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At least 469 records · Page 26

All sources NASA NTRS DOE OSTI

DOE OSTI2023

Efficient Bayesian inference with latent Hamiltonian neural networks in No-U-Turn Sampling

When sampling for Bayesian inference, one popular approach in the computational field is to use Hamiltonian Monte Carlo (HMC) and specifically the No-U-Turn Sampler (NUTS), which automatically decides the end time of the Hamiltonian trajectory. However, HMC and NUTS can require numerous numerical gradients of the target density and can prove slow in practice when relying on computationally expensive forward models. We propose Latent Hamiltonian neural networks (L-HNNs) with HMC and NUTS for solving Bayesian inference problems. Once trained, L-HNNs do not require numerical gradients of the target density during sampling, and hence numerous evaluations of the forward computational model. Moreover, L-HNNs satisfy important properties such as perfect time reversibility and Hamiltonian conservation, making them well-suited for use within HMC and NUTS because stationarity can be shown. We also propose the integration of L-HNNs in an online error monitoring scheme, in which numerical gradients of the target density are used for a few samples whenever the L-HNNs prediction errors are large. This online error monitor scheme prevents sample degeneracy in regions of low probability density and ensures robust uncertainty quantification. We demonstrate L-HNNs in NUTS with online error monitoring on several analytical examples involving complex, heavy-tailed, and high-local-curvature probability densities. We then demonstrate the applicability of L-HNNs in NUTS to two computational case studies, namely the Allen-Cahn stochastic partial differential equation and an elliptic partial differential equation with 25 and 50 inference parameters, respectively. Overall, the L-HNNs in NUTS with online error monitoring satisfactorily inferred these probability densities. In conclusion, compared to traditional NUTS, L-HNNs in NUTS with online error monitoring required 1–2 orders of magnitude fewer numerical gradients of the target density and improved the effective sample size (ESS) per gradient (which is a measure of both the sampling quality and the computational expense) by an order of magnitude.

97 MATHEMATICS AND COMPUTING↗

NASA NTRS1992

Fluid-flow of a row of jets in crossflow - A numerical study

A detailed computer-visualized flow field of a row of jets in a confined crossflow is presented. The Reynolds averaged Navier-Stokes equations are solved using a finite volume method that incorporates a partial differential equation for incremental pressure to obtain a divergence-free flow field. The turbulence is described by a multiple-time-scale turbulence model. The computational domain includes the upstream region of the circular jet so that the interaction between the jet and the crossflow is simulated accurately. It is shown that the row of jets in the crossflow is characterized by a highly complex flow field that includes a horse-shoe vortex and two helical vortices whose secondary velocity components are co-rotating in space. It is also shown that the horse-shoe vortex is a ring of reversed flows located along the circumference of the jet exit.

Kim, S.-W.↗

NASA NTRS1984

A numerical scheme for the identification of hybrid systems describing the vibration of flexible beams with tip bodies

A cubic spline based Galerkin-like method is developed for the identification of a class of hybrid systems which describe the transverse vibration to flexible beams with attached tip bodies. The identification problem is formulated as a least squares fit to data subject to the system dynamics given by a coupled system of ordnary and partial differential equations recast as an abstract evolution equation (AEE) in an appropriate infinite dimensional Hilbert space. Projecting the AEE into spline-based subspaces leads naturally to a sequence of approximating finite dimensional identification problems. The solutions to these problems are shown to exist, are relatively easily computed, and are shown to, in some sense, converge to solutions to the original identification problem. Numerical results for a variety of examples are discussed.

Rosen, I. G.↗

NASA NTRS1976

Finite element solution of turbulent and reacting compressible three-dimensional flows

Results of detailed numerical evaluation of turbulent and reacting three-dimensional compressible flows are presented. These solutions were obtained using a finite element algorithm to solve the system of explicitly non-linear partial differential equations governing three-dimensional, confined unidirectional turbulent flow of a compressible, reacting fluid. Assessments are made of solution accuracy and convergence, including detailed comparisons to experimental data.

Baker, A. J.↗

NASA NTRS1990

Distributed parameter estimation for large beam-like structures for general inputs

In this paper, maximum likelihood estimation for distributed parameter models of large flexible structures has been formulated. Distributed parameter models involve far fewer unknown parameters than independent modal characteristics of finite element models. The closed-form solutions for the partial differential equations with general forced inputs have been derived. The closed-form expressions of the sensitivity functions lead to highly efficient algorithms for analyzing ground or on-orbit test results. Numerical simulations with impulse and step inputs to the NASA Mini-MAST trust have been demonstrated. The estimations of modal properties involve its longitudinal elongation modes, lateral bending modes and torsional modes. The results show that distributed parameter models are promising in the parameter estimation of large flexible structures.

Shen, Ji-Yao↗

DOE OSTI2020

Aeroelastic oscillations of a pitching flexible wing with structural geometric nonlinearities: Theory and numerical simulation

In this paper, we focus on the derivation of an analytical model of the aeroelastic dynamics of an elastically mounted flexible wing. This equations of motion obtained serve to help understand the behaviour of the aeroelastic wind tunnel setup in question, which consists of a rectangular wing with a uniform NACA 0012 airfoil profile, whose base is free to rotate rigidly about a longitudinal axis. Of particular interest are the structural geometric nonlinearities primarily introduced by the coupling between the rigid body pitch degree-of-freedom and the continuous system. A coupled system of partial differential equations (PDEs) coupled with an ordinary differential equation (ODE) describing axial-bending-bending-torsion-pitch motion is derived using Hamilton's principle. A finite dimensional approximation of the system of coupled differential equations is obtained using the Galerkin method, leading to a system of coupled nonlinear ODEs. Subsequently, these nonlinear ODEs are solved numerically using Houbolt's method. The results that are obtained are verified by comparison with the results obtained by direct integration of the equations of motion using a finite difference scheme. Adopting a linear unsteady aerodynamic model, it is observed that the system undergoes coalescence flutter due to coupling between the rigid body pitch rotation dominated mode and the first flapwise bending dominated mode. Finally, the behaviour of the limit cycle oscillations is primarily influenced by the structural geometric nonlinear terms in the coupled system of PDEs and ODE.

42 ENGINEERING↗

NASA NTRS1999

Proper Orthogonal Decomposition in Optimal Control of Fluids

In this article, we present a reduced order modeling approach suitable for active control of fluid dynamical systems based on proper orthogonal decomposition (POD). The rationale behind the reduced order modeling is that numerical simulation of Navier-Stokes equations is still too costly for the purpose of optimization and control of unsteady flows. We examine the possibility of obtaining reduced order models that reduce computational complexity associated with the Navier-Stokes equations while capturing the essential dynamics by using the POD. The POD allows extraction of certain optimal set of basis functions, perhaps few, from a computational or experimental data-base through an eigenvalue analysis. The solution is then obtained as a linear combination of these optimal set of basis functions by means of Galerkin projection. This makes it attractive for optimal control and estimation of systems governed by partial differential equations. We here use it in active control of fluid flows governed by the Navier-Stokes equations. We show that the resulting reduced order model can be very efficient for the computations of optimization and control problems in unsteady flows. Finally, implementational issues and numerical experiments are presented for simulations and optimal control of fluid flow through channels.

Ravindran, S. S.↗

DOE OSTI2025

Gradient flow based phase-field modeling using separable neural networks

Allen–Cahn equation is a reaction–diffusion equation and is widely used for modeling phase separation. Machine learning methods for solving the Allen–Cahn equation in its strong form suffer from inaccuracies in collocation techniques, errors in computing higher-order spatial derivatives, and the large system size required by the space–time approach. To overcome these challenges, we propose solving the gradient flow of the Ginzburg–Landau free energy functional, which is equivalent to the Allen–Cahn equation, thereby avoiding the second-order spatial derivatives associated with the Allen–Cahn equation. A minimizing movement scheme is employed to solve the gradient flow problem, eliminating the complexities of a space–time approach. We utilize a separable neural network that efficiently represents the phase field through low-rank tensor decomposition. As we use the minimizing movement scheme to numerically solve the gradient flow problem, we thus, refer to the proposed method as the Separable Deep Minimizing Movement (SDMM) method. The evaluation of the functional in the minimizing movement scheme using the Gauss quadrature technique bypasses the inaccuracies associated with collocation techniques traditionally used to solve partial differential equations. A hyperbolic tangent transformation is introduced on the phase field prior to the evaluation of the functional to ensure that it remains strictly bounded within the values of the two phases. For this transformation, theoretical guarantee for energy stability of the minimizing movement scheme is established. Our results suggest that this transformation helps to improve the accuracy and efficiency significantly. The proposed method resolves the challenges faced by state-of-the-art machine learning techniques, outperforming them in both accuracy and efficiency. It is also the first machine learning method to achieve an order of magnitude speed improvement over the finite element method. In addition to its formulation and computational implementation, several case studies illustrate the applicability of the proposed method.

42 ENGINEERING↗

DOE OSTI2024

Sparse Cholesky factorization for solving nonlinear PDEs via Gaussian processes

In recent years, there has been widespread adoption of machine learning-based approaches to automate the solving of partial differential equations (PDEs). Among these approaches, Gaussian processes (GPs) and kernel methods have garnered considerable interest due to their flexibility, robust theoretical guarantees, and close ties to traditional methods. They can transform the solving of general nonlinear PDEs into solving quadratic optimization problems with nonlinear, PDE-induced constraints. However, the complexity bottleneck lies in computing with dense kernel matrices obtained from pointwise evaluations of the covariance kernel, and its partial derivatives, a result of the PDE constraint and for which fast algorithms are scarce. The primary goal of this paper is to provide a near-linear complexity algorithm for working with such kernel matrices. We present a sparse Cholesky factorization algorithm for these matrices based on the near-sparsity of the Cholesky factor under a novel ordering of pointwise and derivative measurements. The near-sparsity is rigorously justified by directly connecting the factor to GP regression and exponential decay of basis functions in numerical homogenization. We then employ the Vecchia approximation of GPs, which is optimal in the Kullback-Leibler divergence, to compute the approximate factor. This enables us to compute ϵ-approximate inverse Cholesky factors of the kernel matrices with complexity O(N log d (N/ϵ)) in space and O(N log 2d (N/ϵ)) in time. We integrate sparse Cholesky factorizations into optimization algorithms to obtain fast solvers of the nonlinear PDE. We numerically illustrate our algorithm’s near-linear space/time complexity for a broad class of nonlinear PDEs such as the nonlinear elliptic, Burgers, and Monge-Ampère equations. In summary, we provide a fast, scalable, and accurate method for solving general PDEs with GPs and kernel methods.

97 MATHEMATICS AND COMPUTING↗

DOE OSTI2025

Differentiable lagrangian shock hydrodynamics with application to stable shock acceleration of density interfaces

We develop a gradient based optimization approach for the equations of compressible, Lagrangian hydrodynamics and demonstrate how it can be employed to automatically uncover strategies to control hydrodynamic instabilities arising from shock acceleration of density interfaces. Strategies for controlling the Richtmyer-Meshkov instability (RMI) are of great benefit for inertial confinement fusion (ICF) where shock interactions with many small imperfections in the density interface lead to instabilities which rapidly grow over time. These instabilities lead to mixing which, in the case of laser driven ICF, quenches the runaway fusion process ruining the potential for positive energy return. Here, we demonstrate that control of these instabilities can be achieved by optimization of initial conditions with ( > 100) parameters. Optimizing over a large parameter space like this is not possible with gradient-free optimization strategies. This requires computation of the gradient of the outputs of a numerical solution to the equations of Lagrangian hydrodynamics with respect to the inputs. We show that the efficient computation of these gradients is made possible via a judicious application of (i) adjoint methods, the exact formal representation of sensitivities involving partial differential equations, and (ii) automatic differentiation (AD), the algorithmic calculation of derivatives of functions. Careful regularization of multiple operators including artificial viscosity and timestep control is required. We perform design optimization of > 100 parameter energy field driving the Richtmyer Meshkov instability showing significant suppression while simultaneously enhancing the acceleration of the interface relative to a nominal baseline case.

Hydrophysics↗

DOE OSTI2026

Neural operator transformers capture bifurcating drift-wave turbulence in fusion plasma simulations

Self-consistent modeling of turbulence-driven transport is critical for optimizing confinement in magnetically confined fusion plasmas, such as tokamaks and stellarators. In particular, capturing the long-term co-evolution of turbulence, flow, and background plasma profiles remains computationally challenging. Direct numerical simulation of these multiscale, highly nonlinear processes is often demanding and impractical for real-time control or design optimization. To address this bottleneck, we investigate transformer-based neural operator partial differential equation surrogates for emulating the dynamics of drift-wave turbulence bifurcation mediated by zonal flows, using the modified Hasegawa–Wakatani (MHW) model as a prototypical system. We find that the finetuned neural operator model has excellent performance in capturing the multi-spatiotemporal-scales of MHW turbulence bifurcation and is robust to testing on rare and out-of-distribution dynamics. Specifically, we demonstrate that a single unified model accurately predicts both quasi-steady-state turbulence and a wide range of dynamical transition processes, such as nonlinear saturation, spontaneous suppression of turbulence, and the emergence of macroscopic zonal flows, over time horizons vastly exceeding the local turbulence correlation time. This computationally efficient approach establishes a strong foundation for fast, AI-based modeling of complex, multiscale phenomena in magnetized fusion plasmas.

70 PLASMA PHYSICS AND FUSION TECHNOLOGY↗

NASA NTRS1972

Orbit improvement from satellite imaging data obtainable from the Mariner/Jupiter/Saturn 1977 missions

Equations of motion are established for a dynamic system in which a spacecraft flies close to and interacts with an outer planet and one or more of its satellites. For the computation of the state and mass partials needed in the orbit corrections, a set of variational equations is derived. These assume a notably compact form through the introduction of a matrix operator D. The above system of differential equations is integrated numerically on a computer.

Aksnes, K.↗

NASA NTRS1982

Stability of compressible three-dimensional boundary-layer flows

For compressible three-dimensional flow, the method of multiple scales to formulate the three-dimensional stability problem and determine the partial-differential equations governing variations of the amplitude and complex wavenumbers is used. A method for following one specific wave along its trajectory to ascertain the characteristics of the most unstable disturbance is proposed. Numerical results using the flow over the X-21 wing as calculated from the Kaups-Cebeci code will be presented.

Reed, H. L.↗

NASA NTRS1984

Space structure vibration modes: How many exist? Which ones are important?

This report attempts to shed some light on the two issues raised in the title, namely, how many vibration modes does a real structure have, and which of these modes are important? The surprise-free answers to these two questions are, respectively, an infinite number and the first several modes. The author argues that the absurd subspace (all but the first billion modes) is not a strength of continuum modeling, but, in fact, a weakness. Partial differential equations are not real structures, only mathematical models. This note also explains (1) that the PDE model and the finite element model are, in fact, the same model, the latter being a numerical method for dealing with the former, (2) that modes may be selected on dynamical grounds other than frequency alone, and (3) that long slender rods are useful as primitive cases but dangerous to extrapolate from.

Hughes, P. C.↗

NASA NTRS1992

Linear-stability theory of thermocapillary convection in a model of float-zone crystal growth

Linear-stability theory has been applied to a basic state of thermocapillary convection in a model half-zone to determine values of the Marangoni number above which instability is guaranteed. The basic state must be determined numerically since the half-zone is of finite, O(1) aspect ratio with two-dimensional flow and temperature fields. This, in turn, means that the governing equations for disturbance quantities will remain partial differential equations. The disturbance equations are treated by a staggered-grid discretization scheme. Results are presented for a variety of parameters of interest in the problem, including both terrestrial and microgravity cases.

Neitzel, G. P.↗

NASA NTRS1998

A Posteriori Finite Element Bounds for Sensitivity Derivatives of Partial-Differential-Equation Outputs

We present a Neumann-subproblem a posteriori finite element procedure for the efficient and accurate calculation of rigorous, 'constant-free' upper and lower bounds for sensitivity derivatives of functionals of the solutions of partial differential equations. The design motivation for sensitivity derivative error control is discussed; the a posteriori finite element procedure is described; the asymptotic bounding properties and computational complexity of the method are summarized; and illustrative numerical results are presented.

Lewis, Robert Michael↗

NASA NTRS1998

Active Control of Panel Vibrations Induced by a Boundary Layer Flow

In recent years, active and passive control of sound and vibration in aeroelastic structures have received a great deal of attention due to many potential applications to aerospace and other industries. There exists a great deal of research work done in this area. Recent advances in the control of sound and vibration can be found in the several conference proceedings. In this report we will summarize our research findings supported by the NASA grant NAG-1-1175. The problems of active and passive control of sound and vibration has been investigated by many researchers for a number of years. However, few of the articles are concerned with the sound and vibration with flow-structure interaction. Experimental and numerical studies on the coupling between panel vibration and acoustic radiation due to flow excitation have been done by Maestrello and his associates at NASA/Langley Research Center. Since the coupled system of nonlinear partial differential equations is formidable, an analytical solution to the full problem seems impossible. For this reason, we have to simplify the problem to that of the nonlinear panel vibration induced by a uniform flow or a boundary-layer flow with a given wall pressure distribution. Based on this simplified model, we have been able to study the control and stabilization of the nonlinear panel vibration, which have not been treated satisfactorily by other authors. The vibration suppression will clearly reduce the sound radiation power from the panel. The major research findings will be presented in the next three sections. In Section II we shall describe our results on the boundary control of nonlinear panel vibration, with or without flow excitation. Section III is concerned with active control of the vibration and sound radiation from a nonlinear elastic panel. A detailed description of our work on the parametric vibrational control of nonlinear elastic panel will be presented in Section IV. This paper will be submitted to the Journal of Acoustic Society of America for publication.

Chow, Pao-Liu↗

NASA NTRS1970

Dynamic characteristics of a variable-mass flexible missile

The general motion of a variable mass flexible missile with internal flow and aerodynamic forces is considered. The resulting formulation comprises six ordinary differential equations for rigid body motion and three partial differential equations for elastic motion. The simultaneous differential equations are nonlinear and possess time-dependent coefficients. The differential equations are solved by a semi-analytical method leading to a set of purely ordinary differential equations which are then solved numerically. A computer program was developed for the numerical solution and results are presented for a given set of initial conditions.

Meirovitch, L.↗