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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 OSTI2026

Multiscale Modeling and Experimental Insights into High-Temperature Soil Biodegradation Dynamics of Semi-Crystalline Poly(Lactic Acid) Nonwoven Fabrics

This study investigates the biodegradation of semi-crystalline poly(lactic acid) (PLA) nonwovens (NWs) in soil at 58 °C using both experimental and mathematical modeling approaches. The model utilizes a system of parabolic diffusion-reaction partial differential equations (PDEs) to elucidate chemical transformations over time and in space. It accounts for phenomena such as the diffusion of water and lactic acid monomers through the polymer matrix and into the surrounding soil, along with their microbial breakdown. It also accounts for the initial PLA crystallinity and predicts its evolution in time. The model is solved numerically for a single filament, and the results were used to shed light on PLA NW transformations observed in soil over a 180-day incubation period. Various characterization techniques, including scanning electron microscopy (SEM), differential scanning calorimetry (DSC), and Raman spectroscopy, were employed to assess morphological changes, crystallinity, and molecular changes in the NWs throughout the experiment. By comparing the experimental data with the model predictions, the hydrolysis rate coefficient was found to be 3.37 × 10 -7 s -1 , while the rate of microbial degradation of lactic acid monomers was faster, of the order of 9.63 × 10 -7 s -1 . The findings highlight the significant role of crystallinity in the biodegradation process. The PLA degradation ceases when no amorphous material remains, and the crystallinity reaches 0.8, as observed in the experiments by day 120. Furthermore, this research contributes to a deeper understanding of PLA biodegradation dynamics and offers insights for effectively managing biodegradable materials in environmental settings.

Diffusion−reaction modeling↗

DOE OSTI2021

Efficient quantum algorithm for dissipative nonlinear differential equations

Significance Nonlinear differential equations appear in many domains and are notoriously difficult to solve. Whereas previous quantum algorithms for general nonlinear differential equations have complexity exponential in the evolution time, we give the first quantum algorithm for dissipative nonlinear differential equations that is efficient provided the dissipation is sufficiently strong relative to nonlinear and forcing terms and the solution does not decay too rapidly. We also establish a lower bound showing that differential equations with sufficiently weak dissipation have worst-case complexity exponential in time, giving an almost tight classification of the quantum complexity of simulating nonlinear dynamics. Furthermore, numerical results for the Burgers equation suggest that our algorithm may potentially address complex nonlinear phenomena even in regimes with weaker dissipation.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗

NASA NTRS1993

Analysis of Radiation-natural Convection Interactions in 1-g and low-g Environments using the Discrete Exchange Factor Method

A new numerical method is presented for the analysis of combined natural convection and radiation heat transfer with applications in many engineering situations such as materials processing, combustion and fire research. Because of the recent interest in the low gravity environment of space, attention is devoted to both 1-g and low-g applications. The two-dimensional mathematical model is represented by a set of coupled nonlinear integro-partial differential equations. Radiative exchange is formulated using the Discrete Exchange Factor method (DEF). This method considers point to point exchange and provides accurate results over a wide range of radiation parameters. Numerical results show that radiation significantly influences the flow and heat transfer in both low-g and 1-g applications. In the low-g environment, convection is weak, and radiation can easily become the dominant heat transfer mode. It is also shown that volumetric heating by radiation gives rise to an intricate cell pattern in the top heated enclosure.

Kassemi, M.↗

DOE OSTI2025

Predicting nonequilibrium Green’s function dynamics and photoemission spectra via nonlinear integral operator learning

Understanding the dynamics of nonequilibrium quantum many-body systems is an important research topic in a wide range of fields across condensed matter physics, quantum optics, and high-energy physics. However, numerical studies of large-scale nonequilibrium phenomena in realistic materials face serious challenges due to intrinsic high-dimensionality of quantum many-body problems and the absence of time-invariance. The nonequilibrium properties of many-body systems can be described by the dynamics of the correlator, or the Green's function of the system, whose time evolution is given by a high-dimensional system of integro-differential equations, known as the Kadanoff–Baym equations (KBEs). The time-convolution term in KBEs, which needs to be recalculated at each time step, makes it difficult to perform long-time numerical simulation. In this paper, we develop an operator-learning framework based on recurrent neural networks (RNNs) to address this challenge. We utilize RNNs to learn the nonlinear mapping between Green's functions and convolution integrals in KBEs. By using the learned operators as a surrogate model in the KBE solver, we obtain a general machine-learning scheme for predicting the dynamics of nonequilibrium Green's functions. Besides significant savings per each time step, the new methodology reduces the temporal computational complexity from $O(N_t^3)$ to $O(N_t)$ where N t is the number of steps taken in a simulation, thereby making it possible to study large many-body problems which are currently infeasible with conventional KBE solvers. Through various numerical examples, we demonstrate the effectiveness of the operator-learning based approach in providing accurate predictions of physical observables such as the reduced density matrix and time-resolved photoemission spectra. Moreover, our framework exhibits clear numerical convergence and can be easily parallelized, thereby facilitating many possible further developments and applications.

97 MATHEMATICS AND COMPUTING↗

NASA NTRS1984

Recent advances in methods for numerical solution of O.D.E. initial value problems

In the mathematical modeling of physical systems, it is often necessary to solve an initial value problem (IVP), consisting of a system of ordinary differential equations (ODE). A typical program produces approximate solutions at certain mesh points. Almost all existing codes try to control the local truncation error, while the user is really interested in controlling the true or global error. The present investigation provides a review of recent advances regarding the solution of the IVP, giving particular attention to stiff systems. Stiff phenomena are customarily defined in terms of the eigenvalues of the Jacobian. There are, however, some difficulties connected with this approach. It is pointed out that an estimate of the Lipschitz constant proves to be a very practical way to determine the stiffness of a problem.

Bui, T. D.↗

NASA NTRS1994

A new flux-conserving numerical scheme for the steady, incompressible Navier-Stokes equations

This paper is concerned with the continued development of a new numerical method, the space-time solution element (STS) method, for solving conservation laws. The present work focuses on the two-dimensional, steady, incompressible Navier-Stokes equations. Using first an integral approach, and then a differential approach, the discrete flux conservation equations presented in a recent paper are rederived. Here a simpler method for determining the flux expressions at cell interfaces is given; a systematic and rigorous derivation of the conditions used to simulate the differential form of the governing conservation law(s) is provided; necessary and sufficient conditions for a discrete approximation to satisfy a conservation law in E2 are derived; and an estimate of the local truncation error is given. A specific scheme is then constructed for the solution of the thin airfoil boundary layer problem. Numerical results are presented which demonstrate the ability of the scheme to accurately resolve the developing boundary layer and wake regions using grids which are much coarser than those employed by other numerical methods. It is shown that ten cells in the cross-stream direction are sufficient to accurately resolve the developing airfoil boundary layer.

Scott, James R.↗

DOE OSTI2022

Kernel learning backward SDE filter for data assimilation

In this paper, we develop a kernel learning backward SDE filter method to estimate the state of a stochastic dynamical system based on its partial noisy observations. A system of forward backward stochastic differential equations is used to propagate the state of the target dynamical model, and Bayesian inference is applied to incorporate the observational information. Further, to characterize the dynamical model in the entire state space, we introduce a kernel learning method to learn a continuous global approximation for the conditional probability density function of the target state by using discrete approximated density values as training data. Numerical experiments demonstrate that the kernel learning backward SDE is highly effective.

97 MATHEMATICS AND COMPUTING↗

DOE OSTI2021

A Privacy-Preserving Distributed Control of Optimal Power Flow

Here, we consider a distributed optimal power flow formulated as an optimization problem that maximizes a nondifferentiable concave function. Solving such a problem by the existing distributed algorithms can lead to data privacy issues because the solution information exchanged within the algorithms can be utilized by an adversary to infer the data. To preserve data privacy, in this paper we propose a differentially private projected subgradient (DP-PS) algorithm that includes a solution encryption step. We show that a sequence generated by DP-PS converges in expectation, in probability, and with probability 1. Moreover, we show that the rate of convergence in expectation is affected by a target privacy level of DP-PS chosen by the user. We conduct numerical experiments that demonstrate the convergence and data privacy preservation of DP-PS.

24 POWER TRANSMISSION AND DISTRIBUTION↗

DOE OSTI2022

A gradient-based deep neural network model for simulating multiphase flow in porous media

We report simulation of multiphase flow in porous media is crucial for the effective management of subsurface energy and environment-related activities. The numerical simulators used for modeling such processes rely on spatial and temporal discretization of the governing mass and energy balance partial-differential equations (PDEs) into algebraic systems via finite-difference/volume/element methods. These simulators usually require dedicated software development and maintenance, and suffer low efficiency from a runtime and memory standpoint for problems with multi-scale heterogeneity, coupled-physics processes or fluids with complex phase behavior. Therefore, developing cost-effective, data-driven models can become a practical choice, and in this work, we choose deep learning approaches as they can handle high dimensional data and accurately predict state variables with strong nonlinearity. In this paper, we describe a gradient-based deep neural network (GDNN) constrained by the physics related to multiphase flow in porous media. We tackle the nonlinearity of flow in porous media induced by rock heterogeneity, fluid properties, and fluid-rock interactions by decomposing the nonlinear PDEs into a dictionary of elementary differential operators. We use a combination of operators to handle rock spatial heterogeneity and fluid flow by advection. Since the augmented differential operators are inherently related to the physics of fluid flow, we treat them as first principles prior knowledge to regularize the GDNN training. We use the example of pressure management at geologic CO 2 storage sites, where CO 2 is injected in saline aquifers and brine is produced, and apply GDNN to construct a predictive model that is trained with physics-based simulation data and emulates the physics process. We demonstrate that GDNN can effectively predict the nonlinear patterns of subsurface responses, including the temporal and spatial evolution of the pressure and saturation plumes. We also successfully extend the GDNN to convolutional neural network (CNN), namely gradient-based CNN (GCNN), and validate its capability to improve the prediction accuracy. GDNN has great potential to tackle challenging problems that are governed by highly nonlinear physics and enable the development of data-driven models with higher fidelity.

42 ENGINEERING↗

DOE OSTI2025

Multi-head physics-informed neural networks for learning functional priors and uncertainty quantification

In numerous applications, the integration of prior knowledge and historical information is essential, particularly for tasks requiring the solution of ordinary or partial differential equations (ODEs/PDEs) in data-sparse or noisy environments. For instance, achieving accurate solutions to time-dependent PDEs with limited initial condition measurements necessitates an effective strategy for embedding prior knowledge. Hard-parameter sharing architectures in neural networks (NNs) have demonstrated success in both traditional and scientific machine learning domains, facilitating the learning of informative representations. Here, in this study, we introduce a novel, yet efficient, method to enhance physics-informed neural networks (PINNs) by incorporating a multi-head structure that enables the learning of functional priors from both empirical data and governing physical laws. This prior information can then be used to address data sparsity and high-level noise in solving ODE/PDE problems with uncertainty quantification (UQ). The approach, termed Multi-Head PINN (MH-PINN), consists of a shared body NN and multiple head NNs, each corresponding to an individual PINN instance. Our framework for functional prior learning is carried out in two stages: (1) training the MH-PINNs to develop a shared body NN alongside multiple head NNs, and (2) employing these trained head NNs to estimate a prior distribution through a normalizing flow-based density estimator. The learned functional prior can then be applied as a regularization mechanism in deterministic contexts or as an informative prior within a Bayesian inference framework, aiding in the resolution of subsequent ODE/PDE tasks. We evaluate the efficacy of MH-PINNs across five benchmark problems, including a high-dimensional parametric PDE, all characterized by data sparsity or substantial noise levels. Our findings reveal that MH-PINNs deliver accurate solutions and robust UQ, demonstrating adaptability across a range of complex and challenging scenarios.

Bayesian inference↗

DOE OSTI2022

Nonequilibrium spintronic transport through Kondo impurities

In this work we analyze the nonequilibrium transport through a quantum impurity (quantum dot or molecule) attached to ferromagnetic leads by using a hybrid numerical renormalization group–time-dependent density matrix renormalization group thermofield quench approach. For this, we study the bias dependence of the differential conductance through the system, which shows a finite zero-bias peak, characteristic of the Kondo resonance and reminiscent of the equilibrium local density of states. In the nonequilibrium settings, the resonance in the differential conductance is also found to decrease with increasing the lead spin polarization. The latter induces an effective exchange field that lifts the spin degeneracy of the dot level. Therefore, as we demonstrate, the Kondo resonance can be restored by counteracting the exchange field with a finite external magnetic field applied to the system. Lastly, we investigate the influence of temperature on the nonequilibrium conductance, focusing on the split Kondo resonance. Our work thus provides an accurate quantitative description of the spin-resolved transport properties relevant for quantum dots and molecules embedded in magnetic tunnel junctions.

75 CONDENSED MATTER PHYSICS, SUPERCONDUCTIVITY AND↗

DOE OSTI2022

Integration of Electromagnetic Geophysics Forward Simulation in Coupled Flow and Geomechanics for Monitoring a Gas Hydrate Deposit Located in the Ulleung Basin, East Sea, Korea

We investigate the feasibility of electromagnetic (EM) geophysics methods to detect the dissociation of gas hydrate specifically from a gas hydrate deposit located in the Ulleung Basin, East Sea, Korea via an integrated flow-geomechanics-EM geophysics simulation. To this end, coupled flow and geomechanics simulation is first performed with the multiple porosity model employed, where a mixed formulation with the finite volume (FV) and finite element (FE) methods are taken for the flow and geomechanics, respectively. From the saturation and porosity fields obtained from the coupled flow and geomechanics, the electrical conductivity model is established for the EM simulation. Solving the partial differential equation of electrical diffusion which is linearized using the 3D finite element method (FEM), the EM fields are then computed. For numerical experiments, particularly two approaches in the configuration for the EM methods are compared in this contribution: the surface-to-surface and the surface-to-borehole methods. When the surface-to-surface EM method is employed, the EM is found to be less sensitive, implying low detectability. Especially for the short term of production, the low detectability is attributed to the similarity of electrical resistivity between the dissociated gas (CH4) and hydrate as well as the specific dissociation pattern within the intercalated composites of the field. On the other hand, when the surface-to-borehole EM method is employed, its sensitivity to capture the produced gas flow is improved, confirming its detectability in monitoring gas flow. Hence, the EM geophysics simulation integrated with coupled flow and geomechanics can be a potential tool for monitoring gas hydrate deposits.

depressurization↗

NASA NTRS1977

Ray trajectories in a torus: An application of MACSYMA to a complex numerical computation

The study of ray trajectories of plasma waves in a torodial geometry using MACSYMA is an example of how symbolic, numerical, and graphical facilities can be used in concert to accomplish a complex computational goal. Computational features of this study which are of particular significance include: the derivation of code (i.e. writing functions to generate program fragments), the use of array functions to simplify the specification of a numerical iteration scheme, and the graphical presentation of the results. Mathematically, this study originates in the solution of a linear inhomogeneous partial differential equation in 3 dimensions by the method of characteristics. While it is possible to describe this equation compactly by using vector notation, and by specifying the spatial variation of the coefficients in terms of intermediate parameters, the transformation of the equation into a form amenable to solution is very tedious. A MACSYMA program is presented for obtaining description of the rf field structure excited by a waveguide located at the edge of a toroidal plasma confinement device.

Kulp, J. L.↗

NASA NTRS1977

A user's guide for V174, a program using a finite difference method to analyze transonic flow over oscillating wings

The design and usage of a pilot program using a finite difference method for calculating the pressure distributions over harmonically oscillating wings in transonic flow are discussed. The procedure used is based on separating the velocity potential into steady and unsteady parts and linearizing the resulting unsteady differential equation for small disturbances. The steady velocity potential which must be obtained from some other program, is required for input. The unsteady differential equation is linear, complex in form with spatially varying coefficients. Because sinusoidal motion is assumed, time is not a variable. The numerical solution is obtained through a finite difference formulation and a line relaxation solution method.

Butler, T. D.↗

NASA NTRS1978

On least squares approximations to indefinite problems of the mixed type

A least squares method is presented for computing approximate solutions of indefinite partial differential equations of the mixed type such as those that arise in connection with transonic flutter analysis. The method retains the advantages of finite difference schemes namely simplicity and sparsity of the resulting matrix system. However, it offers some great advantages over finite difference schemes. First, the method is insensitive to the value of the forcing frequency, i.e., the resulting matrix system is always symmetric and positive definite. As a result, iterative methods may be successfully employed to solve the matrix system, thus taking full advantage of the sparsity. Furthermore, the method is insensitive to the type of the partial differential equation, i.e., the computational algorithm is the same in elliptic and hyperbolic regions. In this work the method is formulated and numerical results for model problems are presented. Some theoretical aspects of least squares approximations are also discussed.

Fix, G. J.↗

NASA NTRS1985

Spline-based Rayleigh-Ritz methods for the approximation of the natural modes of vibration for flexible beams with tip bodies

Rayleigh-Ritz methods for the approximation of the natural modes for a class of vibration problems involving flexible beams with tip bodies using subspaces of piecewise polynomial spline functions are developed. An abstract operator theoretic formulation of the eigenvalue problem is derived and spectral properties investigated. The existing theory for spline-based Rayleigh-Ritz methods applied to elliptic differential operators and the approximation properties of interpolatory splines are useed to argue convergence and establish rates of convergence. An example and numerical results are discussed.

Rosen, I. G.↗

NASA NTRS1983

Influence of radiative heating and cumulus convection on development of mean monsoon circulation in July

Numerical simulations of July mean monsoon circulation in the tropics are described. The model used in the simulations was based on a series of primitive equations for the combined effects of variations of solar radiation, radiative diurnal warming, and large-scale and deep cumulus condensation, and the kinematic effects of topography. The initial states of the model were derived from the observed mean distributions of pressure and humidity. Analysis of the numerical results showed that the large-scale features of the mean July monsoon circulation in the tropics are created mainly by differential diabatic heating under the influence of the specific topography. The time necessary to establish the large scale features was only about 5 days when the diurnal variation of solar radiation was taken into account. Graphic illustrations of the simulated mean July flow conditions are provided.

Kuo, H. L.↗