obtained with the dynamic topology optimization.