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NASA NTRS1994

Scaling, elasticity, and CLPT

The first few viewgraphs describe the general solution properties of linear elasticity theory which are given by the following two statements: (1) for stress B.C. on S(sub sigma) and zero displacement B.C. on S(sub u) the altered displacements u(sub i)(*) and the actual stresses tau(sub ij) are elastically dependent on Poisson's ratio nu alone: thus the actual displacements are given by u(sub i) = mu(exp -1)u(sub i)(*); and (2) for zero stress B.C. on S(sub sigma) and displacement B.C. on S(sub u) the actual displacements u(sub i) and the altered stresses tau(sub ij)(*) are elastically dependent on Poisson's ratio nu alone: thus the actual stresses are given by tau(sub ij) = E tau(sub ij)(*). The remaining viewgraphs describe the minimum parameter formulation of the general classical laminate theory plate problem as follows: The general CLT plate problem is expressed as a 3 x 3 system of differential equations in the displacements u, v, and w. The eighteen (six each) A(sub ij), B(sub ij), and D(sub ij) system coefficients are ply-weighted sums of the transformed reduced stiffnesses (bar-Q(sub ij))(sub k); the (bar-Q(sub ij))(sub k) in turn depend on six reduced stiffnesses (Q(sub ij))(sub k) and the material and geometry properties of the k(sup th) layer. This paper develops a method for redefining the system coefficients, the displacement components (u,v,w), and the position components (x,y) such that a minimum parameter formulation is possible. The pivotal steps in this method are (1) the reduction of (bar-Q(sub ij))(sub k) dependencies to just two constants Q(*) = (Q(12) + 2Q(66))/(Q(11)Q(22))(exp 1/2) and F(*) - (Q(22)/Q(11))(exp 1/2) in terms of ply-independent reference values Q(sub ij); (2) the reduction of the remaining portions of the A, B, and D coefficients to nondimensional ply-weighted sums (with 0 to 1 ranges) that are independent of Q(*) and F(*); and (3) the introduction of simple coordinate stretchings for u, v, w and x,y such that the process is neatly completed.

Brunelle, Eugene J.↗

DOE OSTI2022

Long-time integration of parametric evolution equations with physics-informed DeepONets

Ordinary and partial differential equations (ODEs/PDEs) play a paramount role in analyzing and simulating complex dynamic processes across all corners of science and engineering. In recent years machine learning tools are aspiring to introduce new effective ways of simulating such equations, however existing approaches are not able to reliably return stable and accurate predictions across long temporal horizons. We aim to address this challenge by introducing an effective framework for learning evolution operators that map random initial conditions to associated ODE/PDE solutions within a short time interval. Such operators can be parametrized by deep neural networks that are trained in an entirely self-supervised manner without requiring one to generate any paired input-output observations. Global long-time predictions across a range of initial conditions can be then obtained by iteratively evaluating the trained model using each prediction as the initial condition for the next evaluation step. Here, this introduces a new approach to temporal domain decomposition that is shown to be effective in performing accurate long-time simulations for a wide range of parametric ODE and PDE systems, from wave propagation, to reaction-diffusion dynamics and stiff chemical kinetics, introducing a new way of rapidly emulating non-equilibrium processes in science and engineering.

97 MATHEMATICS AND COMPUTING↗

DOE OSTI2021

A scalable exponential-DG approach for nonlinear conservation laws: With application to Burger and Euler equations

In this work, we propose an Exponential DG framework for partial differential equations. We decompose 7 governing equations into linear and nonlinear parts to which we apply the discontinuous Galerkin 8 (DG) spatial discretization. In particular, we construct the linear part using Jacobian that effectively 9 capture stiff characteristics in the system. The former is integrated analytically, whereas the latter 10 is approximated. This approach i) is stable with a large Courant number (Cr > 1); ii) supports 11 high-order solutions both in time and space; iii) is computationally favorable compared to IMEX 12 DG methods with no preconditioner; iv) becomes comparable to explicit RKDG methods on uniform 13 mesh and beneficial on non-uniform grid for Euler equations; v) is scalable in a modern massively 14 parallel computing architecture due to its explicit nature of exponential time integrators and com15 pact communication stencil of DG method. Numerical results demonstrate the performance of our 16 proposed methods through various examples. We also discuss the stability and convergence analysis 17 for our exponential DG scheme in the context of Burgers equation.

42 ENGINEERING↗

NASA NTRS1988

The identification of a distributed parameter model for a flexible structure

A computational method is developed for the estimation of parameters in a distributed model for a flexible structure. The structure we consider (part of the RPL experiment) consists of a cantilevered beam with a thruster and linear accelerometer at the free end. The thruster is fed by a pressurized hose whose horizontal motion effects the transverse vibration of the beam. The Euler-Bernoulli theory is used to model the vibration of the beam and treat the hose thruster assembly as a lumped or point mass dashpot spring system at the tip. Measurements of linear acceleration at the tip are used to estimate the hose parameters (mass, stiffness, damping) and a Voigt-Kelvin viscoelastic structural damping parameter for the beam using a least squares fit to the data. Spline based approximations are considered to the hybrid (coupled ordinary and partial differential equations) systems; theoretical convergence results and numerical studies with both simulation and actual experimental data obtained from the structure are presented and discussed.

Banks, H. T.↗

DOE OSTI2020

On differentiable local bounds preserving stabilization for Euler equations

This work presents the design of nonlinear stabilization techniques for the finite element discretization of Euler equations in both steady and transient form. Implicit time integration is used in the case of the transient form. A differentiable local bounds preserving method has been developed, which combines a Rusanov artificial diffusion operator and a differentiable shock detector. Nonlinear stabilization schemes are usually stiff and highly nonlinear. This issue is mitigated by the differentiability properties of the proposed method. Moreover, in order to further improve the nonlinear convergence, we also propose a continuation method for a subset of the stabilization parameters. The resulting method has been successfully applied to steady and transient problems with complex shock patterns. Numerical experiments show that it is able to provide sharp and well resolved shocks. Furthermore, the importance of the differentiability is assessed by comparing the new scheme with its non-differentiable counterpart. Numerical experiments suggest that, for up to moderate nonlinear tolerances, the method exhibits improved robustness and nonlinear convergence behavior for steady problems. Additionally, in the case of transient problem, we also observe a reduction in the computational cost.

42 ENGINEERING↗

NASA NTRS1983

Linear and Nonlinear Aspects of Rotordynamics

Excessive vibrations of the liquid oxygen pump in the Space Shuttle's Main Engine have been recorded during hot firing ground testing. In order to determine mathematical explanations of this possibility, destructive phenomenon differential equations have been examined which describe the rotordynamics of the pump. Modeling the rotor as a random eigenvalue problem was considered. Analytical expressions were derived for the solution in the case of symmetric damping and stiffness. This enables one to determine accuracy estimates when testing numerical techniques to solve both asymmetric and nonlinear problems. Finally, the rotor model has had nonlinear elements incorporated to improve its simulation of the pump and to expand the corresponding mathematical theory.

Day, W. B.↗

DOE OSTI2020

Data-driven selection of stiff chemistry ODE solver in operator-splitting schemes

Most computational fluid dynamics simulations of practical combustion applications employ operator-splitting schemes, where chemistry and transport are separated and integrated with distinct numerical methods. The changes in composition due to chemistry are evaluated by solving ordinary differential equations (ODE) in each cell of the computational domain, which typically dominates the computational cost when detailed chemistry is considered. In this work, a data-driven approach for the selection of chemistry ODE solvers in operator-splitting schemes is presented. Neural networks are used to predict the ODE solvers CPU times and errors for a given thermochemical state. This allows the selection of an optimal ODE solver on a cell-by-cell, timestep-by-timestep basis. The models are trained using a wide set of thermochemical states generated through partially-stirred reactors and flames simulations. The methodology is validated by quantifying the prediction errors, the classification accuracy, and the computational speedup. The model predicts the optimal ODE solver for 70 to 95% of the validation cases and decreases the computional cost by a factor of 3 or more. The generalizability of the methodology to different chemical mechanisms and different fuels is assessed and it is shown that the model’s performance is only slightly degraded and its applicability is significantly enhanced if the inputs to the neural networks are restricted to a small set of thermochemical state variables present in most chemical mechanisms. In conclusion, the models are used in an homogeneous reactor case and a multi-dimensional CFD simulation of a diesel spray at high pressure where a speedup of more than 3 is achieved.

42 ENGINEERING↗

DOE OSTI2023

Analysis of Neural Networks as Random Dynamical Systems

In this report we present our findings and outcomes of the NNRDS (analysis of Neural Networks as Random Dynamical Systems) project. The work is largely motivated by the analogy of a large class of neural networks (NNs) with a discretized ordinary differential equation (ODE) schemes. Namely, residual NNs, or ResNets, can be viewed as a discretization of neural ODEs (NODEs) where the NN depth plays the role of the time evolution. We employ several legacy tools from ODE theory, such as stiffness, nonlocality, autonomicity, to enable regularization of ResNets thus improving their generalization capabilities. Furthermore, armed with NN analysis tools borrowed from the ODE theory, we are able to efficiently augment NN predictions with uncertainty overcoming wellknown dimensionality challenges and adding a degree of trust towards NN predictions. Finally, we have developed a Python library QUiNN (Quantification of Uncertainties in Neural Networks) that incorporates improved-architecture ResNets, besides classical feed-forward NNs, and contains wrappers to PyTorch NN models enabling several major classes of uncertainty quantification methods for NNs. Besides synthetic problems, we demonstrate the methods on datasets from climate modeling and materials science.

97 MATHEMATICS AND COMPUTING↗

NASA NTRS1990

Dynamics of column stability with partial end restraints

The dynamic behavior of columns with partial end restraints and loads consisting of a dead load and a pulsating load are investigated. The differential equation is solved using a lumped impulse recurrence formula relative to time coupled with a finite difference discretization along the member length. A computer program is written from which the first critical frequencies are found as a function of end stiffness. The case of a pinned ended column compares very well with the exact solution. Also, the natural frequency and buckling load formulas are derived for equal and unequal end restraints.

Gregory, Peyton B.↗

NASA NTRS1984

Response of large space structures with stiffness control

For large space structures, such as the 100-meter-diameter wrap-rib deployable antenna and spinning solar sail, whose out-of-plane stiffness is derived from in-plane tension, the out-of-plane motion can be actively controlled by time-varying in-plane tension. An elastic string is used to demonstrate the proposed approach, which results in a nonlinear ordinary differential equation. An approximation method is outlined from which the magnitude of time-varying tension can be determined based on the efficiency factor, the time factor or the optimal factor.

Chen, J.-C.↗

NASA NTRS1995

Reliability enhancement of Navier-Stokes codes through convergence acceleration

Methods for enhancing the reliability of Navier-Stokes computer codes through improving convergence characteristics are presented. The improving of these characteristics decreases the likelihood of code unreliability and user interventions in a design environment. The problem referred to as a 'stiffness' in the governing equations for propulsion-related flowfields is investigated, particularly in regard to common sources of equation stiffness that lead to convergence degradation of CFD algorithms. Von Neumann stability theory is employed as a tool to study the convergence difficulties involved. Based on the stability results, improved algorithms are devised to ensure efficient convergence in different situations. A number of test cases are considered to confirm a correlation between stability theory and numerical convergence. The examples of turbulent and reacting flow are presented, and a generalized form of the preconditioning matrix is derived to handle these problems, i.e., the problems involving additional differential equations for describing the transport of turbulent kinetic energy, dissipation rate and chemical species. Algorithms for unsteady computations are considered. The extension of the preconditioning techniques and algorithms derived for Navier-Stokes computations to three-dimensional flow problems is discussed. New methods to accelerate the convergence of iterative schemes for the numerical integration of systems of partial differential equtions are developed, with a special emphasis on the acceleration of convergence on highly clustered grids.

Merkle, Charles L.↗

NASA NTRS1981

A Galerkin type finite element method for rotary-wing aeroelasticity in hover and forward flight

A Galerkin finite element method for the spatial discretization of the nonlinear, nonselfadjoint, partial differential equations governing rotary-wing aeroelasticity is presented. This method reduces algebraic manipulative labor significantly when compared to the global Galerkin method based on assumed modes. Furthermore, the Galerkin finite element method is ideally suited to treat rotor blades with discontinuous mass and stiffness distribution and structurally redundant configurations as they appear in bearingless rotors. Implementation of the method is illustrated for the coupled flap-lag aeroelastic problem of hingeless rotor blades in hover and forward flight. Numerical results for stability and response illustrate the numerical properties and convergence behavior of the method. It is concluded that the Galerkin finite element method is a practical tool for solving rotary-wing aeroelastic stability and response problems.

Straub, F. K.↗

DOE OSTI2023

Using Co-Optimized Machine Learned Manifolds for Modeling Chemically Reacting Flows

Chemically reacting flows play a key role in a wide range of engineered systems, from chemical and polymer processing to combustion-based energy conversion technologies. Simulations of these flows involve solving a coupled set of partial differential equations for mass, momentum, energy, and all relevant chemical species in the system. Chemical reaction pathways may be extremely complex and involve hundreds or more intermediate species, with reactions that occur over timescales varying by several orders of magnitude - presenting a significant numerical stiffness challenge. The combination of these factors makes simulation of chemically reacting flows vastly more expensive than nonreactive simulations, and often makes direct solution of the governing equations intractable. It is necessary to apply lower-fidelity models in place of the detailed governing equations in order to reduce computational cost to enable reacting flow simulation tools to be used in the engineering design process. Many of the models employed for this purpose are based on reducing the dimension of the thermochemical state, motivated by the observation that the observed thermochemical states in a system lie on a low-dimensional manifold in thermochemical state space. This behavior occurs due to the fast equilibration of certain reactive and transport processes, and physics-based manifold models rely on idealized assumptions about the balance of timescales and the way in which chemistry and transport are coupled. In this work, we apply a novel method for data-driven manifold-based modeling that can leverage data from high-fidelity reacting flow simulations to improve model accuracy in cases where the physics-based modeling assumptions break down. The approach is designed to be broadly applicable across chemically reacting flow systems but is applied here to turbulent combustion modeling.

machine learning↗

NASA NTRS1983

Application of the method of lines for solutions of the Navier-Stokes equations using a nonuniform grid distribution

The feasibility of the method of lines for solutions of physical problems requiring nonuniform grid distributions is investigated. To attain this, it is also necessary to investigate the stiffness characteristics of the pertinent equations. For specific applications, the governing equations considered are those for viscous, incompressible, two dimensional and axisymmetric flows. These equations are transformed from the physical domain having a variable mesh to a computational domain with a uniform mesh. The two governing partial differential equations are the vorticity and stream function equations. The method of lines is used to solve the vorticity equation and the successive over relaxation technique is used to solve the stream function equation. The method is applied to three laminar flow problems: the flow in ducts, curved-wall diffusers, and a driven cavity. Results obtained for different flow conditions are in good agreement with available analytical and numerical solutions. The viability and validity of the method of lines are demonstrated by its application to Navier-Stokes equations in the physical domain having a variable mesh.

Abolhassani, J. S.↗

NASA NTRS1985

End effects in scarf joints

Scarf joints with small scarf angles are especially sensitive to stiffness mismatch between adherends and to adherend tip bluntness. Pre-assembly breakage of an adherend tip where it is only a few microns thick can cause significant reduction in joint strength. Mathematically, the reason for such sensitivity is that the solutions to the governing differential equation develop boundary layer character when the scarf angles are small. The boundary layers are regions with large adhesive stresses. Experimental strength data for laminated composite adherends agree with the results of this analysis.

Adkins, D. W.↗

DOE OSTI2024

One-sweep moment-based semi-implicit-explicit integration for gray thermal radiation transport

Thermal radiation transport (TRT) is a time dependent, high dimensional partial integro-differential equation. In practical applications such as inertial confinement fusion, TRT is coupled to other physics such as hydrodynamics, plasmas, etc., and the timescales one is interested in capturing are often much slower than the radiation timescale. As a result, TRT is treated implicitly, and due to its stiffness and high dimensionality, is often a dominant computational cost in multiphysics simulations. Here we develop a new approach for implicit-explicit (IMEX) integration of gray TRT in the deterministic SN setting, which requires only one sweep per stage, with the simplest first-order method requiring only one sweep per time step. The partitioning of equations is done via a moment-based high-order low-order formulation of TRT, where the streaming operator and first two moments are used to capture the asymptotic stiff regimes of the streaming limit and diffusion limit. Absorption-reemission is treated explicitly, and although stiff, is sufficiently damped by the implicit solve that we achieve stable accurate time integration without incorporating the coupling of the high order and low order equations implicitly. Due to nonlinear coupling of the high-order and low-order equations through temperature-dependent opacities, to facilitate IMEX partitioning and higher-order methods, we use a semi-implicit integration approach amenable to nonlinear partitions. In conclusion, results are demonstrated on thick Marshak and crooked pipe benchmark problems, demonstrating orders of magnitude improvement in accuracy and wallclock compared with the standard first-order implicit integration typically used.

97 MATHEMATICS AND COMPUTING↗

NASA NTRS1973

Nonlinear Equations for Bending of Rotating Beams with Application to Linear Flap-Lag Stability of Hingeless Rotors

The nonlinear partial differential equations for the flapping and lead-lag degrees of freedom of a torisonally rigid, rotating cantilevered beam are derived. These equations are linearized about an equilibrium condition to study the flap-lag stability characteristics of hingeless helicopter rotor blades with zero twist and uniform mass and stiffness in the hovering flight condition. The results indicate that these configurations are stable because the effect of elastic coupling more than compensates for the destabilizing flap-lag Coriolis and aerodynamic coupling. The effect of higher bending modes on the lead-lag damping was found to be small and the common, centrally hinged, spring restrained, rigid blade approximation for elastic rotor blades was shown to be resonably satisfactory for determining flap-lag stability. The effect of pre-cone was generally stabilizing and the effects of rotary inertia were negligible.

Hodges, D. H.↗

NASA NTRS1984

Analytical results for postbuckling behavior of plates in compression and shear

The postbuckling behavior of long rectangular isotropic and orthotropic plates is determined. By assuming trigonometric functions in one direction, the nonlinear partial differential equations of von Karman large deflection plate theory are converted into nonlinear ordinary differential equations. The ordinary differential equations are solved numerically using an available boundary value problem solver which makes use of Newton's method. Results for longitudinal compression show different postbuckling behavior between isotropic and orthotropic plates. Results for shear show that change in inplane edge constraints can cause large change in postbuckling stiffness.

Stein, M.↗