John T. Hwang

The rise of urban air mobility as a solution to alleviate congestion in metropolitan areas has spurred the development of novel vertical takeoff and landing aircraft, renewing research interest in conceptual design. Adjoint-based multidisciplinary design optimization naturally suits aircraft design by efficiently searching high-dimensional design spaces. However, solving system-level large-scale aircraft design problems reliably remains challenging due to numerical conditioning issues arising from simultaneously modeling many design conditions and disciplines. This paper presents a novel demonstration of system-level large-scale multidisciplinary design optimization of an urban air mobility concept using an integrated, physics-based computational model. We solve a gross weight minimization problem of a lift-plus-cruise reference vehicle with 161 design variables and 96 constraints across 15 design conditions, representing nominal mission and failure conditions. Our computational model integrates physics-based models for aerodynamics, propulsion, structures, aeroacoustics, motor performance, and static stability. The optimized design achieves a 6.5% reduction in gross weight. Parameter sweeps over battery energy density and end-of-mission state of charge confirm the robustness of our approach. Using standard computational resources, the average time for solving the large-scale problem is 1.6 h. These findings highlight the potential of large-scale multidisciplinary design optimization in accelerating aircraft conceptual design.

Surrogate models are widely used in mechanical design and manufacturing process optimization, where high-fidelity computational models may be unavailable or prohibitively expensive. Their effectiveness, however, is often limited by data scarcity, as purely data-driven surrogates struggle to achieve high predictive accuracy in such situations. Subject matter experts (SMEs) frequently possess valuable domain knowledge about functional relationships, yet few surrogate modeling techniques can systematically integrate this information with limited data. We address this challenge with RBF-Gen, a knowledge-guided surrogate modeling framework that combines scarce data with domain knowledge. This method constructs a radial basis function (RBF) space with more centers than training samples and leverages the null space via a generator network, inspired by the principle of maximum information preservation. The introduced latent variables provide a principled mechanism to encode structural relationships and distributional priors during training, thereby guiding the surrogate toward physically meaningful solutions. Numerical studies demonstrate that RBF-Gen significantly outperforms standard RBF surrogates on 1D and 2D structural optimization problems in data-scarce settings, and achieves superior predictive accuracy on a real-world semiconductor manufacturing dataset. These results highlight the potential of combining limited experimental data with domain expertise to enable accurate and practical surrogate modeling in mechanical and process design problems.

Abstract Surrogate models are widely used in mechanical design and manufacturing process optimization, where high-fidelity computational models may be unavailable or prohibitively expensive. Their effectiveness, however, is often limited by data scarcity, as purely data-driven surrogates struggle to achieve high predictive accuracy in such situations. Subject matter experts (SMEs) frequently possess valuable domain knowledge about functional relationships; yet, few surrogate modeling techniques can systematically integrate this information with limited data. We address this challenge with RBF-Gen, a knowledge-guided surrogate modeling framework that combines scarce data with domain knowledge. This method constructs a radial basis function (RBF) space with more centers than training samples and leverages the null space via a generator network, inspired by the principle of maximum information preservation. The introduced latent variables provide a principled mechanism to encode structural relationships and distributional priors during training, thereby guiding the surrogate toward physically meaningful solutions. Numerical studies demonstrate that RBF-Gen significantly outperforms standard RBF surrogates on 1D and 2D structural optimization problems in data-scarce settings and achieves superior predictive accuracy on a real-world semiconductor manufacturing dataset. These results highlight the potential of combining limited experimental data with domain expertise to enable accurate and practical surrogate modeling in mechanical and process design problems.

The aim of this work is to present the first steps of our research to make high-fidelity inviscid compressible flow models computationally tractable for large-scale preliminary aircraft design optimization problems. We take a novel weighted Proper Orthogonal Decomposition (POD) approach and apply it to speed up a Discontinuous Galerkin-based compressible Euler model in an airfoil shape optimization problem. We show how the weighted POD approach improves upon a standard POD approach, especially when predicting drag coefficients, while having lower wall times than the standard POD approach. We intend to extend the current approach with hyper-reduction, in order to obtain further speedups, and switch to gradient-based optimization methods in order to enable more efficient solving of large-scale optimization problems.

Applications of numerical optimization span a wide range of fields, from finance and economics to the natural sciences and engineering. Optimization techniques employed in each field are specialized to exploit the structure of the underlying problems. As optimization problems grow in scale and complexity, they uncover bottlenecks in existing optimization algorithms and necessitate further specialization of the algorithms. Such specialization requires expert knowledge of the underlying mathematical theory and the software implementation of current algorithms. However, currently available optimization libraries lack the modularity, transparency, and accessibility needed for customization and experimentation, as they often provide only monolithic implementations of algorithms. To overcome the challenges posed by this limitation in algorithm development and education, we present modOpt, an open-source Python framework designed to facilitate the construction, customization, and study of optimization algorithms. Its modular architecture enables students and researchers to tailor existing algorithms to new applications by only altering the relevant modules, eliminating the need to understand or reimplement an algorithm in its entirety. The framework is written entirely in Python and supports both novice and advanced users through clear documentation, built-in visualization, and fully transparent implementations of pedagogical algorithms. To facilitate testing and benchmarking of new algorithms, the framework features interfaces to modeling frameworks such as OpenMDAO and CSDL, interfaces to general-purpose optimization algorithms such as SNOPT and SLSQP, and an interface to the CUTEst test problem set. This level of interoperability—spanning 12 external algorithms, 10 pedagogical algorithms, 4 modeling tools, and a benchmark test set—is unique to modOpt and is not available in other optimization libraries. In this paper, we present the software architecture of modOpt, review its various features, discuss several educational and performance-oriented algorithms within modOpt, and present numerical studies illustrating its unique capabilities. modOpt is available as an open-source project on GitHub at https://github.com/lsdolab/modopt , with comprehensive documentation hosted at https://modopt.readthedocs.io/ .

Sequential quadratic programming (SQP) methods have been remarkably successful in solving a broad range of nonlinear optimization problems. These methods iteratively construct and solve quadratic programming (QP) subproblems to compute directions that converge to a local minimum. While numerous open-source and commercial SQP algorithms are available, their implementations lack the transparency and modularity necessary to adapt and fine-tune them for specific applications or to swap out different modules to create a new optimizer. To address this gap, we present OpenSQP, a modular and reconfigurable SQP algorithm implemented in Python that achieves robust performance comparable to leading algorithms. We implement OpenSQP in a manner that allows users to easily modify or replace components such as merit functions, line search procedures, Hessian approximations, and QP solvers. This flexibility enables the creation of tailored variants of the algorithm for specific needs. To demonstrate reliability, we present numerical results using the standard configuration of OpenSQP that employs a smooth augmented Lagrangian merit function for the line search and a quasi-Newton BFGS method for approximating the Hessians. We benchmark this configuration on a comprehensive set of problems from the CUTEst test suite. The results demonstrate performance that is competitive with proven nonlinear optimization algorithms such as SLSQP, SNOPT, and IPOPT.

This paper proposes a conceptual design optimization method for the powertrain of electric vertical takeoff and landing (eVTOL) aircraft. The converter selection, semiconductor device selection, inductor design, capacitor selection, and thermal management design are discussed. The detailed analysis of various losses, including semiconductor losses, magnetic component losses, and auxiliary electronics losses, is illustrated. Moreover, the hybrid optimization algorithm is utilized to address the problem of multiple local optima. A configuration study with a 30km cruise mission range is implemented to analyze the variation of each design variable in the optimization. Additionally, a comparison study is conducted to assess the impact of powertrain system design on eVTOL aircraft. The proposed method achieves up to a 4.3% reduction in gross mass and an 8.8% reduction in required battery energy, highlighting the significance of incorporating powertrain modeling in aircraft-level optimization. Finally, sensitivity analysis is presented with respect to the mission range. This paper bridges the gap in current eVTOL design methodologies from a powertrain perspective, further reducing gross mass and providing guidelines for powertrain design in eVTOL aircraft.

While proper orthogonal decomposition (POD) is widely used for model reduction, its standard form does not take into account any parametric model structure. Extensions to POD have been proposed to address this, but these either require large amounts of solution data, lack online adaptivity, or have limited approximation accuracy. We circumvent these limitations by instead assigning weights to the snapshot matrix columns, and updating these whenever the model is evaluated at a new point in the parameter space. We derive an a posteriori error bound that depends on these snapshot weights, show how these weights can be chosen to tighten the error bound, and present an algorithm to compute the corresponding reduced basis efficiently. We show how this weighted POD approach can be used to naturally generalize the calculation of reduced basis derivatives to situations with multidimensional parameter spaces and snapshots at multiple locations in the parameter space. Lastly, we cover how these approaches can be implemented within an optimization algorithm, without the need for an offline training phase. The proposed weighted POD methods with and without reduced basis derivatives are applied to a gradient-based shell thickness optimization problem with 105 design parameters and a time-dependent partial differential equation. The numerical solutions obtained for this problem attain errors that are several orders of magnitude smaller when using weighted POD than those computed with regular POD and Grassmann manifold interpolation, while having comparable wall times per query and requiring fewer high-dimensional model snapshots to reach an optimal solution.

Comprehensive large-scale multidisciplinary design optimization for aircraft design has become feasible thanks to recent advances in automatic adjoint-based sensitivity analysis. However, a major limitation of existing methods and previous demonstrations is the use of low-fidelity models to address key disciplines. Aerodynamics, one of these key disciplines, is currently restricted to very low or high fidelity methods. Panel methods bridge this gap in modeling fidelity; however, existing tools are not suited for large-scale MDO because they lack adjoint-based sensitivity analysis needed for efficient gradient computation. In addition, panel methods naturally incur a prohibitive ��2 cost. In this paper, we address these needs by presenting a modern, graph-based panel method with automatic adjoint-based sensitivity analysis. We discuss novel methods focused on reducing the prohibitive quadratic cost with respect to grid size by using a new approach we call partitioned vectorization, and present a method incorporating POD that reduces the memory cost for solving the linear system to �� (��). With this, we present results on numerical experiments comparing timing and memory for other standard approaches, showing memory improvements by up to two orders of magnitude with minimal increase in computation time. We then present a large-scale multidisciplinary design optimization under uncertainty (MDOUU) demonstration on the blended wing body aircraft concept using our graph-based panel method. We highlight the importance of robustness to input variation by identifying significant constraint violations of the MDO design in the MDOUU setting.

In this paper, we propose an iterative and distributed method for solving multidisciplinary design optimization problems using a block-coordinate-descent algorithm. The method is motivated by scenarios in which the monolithic formulation of the optimization problem either fails to reliably converge, or cannot be used at all due to implementation challenges. In these situations, we decompose the monolithic problem into multiple optimization sub-problems, each smaller and easier to solve than the original. Our work has three major contributions. First, we introduce a novel distributed architecture for multidisciplinary optimization problems that is solved using block coordinate descent. Second, we present a theoretical result that guarantees global convergence to a stationary point given an unconstrained optimization problem. Third, we demonstrate this theoretical result by solving distributed versions of a control co-design problem and two aero-structural optimization problems. In each of these examples we recover the monolithic solution, and, for the aero-structural problems, we show that distributed warm-starting improves the reliability of convergence from several random initializations.

System-level aircraft conceptual design can be viewed as a high-dimensional, multidisciplinary design optimization (MDO) problem involving potentially thousands of design variables across multiple subsystems and design conditions. Recent advances in modeling and adjoint-based sensitivity analysis have enabled the use of low-fidelity physics-based models to solve this complex problem in a computationally efficient manner using established MDO algorithms. However, integrating higher-fidelity aerodynamic simulations into comprehensive, large-scale MDO of aircraft concepts remains challenging. This is in large part due to the increased computational cost and the large geometric changes that may occur at the conceptual design stage. Since many mid- to high-fidelity aerodynamic solvers require conforming surface meshes, there is a need for a general deformation method that efficiently updates surface mesh nodes based on changes to the design geometry in a differentiable manner. In this paper, we aim to fill this need by developing such a method using a two-step process: 1) recompute intersections between components given large geometric changes and 2) deform the surface mesh using thin plate splines to conform to the updated intersections. The method is reasonably efficient and all steps are differentiable. We test our method on a simple tube-and-wing geometry for which we translate and rotate the wing with respect to the fuselage. The method performs well for all considered rotation angles (±10 degrees) with over 95% of triangles being of acceptable quality. For translations, the performance is similar up to 1.5 times the chord length. We couple the method with a panel-method aerodynamic solver, confirming its applicability in aircraft conceptual design.

Electric vertical takeoff and landing (eVTOL) aircraft have gained significant attention from both industry and academia due to their potential to transform urban transportation and promote sustainability through zero-emission propulsion. Recent work has demonstrated the effectiveness of multidisciplinary design optimization (MDO) in enhancing eVTOL performance at the conceptual design stage. However, most existing studies have focused on deterministic MDO problems neglecting the impact of real-world uncertainties. To address this gap, we formulate and demonstrate the solution of the first large-scale MDO under uncertainty (MDOUU) problem for NASA’s Lift-Plus-Cruise configuration. This problem encompasses 51 design variables and 7 independent uncertain inputs. To keep the computational cost tractable, we introduce a novel sensitivity analysis-based, shared-grid uncertainty quantification framework that (i) screens each quantity of interest (QoI) with first-order Sobol indices and (ii) evaluates the resulting low-dimensional quadrature rules on a single, unioned node set. Numerical results demonstrate that the proposed UQ strategy achieves accurate statistical estimates for all 7 MDOUU quantities of interest (QoIs), with errors all below 5% using only 9 model evaluations. This strategy enables the efficient solution of the MDOUU problem, resulting in a reduction of 10.4% in aircraft empty weight compared to a baseline MDO design that is optimized using conservative safety factors.

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 of the 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 5e-3 and an average surface tangency error of less than 2°. 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.

This paper presents a new method for computing the cross-sectional stiffness matrix of prismatic beam structures from non-matching, overlapping finite element meshes. The approach introduces a variational coupling formulation that enforces interfacial traction compatibility and rigid body motion constraints between independently meshed subdomains using a mortar- style background mesh. A Nitsche-like term is employed to weakly enforce stress equilibrium across interfaces. This formulation enables robust handling of overlapping or non-conforming meshes without the need for remeshing or geometry pre-processing, a current challenge in multi-disciplinary design analysis and optimization. The method is compatible with natural, component-level parameterizations of structural features, such as spar cap thickness, and is designed to integrate directly into state of the art gradient-based optimization workflows critical for tasks like rotor blade shape optimization. Validation against a conformal reference solution demonstrates that the penalty-coupled approach achieves a relative error of less than 1% in the stiffness matrix while producing smooth, physically consistent warping functions for all six fundamental loading modes. The proposed method offers a flexible and optimization-ready framework for high-fidelity beam modeling in the design of advanced aerospace structures.

Endovascular surgeries generally rely on push-based catheters and guidewires, which require significant training to master and can still result in high stress being exerted on the anatomy, especially in tortuous paths. Because these procedures are so technically challenging to perform, many patients have limited access to high-quality treatment. Although various robotic systems have been developed to enhance navigation capabilities, they can also apply high stresses due to sliding against the vascular walls, impeding movement and raising the risk of vascular damage. Soft growing robots offer a promising alternative since their method of movement via eversion minimizes interaction forces with the environment and enables follow-the-leader navigation through tortuous paths. However, reliable steering of small-scale growing robots remains a significant challenge. We propose a robot architecture that combines a hydraulically-actuated, soft growing robot with a soft, tendon-driven notched continuum robot to overcome the challenges of steering for small-scale growing robots in endovascular procedures. The soft notched continuum robot successfully steers around the most difficult aortic arch type, and a 2.67 mm diameter growing robot-comparable in size to current catheters-deploys from the tip, pulling an aspiration catheter through extremely tortuous vessels. We present the design, manufacturing, and control of the notched continuum robot, growing robot, and proximal actuation subsystem. Overall, this robotic architecture facilitates active steering in proximal anatomy and navigation in tortuous distal vessels, with potential to reduce procedure times and expand access to care.

Electric vertical takeoff and landing (eVTOL) aircraft have gained widespread attention in urban air mobility services recently. In the design of eVTOL aircraft, the powertrain system significantly impacts its performance and cost. This paper systematically describes a conceptual design and optimization methodology for eVTOL aircraft from the powertrain perspective, emphasizing its influence on overall aircraft performance and cost. The powertrain modeling methodology is achieved by considering a set of converter topologies and cutting-edge component technology, including wide-bandgap semiconductor devices, high-power-density passive components, and thermal management. Apart from that, a hybrid optimization algorithm is used in this paper to facilitate a rapid global search. Los Angeles has been selected as the example city for this study. A case study with a 30 km cruise mission range is conducted, analyzing the variation of each design variable within the optimization process. Moreover, a comparative study is implemented to illustrate the effects of the powertrain system on the eVTOL aircraft. Finally, a sensitivity analysis of the cruise mission range is presented. This paper addresses the gap in current eVTOL design methodologies from the powertrain perspective, further reducing the mission cost of eVTOL aircraft within the revenue mission profile.

Forward uncertainty quantification in dynamical systems is challenging due to non-smooth or locally oscillating nonlinear behaviors. Spline dimensional decomposition (SDD) addresses such nonlinearity by partitioning input coordinates via knot placement, but its accuracy is highly sensitive to internal knot locations. Optimizing knots using sequential quadratic programming is effective, yet computationally expensive. We propose a computationally efficient, interpolation-based method for optimal knot selection in SDD. The method includes: (1) interpolating input-output profiles, (2) defining subinterval-based reference regions, and (3) selecting knots at maximum gradient points within each region. The resulting knot vector is then applied to SDD for accurate approximation of non-smooth and oscillatory responses. A modal analysis of a lower control arm shows that SDD with the proposed knots yields higher accuracy than SDD with uniformly or randomly spaced knots and a Gaussian process model. In this example, the proposed SDD achieves the lowest relative variance error (2.89%) for the first natural frequency distribution, compared to uniformly spaced knots (12.310%), randomly spaced knots (15.274%), and Gaussian process (5.319%). All surrogates are constructed using the same 401 simulation datasets, and errors are evaluated against a 2000-sample Monte Carlo simulation. Scalability and applicability are demonstrated through stochastic and reliability analyses of one- and three-dimensional benchmark functions, and a ten-dimensional lower control arm model. Results confirm that second-moment statistics and reliability estimates can be accurately obtained with only a few hundred function evaluations or finite element simulations.

Anguilliform-swimming soft fluidic robots hold great promise for a range of underwater applications. However, because they leverage the complex dynamics of soft bodies interacting with fluids, it is challenging to use intuition to determine design parameters. Multidisciplinary design optimization offers a promising solution to this challenge by providing an automated and systematic method for leveraging computational models to find optimal design parameters. This study investigates a method for multidisciplinary control co-design optimization of anguilliform-swimming soft robots, using physics-based models to optimize shape and control parameters. The modeling framework includes a geometry-centric approach for geometry modeling and parameterization, a three-dimensional finite element model for structural mechanics, and an unsteady panel method for fluid dynamics. The approach is tested by applying it to the optimization of a pre-existing eel-inspired soft robot. Model parameters are estimated from existing experimental data, and two control co-design optimizations, with high-level and multilevel shape parameterizations, are performed to minimize energy cost. The optimized actuator module is manufactured, and used to collect additional data for re-estimating the structural model parameters. The optimization is performed again with the updated model parameters, showing a simulated energy cost reduction of 45% and the prescribed 128% speed increase compared to the baseline design with optimized control and updated model parameters. These results demonstrate the potential of the proposed optimization approach to advance the performance of anguilliform-swimming soft fluidic robots.

Abstract We present an open-source Python framework for the shape optimization of complex shell structures using isogeometric analysis (IGA). IGA seamlessly integrates computer-aided design (CAD) and analysis models by employing non-uniform rational B-splines (NURBS) as basis functions, enabling the natural implementation of the Kirchhoff–Love shell model due to their higher order of continuity. We leverage the recently developed FEniCS-based analysis framework, PENGoLINS, for the direct structural analysis of shell structures consisting of a collection of NURBS patches through a penalty-based formulation. This contribution introduces the open-source implementation of gradient-based shape optimization for isogeometric Kirchhoff–Love shells with a modular architecture. Complex shell structures with non-matching intersections are handled using a free-form deformation (FFD) approach and a moving intersections formulation. The symbolic differentiation and code generation capabilities in FEniCS are utilized to compute the analytical derivatives. By integrating FEniCS with OpenMDAO, we build modular components that facilitate gradient-based shape optimization of shell structures. The modular architecture in this work supports future extensions and integration with other disciplines and solvers, making it highly customizable and suitable for a wide range of applications. We validate the design-analysis-optimization workflow through several benchmark problems and demonstrate its application to aircraft wing design optimization. The framework is implemented in a Python library named GOLDFISH (Gradient-based Optimization and Large-scale Design Framework for Isogeometric SHells) and the source code will be maintained at https://github.com/hanzhao2020/GOLDFISH .

Wind gusts, being inherently stochastic, can significantly influence the safety and performance of aircraft. This study investigates a three-dimensional uncertainty quantification (UQ) problem to explore how uncertainties in gust and flight conditions affect the structural response of a Lift-Plus-Cruise eVTOL aircraft wing. The analysis employs an unsteady aeroelastic model with a one-way coupling between a panel method aerodynamic solver and a shell analysis structural solver to predict the wing's response under varying conditions. Additionally, this paper presents a comparative evaluation of commonly used non-intrusive UQ methods, including non-intrusive polynomial chaos, kriging, Monte Carlo, univariate dimension reduction, and gradient-enhanced univariate dimension reduction. These methods are assessed based on their effectiveness in estimating various risk measures-mean, standard deviation, and 95th percentile-of critical structural response outputs such as maximum tip displacement and average strain energy. The numerical results reveal significant variability in the structural response outputs, even under relatively small ranges of uncertain inputs. This highlights the sensitivity of the system to uncertainties in gust and flight conditions. Furthermore, the performance of the implemented UQ methods varies significantly depending on the specific risk measures and the quantity of interest being analyzed.

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 thermo-mechanical 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.

Wind gusts, being inherently stochastic, can significantly influence the safety and performance of aircraft. This study investigates a three-dimensional uncertainty quantification (UQ) problem to explore how uncertainties in gust and flight conditions affect the structural response of a Lift-Plus-Cruise eVTOL aircraft wing. The analysis employs an unsteady aeroelastic model with a one-way coupling between a panel method aerodynamic solver and a shell analysis structural solver to predict the wing's response under varying conditions. Additionally, this paper presents a comparative evaluation of commonly used non-intrusive UQ methods, including non-intrusive polynomial chaos, kriging, Monte Carlo, univariate dimension reduction, and gradient-enhanced univariate dimension reduction. These methods are assessed based on their effectiveness in estimating various risk measures—mean, standard deviation, and 95th percentile—of critical structural response outputs such as maximum tip displacement and average strain energy. The numerical results reveal significant variability in the structural response outputs, even under relatively small ranges of uncertain inputs. This highlights the sensitivity of the system to uncertainties in gust and flight conditions. Furthermore, the performance of the implemented UQ methods varies significantly depending on the specific risk measures and the quantity of interest being analyzed.

Aircraft conceptual design is a high-dimensional, multidisciplinary design optimization (MDO) problem. It involves a large number of design variables (~10-1000s), multiple disciplines, and diverse operating conditions, including nominal on-design and off-design scenarios such as structural sizing and system failure modes. Recent advancements in modeling and adjoint-based sensitivity analysis have facilitated the use of low- to mid-fidelity physics-based models to achieve computationally efficient solutions for this complex, large-scale MDO problem. However, employing higher-fidelity simulations, such as CFD-based aerodynamic analysis, for large-scale MDO of aircraft concepts remains challenging. This is largely due to the high computational cost and the complex mesh movement requirements associated with significant design geometry changes, particularly at the conceptual design stage. In this paper, we propose initial steps toward a mesh movement algorithm designed to bridge the gap between high-fidelity aerodynamic analysis and large-scale MDO of aircraft concepts using mixed-fidelity system models. The algorithm minimizes the projection error volume between the mesh and the central geometry while incorporating a non-uniformity penalty to maintain mesh quality. Additionally, it computes intersection curves between intersecting geometries, ensuring that the mesh conforms to the outer hull of the geometry. We demonstrate the proposed algorithm on a simple test case involving a wing and fuselage configuration.

Soft robotic fish have significant potential for underwater applications, with eel-inspired robots offering notable advantages due to the efficiency of their anguilliform swimming motion. This efficiency enables extended range and endurance, making them ideal for various missions. However, designing eel-inspired robots poses significant challenges due to the nonlinear, dynamic fluid-structure interactions (FSI) that drive system performance. Multidisciplinary design optimization (MDO) offers a systematic approach to exploring this unintuitive design space, though few studies have applied MDO to eel-inspired soft robots. While there have been many studies exploring multidisciplinary design optimization with fluid-structure interaction, gradient-based design optimizations with dynamic fluid-structure interaction remains relatively unexplored. A previous study investigated a method for optimizing eel-inspired robots using a static structural model and a dynamic fluids model. While promising, mapping the static structural solution to the dynamic fluids mesh introduces an unquantified amount of modeling error. This study seeks to build on the prior work by investigating the method of directly modeling the dynamic hydroelasticity for shape optimization. Furthermore, this work seeks to demonstrate dynamic hydroelastic shape optimization with analytic unsteady adjoint computation. The method applies a geometry-centric approach, dynamic Euler-Bernoulli beam theory to model structural dynamics, and an unsteady panel method to model the fluid dynamics. Additionally, the models are implemented using a graph-based modeling language to automate the unsteady adjoint computation. The proposed method is applied to optimize the efficiency of an existing modular, eel-inspired soft robot. Furthermore, the method explored in previous work is applied to the same optimization result to investigate the impact of the difference in modeling approach. The presented method shows a 73.9\% decrease in cost of transport and 130\% increase in swim speed compared to a control optimization of the baseline design. When compared to the optimization result using the static structural model, the optimal designs exhibit similar design trends but significant differences in the scale of the final design. These findings demonstrate the feasibility of automated unsteady hydroelastic adjoint computation and highlight the trade-offs between static and dynamic modeling fidelity in optimizing soft robotic systems.

Joshy et al., (2024). PySLSQP: A transparent Python package for the SLSQP optimization algorithm modernized with utilities for visualization and post-processing. Journal of Open Source Software, 9(103), 7246, https://doi.org/10.21105/joss.07246

Careful design of semiconductor manufacturing equipment is crucial for ensuring the performance, yield, and reliability of semiconductor devices. Despite this, numerical optimization methods are seldom applied to optimize the design of such equipment due to the difficulty of obtaining accurate simulation models. In this paper, we address a practical and industrially relevant electrostatic chuck (ESC) design optimization problem by proposing a novel multi-fidelity surrogate modeling approach. The optimization aims to improve the temperature uniformity of the wafer during the etching process by adjusting seven parameters associated with the coolant path and embossing. Our approach combines low-fidelity (LF) and high-fidelity (HF) simulation data to efficiently predict spatial-field quantities, even with a limited number of data points. We use proper orthogonal decomposition (POD) to project the spatially interpolated HF and LF field data onto a shared latent space, followed by the construction of a multi-fidelity kriging model to predict the latent variables of the HF output field. In the ESC design problem, with hundreds or fewer data, our approach achieves a more than 10% reduction in prediction error compared to using kriging models with only HF or LF data. Additionally, in the ESC optimization problem, our proposed method yields better solutions with improvements in all of the quantities of interest, while requiring 20% less data generation cost compared to the HF surrogate modeling approach.

Recent advances in computing hardware and modeling software have given rise to new applications for numerical optimization. These new applications occasionally uncover bottlenecks in existing optimization algorithms and necessitate further specialization of the algorithms. However, such specialization requires expert knowledge of the underlying mathematical theory and the software implementation of existing algorithms. To address this challenge, we present modOpt, an open-source software framework that facilitates the construction of optimization algorithms from modules. The modular environment provided by modOpt enables developers to tailor an existing algorithm for a new application by only altering the relevant modules. modOpt is designed as a platform to support students and beginner developers in quickly learning and developing their own algorithms. With that aim, the entirety of the framework is written in Python, and it is well-documented, well-tested, and hosted open-source on GitHub. Several additional features are embedded into the framework to assist both beginner and advanced developers. In addition to providing stock modules, the framework also includes fully transparent implementations of pedagogical optimization algorithms in Python. To facilitate testing and benchmarking of new algorithms, the framework features built-in visualization and recording capabilities, interfaces to modeling frameworks such as OpenMDAO and CSDL, interfaces to general-purpose optimization algorithms such as SNOPT and SLSQP, an interface to the CUTEst test problem set, etc. In this paper, we present the underlying software architecture of modOpt, review its various features, discuss several educational and performance-oriented algorithms within modOpt, and present numerical studies illustrating its unique benefits.

We present an open-source Python framework for the shape optimization of complex shell structures using isogeometric analysis (IGA). IGA seamlessly integrates computer-aided design (CAD) and analysis models by employing non-uniform rational B-splines (NURBS) as basis functions, enabling the natural implementation of the Kirchhoff--Love shell model due to their higher order of continuity. We leverage the recently developed FEniCS-based analysis framework, PENGoLINS, for the direct structural analysis of shell structures consisting of a collection of NURBS patches through a penalty-based formulation. This contribution introduces the open-source implementation of gradient-based shape optimization for isogeometric Kirchhoff--Love shells with a modular architecture. Complex shell structures with non-matching intersections are handled using a free-form deformation (FFD) approach and a moving intersections formulation. The symbolic differentiation and code generation capabilities in FEniCS are utilized to compute the analytical derivatives. By integrating FEniCS with OpenMDAO, we build modular components that facilitate gradient-based shape optimization of shell structures. The modular architecture in this work supports future extensions and integration with other disciplines and solvers, making it highly customizable and suitable for a wide range of applications. We validate the design-analysis-optimization workflow through several benchmark problems and demonstrate its application to aircraft wing design optimization. The framework is implemented in a Python library named GOLDFISH (Gradient-based Optimization and Large-scale Design Framework for Isogeometric SHells) and the source code will be maintained at https://github.com/hanzhao2020/GOLDFISH.

Recently, the graph-accelerated non-intrusive polynomial chaos (NIPC) method has been proposed for solving uncertainty quantification (UQ) problems. This method leverages the full-grid integration-based NIPC method to address UQ problems while employing the computational graph transformation approach, AMTC, to accelerate the tensor-grid evaluations. This method exhibits remarkable efficacy on a broad range of low-dimensional (three dimensions or less) UQ problems featuring multidisciplinary models. However, it often does not scale well with problem dimensions due to the exponential increase in the number of quadrature points when using the full-grid quadrature rule. To expand the applicability of this method to a broader range of UQ problems, this paper introduces a new framework for generating a tailored, partially tensor-structured quadrature rule to use with the graph-accelerated NIPC method. This quadrature rule, generated through the designed quadrature approach, possesses a tensor structure that is tailored for the computational model. The selection of the tensor structure is guided by an analysis of the computational graph, ensuring that the quadrature rule effectively capitalizes on the sparsity within the computational graph when paired with the AMTC method. This method has been tested on one 4D and one 6D UQ problem, both originating from aircraft design scenarios and featuring multidisciplinary models. Numerical results show that, when using with graph-accelerated NIPC method, our approach generates a partially tensor-structured quadrature rule that outperforms the full-grid Gauss quadrature and the designed quadrature methods (more than 40% reduction in computational costs) in both of the test problems.

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.

Concentric tube robots (CTRs) are well-suited to address the unique challenges of minimally invasive surgical procedures due to their small size and ability to navigate highly constrained environments. However, uncertainties in the manufacturing process can lead to challenges in the transition from simulated designs to physical robots. In this work, we propose an end-to-end design workflow for CTRs that considers the often-overlooked impact of manufacturing uncertainty, focusing on two primary sources - tube curvature and diameter. This comprehensive approach incorporates a two-step design optimization and an uncertainty-based selection of manufacturing tolerances. Simulation results highlight the substantial influence of manufacturing uncertainties, particularly tube curvature, on the physical robot's performance. By integrating these uncertainties into the design process, we can effectively bridge the gap between simulation and real-world performance. Two hardware experiments validate the proposed CTR design workflow. The first experiment confirms that the performance of the physical robot lies within the simulated probability distribution from the optimization, while the second experiment demonstrates the feasibility of the overall system for use in micro-laryngeal surgical tasks. This work not only contributes to a more comprehensive understanding of CTR design by addressing manufacturing uncertainties, but also creates a new framework for robust design, as illustrated in the context of micro-laryngeal surgery.

While shape optimization using isogeometric shells exhibits appealing features by integrating design geometries and analysis models, challenges arise when addressing computer-aided design (CAD) geometries comprised of multiple non-uniform rational B-splines (NURBS) patches, which are common in practice. The intractability stems from surface intersections within these CAD models. In this paper, we develop an approach for shape optimization of non-matching isogeometric shells incorporating intersection movement. Separately parametrized NURBS surfaces are modeled using Kirchhoff–Love shell theory and coupled using a penalty-based formulation. The optimization scheme allows shell patches to move without preserving relative location with other members during the shape optimization. This flexibility is achieved through an implicit state function, and analytical sensitivities are derived for the relative movement of shell patches. The introduction of differentiable intersections expands the design space and overcomes challenges associated with large mesh distortion, particularly when optimal shapes involve significant movement of patch intersections in physical space. Throughout optimization iterations, all members within the shell structures maintain the NURBS geometry representation, enabling efficient integration of analysis and design models. The optimization approach leverages the multilevel design concept by selecting a refined model for accurate analysis from a coarse design model while maintaining the same geometry. We adopt several example problems to verify the effectiveness of the proposed scheme and demonstrate its applicability to the optimization of the internal stiffeners of an aircraft wing.

PySLSQP is a seamless interface for using the SLSQP algorithm from Python. It wraps the original SLSQP Fortran code sourced from the SciPy repository and provides a host of new features to improve the research utility of the original algorithm. Some of the additional features offered by PySLSQP include auto-generation of unavailable derivatives using finite differences, independent scaling of the problem variables and functions, access to internal optimization data, live-visualization, saving optimization data from each iteration, warm/hot restarting of optimization, and various other utilities for post-processing.