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Search indexed NASA NTRS and DOE OSTI research on propulsion, heat transfer, battery materials and energy systems. Follow report and document links to the original sources.

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At least 541 records · Page 30
All sourcesNASA NTRSDOE OSTI
DOE OSTI2020

Parallel-in-Time Multigrid Methods for Hyperbolic Problems, with a Focus on the Shallow Water Equations

The coming massive parallelism of exascale computing presents a pressing challenge for the many DOE simulations of time-dependent partial differential equations, which typically use traditional sequential time stepping methods. Since this traditional approach is inherently serial, it presents a sequential bottleneck when moving to exascale computing, because future performance gains will come through greater concurrency, not faster clock speeds. Thus, the goal of this work is to research parallelism in time, i.e., methods that compute multiple time values simultaneously, not sequentially. The focus will be on hyperbolic problems of programmatic interest to DOE, with the goal of enabling scalable simulations of time-dependent hyperbolic problems on future architectures. The difficulty lies in the fact that hyperbolic problems are well-known to be difficult for parallel-in-time methods, with the most common parallel-in-time method, Parareal, diverging in many cases. Here, the chosen methodology for scalably solving these hyperbolic space-time equations is multigrid-reductionin-time (MGRIT), because multigrid (when it works) is a powerful, optimal, and scalable solver for discretized PDEs. To further research in this area, this project will explore a model hyperbolic problem (the shallow water equations) in the context of recent advances (e.g., by Wingate and Haut) in constructing improved coarse time-grid time-propagators by using the slow asymptotic structure of the equations. In particular, we will investigate if such coarse time-propagators offer significant advantages in an MGRIT setting, and explore any broader insights gained into parallel-in-time for hyperbolic problems.

97 MATHEMATICS AND COMPUTING↗
NASA NTRS1947

Voltera's Solution of the Wave Equation as Applied to Three-Dimensional Supersonic Airfoil Problems

A surface integral is developed which yields solutions of the linearized partial differential equation for supersonic flow. These solutions satisfy boundary conditions arising in wing theory. Particular applications of this general method are made, using acceleration potentials, to flat surfaces and to uniformly loaded lifting surfaces. Rectangular and trapezoidal plan forms are considered along with triangular forms adaptable to swept-forward and swept-back wings. The case of the triangular plan form in sideslip is also included. Emphasis is placed on the systematic application of the method to the lifting surfaces considered and on the possibility of further application.

Heslet, Max A↗
DOE OSTI2021

Monolithic Multigrid Methods for Magnetohydrodynamics

The magnetohydrodynamics equations model a wide range of plasma physics applications and are characterized by a nonlinear system of partial differential equations that strongly couples a charged fluid with the evolution of electromagnetic fields. After discretization and linearization, the resulting system of equations is generally difficult to solve due to the coupling between variables and the heterogeneous coefficients induced by the linearization process. In this paper, we investigate multigrid preconditioners for this system based on specialized relaxation schemes that properly address the system structure and coupling. Here, three extensions of Vanka relaxation are proposed and applied to problems with up to 170 million degrees of freedom and fluid and magnetic Reynolds numbers up to 400 for stationary problems and up to 20,000 for time-dependent problems.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2026

Structural Aspects of Neutron Survival Probabilities

The neutron survival probability (and related quantities including probabilities of extinction and initiation) is a central element of the broader stochastic theory of neutron populations and finds application in fields including reactor start-up, analysis of reactor power bursts and criticality accidents, and safeguards. In a full neutron transport formulation, the equation governing the single-neutron survival probability is a backward or adjoint-like integro-partial differential equation with the added complexity of being highly nonlinear. Analogous formulations of this equation exist in the context of many approximate theories of neutron transport, with the point kinetics formulation having received significant theoretical attention since the 1940s. This work continues this tradition by providing a novel analysis of the single-neutron survival probability equation using the tools of boundary layer theory. The analysis reveals that the “fully dynamic” solution of the single-neutron survival probability equation—and some key probability distributions derived from it—may be cast as a singular perturbation around the underlying quasi-static single-neutron probability of initiation. In this perturbation solution, the expansion parameter is the ratio of the neutron generation time to a macroscopic time scale characterizing the overall system evolution; this interpretation illuminates some of the fundamental structural aspects of neutron survival phenomena.

97 MATHEMATICS AND COMPUTING↗
NASA NTRS1973

Solution of the stochastic control problem in unbounded domains.

Bellman's dynamic programming equation for the optimal index and control law for stochastic control problems is a parabolic or elliptic partial differential equation frequently defined in an unbounded domain. Existing methods of solution require bounded domain approximations, the application of singular perturbation techniques or Monte Carlo simulation procedures. In this paper, using the fact that Poisson impulse noise tends to a Gaussian process under certain limiting conditions, a method which achieves an arbitrarily good approximate solution to the stochastic control problem is given. The method uses the two iterative techniques of successive approximation and quasi-linearization and is inherently more efficient than existing methods of solution.

Robinson, P.↗
NASA NTRS1985

Parameter identification in distributed systems

This paper describes a method for the identification of the parameters entering into the equations of motion of distributed systems. Because the motion of distributed systems is described in terms of partial differential equations, these parameters are in general continuous functions of the spatial variables. For vibrating systems, these parameters ordinarily represent the mass, stiffness and damping distributions. In this paper, these distributions are expanded in terms of finite series of known functions of the spatial variables multiplied by undetermined coefficients. It is assumed that the nature of the equations of motion is known and that a limited number of eigenvalues and eigenfunctions is identified in advance. Use is then made of the least squares method, in conjunction with the eigenfunctions' orthogonality, to compute the undetermined coefficients, thus identifying the system distributed parameters. A method for the identification of the eigensolution is also presented. The procedure for the identification of the eigensolution and of the system parameters is demonstrated via a numerical example.

Baruh, H.↗
NASA NTRS1986

Nonlinear filtering for spacecraft attitude estimation

Nonlinear filtering techniques are applied to spacecraft attitude estimation using quaternion parameterization for the attitude kinematics. By replacing the angular velocity vector by the gyro output vector, a state dependent noise vector is introduced in the seven-dimensional system equations. The resulting conditional probability density function from the Ito differential rule is governed by the Fokker Planck partial differential equation which is approximated by the second order mean and covariance differential equations. In order to minimize computer loading, the covariance propagation is carried out in six-dimensional state space using a matrix transformation. The star tracker data is used to update the covariance matrix in the seven-dimensional space. The algorithm is simulated for an earth pointing spacecraft mission, using Monte Carlo samples of gyro and star measurements. The performance of the second order filter is compared with the extended Kalman Filter through several simulation runs and drift rates have been identified.

Vathsal, S.↗
NASA NTRS1981

Two-dimensional integrating matrices on rectangular grids

The use of integrating matrices in solving differential equations associated with rotating beam configurations is examined. In vibration problems, by expressing the equations of motion of the beam in matrix notation, utilizing the integrating matrix as an operator, and applying the boundary conditions, the spatial dependence is removed from the governing partial differential equations and the resulting ordinary differential equations can be cast into standard eigenvalue form. Integrating matrices are derived based on two dimensional rectangular grids with arbitrary grid spacings allowed in one direction. The derivation of higher dimensional integrating matrices is the initial step in the generalization of the integrating matrix methodology to vibration and stability problems involving plates and shells.

Lakin, W. D.↗
NASA NTRS2006

Semi-Analytic Reconstruction of Flux in Finite Volume Formulations

Semi-analytic reconstruction uses the analytic solution to a second-order, steady, ordinary differential equation (ODE) to simultaneously evaluate the convective and diffusive flux at all interfaces of a finite volume formulation. The second-order ODE is itself a linearized approximation to the governing first- and second- order partial differential equation conservation laws. Thus, semi-analytic reconstruction defines a family of formulations for finite volume interface fluxes using analytic solutions to approximating equations. Limiters are not applied in a conventional sense; rather, diffusivity is adjusted in the vicinity of changes in sign of eigenvalues in order to achieve a sufficiently small cell Reynolds number in the analytic formulation across critical points. Several approaches for application of semi-analytic reconstruction for the solution of one-dimensional scalar equations are introduced. Results are compared with exact analytic solutions to Burger s Equation as well as a conventional, upwind discretization using Roe s method. One approach, the end-point wave speed (EPWS) approximation, is further developed for more complex applications. One-dimensional vector equations are tested on a quasi one-dimensional nozzle application. The EPWS algorithm has a more compact difference stencil than Roe s algorithm but reconstruction time is approximately a factor of four larger than for Roe. Though both are second-order accurate schemes, Roe s method approaches a grid converged solution with fewer grid points. Reconstruction of flux in the context of multi-dimensional, vector conservation laws including effects of thermochemical nonequilibrium in the Navier-Stokes equations is developed.

Gnoffo, Peter A.↗
DOE OSTI2025

Sylvester-preconditioned adaptive-rank implicit time integrators for advection-diffusion equations with variable coefficients

Here, we consider the adaptive-rank integration of multi-dimensional time-dependent advection-diffusion partial differential equations (PDEs) with variable coefficients. We employ a standard finite-difference method for spatial discretization coupled with high-order diagonally implicit Runge-Kutta temporal schemes. The discrete equation is a generalized Sylvester equation (GSE), which we solve with a projection-based adaptive-rank algorithm structured around two key strategies: (i) constructing dimension-wise subspaces using a novel atypical extended Krylov strategy, and (ii) efficiently solving the basis coefficient matrix with a preconditioned GMRES solver. The low-rank decomposition is performed in 2D using SVD and with high-order SVD (HOSVD) in 3D to represent the tensor in a compressed Tucker format. For d-dimensional problems (here, d = 2 or 3), the computational complexity and memory storage of the approach are found numerically to scale as and $\mathscr{O}(Nr^2) + \mathscr{O} (r^{d+1})$ and $\mathscr{O}(Nr) + \mathscr{O} (r^{d})$, respectively, with the one-dimensional resolution and the maximal rank during the Krylov iteration (which we find to be largely independent of on our numerical examples). We present numerical examples that illustrate the advertised properties of the algorithm.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2021

Deep Learning Approaches to Surrogates for Solving the Diffusion Equation for Mechanistic Real-World Simulations

In many mechanistic medical, biological, physical, and engineered spatiotemporal dynamic models the numerical solution of partial differential equations (PDEs), especially for diffusion, fluid flow and mechanical relaxation, can make simulations impractically slow. Biological models of tissues and organs often require the simultaneous calculation of the spatial variation of concentration of dozens of diffusing chemical species. One clinical example where rapid calculation of a diffusing field is of use is the estimation of oxygen gradients in the retina, based on imaging of the retinal vasculature, to guide surgical interventions in diabetic retinopathy. Furthermore, the ability to predict blood perfusion and oxygenation may one day guide clinical interventions in diverse settings, i.e., from stent placement in treating heart disease to BOLD fMRI interpretation in evaluating cognitive function (Xie et al., 2019; Lee et al., 2020). Since the quasi-steady-state solutions required for fast-diffusing chemical species like oxygen are particularly computationally costly, we consider the use of a neural network to provide an approximate solution to the steady-state diffusion equation. Machine learning surrogates, neural networks trained to provide approximate solutions to such complicated numerical problems, can often provide speed-ups of several orders of magnitude compared to direct calculation. Surrogates of PDEs could enable use of larger and more detailed models than are possible with direct calculation and can make including such simulations in real-time or near-real time workflows practical. Creating a surrogate requires running the direct calculation tens of thousands of times to generate training data and then training the neural network, both of which are computationally expensive. Often the practical applications of such models require thousands to millions of replica simulations, for example for parameter identification and uncertainty quantification, each of which gains speed from surrogate use and rapidly recovers the up-front costs of surrogate generation. We use a Convolutional Neural Network to approximate the stationary solution to the diffusion equation in the case of two equal-diameter, circular, constant-value sources located at random positions in a two-dimensional square domain with absorbing boundary conditions. Such a configuration caricatures the chemical concentration field of a fast-diffusing species like oxygen in a tissue with two parallel blood vessels in a cross section perpendicular to the two blood vessels. To improve convergence during training, we apply a training approach that uses roll-back to reject stochastic changes to the network that increase the loss function. The trained neural network approximation is about 1000 times faster than the direct calculation for individual replicas. Because different applications will have different criteria for acceptable approximation accuracy, we discuss a variety of loss functions and accuracy estimators that can help select the best network for a particular application. We briefly discuss some of the issues we encountered with overfitting, mismapping of the field values and the geometrical conditions that lead to large absolute and relative errors in the approximate solution.

60 APPLIED LIFE SCIENCES↗
NASA NTRS1980

An approximate factorization solution of the Navier-Stokes equations for transonic flow using body-fitted coordinates with application to NACA 64A010 airfoils

The implementation of the approximate factorization algorithm and its ability to efficiently and accurately describe transonic flow about an NACA 64A010 airfoil section is examined. The approximate factorization algorithm is developed from the nondimensional, conservative, vectorized Navier-Stokes equations expressed in curvilinear coordinates. Equations of state and transport coefficient relations appropriate to atmospheric air are appended to close the system of partial differential equations. An algebraic turbulence model is also incorporated into the equation set. This algorithm was verified by investigating the flow about an NACA 64A010 airfoil at 0, 2, and 3.5 deg angle of attack for free-stream conditions of 2,000,000 Reynolds number and 0.8 Mach number. Overall results were in good qualitative agreement with wind tunnel data sets. However, while nondimensional times of six were attained, numerical difficulties prevented any case from reaching a true steady state.

Copper, G. K.↗
NASA NTRS1982

A domain segmentation approach to modeling distributed parameter systems

Existing modeling methods in distributed parameter systems are directed to field problems for which a single equation gives an adequate representation. As a result, the techniques are severely taxed when coefficients of the governing partial differential equation (PDE) have relatively complex functional forms. The field is postulated to consist of contiguous regions that are easily modeled. During the modeling process itself, the regional boundaries are identified using a method based on pattern recognition (PR) techniques. The method was tested on computer simulations of a water quality resource system. Quantitative measures are developed and used to demonstrate the feasibility of applying the methodology to realistic distributed parameter systems (DPS's).

Apenyo, K.↗
NASA NTRS1993

Numerical solution of flame sheet problems with and without multigrid methods

Flame sheet problems are on the natural route to the numerical solution of multidimensional flames, which, in turn, are important in many engineering applications. In order to model the structure of flames more accurately, we use the vorticity-velocity formulation of the fluid flow equations, as opposed to the streamfunction-vorticity approach. The numerical solution of the resulting nonlinear coupled elliptic partial differential equations involves a pseudo transient process and a steady state Newton iteration. Rather than working with dimensionless variables, we introduce scale factors that can yield significant savings in the execution time. In this context, we also investigate the applicability and performance of several multigrid methods, focusing on nonlinear damped Newton multigrid, using either one way or correction schemes.

Douglas, Craig C.↗
DOE OSTI2020

The extended auxiliary equation mapping method to determine novel exact solitary wave solutions of the nonlinear fractional PDEs

Abstract In this paper, some new nonlinear fractional partial differential equations (PDEs) have been considered.Three models are including the space-time fractional-order Boussinesq equation, space-time (2 + 1)-dimensional breaking soliton equations, and space-time fractional-order SRLW equation describe the behavior of these equations in the diverse applications. Meanwhile, the fractional derivatives in the sense of β -derivative are defined. Some fractional PDEs will convert to the considered ordinary differential equations by the help of transformation of β -derivative. These equations are analyzed utilizing an integration scheme, namely, the extended auxiliary equation mapping method. The different kinds of traveling wave solutions, solitary, topological, dark soliton, periodic, kink, and rational, fall out as a by-product of this scheme. Finally, the existence of the solutions for the constraint conditions is also shown. The outcome indicates that some fractional PDEs are used as a growing finding in the engineering sciences, mathematical physics, and so forth.

Engineering↗
NASA NTRS1989

Stable explicit schemes for equations of Schroedinger type

A method for constructing explicit finite-difference schemes which can be used to solve Schroedinger-type partial-differential equations is presented. A forward Euler scheme that is conditionally stable is given by the procedure. The results presented are based on the analysis of the simplest Schroedinger type equation.

Mickens, Ronald E.↗
NASA NTRS2019

On the Calculation of Exact Cumulative Distribution Statistics for Burgers Equation

A mathematical procedure is presented for the calculation of exact cumulative distribution statistics for a viscosity-free variant of Burgers nonlinear partial differential equation (PDE) in one space dimension and time subject to sinusoidal initial data with uncertain (random variable) amplitude or phase shift. Analytical solutions of nonlinear PDEs with uncertain initial and/or boundary data are invaluable benchmarks in assessing approximate uncertainty quantification techniques. The Burgers equation solution with uncertain initial data results in nonsmooth solution behavior in both physical and random variable dimensions which provides a severe test for approximate uncertainty quantification techniques. Mathematical proofs are provided to verify that exact cumulative distribution statistics can be systematically and robustly obtained for all forward time.

Burgers↗