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DOE OSTI2025

Aerodynamic Sensitivities over Separable Shape Tensors

Here, we present a comprehensive aerodynamic sensitivity analysis of airfoil parameterization informed by separable shape tensors. This parameterization approach uniquely benefits the design process by isolating various well-studied shape characteristics, such as airfoil thickness, and providing a well-regulated low-dimensional parameter domain for aerodynamic designs. Exploring the aerodynamic sensitivities of this novel parameterization can provide valuable insights for more robust designs and future manufacturing efforts. We construct a data-driven parameter space of airfoils using principal geodesic analysis of separable shape tensors informed by a curated database containing almost 20,000 suitable engineering airfoils. Analyzing the shape reconstruction error and the maximum mean discrepancy between joint distributions of aerodynamic quantities, we study the dimensionality of the learned parameter space. This simple numerical experiment demonstrates a dramatic dimension reduction that retains design effectiveness and promotes regularity of the shape representations. Finally, we generate new airfoils and use the HAM2D Reynolds-averaged Navier–Stokes solver to predict lift, drag, and moment coefficients. We compute multiple sensitivity metrics to quantify and assert the consistency of parameter influence on the aerodynamic quantities. We also explore low-dimensional polynomial ridge approximations to motivate physical intuitions and offer explanations of the approximated sensitivities.

17 WIND ENERGY↗

DOE OSTI2023

Separable shape tensors for aerodynamic design

Airfoil shape design is a classical problem in engineering and manufacturing. In this work, we combine principled physics-based considerations for the shape design problem with modern computational techniques using a data-driven approach. Modern and traditional analyses of two-dimensional (2D) and three-dimensional (3D) aerodynamic shapes reveal a flow-based sensitivity to specific deformations that can be represented generally by affine transformations (rotation, scaling, shearing, and translation). We present a novel representation of shapes that decouples affine-style deformations over a submanifold and a product submanifold principally of the Grassmannian. As an analytic generative model, the separable representation, informed by a database of physically relevant airfoils, offers: (i) a rich set of novel 2D airfoil deformations not previously captured in the data, (ii) an improved low-dimensional parameter domain for inferential statistics informing design/manufacturing, and (iii) consistent 3D blade representation and perturbation over a sequence of nominal 2D shapes.

42 ENGINEERING↗

DOE OSTI2023

G2Aero: A Python package for separable shape tensors

G2Aero is a Python package for the design and deformation of discrete planar curves and tubular surfaces using a geometric data-driven approach. G2Aero utilizes a topology of product manifolds: the Grassmannian, $\mathcal{G}$($\mathcal{n}$, 2) - the set of 2-dimensional subspaces in $\mathbb{R}$ $\mathcal{n}$ - and the symmetric positive-definite (SPD) manifold, $\mathcal{S}^{2}_{++}$ - the set of 2x2 SPD matrices. The package provides a novel framework for representing separable deformations to shapes, which consist of stretching, scaling, rotating, and translating - also known as affine deformations - and a set of complementary deformations - which we refer to as undulation-type deformations. We focus on airfoil and blade design applications to emphasize the utility of the methods in an environment where the separation of affine and undulation-type deformations is critical. Notable functionalities of the framework for blade design include: 1) generating novel 2D (airfoil) shapes informed by a database of physically relevant airfoils, 2) building 3D blades by interpolating sequences of 2D airfoil cross-sections, and 3) generating blades with consistent perturbations along the blade span. We discuss the framework and provide examples in the context of wind energy applications, specifically wind turbine blade design. Figure 1 shows the wire frame obtained by interpolating airfoils defining the IEA 15-MW wind turbine blade and applying affine transformations corresponding to twist, chordal scaling, and bending. This, and all other figures in the paper, can be reproduced following examples and referencing supporting documentation provided in the G2Aero package.

97 MATHEMATICS AND COMPUTING↗

DOE OSTI2023

Aerodynamic Sensitivity of a Novel Data-Driven Airfoil Shape Representation Framework

We explore the aerodynamic implications of a novel data-driven separable shape tensor framework used to represent discrete airfoil shapes. In this study, we construct a data-driven parameter space defined by separable shape tensors and informed by tens of thousands of distinct airfoils. We use this design space to generate new airfoil designs to study parametric sensitivities with respect to various aerodynamic responses. We use a HAM2D RANS solver to approximate the lift, drag, and moment coefficients for the generated airfoils at two different angles-of-attack. We analyze the robustness and sensitivities of using the separable shape tensor design space by examining the coverage of the aerodynamic response space, uncovering low-dimensional polynomial ridge approximations, and computing various sensitivity metrics. The results show that the data-driven design space produce significant variation in target aerodynamic quantities and facilitate highly accurate approximations (R^2 > 0.96) of one- and two-dimensional structures in each aerodynamic response. This further reduces the effective dimension to enable simplified design and optimization tasks.

aerodynamics↗

DOE OSTI2024

G2Aero Database of Airfoils - Curated Airfoils

This dataset contains a curated set of 19,164 airfoil shapes from various applications and the data-driven design space of separable shape tensors (PGA space), which can be used as a parameter space for machine-learning applications focused on airfoil shapes. We constructed the airfoil dataset in two main stages. First, we identified 13 baseline airfoils from the NREL 5MW and IEA 15MW reference wind turbines. We reparameterized these shapes using least-squares fits of 8-order CST parametrizations, which involve 18 coefficients. By uniformly perturbing all 18 CST coefficients by +/-20% around each baseline airfoil, we generated 1,000 unique airfoils. Each airfoil was sampled with 1,001 shape landmarks whose x-coordinates followed a cosine distribution along the chord. This process resulted in a total of 13,000 airfoil shapes, each with 1,001 landmarks. In the second phase, we gathered additional airfoils from the extensive BigFoil database, which consolidates data from sources such as the University of Illinois Urbana-Champaign (UIUC) airfoil database, the JavaFoil database, the NACA-TR-824 database, and others. We undertook a thorough pre-processing step to filter out shapes with sparse, noisy, or incomplete data. We also removed airfoils with sharp leading edge and those exceeding our threshold for trailing edge thickness. Additionally, we thinned out the collection of NACA airfoils-- parametric sweeps of NACA airfoils with increasing thickness and camber present in BigFoil database-- by selecting every fourth step in the parameter sweeps. Finally, we regularized the airfoils by reparametrizing them with an 8-order CST parametrization (with 1,001 shape landmarks with x coordinated following cosine distribution along the chord) and removing airfoils with high reconstruction errors. This data pre-processing resulted in a set of 6,164 airfoils. In total, our curated airfoil dataset comprises 19,164 airfoils, each with 1,001 landmarks, and is stored in the curated_airfoils.npz file. Using this curated airfoil dataset, we utilized the separable shape tensors framework to develop a data-driven parameterization of airfoils based on principal geodesic analysis (PGA) of separable shape tensors. This PGA space is provided in PGAspace.npz file.

airfoils↗

DOE OSTI2023

Airfoil Computational Fluid Dynamics - 2k shapes, 25 AoA's, 3 Re numbers

This dataset contains aerodynamic quantities - including flow field values (momentum, energy, and vorticity) and summary values (coefficients of lift, drag, and momentum) - for 1,830 airfoil shapes computed using the HAM2D CFD (computational fluid dynamics) model. The airfoil shapes were designed using the separable shape tensor parameterization that encodes two-dimensional shapes as elements of the Grassmann manifold. This data-driven approach learns two independent spaces of parameter from a collection of sample airfoils. The first captures large-scale, linear perturbations, and the second defines small-scale, higher-order perturbations. For this dataset, we used the G2Aero database of over 19,000 airfoil shapes to learn a parameter space that captured a wide array of shape characteristics. We sampled airfoil designs over both parameter spaces to explore the full range of possible shape variations. The aerodynamic quantities for the generated airfoil were obtained using the HAM2D code, which is a finite-volume Reynolds-averaged Navier-Stokes (RANS) flow solver. We employ a fifth-order WENO scheme for spatial reconstruction with Roe's flux difference scheme for inviscid flux and second-order central differencing for viscous flux. A preconditioned GMRES method is applied for implicit integration. The Spalart-Allmaras 1-eq turbulence model is used for the turbulence closure, and the Medida-Baeder 2-eq transition model is applied to account for the effects of laminar turbulent transition. The airfoil grid is generated with a total of 400 points on the airfoil surface, the initial wall-normal spacing of y+ = 1, and an outer boundary located at 300 chord lengths away from the wall. The CFD simulations are performed at a freestream Mach number of 0.1, for or three different Reynolds' numbers (3M, 6M, and 9M), and for 25 angles of attack from -4 deg. to 20 deg. with 1 degree increments. Across all these various parameters, this dataset includes the results from over 250,000 CFD simulations. The simulations were performed using the Bridges-2 system at the Pittsburgh Supercomputing Center in February 2023 as part of the INTEGRATE project funded by the Advanced Research Projects Agency - Energy, in the U.S. Department of Energy. The data was collected, reformatted, and preprocessed for this OEDI submission in July 2023 under the Foundational AI for Wind Energy project funded by the U.S. Department of Energy Wind Energy Technologies Office. This dataset is intended to serve as a benchmark against which new artificial intelligence (AI) or machine learning (ML) tools may be tested. Baseline AI/ML methods for analyzing this dataset have been implemented, and a link to their repository containing those models has been provided. The .h5 data file structure can be found in the GitHub Repository resource under explore_airfoil_2k_data.ipynb.

2k↗

DOE OSTI2023

Airfoil Computational Fluid Dynamics - 9k shapes, 2 AoA's

This dataset contains aerodynamic quantities - including flow field values (momentum, energy, and vorticity) and summary values (coefficients of lift, drag, and momentum) - for 8,996 airfoil shapes, computed using the HAM2D CFD (computational fluid dynamics) model. The airfoil shapes were designed using the separable shape tensor parameterization that encodes two-dimensional shapes as elements of the Grassmann manifold. This data-driven approach learns two independent spaces of parameter from a collection of sample airfoils. The first captures large-scale, linear perturbations, and the second defines small-scale, higher-order perturbations. For this data, we used the G2Aero database of over 19,000 airfoil shapes to learn a parameter space that captured a wide array of shape characteristics. We fixed the linear deformations to be the mean over the database and sampled new shapes over a four-dimensional parameter space of higher-order perturbation. This sampling approaches allows for isolated analysis of non-linear airfoil shape deformations while holding other aspects (e.g., airfoil thickness) approximately constant. The aerodynamic quantities for the generated airfoil were obtained using the HAM2D code, which is a finite-volume Reynolds-averaged Navier-Stokes (RANS) flow solver. We employ a fifth-order WENO scheme for spatial reconstruction with Roe's flux difference scheme for inviscid flux and second-order central differencing for viscous flux. A preconditioned GMRES method is applied for implicit integration. The Spalart-Allmaras 1-eq turbulence model is used for the turbulence closure, and the Medida-Baeder 2-eq transition model is applied to account for the effects of laminar turbulent transition. The airfoil grid is generated with a total of 400 points on the airfoil surface, the initial wall-normal spacing of y+ = 1, and an outer boundary located at 300 chord lengths away from the wall. The CFD simulations are performed at a freestream Mach number of 0.1, Reynolds number of 9M, and at two angles of attack, 4 deg. and 12 deg. The simulations were performed using the Bridges-2 system at the Pittsburgh Supercomputing Center in February 2023 as part of the INTEGRATE project funded by the Advanced Research Projects Agency - Energy in the U.S. Department of Energy. The data was collected, reformatted, and preprocessed for this OEDI submission in July 2023 under the Foundational AI for Wind Energy project funded by the U.S. Department of Energy Wind Energy Technologies Office. This dataset is intended to serve as a benchmark against which new artificial intelligence (AI) or machine learning (ML) tools may be tested. Baseline AI/ML methods for analyzing this dataset have been implemented, and a link to their repository containing those models has been provided. The .h5 data file structure can be found in the GitHub Repository resource under explore_airfoil_9k_data.ipynb.

9k↗

DOE OSTI2023

Preserving Superconvergence of Spectral Elements for Curved Domains via $h$ and $p$-Geometric Refinement

Spectral element methods (SEM), extensions of finite element methods (FEM), have emerged as significant techniques for solving partial differential equations in physics and engineering. SEM can potentially deliver superior accuracy due to the potential superconvergence in nodal solutions for well-shaped tensor-product elements. However, the accuracy of SEM often degrades in complex geometries due to geometric inaccuracies near curved boundaries and the loss of superconvergence with simplicial or non-tensor-product elements. To overcome the first issue, we propose using h- and p-geometric refinement, which refines the mesh near high-curvature regions and increases the degree of geometric basis functions, respectively. We show that when using mixed-element meshes with tensor-product elements in the interior of the domain, curvature-based geometric refinement near boundaries can improve the accuracy of the interior elements by reducing pollution errors and preserving the superconvergence in nodal solutions. To address the second issue, we introduce a post-processing technique using the adaptive extended stencil finite element method (AES-FEM) to recover the accuracy near the curved boundaries. The combination of curvature-based geometric refinement and accurate post-processing offers an effective and easier-to-implement alternative to methods reliant on exact geometries. We demonstrate our techniques by solving the convection-diffusion equation in 2D and show up to two orders of magnitude of improvement in the solution accuracy, even when the elements are poorly shaped near boundaries.

97 MATHEMATICS AND COMPUTING↗

DOE OSTI2024

Preserving Superconvergence of Spectral Elements for Curved Domains

Spectral element methods (SEM), extensions of finite element methods (FEM), have emerged as significant techniques for solving partial differential equations in physics and engineering. SEM can potentially deliver superior accuracy due to the potential superconvergence in nodal solutions for well-shaped tensor-product elements. However, the accuracy of SEM often degrades in complex geometries due to geometric inaccuracies near curved boundaries and the loss of superconvergence with simplicial or non-tensor-product elements. To overcome the first issue, we propose using geometric refinement, which both refines the mesh near high-curvature regions and increases the degree of geometric basis functions. We show that when using mixed-element meshes with tensor-product elements in the interior of the domain, curvature-based geometric refinement near boundaries can improve the accuracy of the interior elements by reducing pollution errors and preserving the superconvergence in nodal solutions. To address the second issue, we introduce ApSEM, a post-processing technique using the adaptive extended stencil finite element method (AES-FEM) to recover the accuracy near the curved boundaries. The combination of curvature-based geometric refinement and accurate post-processing offers an effective and easier-to-implement alternative to methods reliant on exact geometries. We demonstrate our techniques by solving the convection-diffusion equation in 2D and 3D and show up to two orders of magnitude of improvement in the solution accuracy, even when the elements are poorly shaped near boundaries. We also show the efficiency of ApSEM as it can recover superconvergence in nodal solutions without drastically increasing the computational cost.

97 MATHEMATICS AND COMPUTING↗

NASA NTRS1983

The shape, propagation and mean-flow interaction of large-scale weather systems

In investigations concerned with obtaining an understanding of the general circulation of the atmosphere, the determination of the role of the large-scale eddies presents a special problem. In the present study,attention is given to a theory which, in some ways, provides an extension of the Eliassen-Palm flux concept so that it can be applied in particular to the time-averaged three space dimension problem. Particular emphasis is given to the understanding of the feedback of the eddies onto the mean flow. However, the behavior of the eddies themselves is also discussed. It is shown that, in simple situations, the anisotropic eddy horizontal velocity correlation tensor implies the shape and propagation of eddies and the feedback of the eddies onto the mean flow.

Hoskins, B, J.↗

DOE OSTI2024

A Variable Eddington Factor Model for Thermal Radiative Transfer with Closure Based on Data-Driven Shape Function

Here, a new variable Eddington factor (VEF) model is presented for nonlinear problems of thermal radiative transfer (TRT). The VEF model is data-driven and acts on known (a-priori) radiation-diffusion solutions for material temperatures in the TRT problem. A linear auxiliary problem is constructed for the radiative transfer equation (RTE) whose emission source and opacities are evaluated at these known material temperatures. The solution to this RTE approximates the specific intensity distribution in phase-space and time. It is applied as a shape function to define the Eddington tensor for the presented VEF model. The shape function computed via the auxiliary RTE problem will capture some degree of transport effects within the TRT problem. The VEF moment equations closed with this approximate Eddington tensor will thus carry with them these captured transport effects. In this study, the temperature data comes from multigroup P 1 , P 1/3 , and flux-limited diffusion radiative transfer models. The proposed VEF model can be interpreted as a transport-corrected diffusion reduced-order model. Numerical results are presented on the Fleck-Cummings test problem which models a supersonic wavefront of radiation. The VEF model is shown to improve accuracy by 1–2 orders of magnitude compared to the considered radiation-diffusion model solutions to the TRT problem.

73 NUCLEAR PHYSICS AND RADIATION PHYSICS↗

NASA NTRS1993

Microwave Scattering Model for Grass Blade Structures

In this paper, the electromagnetic scattering solution for a grass blade with complex cross-section geometry is considered. It is assumed that the blade cross section is electrically small, but its length is large compared to the incident wavelength. In a recent study it has been shown that the scattering solution for such problems, in the form of a polarizability tensor, can be obtained using the low-frequency approximation in conjunction with the method of moments. In addition, the study shows that the relationship between the polarizability tensor of a dielectric cylinder and its dielectric constant can be approximated by a simple algebraic expression. The results of this study are used to show that this algebraic approximation is valid also for cylinders with cross sections the shape of grass blades, providing that proper values am selected for each of three constants appearing in the expression. These constants are dependent on cylinder shape, and if the relationship between the constants and the three parameters describing a grass blade shape can be determined, an algebraic approximation relating polarizability tensor to blade shape, as well as dielectric constant, can be formed. Since the elements of the polarizability tensor are dependent on only these parameters, this algebraic approximation can replace the cumbersome method of moments model. A conjugate gradient method is then implemented to correctly determine the three constants of the algebraic approximation for each blade shape. A third-order polynomial fit to the data is then determined for each constant, thus providing a complete analytic replacement to the numerical (moment method) scattering model. Comparisons of this approximation to the numerical model show an average error of less than 3%.

Stiles, James M.↗

DOE OSTI2025

Kinetic study of strong shock waves in fully ionized plasmas

Fully kinetic simulations are used in this work to study strong shock waves in fully ionized plasmas for charge state 𝑍 ranging from 1 to 18. The ion stress tensor amplitude agrees well with classical predictions within the compression layer; however, its spatial structure is noticeably shaped by nonlocal transport effects. The electron stress tensor remains significantly smaller than that of the ions for all values of 𝑍 considered. Ion overheating in the compression layer is observed to scale approximately linearly with 𝑍. Both findings diverge from hydrodynamic models based on classical electron stress formulations. In conclusion, the electrons exhibit strong nonlocal behavior in the compression layer, with Knudsen numbers approaching unity for 𝑍 ≥ 6.

Physics - Plasma physics↗

NASA NTRSDate not supplied

Solar Sail Torque Model Characterization for the Near Earth Asteroid Scout Mission

Near Earth Asteroid Scout (NEA Scout) was a mission to test solar sail propulsion for orbital transfer from cislunar space to flyby and image an asteroid. Had it succeeded, one of the mission goals was to characterize the solar torque on the sail to ensure successful attitude control for the orbit transfer and imaging the asteroid. The simulation used to develop the flight attitude control software uses the generalized model for solar sails, a tensor equation of the forces and torques on sails of arbitrary shape. Rios-Reyes and Scheeres developed a general process to update the torque tensor coefficients using estimates of sail torque over a range of directions to the sun. Their process was adapted and implemented for the specific case of NEA Scout using spacecraft telemetry collected during sail characterization maneuvers in combination with simulation models and parameters. The NEA Scout maneuvers were limited to the operating range of the mission and constraints of the control hardware and allowed safe testing of each attitude before proceeding to the next. The NEA Scout reaction wheel speeds are used to measure accumulated momentum, while the Active Mass Translator (AMT) position is used to subtract out the torque from the center of mass crossed with the sail force and isolate the torque from only the sail shape. The process was tested by running attitude control simulations of the characterization maneuvers, generating simulated telemetry, estimating the solar torques, then using a least squares estimating the solar torque coefficients using least-squares and then performing a least-squares fit to the solar torque tensor coefficients. These estimated coefficients were tested by evaluating the solar torques under the same conditions as the simulated telemetry and comparing to the true simulated torques. Solar force model updates can be performed separately by observing the effect of the sail on the trajectory, and the torque model can be refined using those solar force updates. This process met the needs of the NEA Scout mission and can be adapted to characterize the solar torque for other missions with different sails.

solar sail↗

NASA NTRSDate not supplied

Solar Sail Torque Model Characterization for the Near Earth Asteroid Scout Mission

Near Earth Asteroid Scout (NEA Scout) was a mission to test solar sail propulsion for orbital transfer from cislunar space to flyby and image an asteroid. One of the goals of the mission was to characterize the solar torque on the sail to ensure successful attitude control for the orbit transfer and imaging the asteroid. The simulation used to develop the flight attitude control software uses the generalized model for solar sails, a tensor equation of the forces and torques on sails of arbitrary shape. Rios-Reyes and Scheeres developed a general process to update the torque tensor coefficients using estimates of sail torque over a range of directions to the sun. Their process was adapted and implemented for the specific case of NEA Scout using spacecraft telemetry collected during sail characterization maneuvers in combination with simulation models and parameters. The NEA Scout maneuvers were limited to the operating range of the mission and constraints of the control hardware and allowed safe testing of each attitude before proceeding to the next. The NEA Scout reaction wheel speeds are used to measure accumulated momentum, while the Active Mass Translator (AMT) position is used to subtract out the torque from the center of mass crossed with the sail force and isolate the torque from only the sail shape. The process was tested by running attitude control simulations of the characterization maneuvers, generating simulated telemetry, estimating the solar torques, then using a least squares estimating the solar torque coefficients using least-squares and then performing a least-squares fit to the solar torque tensor coefficients. These estimated coefficients were tested by evaluating the solar torques under the same conditions as the simulated telemetry and comparing to the true simulated torques. Solar force model updates can be performed separately by observing the effect of the sail on the trajectory, and the torque model can be refined using those solar force updates. This process met the needs of the NEA Scout mission and can be adapted to characterize the solar torque for other missions with different sails.

solar sail↗

NASA NTRS2003

Inertial sensor and method of use

The inertial sensor of the present invention utilizes a proof mass suspended from spring structures forming a nearly degenerate resonant structure into which a perturbation is introduced, causing a split in frequency of the two modes so that the mode shape become uniquely defined, and to the first order, remains orthogonal. The resonator is provided with a mass or inertia tensor with off-diagonal elements. These off-diagonal elements are large enough to change the mode shape of the two nearly degenerate modes from the original coordinate frame. The spring tensor is then provided with a compensating off-diagonal element, such that the mode shape is again defined in the original coordinate frame. The compensating off-diagonal element in the spring tensor is provided by a biasing voltage that softens certain elements in the spring tensor. Acceleration disturbs the compensation and the mode shape again changes from the original coordinate frame. By measuring the change in the mode shape, the acceleration is measured.

Gutierrez, Roman C.↗

NASA NTRS1984

A convective model for turbulent mixing in rotating convection zones

The effects of rotation are included in an analytical model for the convective motions in a plane-parallel layer of an ideal fluid. The turbulent stress tensor, formed by taking products and averages of the various velocity components, is calculated for an arbitrary eddy size and shape. Heuristic formulae presented for determining the size and shape of the dominant eddy then give a fully specified stress tensor. Applications for this stress tensor in problems of stellar internal dynamics, heat flow, scalar diffusion, and dynamo theory are suggested. The resultant stresses tend to produce differential rotation profiles with rapidly rotating equators and interiors. The dynamo activity associated with these convective motions tends to occur near the lower boundary of the convection zone.

Hathaway, D. H.↗

DOE OSTI2024

Resonant Ultrasound Spectroscopy for Irregularly Shaped Samples and Its Application to Uranium Ditelluride

Resonant ultrasound spectroscopy (RUS) is a powerful technique for measuring the full elastic tensor of a given material in a single experiment. Previously, this technique was practically limited to regularly shaped samples such as rectangular parallelepipeds, spheres, and cylinders [W. M. Visscher et al. J. Acoust. Soc. Am. 90, 2154 (1991)]. We demonstrate a new method for determining the elastic moduli of irregularly shaped samples, extending the applicability of RUS to a much larger set of materials. Here, we apply this new approach to the recently discovered unconventional superconductor UTe 2 and provide its elastic tensor at both 300 and 4 kelvin.

75 CONDENSED MATTER PHYSICS, SUPERCONDUCTIVITY AND↗