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All sourcesNASA NTRSDOE OSTI
DOE OSTI2024

Theory of nonlinear terahertz susceptibility in ferroelectrics

An analytical theory is developed for predicting the nonlinear susceptibility of ionic polarization to continuous electromagnetic waves in both bulk and strained thin film ferroelectrics. Using a perturbation method for solving the nonlinear equation of motion for ionic polarization within the framework of Landau-Ginzburg-Devonshire theory, the full second-order nonlinear susceptibility tensor is derived as a function of frequency, temperature, and strain. Here, the theory predicts the coexistence of a significantly enhanced second-order dielectric susceptibility and a relatively low dielectric loss in BaTiO 3 films with a strain-stabilized monoclinic ferroelectric phase and in a strained SrTiO 3 film near its temperature-driven second-order ferroelectric-to-paraelectric phase transition. In this paper, we establish a theoretical framework for predicting and exploiting nonlinear interactions between terahertz waves and ferroelectric materials and, more generally, suggest exciting opportunities to strain-engineer nonlinear dynamical properties of ferroelectrics beyond the static and quasistatic limits.

36 MATERIALS SCIENCE↗
DOE OSTI2023

Physics-Informed Deep Learning-Based Proof-of-Concept Study of a Novel Elastohydrodynamic Seal for Supercritical CO 2 Turbomachinery

Supercritical carbon dioxide (sCO 2 ) power cycles show promising potential of higher plant efficiencies and power densities for a wide range of power generation applications such as fossil fuel power plants, nuclear power production, solar power, and geothermal power generation. sCO 2 leakage through the turbomachinery has been one of the main concerns in such applications. To offer a potential solution, we propose an elastohydrodynamic (EHD) seal that can work at elevated pressures and temperatures with low leakage and minimal wear. The EHD seal has a very simple, sleeve-like structure, wrapping on the rotor with minimal initial clearance at micron levels. Here, in this work, a proof-of-concept study for the proposed EHD seal was presented by using the simplified Reynolds equation and Lame’s formula for the fluid flow in the clearance and for seal deformation, respectively. The set of nonlinear equations was solved by using both the conventional Prediction–Correction (PC) method and modern Physics-Informed Neural Network (PINN). It was shown that the physics-informed deep learning method provided good computational efficiency in resolving the steep pressure gradient in the clearance with good accuracy. The results showed that the leakage rates increased quadratically with working pressures and reached a steady-state at high-pressure values of 15~20 MPa, where Q = 300 g/s at 20 MPa for an initial seal clearance of 255 μm. This indicates that the EHD seal could be tailored to become a potential solution to minimize the sCO 2 discharge in power plants.

30 DIRECT ENERGY CONVERSION↗
DOE OSTI2023

Modeling the Yaw Behavior of Tail Fins for Small Wind Turbines: November 22, 2021 - May 21, 2024

This report describes the aerodynamic modelling of the yaw behavior of tail fins for small wind turbines (SWTs). The analysis is based on unsteady slender body theory (USBT) and is formulated to be used with the proposed tail fin module to be added to OpenFAST. Linearized USBT, which has been applied in the past to SWT tail fins, is developed in three main ways. First, USBT is extended to high yaw angles and aspect ratios by modelling the associated nonlinear vortex dynamics and the chordwise load distribution respectively. Second, we consider the effect of time-varying wind speed. The extended theory is compared to recent unpublished measurements at the University of Perugia, Italy, for the yaw behavior of delta, elliptical, and rectangular tail fins without a rotor and nacelle. The fins were released from initial yaw angles of -40 degrees and -80 degrees ; the latter is of sufficient magnitude to show the importance of the nonlinear yaw dynamics. Further, the friction in the model tail fin bearings was measured and modelled. The constants in the aerodynamic models were taken from the literature but significant improvement in accuracy is achieved by using system identification techniques to optimize the values. This leads to the third main finding: extended USBT is accurate for all generic tail fin shapes that we considered and is, therefore, very suitable for inclusion in aeroelastic codes for SWTs. The final development is of a general nonlinear equation for yaw response that is simpler than the extended USBT and is also suitable for inclusion in OpenFAST. The two nonlinear models are used to highlight the geometric requirements for good tail fin design.

distributed wind↗
DOE OSTI2026

A Tensor Network-Based Quantum Algorithm for the Nonlinear 1D Burgers' Equation

In this work, we implement a tensor network-based quantum algorithm to solve unsteady, nonlinear partial differential equations (PDEs). The challenge lies in how to effectively represent, encode, process, and evolve the nonlinear system of PDEs on quantum computers. We will discuss the new techniques using the compressible 1-dimensional (1D) Burgers' equation as an example, because it represents the fundamental nonlinear feature and yet removes certain complexity in physics, allowing us to focus on the design of quantum algorithms. Previous attempts to solve nonlinear PDEs in quantum computation have often involved storing multiple copies of solutions or employing linearizations. Neither is practical due to exponential scaling with evolution time or insufficient solution accuracy. Our framework is based on matrix product states (MPSs) and matrix product operators (MPOs). For example, the velocity field is represented by MPS, whereas the linear and nonlinear spatial differential terms of the velocity field are processed by MPOs. Our primary focus herein is to verify and validate the various tensor network components of the algorithm using solutions obtained by the classical algorithms on high performance computing (HPC) architectures. We use a classical time marching method to demonstrate the functionality of the tensor network operations to model the PDE and their robustness with the time evolution of the system. Our classical simulation results demonstrate the utility of tensor network-based operations in modeling nonlinear PDEs and highlight the necessity as well as potential advantages of using quantum simulations for these techniques.

Gopalakrishnan Meena, Murali [ORNL] (ORCID:0000000↗
DOE OSTI2020

Inelastic equation of state for solids

In this study, a complete inelastic equation of state (IEOS) for solids is developed based on a superposition of thermodynamic energy potentials. The IEOS allows for a tensorial stress state by including an isochoric hyperelastic Helmholtz potential in addition to the zero-kelvin isotherm and lattice vibration energy contributions. Inelasticity is introduced through the nonlinear equations of finite strain plasticity which utilize the temperature dependent Johnson–Cook yield model. Material failure is incorporated into the model by a coupling of the damage history variable to the energy potentials. The numerical evaluation of the IEOS requires a nonlinear solution of stress, temperature and history variables associated with elastic trial states for stress and temperature. The model is implemented into the ALEGRA shock and multi-physics code and the applications presented include single element deformation paths, the Taylor anvil problem and an energetically driven thermo-mechanical problem.

42 ENGINEERING↗
DOE OSTI2021

A moving discontinuous Galerkin finite element method with interface condition enforcement for compressible flows

A variation of moving discontinuous Galerkin finite element method with interface condition enforcement (MDG-ICE) is developed for solving the compressible Euler equations. The MDG-ICE method, originating from the work of Corrigan et al. [1], [2], [3], [4], is based on the space-time DG formulation, where both flow field and grid geometry are considered as independent variables and the conservation laws are enforced both on discrete elements and element interfaces. The element conservation laws are solved in the standard discontinuous solution space to determine conservative quantities, while the interface conservation is enforced using a variational formulation in a continuous space to determine discrete grid geometry. The resulting over-determined system of nonlinear equations arising from the MDG-ICE formulation can then be solved in a least-squares sense, leading to an unconstrained nonlinear least-squares problem that is regularized and solved by Levenberg-Marquardt method. A number of numerical experiments for both 1D unsteady and 2D steady state compressible flow problems are conducted to assess the accuracy and robustness of the MDG-ICE method. Numerical results obtained indicate that the MDG-ICE method is able to implicitly detect and track all types of discontinuities via interface conservation enforcement and satisfy the conservation law on both elements and interfaces via grid movement and grid management, demonstrating that an exponential rate of convergence for Sod and Lax-Harden shock tube problems can be achieved and highly accurate solutions without overheating to both double-rarefaction wave and Noh problems can be obtained.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2021

Implicit Multirate GARK Methods

This article considers multirate generalized-structure additively partitioned Runge–Kutta methods for solving stiff systems of ordinary differential equations with multiple time scales. These methods treat different partitions of the system with different timesteps for a more targeted and efficient solution compared to monolithic single rate approaches. With implicit methods used across all partitions, methods must find a balance between stability and the cost of solving nonlinear equations for the stages. In order to characterize this important trade-off, we explore multirate coupling strategies, problems for assessing linear stability, and techniques to efficiently implement Newton iterations for stage equations. Unlike much of the existing multirate stability analysis which is limited in scope to particular methods, we present general statements on stability and describe fundamental limitations for certain types of multirate schemes. New implicit multirate methods up to fourth order are derived, and their accuracy and efficiency properties are verified with numerical tests.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2023

Thermo4PFM: Facilitating Phase-field simulations of alloys with thermodynamic driving forces

Phase-field modeling is a popular front-tracking approach used to model solidification. Its time-evolution equations are often coupled to alloy composition and/or thermal diffusion in high-resolution multiphysics approaches. Materials thermodynamic properties tabulated in CALPHAD databases can be used for phase-field modeling to parameterize bulk energies of alloys. In addition, they can be naturally integrated into models such as the Kim-Kim-Suzuki (KKS) model where driving forces depend on the differences between chemical potentials of co-existing phases. In that case, a small system of coupled nonlinear equations needs to be solved at every point in space where the phase-field order parameter is to be updated and evolved in time. Here we present Thermo4PFM, a solver for the KKS equations for binary and ternary alloys, with two or three phases, and parameterized with CALPHAD models. Thermo4PFM is open source, written in C++, and can take advantage of Graphics Processing Units (GPU) accelerators. Using OpenMP offload capabilities for C++ classes, an excellent performance is demonstrated on GPU using the LLVM compiler. CALPHAD data is read from simple JSON files using an open source parser from the boost library.

36 MATERIALS SCIENCE↗
DOE OSTI2020

DESC: A stellarator equilibrium solver

In this paper, the new code DESC is presented to solve for fixed-boundary ideal magnetohydrodynamic equilibria in stellarators. The approach directly solves the equilibrium force balance as a system of nonlinear equations in the form f(x) = 0. The independent variables x represent nested magnetic flux surfaces expressed in the inverse representation with toroidal flux coordinates, and the equations f(x) quantify equilibrium force balance errors at discrete points in real space. Discretizing with global Fourier–Zernike basis functions properly treats the magnetic axis and minimizes the number of coefficients needed to describe the flux surfaces. The pseudospectral method provides great flexibility in where the errors are evaluated, and the system of equations is efficiently solved with a Newton–Raphson iteration. Equilibria are computed and compared against VMEC for both axisymmetric and non-axisymmetric examples. The results show fast convergence rates and solutions with low errors throughout the plasma volume.

70 PLASMA PHYSICS AND FUSION TECHNOLOGY↗
DOE OSTI2022

Entropy–Preserving and Entropy–Stable Relaxation IMEX and Multirate Time–Stepping Methods

In this work, we propose entropy-preserving and entropy-stable partitioned Runge–Kutta (RK) methods. In particular, we extend the explicit relaxation Runge–Kutta methods to IMEX–RK methods and a class of explicit second-order multirate methods for stiff problems arising from scale-separable or grid-induced stiffness in a system. The proposed approaches not only mitigate system stiffness but also fully support entropy-preserving and entropy-stability properties at a discrete level. The key idea of the relaxation approach is to adjust the step completion with a relaxation parameter so that the time-adjusted solution satisfies the entropy condition at a discrete level. The relaxation parameter is computed by solving a scalar nonlinear equation at each timestep in general; however, as for a quadratic entropy function, we theoretically derive the explicit form of the relaxation parameter and numerically confirm that the relaxation parameter works the Burgers equation. Several numerical results for ordinary differential equations and the Burgers equation are presented to demonstrate the entropy-conserving/stable behavior of these methods. We also compare the relaxation approach and the incremental direction technique for the Burgers equation with and without a limiter in the presence of shocks.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2021

Sensitive parameter identification and uncertainty quantification for the stability of pipeline conveying fluid

In this study, several uncertainty quantification and sensitivity analysis methods are used to determine the most sensitive geometric and material input parameters of a cantilevered pipeline conveying fluid when uncertainty is introduced to the system at the onset of instability. The full nonlinear equations of motion are modeled using the extended Hamilton’s principle and then discretized using Galerkin’s method. A parametric study is first performed, and the Morris elementary effects are calculated to obtain a preliminary understanding of how the onset speed changes when each parameter is introduced to a ± 5% uncertainty. Then, four different input uncertainty distributions, mainly, uniform and Gaussian distribution, are chosen to investigate how input distributions affect uncertainty in the output. A convergence analysis is used to determine the number of samples needed to maintain simulation accuracy while saving the most computational time. Then, Monte Carlo simulations are run, and the output distributions for each input distribution at ± 1%, ± 3% and ± 5% input uncertainty range are found and discussed. Additionally, the Pearson correlation coefficients are evaluated for different uncertainty ranges. A final Monte Carlo study is performed in which single parameters are held constant while all others still have uncertainty. Overall, the flow speed at the onset of instability is the most sensitive to changes in the outer diameter of the pipe.

36 MATERIALS SCIENCE↗
DOE OSTI2021

Recurrent neural network-based multiaxial plasticity model with regularization for physics-informed constraints

We report a recurrent neural network (RNN) based model is developed as a surrogate to predict nonlinear plastic response under multiaxial loading. The RNN-based model is trained and tested on stress versus strain curves generated using a numerical solution based on the classical radial return method. Besides simply learning the basic constitutive relationship, a novel approach is taken to enforce certain physical conditions. Specifically, regularization is employed to maintain non-negative plastic power density throughout the loading history thereby ensuring monotonically increasing plastic work and thermodynamic consistency. Enforcing physics in this manner permits coupling of the data-driven RNN approach with physics-based knowledge and laws. This has the effect of reducing the necessary amount of data and ensuring known physical laws are not violated. Since, once trained, the model need not perform the expensive task of solving nonlinear equations, its efficiency is orders of magnitude greater than its numerical counterpart. The RNN-based model has been trained on varied sets of data and the accuracy on test datasets validated. The developed model is general and robust and has widespread application such as in the simulation of metal forming, large scale plasticity, and part life prediction.

42 ENGINEERING↗
DOE OSTI2023

Microbial Models for Simulating Soil Carbon Dynamics: A Review

Abstract Soils store the largest amount of carbon (C) in the biosphere, and the C pool in soil is critical to the global C balance. Numerous microbial models have been developed over the last few decades to represent microbial processes that regulate the responses of soil organic carbon (SOC) to climate change. However, the representation of microbial processes varies, and how microbial processes are incorporated into SOC models has not been well explored. Here, we reviewed 71 microbial models to characterize the microbial processes incorporated into SOC models and analyzed variations in mechanistic complexity. We revealed that (a) four processes (microbial‐mediated decomposition, mineral interaction, microbial necromass recycling, and active and dormant microbial dynamics) are commonly incorporated in microbial models, (b) ∼48% of models simulate only one microbial process (i.e., microbial‐mediated decomposition) and 35% of models simulate two microbial processes: for example, microbial‐mediated decomposition and mineral interaction, (c) more than 80% microbial models use nonlinear equations, such as forward Michaelis‐Menten kinetics, to represent SOC decomposition, (d) the concept of persistence of SOC due to its intrinsic properties has been replaced by organo‐mineral interaction (∼39% of microbial models) that protects SOC from decomposition, and (e) various temperature and moisture modifiers and pH effects have been used to explain the environmental effect on microbial processes. In the future, to realistically incorporate microbial processes into Earth System Models, it is imperative to identify experimental evidence on rate limitation processes and firmly ground model structure on the field and laboratory data.

Environmental Sciences & Ecology↗
DOE OSTI2021

Hamiltonian systems, Toda lattices, solitons, Lax pairs on weighted Z -graded graphs

In this study, we consider discrete one-dimensional nonlinear equations and present the procedure of lifting them to Z -graded graphs. We identify conditions that allow one to lift one-dimensional solutions to solutions on graphs. In particular, we prove the existence of solitons for static potentials on graded fractal graphs. We also show that even for a simple example of a topologically interesting graph, the corresponding non-trivial Lax pairs and associated unitary transformations do not lift to a Lax pair on the Z -graded graph.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗
DOE OSTI2021

Continuous Integration, In-Code Documentation, and Automation for Nuclear Quality Assurance Conformance

The Multiphysics Object Oriented Simulation Environment (MOOSE) is an open-source, finite element framework for solving highly coupled sets of nonlinear equations. The development of the framework and applications occurs concurrently using an agile, continuous-integration software package. Included in the framework is an in-code, extensible documentation system. Using these two tools in union with the repository management tools GitHub and GitLab, a software quality plan was created and followed such that MOOSE and a MOOSE-based application (BISON) have been shown to meet the American Society of Mechanical Engineers’ Nuclear Quality Assurance-1 standard. The approach relies heavily on automation for both testing and documentation. The resulting effort demonstrates that a rigorous software quality plan may be implemented that incurs a minimal impact on day-to-day development of the software, satisfying the stringent guidelines necessary to operate the software in a safety function within a nuclear facility.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2025

Importance of $\delta B_{\|}$ on ETG stability, turbulence, and transport in NSTX

This study employs electron-scale gyrokinetic simulations to investigate the electron temperature gradient (ETG) driven instabilities, turbulence, and transport in the pedestal region of the National Spherical Torus Experiment, comparing non-lithiated (narrow pedestal) and lithiated (wide pedestal) scenarios. Our findings reveal that, in the non-lithiated case, a branch of strongly unstable ETG modes exhibiting finite parallel magnetic field fluctuations ($\delta B_{\parallel} \neq 0$) emerges at the pedestal top and upper density pedestal region. This branch is uncovered only when $\delta B_{\parallel}$ is retained in the simulations and is associated with substantial electrostatic electron heat flux. This region of strong ETG transport corresponds to the only region in the plasma where the pressure gradient is far below the critical gradient for kinetic ballooning modes. We investigated the origin of this finite $\delta B_{\parallel}$ ETG branch by analyzing the gyrokinetic field equations. Nonlinear saturation is also analyzed and contrasted for simulations with and without $\delta B_{\parallel}$. In contrast with the nonlithiated case, ETG modes in the lithiated case produce substantial transport in the steep gradient region, but are negligible at the pedestal top.

ETG↗
DOE OSTI2021

Theoretical methods to design and test quantum simulators for the compact Abelian Higgs model

The lattice compact Abelian Higgs model is a nonperturbative regularized formulation of low-energy scalar quantum electrodynamics. In 1+1 dimensions, this model can be quantum simulated using a ladder-shaped optical lattice with Rydberg-dressed atoms. In this setup, one spatial dimension is used to carry the angular momentum of the quantum rotors. One can use truncations corresponding to spin-2 and spin-1 to build local Hilbert spaces associated with the links of the lattice. We argue that ladder-shaped configurable arrays of Rydberg atoms can be used for the same purpose. We make concrete proposals involving two and three Rydberg atoms to build one local spin-1 space (a qutrit). We show that the building blocks of the Hamiltonian calculations are models with one and two spins. We compare target and simulators using perturbative and numerical methods. The two-atom setup provides an easily controllable simulator of the one-spin model while the three-atom setup involves solving nonlinear equations. We discuss approximate methods to couple two spin-1 spaces. The article provides analytical and numerical tools necessary to design and build the proposed simulators with current technology.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗
DOE OSTI2025

Energy-momentum-conserving stochastic differential equations and algorithms for the nonlinear Landau-Fokker-Planck equation

Coulomb collision is a fundamental diffusion process in plasmas that can be described by the Landau-Fokker-Planck (LFP) equation or the stochastic differential equation (SDE). While energy and momentum are conserved exactly in the LFP equation, they are conserved only on average by the conventional corresponding SDEs, suggesting that the underlying stochastic process may not be well defined by such SDEs. Here, in this study, we derive new SDEs with exact energy-momentum conservation for the Coulomb collision by factorizing the collective effect of field particles into individual particles and enforcing Newton's third law. These SDEs, when interpreted in the Stratonovich sense, have a particularly simple form that represents pure diffusion between particles without drag. To demonstrate that the new SDEs correspond to the LFP equation, we develop numerical algorithms that converge to the SDEs and preserve discrete conservation laws. Simulation results are presented in a benchmark of various relaxation processes.

70 PLASMA PHYSICS AND FUSION TECHNOLOGY↗