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Results for “Numerical partial differential equations”

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 361 records · Page 20
All sourcesNASA NTRSDOE OSTI
NASA NTRS1992

Using adaptive grid in modeling rocket nozzle flow

The mechanical behavior of a rocket motor internal flow field results in a system of nonlinear partial differential equations which cannot be solved analytically. However, this system of equations called the Navier-Stokes equations can be solved numerically. The accuracy and the convergence of the solution of the system of equations will depend largely on how precisely the sharp gradients in the domain of interest can be resolved. With the advances in computer technology, more sophisticated algorithms are available to improve the accuracy and convergence of the solutions. An adaptive grid generation is one of the schemes which can be incorporated into the algorithm to enhance the capability of numerical modeling. It is equivalent to putting intelligence into the algorithm to optimize the use of computer memory. With this scheme, the finite difference domain of the flow field called the grid does neither have to be very fine nor strategically placed at the location of sharp gradients. The grid is self adapting as the solution evolves. This scheme significantly improves the methodology of solving flow problems in rocket nozzles by taking the refinement part of grid generation out of the hands of computational fluid dynamics (CFD) specialists and place it into the computer algorithm itself.

Chow, Alan S.↗
NASA NTRS1993

An analytic study of a two-phase laminar airfoil in simulated heavy rain

A mathematical model for a two-phase flow laminar airfoil in simulated heavy rain has been established. The set of non-linear partial differential equations has been converted into a set of finite difference equations; appropriate initial and boundary conditions are provided. The numerical results are compared with the experimental measurements. They show good agreement in quality.

Hsu, Yu-Kao↗
DOE OSTI2024

Quantifying local and global mass balance errors in physics-informed neural networks

Physics-informed neural networks (PINN) have recently become attractive for solving partial differential equations (PDEs) that describe physics laws. By including PDE-based loss functions, physics laws such as mass balance are enforced softly in PINN. This paper investigates how mass balance constraints are satisfied when PINN is used to solve the resulting PDEs. We investigate PINN’s ability to solve the 1D saturated groundwater flow equations (diffusion equations) for homogeneous and heterogeneous media and evaluate the local and global mass balance errors. We compare the obtained PINN’s solution and associated mass balance errors against a two-point finite volume numerical method and the corresponding analytical solution. We also evaluate the accuracy of PINN in solving the 1D saturated groundwater flow equation with and without incorporating hydraulic heads as training data. We demonstrate that PINN’s local and global mass balance errors are significant compared to the finite volume approach. Tuning the PINN’s hyperparameters, such as the number of collocation points, training data, hidden layers, nodes, epochs, and learning rate, did not improve the solution accuracy or the mass balance errors compared to the finite volume solution. Mass balance errors could considerably challenge the utility of PINN in applications where ensuring compliance with physical and mathematical properties is crucial.

54 ENVIRONMENTAL SCIENCES↗
DOE OSTI2020

An extended trajectory-mechanics approach for calculating two-phase flow paths

A technique originating in quantum dynamics is used to derive a trajectory-based, semi-analytical solution for two-phase flow. The partial differential equation governing the evolution of the aqueous phase is equivalent to a family of ordinary differential equations defined along a path through the porous medium. The trajectories may be found by solving the differential equations directly or by post-processing the output of a numerical solution to the full set of governing equations. The trajectories, which differ from conventional streamlines, are found to bend downward in response to gravitational forces. The curvature is more pronounced as the dip of the porous layer containing the flow increases. Subtle changes in the relative permeability curve can lead to significant variations in the trajectories. The ordinary differential equation for the trajectory provides an expression for the travel time along the path. The expression produces a semi-analytical approximation to the model parameter sensitivities, the partial derivatives of the travel times with respect to changes in the permeability model. The semi-analytical trajectory-based sensitivities generally agree with those computed using a numerical reservoir simulator and a perturbation approach. The sensitivities are useful in tomographic imaging algorithms designed to estimate the spatial variation in permeability within a porous medium using multiphase observations.

58 GEOSCIENCES↗
NASA NTRS1977

TOMCAT - A code for numerical generation of boundary-fitted curvilinear coordinate systems on fields containing any number of arbitrary two-dimensional bodies

A method for automatic generation of boundary-fitted curvilinear coordinate systems, where the transformed coordinates are solutions of an elliptic differential system in the physical plane, and where the coordinate lines are coincident with all boundaries of a general multiply-connected, two-dimensional region containing any number of arbitrarily shaped bodies, and is described along with a suitable computer code for implementing the method. Any partial differential system can be solved on the boundary-fitted coordinate system by appropriate transformations. The transformed equations are approximated by finite differences and solved numerically in the transformed plane. All computations, whether for generating coordinate system or then solving the transformed equations, can be done on a rectangular field with square mesh with no interpolation required on the boundaries. The physical boundaries may even be time-dependent.

Thompson, J. F.↗
NASA NTRS2004

A Model for the Oxidation of Carbon Silicon Carbide Composite Structures

A mathematical theory and an accompanying numerical scheme have been developed for predicting the oxidation behavior of carbon silicon carbide (C/SiC) composite structures. The theory is derived from the mechanics of the flow of ideal gases through a porous solid. The result of the theoretical formulation is a set of two coupled nonlinear differential equations written in terms of the oxidant and oxide partial pressures. The differential equations are solved simultaneously to obtain the partial vapor pressures of the oxidant and oxides as a function of the spatial location and time. The local rate of carbon oxidation is determined using the map of the local oxidant partial vapor pressure along with the Arrhenius rate equation. The nonlinear differential equations are cast into matrix equations by applying the Bubnov-Galerkin weighted residual method, allowing for the solution of the differential equations numerically. The numerical method is demonstrated by utilizing the method to model the carbon oxidation and weight loss behavior of C/SiC specimens during thermogravimetric experiments. The numerical method is used to study the physics of carbon oxidation in carbon silicon carbide composites.

Sullivan, Roy M.↗
DOE OSTI2025

Thermomechanics coupling to Monte Carlo particle transport on unstructured mesh geometries using Cardinal

Geometry deformation due to thermal expansion influences neutron transport in many systems. Studying this phenomenon involves coupling models for neutronics, thermal hydraulics, and solid mechanics. To enable high fidelity modeling of these coupled physics, new capabilities were introduced in Cardinal, coupling OpenMC Monte Carlo particle transport models with MOOSE thermomechanical physics on unstructured moving-mesh geometries. In this work, we present a fully open-source capability leveraging on-the-fly mesh skinning to automatically regenerate OpenMC geometry, which allows multiphysics feedback from temperature, density, and geometry changes. The new capability is verified using an analytic benchmark slab problem, which couples S 2 neutron transport with thermal conduction, convective boundary conditions, Doppler-broadened cross sections, and nonlinear thermal expansion effects along the heated slab. Cardinal reproduces the analytic solutions for the neutron flux, heating, k eff , and temperature with demonstrated convergence in various error terms including mesh resolution and cross section temperature library spacing. For the nominal benchmark conditions and with a fine mesh, maximum relative errors for neutron flux, temperature, and heating are lower than 1%, while errors in integral quantities such as k eff and slab length are within 1 pcm and 48 µm, respectively. This work (i) presents a new numerical approach to thermomechanics coupling with OpenMC models, (ii) is the first (to our knowledge) to utilize a mechanical partial differential equation (PDE) solution to solve the (Griesheimer and Kooreman, 2022) analytic benchmark, and (iii) develops this verified capability within an open-source package.

97 - MATHEMATICS AND COMPUTING↗
NASA NTRS1983

Cubic spline approximation techniques for parameter estimation in distributed systems

Approximation schemes employing cubic splines in the context of a linear semigroup framework are developed for both parabolic and hyperbolic second-order partial differential equation parameter estimation problems. Convergence results are established for problems with linear and nonlinear systems, and a summary of numerical experiments with the techniques proposed is given.

Banks, H. T.↗
NASA NTRS1983

Second-order explicit finite-difference methods for transient-flow analysis

Three second-order accurate numerical methods - MacCormack's method, Lambda scheme and Gabutti scheme - are introduced to solve the quasi-linear, hyperbolic partial differential equations describing transient flows in closed conduits. The details of these methods and the treatment of boundary conditions are presented and the results computed by using these methods for a typical piping system are compared. It is shown that for the same accuracy, second-order methods require considerably lesser number of computational nodes and computer time as compared to those required by the first-order methods.

Chaudhry, M. H.↗
DOE OSTI2023

Leverage Score Sampling for Parametric PDEs (Final Technical Report)

This final technical report summarizes the accomplishments of work performed under DOE Office of Science Award DE-SC0022266, which is titled “Leverage Score Sampling for Parametric PDEs”. The goal of the project was to extend methods from Randomized Numerical Linear Algebra (RandNLA) to tackle central computational challenges in model order reduction and uncertainty quantification (UQ) for parametric partial differential equations (PDEs). In particular, we sought to use importance sampling methods originally developed for RandNLA to develop sample efficient active learning algorithms for approximating high-dimensional scalar functions, e.g. by polynomials, Gaussian process models, and simple neural networks. Such methods can be immediately applied to developing surrogate models or to approximating quantity of interest (QoI) surfaces. In the context of PDEs, each sample used for learning equates to the solution of the differential equation for a particular set of parameters, so sample efficiency translates to improved computational efficiency for a variety of downstream tasks.

97 MATHEMATICS AND COMPUTING↗
NASA NTRS1980

Thermal Pollution Mathematical Model: Verification of One-Dimensional Numerical Model at Lake Keowee - Volume 2

A one dimensional model for studying the thermal dynamics of cooling lakes was developed and verified. The model is essentially a set of partial differential equations which are solved by finite difference methods. The model includes the effects of variation of area with depth, surface heating due to solar radiation absorbed at the upper layer, and internal heating due to the transmission of solar radiation to the sub-surface layers. The exchange of mechanical energy between the lake and the atmosphere is included through the coupling of thermal diffusivity and wind speed. The effects of discharge and intake by power plants are also included. The numerical model was calibrated by applying it to Cayuga Lake. The model was then verified through a long term simulation using Lake Keowee data base. The comparison between measured and predicted vertical temperature profiles for the nine years is good. The physical limnology of Lake Keowee is presented through a set of graphical representations of the measured data base.

Lee, S. S.↗
DOE OSTI2021

Parallel-In-Time Integration for Hyperbolic Problems (Final Report)

The work for the subcontract was situated in the area of parallel-in-time integration for advection-dominated and hyperbolic partial differential equations (PDEs). Parallel-in-time integration is an active area of research due to its ability to enable faster numerical simulations for applications throughout many areas of science. Over the past two decades, much progress has been made in this area; however, this progress has largely been limited to non-hyperbolic, diffusion-dominated PDEs. Given the ubiquity of numerical simulations of advection-dominated hyperbolic PDEs throughout the sciences, there is a strong need to develop efficient parallel-in-time techniques for hyperbolic problems, which was the main topic of this subcontract.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2022

Parallel-in-Time Integration for Nonlinear Hyperbolic Problems (Final Report)

The work for the subcontract was situated in the area of parallel-in-time integration for hyperbolic partial differential equations (PDEs). Parallel-in-time integration is an active area of research due to its ability to enable faster numerical simulations for applications throughout many areas of science. Over the past two decades, much progress has been made in this area; however, this progress has largely been limited to diffusion-dominated PDEs, with some recent success in scalar linear hyperbolic PDEs. Given the ubiquity of numerical simulations of hyperbolic PDEs throughout the sciences, in particular, nonlinear hyperbolic systems, there is a strong need to develop efficient parallel-in-time techniques for hyperbolic problems beyond simple scalar and linear cases, which is the main topic of this subcontract. The main focus of the work was to further develop and perfect coarse-grid operators for the Multigrid Reduction-inTime (MGRIT) method applied to hyperbolic PDEs that were recently proposed in PhD thesis, based on a modified semi-Lagrangian approach.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2024

Parallel-in-Time Methods for Method-of-Lines Discretizations of Nonlinear Hyperbolic PDEs and Systems (Final Report)

The work for the subcontract is situated in the area of parallel-in-time integration for hyperbolic partial differential equations (PDEs). Parallel-in-time integration is an active area of research due to its ability to enable faster numerical simulations for applications throughout many areas of science. The work in this subcontract builds on a variety of results that were obtained, as part of the work performed for Subcontract No. B648355, for the Multigrid Reduction-in-Time (MGRIT) method from [1] applied to hyperbolic PDEs. This subcontract extends these results further to more efficient methods and to the case of method-of-lines discretizations for nonlinear hyperbolic PDES and systems of PDEs. The following is a summary of the research performed and results achieved during milestone periods 1, 2 and 3 by the PI (Hans De Sterck) and Postdoctoral Research Associate (Oliver Krzysik), for required tasks 1-4 (as listed in the Statement of Work): Research over the previous year has been split into three main projects: (i) solution of acoustic equation system; (ii) solution of nonlinear scalar hyperbolic PDEs; (iii) solution of nonlinear hyperbolic systems of PDEs.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2024

Tensor Network Space-Time Spectral Collocation Method for Time-Dependent Convection-Diffusion-Reaction Equations

Emerging tensor network techniques for solutions of partial differential equations (PDEs), known for their ability to break the curse of dimensionality, deliver new mathematical methods for ultra-fast numerical solutions of high-dimensional problems. Here, we introduce a Tensor Train (TT) Chebyshev spectral collocation method, in both space and time, for the solution of the time-dependent convection-diffusion-reaction (CDR) equation with inhomogeneous boundary conditions, in Cartesian geometry. Previous methods for numerical solution of time-dependent PDEs often used finite difference for time, and a spectral scheme for the spatial dimensions, which led to a slow linear convergence. Spectral collocation space-time methods show exponential convergence; however, for realistic problems they need to solve large four-dimensional systems. We overcome this difficulty by using a TT approach, as its complexity only grows linearly with the number of dimensions. We show that our TT space-time Chebyshev spectral collocation method converges exponentially, when the solution of the CDR is smooth, and demonstrate that it leads to a very high compression of linear operators from terabytes to kilobytes in TT-format, and a speedup of tens of thousands of times when compared to a full-grid space-time spectral method. These advantages allow us to obtain the solutions at much higher resolutions.

97 MATHEMATICS AND COMPUTING↗
NASA NTRS1974

Finite element solution theory for three-dimensional boundary flows

A finite element algorithm is derived for the numerical solution of a three-dimensional flow field described by a system of initial-valued, elliptic boundary value partial differential equations. The familiar three-dimensional boundary layer equations belong to this description when diffusional processes in only one coordinate direction are important. The finite element algorithm transforms the original description into large order systems of ordinary differential equations written for the dependent variables discretized at node points of an arbitrarily irregular computational lattice. The generalized elliptic boundary conditions is piecewise valid for each dependent variable on boundaries that need not explicitly coincide with coordinate surfaces. Solutions for sample problems in laminar and turbulent boundary flows illustrate favorable solution accuracy, convergence, and versatility.

Baker, A. J.↗
NASA NTRS1982

An approximation theory for nonlinear partial differential equations with applications to identification and control

Approximation results from linear semigroup theory are used to develop a general framework for convergence of approximation schemes in parameter estimation and optimal control problems for nonlinear partial differential equations. These ideas are used to establish theoretical convergence results for parameter identification using modal (eigenfunction) approximation techniques. Results from numerical investigations of these schemes for both hyperbolic and parabolic systems are given.

Banks, H. T.↗