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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 1,189 records · Page 66
All sourcesNASA NTRSDOE OSTI
DOE OSTI2018

A new computational approach for the solutions of generalized pantograph-delay differential equations

In this study, a new computational approach is presented to solve the generalized pantograph-delay differential equations (PDDEs). The solutions obtained by our scheme represent by a linear combination of a special kind of basis functions, and can be deduced in a straightforward manner. Firstly, using the least squares approximation method and the Lagrange-multiplier method, the given PDDE is converted to a linear system of algebraic equations, and those unknown coefficients of the solution of the problem are determined by solving this linear system. Secondly, a PDDE related to the error function of the approximate solution is constructed based on the residual error function technique, and error estimation is presented for the suggested method. The convergence of the approximate solution is proved. Several numerical examples are given to demonstrate the accuracy and efficiency. Comparisons are made between the proposed method and other existing methods.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2018

On differential singular perturbation problems: a simple variational approach

An alternative approach for the analysis and the numerical approximation of singular perturbation problems, using a variational framework, is presented. It is based on the natural idea of minimizing the residual of the differential equation measured in the L{sup 2} norm. By this strategy, the approximation is based in the solution of a set of linear problems giving the descent step directions of the problem. This is the main advantage of our approach, since we can use stable and convergent methods for linear problems (without assuming the knowledge of good initial guesses used in the approximation of the associated non-linear systems necessary in Newton-type methods). Remember that for this type of problems, we should use implicit methods. We prove that our procedure can never get stuck in local minima, and the error decreases until getting to the original solution independent of the perturbation parameter. Finally, we include some numerical examples.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2018

Fifth-kind orthonormal Chebyshev polynomial solutions for fractional differential equations

The principal aim of the current paper is to present and analyze two new spectral algorithms for solving some types of linear and nonlinear fractional-order differential equations. The proposed algorithms are obtained by utilizing a certain kind of shifted Chebyshev polynomials called the shifted fifth-kind Chebyshev polynomials as basis functions along with the application of a modified spectral tau method. The class of fifth-kind Chebyshev polynomials is a special class of a basic class of symmetric orthogonal polynomials which are constructed with the aid of the extended Sturm–Liouville theorem for symmetric functions. An investigation for the convergence and error analysis of the proposed Chebyshev expansion is performed. For this purpose, a new connection formulae between Chebyshev polynomials of the first and fifth kinds are derived. The obtained numerical results ascertain that our two proposed algorithms are applicable, efficient and accurate.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2018

A computationally efficient method for tempered fractional differential equations with application

This paper investigates the numerical approximation of the tempered fractional integral by using the Sinc-collocation scheme. The algorithm is extended to solve a class of tempered fractional differential equations that converges to the solution with exponential rate. Several numerical examples compare the numerical approximations with the exact solutions. The behavioral responses of the lumped capacitance model with tempered fractional order for transient conduction are investigated. The efficiency and accuracy of the proposed scheme are analyzed in the perspective of the L{sub 2}-norm error and convergence order.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2018

Variational multiscale element-free Galerkin method combined with the moving Kriging interpolation for solving some partial differential equations with discontinuous solutions

Solving partial differential equations with discontinuous solutions is an important challenging problem in numerical analysis. To this end, there are some methods such as finite volume method, discontinuous Galerkin approach and particle technique that are able to solve these problems. In the current paper, the moving Kriging element-free Galerkin method has been combined with the variational multiscale algorithm to obtain acceptable and high-resolution solutions. For testing this technique, we select some PDEs with discontinuous solution such as Burgers’, Sod’s shock tube, advection–reaction–diffusion, Kuramoto–Sivashinsky, Boussinesq and shallow water equations. First, we obtain a time-discrete scheme by approximating time derivative via finite difference technique. Then we introduce the moving Kriging interpolation and also obtain their shape functions. We use the element-free Galerkin method for approximating the spatial derivatives. This method uses a weak form of the considered equation that is similar to the finite element method with the difference that in the classical element-free Galerkin method test and trial functions are moving least squares approximation (MLS) shape functions. Since the shape functions of moving least squares (MLS) approximation do not have Kronecker delta property, we cannot implement the essential boundary condition, directly. Thus, we employ the shape functions of moving Kriging interpolation and radial point interpolation technique which have the mentioned property. Also, in the element-free Galekin method, we do not use any triangular, quadrangular or other type of meshes. The element-free Galerkin method is a global method while finite element method is a local one. This technique employs a background mesh for integration which makes it different from the truly mesh procedures. Several test problems are solved and numerical simulations are reported which confirm the efficiency of the proposed schemes.

97 MATHEMATICS AND COMPUTING↗
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 OSTI2018

On spectral methods for solving variable-order fractional integro-differential equations

This paper applies the shifted Jacobi–Gauss collocation (SJ–G-C) method for solving variable-order fractional integro-differential equations (VO-FIDE) with initial conditions. The Riemann–Liouville fractional derivative, D{sup ν(x)}, and integral, I{sup ν(x)}, of variable order are combined, and the SJ–G-C applied to produce a system of algebraic equations. Numerical experiments demonstrate the applicability and reliability of the algorithm when compared with current methods.

97 MATHEMATICS AND COMPUTING↗
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 OSTI2018

A Müntz wavelets collocation method for solving fractional differential equations

In this paper, we introduce a new set of wavelets, namely Müntz wavelets. These wavelets are defined based on Müntz Legendre polynomials which have the property of orthogonality. We stabilize the wavelet coefficients by utilizing Jacobi polynomials to construct the modified Müntz wavelets. Next, we present a collocation scheme based on Müntz wavelets for solving fractional differential equations. The fractional derivative is described in the Caputo sense. Some error estimates are given and numerical examples are included to demonstrate the applicability and accuracy of the proposed method. This includes the elastic-dissipative mathematical model of linear mechanical devices known as Boltzman operator.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2018

Two new fractional derivatives of variable order with non-singular kernel and fractional differential equation

In this paper, we introduce two new fractional derivatives with non-singular kernel and present a series of other fractional derivatives derived from the new formulations. We present some important results on uniformly convergent sequences of continuous functions with the help of the comparison principle and other results that allow the study of the limitation of the fractional nonlinear differential equation.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2018

Numerical multi-scaling method to solve the linear stochastic partial differential equations

In this paper, we propose a multi-scaling method to solve a class of the stochastic partial differential equations (SPDEs) of Itô-type. By this method, we approximate the solution of the SPDEs with strongly elliptic operator on a finite domain. The method is based on B-spline wavelet approximations; some of these functions are reshaped to satisfy on boundary conditions exactly. We prove the consistency and stability of the method. Illustrative examples are included to demonstrate the validity and applicability of our proposed technique.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2024

Efficient sensitivity analysis of the thermal profile in powder bed fusion of metals using hypercomplex automatic differentiation finite element method

Rapid cyclic temperature fluctuation occurring in powder bed fusion of metals using a laser beam (PBF-LB/M) influences the formation of flaws in printed parts. Consequently, there is a pressing need to enhance the quality of printed parts by developing innovative methodologies that can predict thermal histories and help uncover the intricate relationships between process parameters and thermal profiles. Sensitivity Analysis (SA) emerges as an essential tool for this, offering the potential for process optimization and enhanced quality control. Nonetheless, conventional SA methodologies often incur in excessive computational costs and potential numerical approximation errors. Here, to address this technical challenge, we present a novel method for SA that integrates the HYPercomplex-based Automatic Differentiation (HYPAD) technique with transient thermal simulations conducted via the finite element method (FEM). Leveraging this methodology, we efficiently and accurately perform SA for PBF-LB/M processes in a post-processing step. Compared to traditional methods like Finite Differences (FD), HYPAD-FEM required 96 % less computational time for obtaining sensitivities for 22 process parameters, under a comparative study conducted within the context of the 2018–02 AM benchmark of the National Institute of Standards and Technology. In summary, HYPAD-FEM offers superior efficiency and accuracy in SA over conventional methods, delivering the best sensitivity of a model without the need for step-size selection and problem or parameter-based implementations.

36 MATERIALS SCIENCE↗
DOE OSTI2019

Shape functions and singly differential cross sections of bremsstrahlung at electron energies from 10 eV to 3 MeV for Z = 1–100

This work describes a library of shape functions and singly differential cross sections of bremsstrahlung due to interaction of the electrons with energies from 10 eV to 3 MeV with neutral atoms. The calculations have been performed using an updated version of the set of codes BREMS, which was described in a previous publication. The calculation method is based on the relativistic partial-wave formulation describing interaction of the incident electron with a central potential of the neutral target atom. The interaction potential is obtained from the Kohn–Sham density functional theory. The Dirac equation for the continuum wave function is solved by the power-series method. The mentioned library uses a comparatively dense grid of the incident electron energy (ten points per decade), and it covers all chemical elements with atomic numbers from 1 to 100. The agreement of the present data with the previously-published tabulations is satisfactory.

73 NUCLEAR PHYSICS AND RADIATION PHYSICS↗
DOE OSTI2021

Differentiating densities on smooth manifolds

Lebesgue integration of derivatives of strongly-oscillatory functions is a recurring challenge in computational science and engineering. Integration by parts is an effective remedy for huge computational costs associated with Monte Carlo integration schemes. In case of Lebesgue integrals over a smooth manifold, however, integration by parts gives rise to a derivative of the density implied by charts describing the domain manifold. This paper focuses on the computation of that derivative, which we call the density gradient function, on general smooth manifolds. We analytically derive formulas for the density gradient and present examples of manifolds determined by popular differential equation-driven systems. Furthermore, we highlight the significance of the density gradient by demonstrating a numerical example of Monte Carlo integration involving oscillatory integrands.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2021

Quasi-differential neutron induced neutron emissions from 235 U, and 239 Pu

Uncertainty in nuclear reaction cross sections, angular distributions, and other nuclear data directly impact how well simulations of nuclear systems represent physical observations. Here, to determine how well the nuclear data in ENDF/B-VIII.0, JEFF-3.3, and JENDL-4.0 evaluations describe the physical behaviour of 235 U, and 239 Pu when subjected to a neutron flux, the neutron emission spectrum was measured for carbon, 93.0% 235 U, and 93.9% 239 Pu samples, and compared against detailed MCNP6 simulations. The measurements were performed at the Los Alamos Neutron Science Center using a quasi-differential method previously developed at Rensselaer Polytechnic Institute. The measurement spanned 0.82–20 MeV and 30–150 degrees. The measurements show there are a significant number of discrepancies between library predictions of the neutron yield and physical observation. A few of the main discrepancies found are described in this paper. Based on these results a new evaluation utilizing these results for carbon, 235 U, and 239 Pu is recommended.

235U↗
DOE OSTI2017

Generalizing global error estimation for ordinary differential equations by using coupled time-stepping methods

Here, this study introduces new time-stepping strategies with built-in global error estimators. The new methods propagate the defect along with the numerical solution much like solving for the correction or Zadunaisky’s procedure; however, the proposed approach allows for overlapped internal computations and, therefore, represents a generalization of the classical numerical schemes for solving differential equations with global error estimation. The resulting algorithms can be effectively represented as general linear methods. Finally, several explicit self-starting schemes akin to Runge–Kutta methods with global error estimation are introduced, and the theoretical considerations are illustrated in several examples.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2021

Probabilistic learning on manifolds constrained by nonlinear partial differential equations for small datasets

A novel extension of the Probabilistic Learning on Manifolds (PLoM) is presented. It makes it possible to synthesize solutions to a wide range of nonlinear stochastic boundary value problems described by partial differential equations (PDEs) for which a stochastic computational model (SCM) is available and which depend on a vector-valued random control parameter. The cost of a single numerical evaluation of this SCM is assumed to be such that only a limited number of points can be computed for constructing the training dataset (small data). Each point of the training dataset is made up of realizations from a vector-valued stochastic process (the stochastic solution) and the associated random control parameter on which it depends. The presented PLoM constrained by PDE allows for generating a large number of learned realizations of the stochastic process and its corresponding random control parameter. These learned realizations are generated so as to minimize the vector-valued random residual of the PDE in the mean-square sense. Appropriate novel methods are developed to solve this challenging problem. Three applications are presented. The first one is a simple uncertain nonlinear dynamical system with a nonstationary stochastic excitation. The second one concerns the 2D nonlinear unsteady Navier–Stokes equations for incompressible flows in which the Reynolds number is the random control parameter. Here, the last one deals with the nonlinear dynamics of a 3D elastic structure with uncertainties. The results obtained make it possible to validate the PLoM constrained by stochastic PDE but also provide further validation of the PLoM without constraint.

Machine learning↗
DOE OSTI2024

Divide and conquer: Learning chaotic dynamical systems with multistep penalty neural ordinary differential equations

Forecasting high-dimensional dynamical systems is a fundamental challenge in various fields, such as geosciences and engineering. Neural Ordinary Differential Equations (NODEs), which combine the power of neural networks and numerical solvers, have emerged as a promising algorithm for forecasting complex nonlinear dynamical systems. However, classical techniques used for NODE training are ineffective for learning chaotic dynamical systems. In this work, we propose a novel NODE-training approach that allows for robust learning of chaotic dynamical systems. Here, our method addresses the challenges of non-convexity and exploding gradients associated with underlying chaotic dynamics. Training data trajectories from such systems are split into multiple, non-overlapping time windows. In addition to the deviation from the training data, the optimization loss term further penalizes the discontinuities of the predicted trajectory between the time windows. The window size is selected based on the fastest Lyapunov time scale of the system. Multi-step penalty(MP) method is first demonstrated on Lorenz equation, to illustrate how it improves the loss landscape and thereby accelerates the optimization convergence. MP method can optimize chaotic systems in a manner similar to least-squares shadowing with significantly lower computational costs. Our proposed algorithm, denoted the Multistep Penalty NODE, is applied to chaotic systems such as the Kuramoto-Sivashinsky equation, the two-dimensional Kolmogorov flow, and ERA5 reanalysis data for the atmosphere. It is observed that MP-NODE provide viable performance for such chaotic systems, not only for short-term trajectory predictions but also for invariant statistics that are hallmarks of the chaotic nature of these dynamics.

Chaotic dynamical systems↗