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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 37 records · Page 2
All sourcesNASA NTRSDOE OSTI
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↗
DOE OSTI2024

Nonlinear Poisson–Boltzmann solutions for charged parallel plates: When opposite charges repel

I present an exact solution of the Poisson–Boltzmann equation for two parallel plates and discuss the solution properties. I discuss in more detail plates with opposite charges: In this case, there are two critical separations, L c,1 < L c,2 . For separations less than L c,1 , the force between plates is repulsive. It switches to attractive at L c,1 , but with the electric potential having the same sign on both plates. For L > L c,2 , the force remains attractive, and the potential at the plates has the same sign as the charge on each plate. I also describe charge regulation, determined by pK a , and provide formulas for both the critical distance where oppositely charged plates repel and their charging process. Finally, the implications of these results for the nanoparticle assembly, as driven by electrostatic interactions, are also discussed.

37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CH↗
DOE OSTI2025

Hierarchical Network Partitioning for Solution of Potential-Driven, Steady-State Nonlinear Network Flow Equations

The solution of potential-driven steady-state flow in large networks is a task which manifests in various engineering applications, such as transport of natural gas or water through pipeline networks. The resultant system of nonlinear equations depends on the network topology, and in general, there is no numerical algorithm that offers guaranteed convergence to the solution (assuming a solution exists). Some methods offer guarantees in cases where the network topology satisfies certain assumptions, but these methods fail for larger networks. On the other hand, the Newton-Raphson algorithm offers a convergence guarantee if the starting point lies close to the (unknown) solution. It would be advantageous to compute the solution of the large nonlinear system through the solution of smaller nonlinear sub-systems wherein the solution algorithms (Newton-Raphson or otherwise) are more likely to succeed. Here, this letter proposes and describes such a procedure, a hierarchical network partitioning algorithm that enables the solution of large nonlinear systems corresponding to potential-driven steady-state network flow equations.

42 ENGINEERING↗
DOE OSTI2022

Novel superposed kinklike and pulselike solutions for several nonlocal nonlinear equations

In this work, we show that a number of nonlocal nonlinear equations, including the Ablowitz–Musslimani and Yang variant of the nonlocal nonlinear Schrödinger (NLS) equation, the nonlocal modified Korteweg de Vries (mKdV) equation, and the nonlocal Hirota equation, admit novel kinklike and pulselike superposed periodic solutions. Furthermore, we show that the nonlocal mKdV equation also admits the superposed (hyperbolic) kink–antikink solution. In addition, we show that while the nonlocal Ablowitz–Musslimani variant of the NLS admits complex parity-time reversal-invariant kink and pulse solutions, neither the local NLS nor the Yang variant of the nonlocal NLS admits such solutions. Finally, except for the Yang variant of the nonlocal NLS, we show that the other three nonlocal equations admit both the kink and pulse solutions in the same model.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2022

Exponential Decay of Sensitivity in Graph-Structured Nonlinear Programs

We study solution sensitivity for nonlinear programs (NLPs) whose structures are induced by graphs. These NLPs arise in many applications such as dynamic optimization, stochastic optimization, optimization with partial differential equations, and network optimization. We illustrate that for a given pair of nodes, the sensitivity of the primal-dual solution at one node against a data perturbation at the other node decays exponentially with respect to the distance between these two nodes on the graph. In other words, the solution sensitivity decays as one moves away from the perturbation point. This result, which we call exponential decay of sensitivity, holds under the strong second-order sufficiency condition and the linear independence constraint qualification. We also present conditions under which the decay rate remains uniformly bounded; this allows us to characterize the sensitivity behavior of NLPs defined over subgraphs of infinite graphs. The theoretical developments are illustrated with numerical examples.

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 OSTI2025

First‐Order Empirical Interpolation Method for Real‐Time Solution of Parametric Time‐Dependent Nonlinear PDEs

ABSTRACT We present a model reduction approach for the real‐time solution of time‐dependent nonlinear partial differential equations (PDEs) with parametric dependencies. A major challenge in constructing efficient and accurate reduced‐order models for nonlinear PDEs is the efficient treatment of nonlinear terms. We address this by unifying the implementation of hyperreduction methods to deal with nonlinear terms. Furthermore, we introduce a first‐order empirical interpolation method (EIM) to provide an efficient approximation of the nonlinear terms in time‐dependent PDEs. We demonstrate the effectiveness of our approach on the Allen–Cahn equation, which models phase separation, and the Buckley–Leverett equation, which describes two‐phase fluid flow in porous media. Numerical results highlight the accuracy, efficiency, and stability of the proposed method compared with both the Galerkin–Newton approach and hyper‐reduced models using the standard EIM.

Nguyen, Ngoc Cuong [Center for Computational Engin↗
DOE OSTI2024

DG-IMEX method for a two-moment model for radiation transport in the $\mathscr{O}$($v$/$c$) limit

Here, we consider neutral particle systems described by moments of a phase-space density and propose a realizability-preserving numerical method to evolve a spectral two-moment model for particles interacting with a background fluid moving with nonrelativistic velocities. The system of nonlinear moment equations, with special relativistic corrections to $\mathscr{O}$($v$/$c$), expresses a balance between phase-space advection and collisions and includes velocity-dependent terms that account for spatial advection, Doppler shift, and angular aberration. The model is conservative for the correct $\mathscr{O}$($v$/$c$) Eulerian-frame number density and is consistent, to $\mathscr{O}$($v$/$c$), with Eulerian-frame energy and momentum conservation. This model is closely related to the one promoted by Lowrie et al. and similar to models currently used to study transport phenomena in large-scale simulations of astrophysical environments. The proposed numerical method is designed to preserve moment realizability, which guarantees that the moments correspond to a nonnegative phase-space density. The realizability-preserving scheme consists of the following key components: (i) a strong stability-preserving implicit-explicit (IMEX) time-integration method; (ii) a discontinuous Galerkin (DG) phase-space discretization with carefully constructed numerical uxes; (iii) a realizability-preserving implicit collision update; and(iv) a realizability-enforcing limiter. In time integration, nonlinearity of the moment model necessitates solution of nonlinear equations, which we formulate as fixed-point problems and solve with tailored iterative solvers that preserve moment realizability with guaranteed global convergence. We also analyze the simultaneous Eulerian-frame number and energy conservation properties of the semi-discrete DG scheme and propose a "spectral redistribution" scheme that promotes Eulerian-frame energy conservation. Through numerical experiments, we demonstrate the accuracy and robustness of this DG-IMEX method and investigate its Eulerian-frame energy conservation properties.

73 NUCLEAR PHYSICS AND RADIATION PHYSICS↗
DOE OSTI2020

Nonlinear dynamic analysis of solution multiplicity of buoyancy ventilation in a typical underground structure

Buoyancy ventilation is widely used in underground buildings, such as underground hydropower stations. Multiple solutions of buoyancy ventilation may exist in those underground structures. In this study, we developed a transient model comprising an ordinary differential equation system to describe buoyancy ventilation patterns in typical two-zone underground structures. Additionally, the accuracy of the model was validated. Nonlinear dynamical analysis was conducted to study multiple steady-state airflow. According to mathematical derivation, the configuration of one local heat source at the bottom corner introduces two stable solutions. The criterion to determine the stability and existence of solutions for more general scenarios was developed. Using this criterion, we obtained the multiple steady states of any two-zone underground buildings for different stack height ratios and the strength ratios of the heat sources. Furthermore, this criterion can be adopted for the design of buoyancy ventilation or natural smoke ventilation systems. Designers can change the height ratio of the stack or the heat ratio of two zones to induce the desired ventilation patterns. Finally, a case study was conducted with field measurements to demonstrate the use of the nonlinear dynamical analysis method to investigate the multiple steady states of buoyancy ventilation. Through the case study, we validated that the proposed criterion could produce the same result as the nonlinear dynamical analysis.

32 ENERGY CONSERVATION, CONSUMPTION, AND UTILIZATI↗
DOE OSTI2023

Solution and sensitivity analysis of nonlinear equations using a hypercomplex-variable Newton-Raphson method

Here, the classical Newton-Raphson (NR) method for solving nonlinear equations is enhanced in two ways through the use of hypercomplex variables and algebra. In particular, i) the Jacobian is computed in a highly accurate and automated way, and ii) the derivative of the solution to the nonlinear equations is computed with respect to any parameter contained within the system of equations. These advances provide two significant enhancements in that it is straightforward to provide an accurate Jacobian and to construct a reduced order model (ROM) of arbitrary order with respect to any parameter of the system. The ROM can then be used to approximate the solution for other parameter values without requiring additional solutions of the nonlinear equations. Several case studies are presented including 1D and 2D academic examples with fully functioning Python code provided. Additionally, a case of study of the catenary of an elastic cable subject to its own weight and a vertical point load. Derivatives up to 10th order were computed with respect to material, loading, and geometrical parameters. The derivatives were used to generate reduced order models of the cable deformation and reaction forces at its ends with respect to multiple input parameters. Results show that from a single hypercomplex evaluation of the cable under a single vertical point load, it is possible to generate an accurate reduced order model capable of predicting the cable deformation with 1.5 times the load in the opposite direction and with 3.5 times the load in the same direction without resolving the system of equations.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2025

An eigenvalue-based method for computing the relaxed pressure in compressible multiphase flow with N phases

The modeling of compressible multiphase flows is a decades-old area of study with many applications across various fields. Many of these application areas use stiff pressure relaxation. This process involves the solution of a nonlinear system with N + 1 equations and N + 1 unknowns, where N is the number of phases. The resolution of this system with general equations of state (EOSs) is difficult. Furthermore, nonlinear systems can admit multiple solutions, and current solution methods do not address this possibility. Very recently, a thermodynamic relaxation method was introduced, which effectively maps a relatively simple predictor equation of state onto a more complex target equation of state. In this context, the target EOSs are the chosen EOSs for the thermodynamic model. Furthermore, this thermodynamic relaxation has the benefit of simplifying the stiff pressure relaxation system of equations. In this article, we show this system reduces to a polynomial of degree N, which can be recast as an eigenvalue problem through the use of the associated companion matrix. We show that although this eigenvalue method is generally less efficient than Newton–Raphson iteration, it does not suffer from convergence issues and finds all N roots of the polynomial. Hence, the method provides a fail-safe for root-finding iterative methods and a way to address the issue of multiple solutions to the nonlinear system of equations in stiff pressure relaxation.

Eigenvalue algorithm↗
DOE OSTI2022

Asynchronous domain decomposition methods for nonlinear PDEs

One- and two-level parallel asynchronous methods for the numerical solution of nonlinear systems of equations, especially those arising from (nonlinear) partial differential equations, are studied. The proposed methods are based on domain decomposition techniques. Local convergence theorems are presented in several cases, with appropriate hypotheses. Computational results on a shared memory multiprocessor machine for various problems exhibiting nonlinearities are reported, illustrating the potential of these asynchronous methods, especially for heterogeneous clusters.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2024

TNet: A Model-Constrained Tikhonov Network Approach for Inverse Problems

Deep learning (DL), in particular deep neural networks, by default is purely data-driven and in general does not require physics. This is the strength of DL but also one of its key limitations when applied to science and engineering problems in which underlying physical properties—such as stability, conservation, and positivity—and accuracy are required. DL methods in their original forms are often not capable of respecting the underlying mathematical models or achieving desired accuracy even in big-data regimes. On the other hand, many data-driven science and engineering problems, such as inverse problems, typically have limited experimental or observational data, and DL would overfit the data in this case. Leveraging information encoded in the underlying mathematical models, we argue, not only compensates for missing information in low data regimes but also provides opportunities to equip DL methods with the underlying physics, hence promoting better generalization. This paper develops a model-constrained DL approach and its variant TNet—a Tikhonov neural network—which are capable of learning not only information hidden in the training data but also in the underlying mathematical models to solve inverse problems governed by partial differential equations in low data regimes. We provide the constructions and some theoretical results for the proposed approaches for both linear and nonlinear inverse problems. Since TNet is designed to learn inverse solutions with Tikhonov regularization, it is interpretable: in fact it recovers Tikhonov solutions for linear cases while potentially approximating Tikhonov solutions for nonlinear inverse problems. We also prove that data randomization can enhance not only the smoothness of the networks but also their generalizations. Comprehensive numerical results confirm the theoretical findings and show that with even as little as 1 training data sample for one-dimensional (1D) deconvolution, 5 for an inverse 2D heat conductivity problem, 100 for inverse initial conditions for a time-dependent 2D Burgers’s equation, and 50 for inverse initial conditions for 2D Navier–Stokes equations, TNet solutions can be as accurate as Tikhonov solutions while being several orders of magnitude faster. Furthermore, this is possible owing to the model-constrained term, replications, and randomization.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2020

Variational nonlinear WKB in the Eulerian frame

Nonlinear WKB is a multiscale technique for studying locally plane-wave solutions of nonlinear partial differential equations (PDEs). Its application comprises two steps: (1) replacement of the original PDE with an extended system separating the large scales from the small and (2) reduction of the extended system to its slow manifold. In the context of variational fluid theories with particle relabeling symmetry, nonlinear WKB in the mean Eulerian frame is known to possess a variational structure. This much has been demonstrated using, for instance, the theoretical apparatus known as the generalized Lagrangian mean. On the other hand, the variational structure of nonlinear WKB in the conventional Eulerian frame remains mysterious. By exhibiting a variational principle for the extended equations from step (1) above, here we demonstrate that nonlinear WKB in the Eulerian frame is in fact variational. Remarkably, the variational principle for the extended system admits loops of relabeling transformations as a symmetry group. Noether’s theorem therefore implies that the extended Eulerian equations possess a family of circulation invariants parameterized by S 1 . As an illustrative example, we use our results to systematically deduce a variational model of high-frequency acoustic waves interacting with a larger-scale compressible isothermal flow.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2020

Stretched and Filtered Multigroup Pn Transport for Improved Positivity and Accuracy

One of the common methods for solving the radiation transport equation is to use a polynomial expansion for the angle variable(s). Recent research that reduces the oscillations and improves the positivity of the gray transport equation solutions is here applied to the multigroup transport equations. Constant scale factors that stretch the time axis and constant scattering opacities that filter the solution greatly increase the accuracy of the solution with no added nonlinearities. No new solution techniques are required. Test problems are presented in one and two dimensions.

73 NUCLEAR PHYSICS AND RADIATION PHYSICS↗
DOE OSTI2020

Adaptive Activation Functions Accelerate Convergence in Deep and Physics-informed Neural Networks

We employ adaptive activation functions for regression in deep and physics-informed neural networks (PINNs) to approximate smooth and discontinuous functions as well as solutions of linear and nonlinear partial differential equations. In particular, we solve the nonlinear Klein-Gordon equation, which has smooth solutions, the nonlinear Burgers equation, which can admit high gradient solutions, and the Helmholtz equation. We introduce a scalable hyper-parameter in the activation function, which can be optimized to achieve best performance of the network as it changes dynamically the topology of the loss function involved in the optimization process. The adaptive activation function has better learning capabilities than the traditional one (fixed activation) as it improves greatly the convergence rate, especially at early training, as well as the solution accuracy. To better understand the learning process, we plot the neural network solution in the frequency domain to examine how the network captures successively different frequency bands present in the solution. We consider both forward problems, where the approximate solutions are obtained, as well as inverse problems, where parameters involved in the governing equation are identified. Our simulation results show that the proposed method is a very simple and effective approach to increase the efficiency, robustness and accuracy of the neural network approximation of nonlinear functions as well as solutions of partial differential equations, especially for forward problems. We theoretically prove that in the proposed method, gradient descent algorithms are not attracted to suboptimal critical points or local minima.

machine leaning, Bad minima, Inverse problems, Phy↗