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DOE OSTI2016

NONLINEAR MULTIGRID SOLVER EXPLOITING AMGe COARSE SPACES WITH APPROXIMATION PROPERTIES

The paper introduces a nonlinear multigrid solver for mixed nite element discretizations based on the Full Approximation Scheme (FAS) and element-based Algebraic Multigrid (AMGe). The main motivation to use FAS for unstruc- tured problems is the guaranteed approximation property of the AMGe coarse spaces that were developed recently at Lawrence Livermore National Laboratory. These give the ability to derive stable and accurate coarse nonlinear discretization problems. The previous attempts (including ones with the original AMGe method, [5, 11]), were less successful due to lack of such good approximation properties of the coarse spaces. With coarse spaces with approximation properties, our FAS approach on un- structured meshes should be as powerful/successful as FAS on geometrically re ned meshes. For comparison, Newton's method and Picard iterations with an inner state-of-the-art linear solver is compared to FAS on a nonlinear saddle point problem with applications to porous media ow. It is demonstrated that FAS is faster than Newton's method and Picard iterations for the experiments considered here. Due to the guaranteed approximation properties of our AMGe, the coarse spaces are very accurate, providing a solver with the potential for mesh-independent convergence on general unstructured meshes.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2018

Nonlinear Modeling of Macroscopic Plasma Dynamics Final Scientific Report May 1, 2006 – April 30, 2018

The project entitled, "Nonlinear Modeling of Macroscopic Plasma Dynamics," applied numerical computation to solve mathematical models of plasma behavior in laboratory systems. Plasma is the state of matter with sufficiently large internal energy density such that electrons dissociate from atoms to form gas of electrically charged particles. It occurs naturally in stars, planetary magnetospheres, and interstellar and intergalactic media. It is also used for industrial purposes and in lighting. Fusion energy science seeks to create and sustain plasma conditions where atomic nuclei fuse to create energy that can be captured for power production. Many laboratory experiments for this effort use magnetic field to contain plasma and to thermally insulate it from its surroundings. Possible plasma dynamics in magnetized systems include collective behavior over spatial scales that are comparable to the configuration size, and these dynamics often include evolution of the magnetic field that is used for confinement. Solving the mathematical models that describe these dynamics in specific experiments often requires computer simulations, as done in this project. This project analyzed plasma magnetic relaxation dynamics in the laboratory configurations known as the reversed-field pinch (RFP), the spheromak, and the spherical torus (ST). It also developed capabilities for modeling disruptive plasma dynamics in the tokamak, which has achieved the highest performance for magnetized plasma and is the design of the large international ITER experiment (https://www.iter.org) that is under construction and is expected to be the first to achieve self-heating conditions. Dynamic evolution to states of low energy, the aforementioned magnetic relaxation, is known to be important for generating the RFP and spheromak configurations. Nonlinear computational modeling with the Non-Ideal Magnetohydrodynamics with Rotation, Open Discussion code (NIMROD, https://nimrodteam.org) was applied to understand specific aspects of magnetic relaxation in RFPs, spheromaks, and STs. The RFP study considered how the nonlinear transfer of power among different fluctuation harmonics is altered due to the application of pulses of electric field, as done experimentally in the Madison Symmetric Torus RFP device at the University of Wisconsin-Madison. It also investigated the separation of electron and ion dynamics, known as two-fluid effects, including effects on the evolution of the plasma flow profile. The spheromak study considered the influence of these two-fluid effects in conditions relevant to the Sustained Spheromak Experiment (SSPX), formerly at Lawrence Livermore National Laboratory. Our study found that two-fluid effects enhance stability with respect to interchange in SSPX-relevant conditions, but having a complete model with two-fluid effects in separate electron and ion temperature equations is important. The study for STs focused on plasma startup in the Pegasus Toroidal Experiment at UW-Madison, which uses DC voltage applied through a small plasma gun. The simulations show how the resulting helical current stream relaxes through repetitive merging, forming current rings that accumulate into a tokamak-like configuration. Analytical theory was also developed to help explain behavior that was observed in the RFP and spheromak computations. Although the tokamak configuration has outperformed other magnetic confinement configurations, it has a tendency to undergo discharge-terminating dynamics that release energy onto plasma-facing surfaces and exert electromechanical forces on magnetic coils and structures. Our efforts have improved NIMROD's capabilities to model tokamak disruption by incorporating a resistive wall model that can be applied to axisymmetric, as well as asymmetric, distortions of the magnetic field configuration. Computations addressed whether mathematical conditions applied to plasma flow at the wall influence disruptive dynamics.

70 PLASMA PHYSICS AND FUSION TECHNOLOGY↗
DOE OSTI2016

The 2015 Nonlinear Mechanics and Dynamics Research Institute

The Nonlinear Mechanics and Dynamics (NOMAD) Research Institute is a six week long collaborative research program for graduate students from across the world. The 2015 NOMAD Research Institute was hosted jointly by Sandia National Laboratories and the University of New Mexico, and featured 24 graduate students working on seven different research projects. These projects included: developing experimental strategies for studying the dynamics of nonlinear systems, a numerical round robin for predicting the response of a jointed system, quantification of uncertainty in a lap joint, assessment of experimental substructuring methods, a study of stress waves propagating through jointed interfaces, structural design optimization with joints, and the nonlinear system identification of MEMS devices. This report details both the technical research and the programmatic organization of the 2015 NOMAD Research Institute.

42 ENGINEERING↗
DOE OSTI2015

The 2014 Sandia Nonlinear Mechanics and Dynamics Summer Research Institute

A collaborative research institute was organized and held at Sandia Albuquerque for a period of six weeks. This research institute brought together researchers from around the world to work collaboratively on a set of research projects. These research projects included: developing experimental guidelines for studying variability and repeatability of nonlinear structures; decoupling aleatoric and epistemic uncertainty in measurements to improve dynamic predictions; a numerical round robin to assess the performance of five different numerical codes for modeling systems with strong nonlinearities; and an assessment of experimentally derived and numerically derived reduced order models. In addition to the technical collaborations, the institute also included a series of seminars given by both Sandians and external experts, as well as a series of tours and field trips to local places of scientific and engineering importance. This report details both the technical research and the programmatic organization of the 2014 Sandia Nonlinear Mechanics and Dynamics Summer Research Institute.

36 MATERIALS SCIENCE↗
DOE OSTI2021

Equations of Motion for the Vertical Rigid-Body Rotor: Linear and Nonlinear Cases

Centuries ago, the prolific mathematician Leonhard Euler (1707–1783) wrote down the equations of motion (EOM) for the heavy symmetrical top with one point fixed. The resulting set of equations turned out to be nonlinear and had a limited number of closed-form solutions. Today, tools such as transfer matrix and finite elements enable the calculation of the rotor dynamic properties for rotor-bearing systems. Some of these tools rely on the “linearized” version of the EOM to calculate the eigenvalues, unbalance response, or transients in these systems. In fact, industry standards mandate that rotors be precisely balanced to have safe operational characteristics. However, in some cases, the nonlinear aspect of the EOM should be considered. The purpose of this paper is to show examples of how the linear vs. nonlinear formulations differ. This paper will also show how excessive unbalance is capable of dramatically altering the behavior of the system and can produce chaotic motions associated with the “jump” phenomenon.

42 ENGINEERING↗
DOE OSTI2024

Approximate Entropy Analysis for Nonlinear Beam Dynamics

In this paper, we applied an approximate entropy (ApEn) analysis to nonlinear beam dynamics. Due to the presence of strong nonlinear magnets, the degree of chaos in beam motion gradually increases proportionally with its amplitude. Such chaos can be quantitatively characterized with ApEn of beam turn-by-turn readings. ApEn analysis is a technique used to quantify the amount of regularity and the unpredictability of fluctuations with respect to time-series data. The ApEn, when applied as a chaos indicator, can then be used for nonlinear lattice optimization and analysis. The National Synchrotron Light Source II (NSLS-II) electron storage ring lattice was used as an example of a real-world application of our technique.

43 PARTICLE ACCELERATORS↗
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 OSTI2024

Developing nonlinear laser-plasma instability models for high-fidelity, multi-physics simulation capability for ICF/HED (IC Report) [Slides]

Modeling nonlinear effects of laser plasma instabilities (LPI) is critically important for ICF/HED experiments. Our project has the following goals: • Develop physics-based nonlinear LPI models using PIC simulations • Couple the nonlinear LPI effects to macroscopic modeling of ICF/HED exp. through laser ray-tracing (LRT).

70 PLASMA PHYSICS AND FUSION TECHNOLOGY↗
DOE OSTI2025

Toroidal Alfven Wave Coupling (Nonlinear Wave-Wave Interactions) on DIII-D

Connect DIII-D physics with space plasma phenomena. In this case of using the toroidal Alfvén eigenmodes and frequency-chirping Reversed-Shear Alfvén eignmodes in DIII-D, we will document how the nonlinear interactions among dipolar Kinetic Alfvén Wave eigenmodes in space plasmas may determine saturation levels of these fluctuations. We seek evidence of nonlinear energy transfer and wave-wave coupling during 3-wave interactions mediated by a much lower-frequency mode. In FY2019, we found evidence of nonlinear “wave-wave” interactions in 175 relevant shots of archival DIII-D data. Toroidal mode number was identified and spectrograms were produced from each shot’s Mirnov coil data. Bispectral analysis was performed using a preliminary version of a new user friendly code derived from a 1995 M.S. thesis at WVU. These results formed a part of a May 2019 M.S. thesis at WVU.

70 PLASMA PHYSICS AND FUSION TECHNOLOGY↗
DOE OSTI2018

Decomposition of Nonlinear Dynamical Systems Using Koopman Gramians

In this paper we propose a new Koopman operator approach for nonlinear model decomposition using Koopman gramians. We introduce Koopman operators and discuss the notion of a input-Koopman operator. We show how input-Koopman operators can be used to cast a nonlinear system into the classical state-space form, as well as conditions under which input and state observable functions are well separated. We extend an existing method of dynamic mode decomposition for learning Koopman operators from data known as deep dynamic mode decomposition, for systems with control or disturbances. We illustrate the accuracy of the method on learning an input-state separable Koopman operator for an example system, even when the underlying system exhibits mixed state-input terms. Finally, we introduce a nonlinear decomposition algorithm, based on Koopman gramians, that maximizes internal subsystem observability and disturbance rejection from unwanted noise in other subsystems. To relieve the NP-hardness of the decomposition problem, we derive a relaxation based on Koopman gramians and multi-way partitioning. We illustrate our algorithm on a swing dynamics model for a IEEE 39 bus and discuss our findings.

Learning, Control, Distributed Control, koopman op↗
DOE OSTI2020

Closure Learning for Nonlinear Model Reduction Using Deep Residual Neural Network

Developing accurate, efficient, and robust closure models is essential in the construction of reduced order models (ROMs) for realistic nonlinear systems, which generally require drastic ROM mode truncations. We propose a deep residual neural network (ResNet) closure learning framework for ROMs of nonlinear systems. The novel ResNet-ROM framework consists of two steps: (i) In the first step, we use ROM projection to filter the given nonlinear system and construct a spatially filtered ROM. This filtered ROM is low-dimensional, but is not closed. (ii) In the second step, we use ResNet to close the filtered ROM, i.e., to model the interaction between the resolved and unresolved ROM modes. We emphasize that in the new ResNet-ROM framework, data is used only to complement classical physical modeling (i.e., only in the closure modeling component), not to completely replace it. We also note that the new ResNet-ROM is built on general ideas of spatial filtering and deep learning and is independent of (restrictive) phenomenological arguments, e.g., of eddy viscosity type. The numerical experiments for the 1D Burgers equation show that the ResNet-ROM is significantly more accurate than the standard projection ROM. The new ResNet-ROM is also more accurate and significantly more efficient than other modern ROM closure models.

Xie, Xuping↗
DOE OSTI2016

Responses of two nonlinear microbial models to warming and increased carbon input

A number of nonlinear microbial models of soil carbon decomposition have been developed. Some of them have been applied globally but have yet to be shown to realistically represent soil carbon dynamics in the field. A thorough analysis of their key differences is needed to inform future model developments. In this paper, we compare two nonlinear microbial models of soil carbon decomposition: one based on reverse Michaelis–Menten kinetics (model A) and the other on regular Michaelis–Menten kinetics (model B). Using analytic approximations and numerical solutions, we find that the oscillatory responses of carbon pools to a small perturbation in their initial pool sizes dampen faster in model A than in model B. Soil warming always decreases carbon storage in model A, but in model B it predominantly decreases carbon storage in cool regions and increases carbon storage in warm regions. For both models, the CO 2 efflux from soil carbon decomposition reaches a maximum value some time after increased carbon input (as in priming experiments). This maximum CO 2 efflux (F max ) decreases with an increase in soil temperature in both models. However, the sensitivity of F max to the increased amount of carbon input increases with soil temperature in model A but decreases monotonically with an increase in soil temperature in model B. These differences in the responses to soil warming and carbon input between the two nonlinear models can be used to discern which model is more realistic when compared to results from field or laboratory experiments. Lastly, these insights will contribute to an improved understanding of the significance of soil microbial processes in soil carbon responses to future climate change.

58 GEOSCIENCES↗
DOE OSTI2016

Inversion of geothermal heat flux in a thermomechanically coupled nonlinear Stokes ice sheet model

We address the inverse problem of inferring the basal geothermal heat flux from surface velocity observations using a steady-state thermomechanically coupled nonlinear Stokes ice flow model. This is a challenging inverse problem since the map from basal heat flux to surface velocity observables is indirect: the heat flux is a boundary condition for the thermal advection–diffusion equation, which couples to the nonlinear Stokes ice flow equations; together they determine the surface ice flow velocity. This multiphysics inverse problem is formulated as a nonlinear least-squares optimization problem with a cost functional that includes the data misfit between surface velocity observations and model predictions. A Tikhonov regularization term is added to render the problem well posed. We derive adjoint-based gradient and Hessian expressions for the resulting partial differential equation (PDE)-constrained optimization problem and propose an inexact Newton method for its solution. As a consequence of the Petrov–Galerkin discretization of the energy equation, we show that discretization and differentiation do not commute; that is, the order in which we discretize the cost functional and differentiate it affects the correctness of the gradient. Using two- and three-dimensional model problems, we study the prospects for and limitations of the inference of the geothermal heat flux field from surface velocity observations. The results show that the reconstruction improves as the noise level in the observations decreases and that short-wavelength variations in the geothermal heat flux are difficult to recover. We analyze the ill-posedness of the inverse problem as a function of the number of observations by examining the spectrum of the Hessian of the cost functional. Motivated by the popularity of operator-split or staggered solvers for forward multiphysics problems – i.e., those that drop two-way coupling terms to yield a one-way coupled forward Jacobian – we study the effect on the inversion of a one-way coupling of the adjoint energy and Stokes equations. Here, we show that taking such a one-way coupled approach for the adjoint equations can lead to an incorrect gradient and premature termination of optimization iterations. This is due to loss of a descent direction stemming from inconsistency of the gradient with the contours of the cost functional. Nevertheless, one may still obtain a reasonable approximate inverse solution particularly if important features of the reconstructed solution emerge early in optimization iterations, before the premature termination.

15 GEOTHERMAL ENERGY↗
NASA NTRS1966

Nonlinear Aspects of Combustion Instability in Liquid Propellant Rocket Motors Sixth Yearly Progress Report, 1 Jun. 1965 - 31 May 1966

This is the sixth yearly report on the research at Princeton University on the problem of combustion instability in liquid-propellant rocket motors. These investigations have been sponsored by the National Aeronautics and Space Administration under NASA Contract NASr-217 and, in the past, under NASA Grant NsG-99-60. The nonlinear aspects of the problem have been emphasized in both the theoretical and experimental portions of the program. Presently, the theoretical research primarily involves studies of possible mechanisms or forcing functions of the instabilities, the nonlinear gasdynamic oscillations associated with instability in rocket motors, the effect of the exhaust nozzle on the stability, and the effect of damping devices on the stability. In conjunction with the theoretical program, an experimental program of an equally broad range is being conducted. This program includes basic experiments both with rocket motors and "cold" chambers which have been designed to study various combustion processes in order to provide understanding which would be useful in the construction of a theoretical model. It also includes studies of the effects of exhaust nozzles and damping devices upon the instability as well as controlled testing of liquid rocket motors with parametric variations intended to determine the factors which strongly influence the incidence of nonlinear combustion instability.

Combustion instability↗
NASA NTRS1971

The effect of system nonlinearities on system noise statistics

The effects are studied of nonlinearities in a baseline communications system on the system noise amplitude statistics. So that a meaningful identification of system nonlinearities can be made, the baseline system is assumed to transmit a single biphase-modulated signal through a relay satellite to the receiving equipment. The significant nonlinearities thus identified include square-law or product devices (e.g., in the carrier reference recovery loops in the receivers), bandpass limiters, and traveling wave tube amplifiers.

Robinson, L. H., Jr.↗
NASA NTRS1971

Observations of nonlinear scattering from a plasma column.

Nonlinear interaction effects are considered for signals incident on an unmagnetized cold plasma column, without making the quasi-static approximation. It is found that the nonlinearly generated quadrupole component is only modified slightly and that no multipolar component can give a forward- or back-scattered signal. However, strong nonlinear forward scattering was observed in previous experiments. This spurious result has been traced to the effect of the earth's magnetic field. When this is cancelled, predominantly quadrupolar radiation is observed, in agreement with theory.

Bruce, R. L.↗
NASA NTRS1971

Estimation in nonlinear systems with transport delay.

The problem of estimation of state in nonlinear dynamical systems containing time delays is studied. The plant is specified by a set of nonlinear differential-difference equations. Observations are a nonlinear function of current and/or delayed states. Both contain additive disturbances. The criterion used for the optimal estimates is the integral of the weighted squared error. Using the theory of the calculus of variations, equations are developed for the estimation. They are first expressed in the form of a split boundary value problem, which is then converted to an initial value problem for on-line estimation. The result yields a sequential estimation scheme in which filtered and smoothed estimates are computed in a sequential manner. The applicability of the procedure is demonstrated by a practical example.

Stoller, R. L.↗
NASA NTRS1973

Analysis of nonlinear vibrations of cylinders

In study of geometric nonlinear vibrations, infinite, long, thin-walled cylinder was analyzed under periodic, dynamic loading to demonstrate that some nonlinear phenomena cannot be obtained by straight-forward numerical solution methods. Results demonstrate that nonlinear phenomenon in large amplitude vibration traveling-wave condition can be predicted by analysis.

Chen, J. C.↗