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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 19 records
All sourcesNASA NTRSDOE OSTI
DOE OSTI2018

Convergence and asymptotical stability of numerical solutions for neutral stochastic delay differential equations driven by G-Brownian motion

There are few numerical analysis results for nonlinear neutral stochastic delay differential equations driven by G-Brownian motion (G-NSDDEs). This paper is concerned with the numerical solutions of the G-NSDDEs to fill this gap. In this paper, we first devote to show that the stochastic theta numerical solution converges to the exact solution for the G-NSDDE. We then prove that the backward Euler–Maruyama numerical solution for the G-NSDDE is asymptotically mean-square stable under suitable conditions. Moreover, a numerical example is demonstrated to illustrate the effectiveness of our obtained results.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2021

Numerical solution of singular Lyapunov equations

We consider the numerical solution of large scale singular (continuous-time) Lyapunov equations of the form AX + XA T + BB T = 0, where A is semistable, that is, its spectrum is contained in the left half plane, with the exception of a few semisimple eigenvalues at zero. We also consider the case of a few semisimple eigenvalues on the imaginary axis. We assume that we know these few eigenvalues (zero or imaginary), and that we have or can compute the corresponding invariant subspaces. We use this information to build an appropriate newly proposed subspace on which to project the Lyapunov equations, and then compute a low-rank approximation to the least squares solution. Selected illustrative numerical examples are provided.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2022

Benchmark of few-level quantum theory vs ab initio numerical solutions for the strong-field Autler–Townes effect in photoionization of hydrogen

Abstract The temporal and spectral consequences of an intermediate resonance en route to photoionization are investigated theoretically in two ways: by solving few-level model equations and by ab initio numerical solution of the time-dependent Schrödinger equation, in both cases for hydrogen in three dimensions. The model consists of atomic states resonantly field-dressed in a three-level reduction of the hydrogen atom that consists of the 2 p –3 d (Balmer) transition and one energetically-distant continuum state. The model’s level occupation probabilities are derived from three Schrödinger amplitude equations and are benchmarked against an ab initio numerical solution for the hydrogen electron’s wavefunction under the same field. We examine contrasts between the results of the two approaches with a particular focus on Autler–Townes doublets that appear in the photoelectron spectrum.

74 ATOMIC AND MOLECULAR PHYSICS↗
DOE OSTI2018

Discrete Hahn polynomials for numerical solution of two-dimensional variable-order fractional Rayleigh–Stokes problem

The main aim of this paper is to find the numerical solutions of 2D Rayleigh–Stokes problem with the variable-order fractional derivatives in the Riemann–Liouville sense. The presented method is based on collocation procedure in combination with the new operational matrix of the variable-order fractional derivatives, in the Caputo sense, for the discrete Hahn polynomials. The main advantage of the proposed method is obtaining a global approximation for spatial and temporal discretizations, and it reduced the problem to an algebraic system, which is easier to solve. Also, the profit of approximating a continuous function by Hahn polynomials is that for computing the coefficients of the expansion, we only have to compute a summation and the calculation of coefficients is exact. The error bound for the approximate solution is estimated. Finally, we evaluate results of the presented method with other numerical methods.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2022

Impact of the numerical solution approach of a plant hydrodynamic model (v0.1) on vegetation dynamics

Abstract. Numerous plant hydrodynamic models have started to be implemented in vegetation dynamics models, reflecting the central role of plant hydraulic traits in driving water, energy, and carbon cycles, as well as plant adaptation to climate change. Different numerical approximations of the governing equations of the hydrodynamic models have been documented, but the numerical accuracy of these models and its subsequent effects on the simulated vegetation function and dynamics have rarely been evaluated. Using different numerical solution methods (including implicit and explicit approaches) and vertical discrete grid resolutions, we evaluated the numerical performance of a plant hydrodynamic module in the Functionally Assembled Terrestrial Ecosystem Simulator (FATES-HYDRO version 0.1) based on single-point and global simulations. Our simulation results showed that when near-surface vertical grid spacing is coarsened (grid size >10 cm), the model significantly overestimates aboveground biomass (AGB) in most of the temperate forest locations and underestimates AGB in the boreal forest locations, as compared to a simulation with finer vertical grid spacing. Grid coarsening has a small effect on AGB in the tropical zones of Asia and South America. In particular, coarse surface grid resolution should not be used when there are large and prolonged water content differences among soil layers at depths due to long dry-season duration and/or well-drained soil or when soil evaporation is a dominant fraction of evapotranspiration. Similarly, coarse surface grid resolution should not be used when there is lithologic discontinuity along the soil depth. This information is useful for uncertainty quantification, sensitivity analysis, or the training of surrogate models to design the simulations when computational cost limits the use of ensemble simulations.

54 ENVIRONMENTAL SCIENCES↗
DOE OSTI2018

Semi-numerical solution to a fractal telegraphic dual-porosity fluid flow model

We present a semi-numerical solution to a fractal telegraphic dual-porosity fluid flow model. It combines Laplace transform and finite difference schemes. The Laplace transform handles the time variable whereas the finite difference method deals with the spatial coordinate. This semi-numerical scheme is not restricted by space discretization and allows the computation of a solution at any time without compromising numerical stability or the mass conservation principle. Our formulation results in a non-analytically-solvable second-order differential equation whose spatial discretization yields a tridiagonal linear algebraic system. Moreover, we describe comparisons between semi-numerical and semi-analytical solutions for particular cases. Results agree well with those from semi-analytic solutions. Furthermore, we expose a parametric analysis from the coupled model in order to show the effects of relevant parameters on pressure profiles and flow rates for the case where neither analytic nor semi-analytic solutions are available.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2018

Numerical solution of linear and nonlinear hyperbolic telegraph type equations with variable coefficients using shifted Jacobi collocation method

The telegraph equation is one of the important equations of mathematical physics. In this work, a spectral collocation scheme is proposed for the numerical solutions of one- and two-dimensional linear telegraph equations and telegraph equations with nonlinear forcing term. The homogeneous initial and boundary conditions are satisfied exactly by expanding the unknown variable using polynomial bases of functions which are built upon the Jacobi polynomials. The suggested scheme is successfully developed for the aforementioned problem with nonhomogeneous data. Extensive numerical experiments are presented to verify the efficiency of the proposed scheme.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2018

Numerical solution of the electron transport equation in the upper atmosphere

A new approach for solving the electron transport equation in the upper atmosphere is derived. The problem is a very stiff boundary value problem, and to obtain an accurate numerical solution, matrix factorizations are used to decouple the fast and slow modes. A stable finite difference method is applied to each mode. This solver is applied to a simplified problem for which an exact solution exists using various versions of the boundary conditions that might arise in a natural auroral display. The numerical and exact solutions are found to agree with each other, verifying the method.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2022

Numerical Solution of the Steady-State Network Flow Equations for a Non-Ideal Gas

Herein we formulate a steady-state network flow problem for non-ideal gas that relates injection rates and nodal pressures in the network to flows in pipes. For this problem, we present and prove a theorem on uniqueness of generalized solution for a broad class of non-ideal pressure-density relations that satisfy a monotonicity property. Further, we develop a Newton-Raphson algorithm for numerical solution of the steady-state problem, which is made possible by a systematic non-dimensionalization of the equations. The developed algorithm has been extensively tested on benchmark instances and shown to converge robustly to a generalized solution. Previous results [1]-[4], indicate that the steady-state network flow equations for an ideal gas are difficult to solve by the Newton-Raphson method because of its extreme sensitivity to the initial guess. In contrast, we find that non-dimensionalization of the steady-state problem is key to robust convergence of the Newton-Raphson method. We identify criteria based on the uniqueness of solutions under which the existence of a non-physical generalized solution found by a non-linear solver implies non-existence of a physical solution, i.e., infeasibility of the problem. Finally, we compare pressure and flow solutions based on ideal and non-ideal equations of state to demonstrate the need to apply the latter in practice. The solver developed in this article is open-source and is made available for both the academic and research communities as well as the industry.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2024

Computationally efficient and error aware surrogate construction for numerical solutions of subsurface flow through porous media

Limiting the injection rate to restrict the pressure below a threshold at a critical location can be an important goal of simulations that model the subsurface pressure between injection and extraction wells. The pressure is approximated by the solution of Darcy’s partial differential equation for a given permeability field. The subsurface permeability is modeled as a random field since it is known only up to statistical properties. This induces uncertainty in the computed pressure. Solving the partial differential equation for an ensemble of random permeability simulations enables estimating a probability distribution for the pressure at the critical location. These simulations are computationally expensive, and practitioners often need rapid online guidance for real-time pressure management. An ensemble of numerical partial differential equation solutions is used to construct a Gaussian process regression model that can quickly predict the pressure at the critical location as a function of the extraction rate and permeability realization. The Gaussian process surrogate analyzes the ensemble of numerical pressure solutions at the critical location as noisy observations of the true pressure solution, enabling robust inference using the conditional Gaussian process distribution. Our first novel contribution is to identify a sampling methodology for the random environment and matching kernel technology for which fitting the Gaussian process regression model scales as O ( n log n ) instead of the typical O ( n 3 ) rate in the number of samples n used to fit the surrogate. The surrogate model allows almost instantaneous predictions for the pressure at the critical location as a function of the extraction rate and permeability realization. Our second contribution is a novel algorithm to calibrate the uncertainty in the surrogate model to the discrepancy between the true pressure solution of Darcy’s equation and the numerical solution. Finally, although our method is derived for building a surrogate for the solution of Darcy’s equation with a random permeability field, the framework broadly applies to solutions of other partial differential equations with random coefficients.

54 ENVIRONMENTAL SCIENCES↗
DOE OSTI2018

Numerical solution of systems of fractional delay differential equations using a new kind of wavelet basis

In the present paper, a new orthonormal wavelet basis, called Chelyshkov wavelet, is constructed from a class of orthonormal polynomials. These wavelet basis and their properties are utilized to obtain their operational matrix of fractional integration in the Riemann–Liouville sense and delay operational matrix. Convergence and error bound of the expansion by this kind of wavelet functions are investigated. Then, these operational matrices along with the Galerkin approach have been implemented to solve systems of fractional delay differential equations (SFDDEs). The main superiority of the proposed technique is that it reduces SFDDEs to a system of algebraic equations. Moreover, accuracy and efficiency of the suggested Chelyshkov wavelet approach are verified through some linear and nonlinear SFDDEs. Finally, the obtained numerical results are compared with those previously reported in the literature.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2022

A fourth-order phase-field fracture model: Formulation and numerical solution using a continuous/discontinuous Galerkin method

Modeling crack initiation and propagation in brittle materials is of great importance to be able to predict sudden loss of load-carrying capacity and prevent catastrophic failure under severe dynamic loading conditions. Second-order phase-field fracture models have gained wide adoption given their ability to capture the formation of complex fracture patterns, e.g. via crack merging and branching, and their suitability for implementation within the context of the conventional finite element method. Higher-order phase-field models have also been proposed to increase the regularity of the exact solution and thus increase the spatial convergence rate of its numerical approximation. However, they require special numerical techniques to enforce the necessary continuity of the phase field solution. In this paper, we derive a fourth-order phase-field model of fracture in two independent ways; namely, from Hamilton’s principle and from a higher-order micromechanics-based approach. The latter approach is novel, and provides a physical interpretation of the higher-order terms in the model. In addition, we propose a continuous/discontinuous Galerkin (C/DG) method for use in computing the approximate phase-field solution. This method employs Lagrange polynomial shape functions to guarantee -continuity of the solution at inter-element boundaries, and enforces the required regularity with the aid of additional variational and interior penalty terms in the weak form. Finally, the phase-field equation is coupled with the momentum balance equation to model dynamic fracture problems in hyper-elastic materials. Two benchmark problems are presented to compare the numerical behavior of the C/DG method with mixed finite element methods.

42 ENGINEERING↗
DOE OSTI2019

On the numerical solution of the exact factorization equations

The exact factorization (EF) approach to coupled electron-ion dynamics recasts the time-dependent molecular Schrödinger equation as two coupled equations, one for the nuclear wavefunction and one for the conditional electronic wavefunction. The potentials appearing in these equations have provided insight into non-adiabatic processes, and new practical non-adiabatic dynamics methods have been formulated starting from these equations. We provide a first demonstration of a self-consistent solution of the exact equations, with a preliminary analysis of their stability and convergence properties. The equations have an unprecedented mathematical form, involving a Hamiltonian outside the class of Hermitian Hamiltonians usually encountered in time-propagation, and so the usual numerical methods for time-dependent Schrödinger fail when applied in a straightforward way to the EF equations. We find an approach that enables stable propagation long enough to witness non-adiabatic behavior in a model system before non-trivial instabilities take over. Implications for the development and analysis of EF-based methods are discussed.

37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CH↗
DOE OSTI2019

A Classical Field Theory Formulation for the Numerical Solution of Time Harmonic Electromagnetic Fields

Finite element representations of Maxwell's equations pose unusual challenges inherent to the variational representation of the “curl-curl” equation for the fields. We present a variational formulation which solves for the four-potential instead, based on classical field theory. Borrowing from quantum electrodynamics, we modify the Lagrangian by adding an implicit gauge-fixing term. This reformulation explicitly accounts for Gauss’ law through the coupling between $\phi$ and $\rho$ , and enables the use of nodal basis functions instead of edge elements for time-harmonic problems. Here, we demonstrate how this formulation, adhering to the deeper underlying symmetries of the four-dimensional covariant field description, provides a highly general, robust numerical framework.

43 PARTICLE ACCELERATORS↗
DOE OSTI2023

11-th order of accuracy for numerical solution of 3-D Poisson equation with irregular interfaces on unfitted Cartesian meshes

For the first time the optimal local truncation error method (OLTEM) with 125-point stencils and unfitted Cartesian meshes has been developed in the general 3-D case for the Poisson equation for heterogeneous materials with smooth irregular interfaces. The 125-point stencils equations that are similar to those for quadratic finite elements are used for OLTEM. The interface conditions for OLTEM are imposed as constraints at a small number of interface points and do not require the introduction of additional unknowns, i.e., the sparse structure of global discrete equations of OLTEM is the same for homogeneous and heterogeneous materials. The stencils coefficients of OLTEM are calculated by the minimization of the local truncation error of the stencil equations. These derivations include the use of the Poisson equation for the relationship between the different spatial derivatives. Such a procedure provides the maximum possible accuracy of the discrete equations of OLTEM. In contrast to known numerical techniques with quadratic elements and third order of accuracy on conforming and unfitted meshes, OLTEM with the 125-point stencils provides 11-th order of accuracy, i.e., an extremely large increase in accuracy by 8 orders for similar stencils. The numerical results show that OLTEM yields much more accurate results than high-order finite elements with much wider stencils. The increased numerical accuracy of OLTEM leads to an extremely large increase in computational efficiency. Additionally, a new post-processing procedure with the 125-point stencil has been developed for the calculation of the spatial derivatives of the primary function. The post-processing procedure includes the minimization of the local truncation error and the use of the Poisson equation. It is demonstrated that the use of the partial differential equation (PDE) for the 125-point stencils improves the accuracy of the spatial derivatives by 6 orders compared to post-processing without the use of PDE as in existing numerical techniques. At an accuracy of 0.1% for the spatial derivatives, OLTEM reduces the number of degrees of freedom by 900 - 4∙10 6 times compared to quadratic finite elements. The developed post-processing procedure can be easily extended to unstructured meshes and can be independently used with existing post-processing techniques (e.g., with finite elements).

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2025

10-th order of accuracy for numerical solution of 3-D elasticity equations for heterogeneous materials on unfitted Cartesian meshes

We have developed the Optimal Local Truncation Error Method (OLTEM) with 10-th order of accuracy on unfitted Cartesian meshes for a system of 3-D elasticity equations with smooth irregular interfaces. 5 x 5 x 5 = 125-point stencils (similar to those for quadratic finite elements) for elastic heterogeneous materials are used for OLTEM. There are no unknowns at the interface points between different materials; the structure of the global discrete equations is the same for homogeneous and heterogeneous materials. The calculation of unknown stencil coefficients is based on the minimization of the local truncation error of the stencil equations and yields the optimal 10-th order of accuracy for OLTEM on unfitted Cartesian meshes, i.e., the increase by 7 orders in accuracy compared to quadratic finite elements on conformal meshes. A new post-processing procedure provides the 9-th order of accuracy for stresses in the 3-D case. Similar to basic computations it uses OLTEM with the 125-point stencils, the interface conditions and the elasticity equations. It was shown that the use of the elasticity equations for post-processing improves the accuracy of 0.1% stresses by 6 orders compared to post-processing without the use of PDEs. At an accuracy of for stresses, OLTEM with the new post-processing procedure reduces the number of degrees of freedom by 360 - 8000 times compared to quadratic finite elements with similar stencils. OLTEM with the 125-point stencils yields even more accurate results than high-order finite elements with much wider stencils. OLTEM provides accurate numerical results for compressible and nearly incompressible materials.

elasticity equations↗
DOE OSTI2019

Effect Laws of Different Factors on Levitation Characteristics of High-T{sub c} Superconducting Maglev System with Numerical Solutions

The interaction between the high-temperature superconducting (HTS) bulk and permanent magnet guideway (PMG) has been a topic among the existing HTS magnetic levitation systems. Up to now, the theories about the electromagnetic and force characteristics have been relatively consummated. Here, using the finite element model by COMSOL Multiphysics 5.3a, we investigated the impact of the intrinsic and external factors on the levitation characteristics of HTS maglev systems, the application laws are concluded. Factors including the critical current density and geometric dimensions of the bulk superconductor, as well as the moving velocity, are discussed. The effects of the multiple back and forth movements are also analyzed. By comparing calculation results of the distributions of the induced current and the levitation force of the bulk superconductor, we can clearly confirm the effect laws of various factors on the electromagnetic characteristics and levitation force. These observations support the physical parameters that are difficult to be tested by experiments. And they also have a significant contribution to the experimental condition selections and optimization of the high-temperature superconducting magnetic levitation.

75 CONDENSED MATTER PHYSICS, SUPERCONDUCTIVITY AND↗