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NASA NTRS1996

A dual-input nonlinear system analysis of autonomic modulation of heart rate

Linear analyses of fluctuations in heart rate and other hemodynamic variables have been used to elucidate cardiovascular regulatory mechanisms. The role of nonlinear contributions to fluctuations in hemodynamic variables has not been fully explored. This paper presents a nonlinear system analysis of the effect of fluctuations in instantaneous lung volume (ILV) and arterial blood pressure (ABP) on heart rate (HR) fluctuations. To successfully employ a nonlinear analysis based on the Laguerre expansion technique (LET), we introduce an efficient procedure for broadening the spectral content of the ILV and ABP inputs to the model by adding white noise. Results from computer simulations demonstrate the effectiveness of broadening the spectral band of input signals to obtain consistent and stable kernel estimates with the use of the LET. Without broadening the band of the ILV and ABP inputs, the LET did not provide stable kernel estimates. Moreover, we extend the LET to the case of multiple inputs in order to accommodate the analysis of the combined effect of ILV and ABP effect on heart rate. Analyzes of data based on the second-order Volterra-Wiener model reveal an important contribution of the second-order kernels to the description of the effect of lung volume and arterial blood pressure on heart rate. Furthermore, physiological effects of the autonomic blocking agents propranolol and atropine on changes in the first- and second-order kernels are also discussed.

NASA Discipline Regulatory Physiology↗

NASA NTRS2004

Observers for Systems with Nonlinearities Satisfying an Incremental Quadratic Inequality

We consider the problem of state estimation for nonlinear time-varying systems whose nonlinearities satisfy an incremental quadratic inequality. These observer results unifies earlier results in the literature; and extend it to some additional classes of nonlinearities. Observers are presented which guarantee that the state estimation error exponentially converges to zero. Observer design involves solving linear matrix inequalities for the observer gain matrices. Results are illustrated by application to a simple model of an underwater.

observers↗

DOE OSTI2015

DEVELOPMENT OF A NONLINEAR TIME DOMAIN METHODOLOGY, RESULTS (PART II)

A complementary paper, “Development of a Nonlinear Time Domain Methodology (Part I),” lays out a methodology to follow when performing nonlinear soil-structure interaction (NLSSI) analysis. This paper presents results from numerical analyses that compare a recently verified and validated version of SASSI with those from a NLSSI code using increasing levels of earthquake ground motion. Comparisons will be presented at multiple locations inside a generic nuclear power plant structure. The generic NPP is placed on a well-studied soil site. NLSSI results are verified by comparing the recently V&V version of SASSI with NLSSI. At low levels of ground motion the linear analysis and NLSSI produce similar results with results diverging for higher levels of ground motion. Results show that differences in the spectral accelerations calculated using the equivalent linear and nonlinear methods are observed at significant levels of ground motion shaking. For the NPP analyse, the differences are associated with gapping and sliding, which cannot be captured explicitly with equivalent linear analysis.

42 ENGINEERING↗

DOE OSTI2021

Variational Autoencoders for Learning Nonlinear Dynamics of Physical Systems

We develop data-driven methods for incorporating physical information for priors to learn parsimonious representations of nonlinear systems arising from parameterized PDEs and mechanics. Our approach is based on Variational Autoencoders (VAEs) for learning nonlinear state space models from observations. We develop ways to incorporate geometric and topological priors through general manifold latent space representations. We investigate the performance of our methods for learning low dimensional representations for the nonlinear Burgers equation and constrained mechanical systems.

97 MATHEMATICS AND COMPUTING↗

DOE OSTI2022

Enabling New Flexibility in the SUNDIALS Suite of Nonlinear and Differential/Algebraic Equation Solvers

In recent years, the SUite of Nonlinear and DIfferential/ALgebraic equation Solvers (SUNDIALS) has been redesigned to better enable the use of application-specific and third-party algebraic solvers and data structures. Throughout this work, we have adhered to specific guiding principles that minimized the impact to current users while providing maximum flexibility for later evolution of solvers and data structures. The redesign was done through the addition of new linear and nonlinear solvers classes, enhancements to the vector class, and the creation of modern Fortran interfaces. The vast majority of this work has been performed “behind-the-scenes,” with minimal changes to the user interface and no reduction in solver capabilities or performance. These changes allow SUNDIALS users to more easily utilize external solver libraries and create highly customized solvers, enabling greater flexibility on extreme-scale, heterogeneous computational architectures.

97 MATHEMATICS AND COMPUTING↗

DOE OSTI2014

Nonlinear electric field structures in the inner magnetosphere

Recent observations by the Van Allen Probes spacecraft have demonstrated that a variety of electric field structures and nonlinear waves frequently occur in the inner terrestrial magnetosphere, including phase space holes, kinetic field-line resonances, nonlinear whistler-mode waves, and several types of double layer. However, it is nuclear whether such structures and waves have a significant impact on the dynamics of the inner magnetosphere, including the radiation belts and ring current. To make progress toward quantifying their importance, this study statistically evaluates the correlation of such structures and waves with plasma boundaries. A strong correlation is found. These statistical results, combined with observations of electric field activity at propagating plasma boundaries, are consistent with the identification of these boundaries as the source of free energy responsible for generating the electric field structures and nonlinear waves of interest. Therefore, the ability of these structures and waves to influence plasma in the inner magnetosphere is governed by the spatial extent and dynamics of macroscopic plasma boundaries in that region.

70 PLASMA PHYSICS AND FUSION TECHNOLOGY↗

DOE OSTI2017

Growth and nonlinear saturation of electromagnetic ion cyclotron waves in multi-ion species magnetospheric plasma

Here, the growth and saturation of electromagnetic ion cyclotron (EMIC) waves is essential to the magnetospheric dynamics. Determining and isolating the effects of multiple ion parameters such as temperatures, anisotropies, and relative abundances is important for quantifying these processes in the magnetospheric plasma. In order to study these process, we utilize a 2.5–D hybrid model (where ions are modeled with the particle–in–cell (PIC) method, and electrons are modeled as background neutralizing fluid) to study the nonlinear electromagnetic wave–particle interactions of hot H + , cold H + , cold He + , and cold or hot O + ions for a broad range of typical magnetospheric parameters. The excitation of EMIC waves is driven by the temperature anisotropy of hot H + in our model. As a result, we quantify the parametric dependence of the linear growth, the nonlinear saturation level of perpendicular magnetic fluctuations, and the temporal evolution of the ion temperature anisotropies. We establish the relation between key plasma parameters and the saturated EMIC wave power, using either power law fits or a nonlinear regression method. We construct the dispersion relation of the waves using the results of the model and investigate the energy content in the various branches of the dispersion ( k ∥ –ω space), showing that the different modes can generate wave power in different regions of k space. We find that large O + concentration reduces the growth and saturated amplitude of the waves; but the waves are less sensitive to the temperature of the O + in the temperature range relevant to the magnetosphere.

79 ASTRONOMY AND ASTROPHYSICS↗

DOE OSTI2016

Modeling nonlinear response of fractured rocks and reservoirs

Summary Since the early work of Athey (1930), there have been many attempts to describe the various nonlinear behaviors of rocks and soils in terms of functionals having only a few parameters, while nevertheless being able to fit the complicated available data with satisfactory accuracy. Such approaches have not been universally applied however, and the present analyses are intended to draw attention to the possibility of using such nonlinear fitting methods on old as well as new data sets. In particular, some special emphasis is placed here on re‐examining the well‐known laboratory data of Coyner (1984) on rocks in light of such modeling tools, and we find that the nonlinear approach again has several clear advantages – especially in terms of reducing the number of variables needed to describe the observed behavior of both bulk modulus and porosity of rocks undergoing large changes in pressure. Copyright © 2016 John Wiley & Sons, Ltd.

Berryman, James G.↗

DOE OSTI2019

Transient dynamics, damping, and mode coupling of nonlinear systems with internal resonances

The study of both linear and nonlinear structural vibrations routinely circles the concise yet complex problem of choosing a set of coordinates which yield simple equations of motion. In both experimental and mathematical methods, that choice is a difficult one because of measurement, computational, and interpretation difficulties. Often times, researchers choose to solve their problems in terms of linear, undamped mode shapes because they are easy to obtain; however, this is known to give rise to complicated phenomena such as mode coupling and internal resonance. This work considers the nature of mode coupling and internal resonance in systems containing non-proportional damping, linear detuning, and cubic nonlinearities through the method of multiple scales as well as instantaneous measures of effective damping. The energy decay observed in the structural modes is well approximated by the slow-flow equations in terms of the modal amplitudes, and it is shown how mode coupling enhances the damping observed in the system. Moreover, in the presence of a 3:1 internal resonance between two modes, the nonlinearities not only enhance the dissipation, but can allow for the exchange and transfer of energy between the resonant modes. However, this exchange depends on the resonant phase between the modes and is proportional to the energy in the lowest mode. The results of the analysis tie together interpretations used by both experimentalists and theoreticians to study such systems and provide a more concrete way to interpret these phenomena.

42 ENGINEERING↗

DOE OSTI2026

Dynamics of McMillan mappings III. Symmetric map with mixed nonlinearity

This article extends the study of the dynamical properties of the symmetric McMillan map, emphasizing its utility in understanding and modeling complex nonlinear systems. Although the map features six parameters, we demonstrate that only two are irreducible: the linearized rotation number at the fixed point and a nonlinear parameter representing the ratio of terms in the biquadratic invariant. Through a detailed analysis, we classify regimes of stable motion, provide exact solutions to the mapping equations, and derive a canonical set of action-angle variables, offering analytical expressions for the rotation number and nonlinear tune shift. We further establish connections between general standard-form mappings and the symmetric McMillan map, using the area-preserving Hénon map and accelerator lattices with thin sextupole magnet as representative case studies. Our results show that, despite being a second-order approximation, the symmetric McMillan map provides a highly accurate depiction of dynamics across a wide range of system parameters, demonstrating its practical relevance in both theoretical and applied contexts.

43 PARTICLE ACCELERATORS↗

DOE OSTI2026

Extremized nonlinear and linearized responses in soft metamaterials enabled by gradient-based design and grayscale digital light processing

In this study, we develop a gradient-based design approach that exploits grayscale digital light processing (DLP) 3D printing for extremizing the nonlinear and linearized response of soft metamaterials — materials that harness engineered geometric instabilities to undergo large and programmable changes in configuration. Grayscale DLP approaches modulate local mechanical properties at the pixel scale by tuning the light intensity within a single grayscale image, unlocking an exceptionally large design space. To effectively navigate this space, we develop smooth mappings between local light intensity values and global quantities of interest that characterize the behavior of soft metamaterials. Enabling these smooth mappings are robust and differentiable nonlinear finite element simulations powered by a trust region solver. A PDE-constrained optimization problem is then solved to invert these mappings and produce light intensity distributions that endow the printed part with varying stiffness and flexibility in distinctive regions. It is shown that optimizing the distribution of soft and stiff phases throughout a metamaterial structure results in markedly different buckling and self-contact configurations to drive extremized nonlinear compression and linearized vibration responses. Optimized light intensity distributions are translated to grayscale images and directly used to print soft metamaterial samples, showing remarkable agreement between the buckling and self-contact response in simulated and measured deformed configurations.

Additive manufacturing↗

DOE OSTI2019

Coupled nonlinear elasticity, plastic slip, twinning, and phase transformation in single crystal titanium for plate impact loading

High purity single crystal titanium (Ti) under shockwave loading in plate impact experiment is modelled and simulated, with help of the finite element. A thermodynamically consistent system of equations is formulated in the frame of large deformation to consider material anisotropy, rate dependence, and multi-physics including nonlinear elasticity, dislocation based plastic slip, deformation twinning, and phase transformation. A novel kinematics is proposed to consider phase transformation in the α twin variants and also to consider the dislocation based slip in all of components of parent material, primary twins and the ω phase. The thermodynamically consistent driving forces for plastic slip, twinning, and phase transformation are developed based upon the second law of thermodynamics. For nonlinear elasticity, the stiffness matrix is dependent on the volume fractions of all components, and the volumetric part of Cauchy stress is obtained from the nonlinear equation of state. Dislocation based plastic slip in multi-variant and multiphase heterogeneous materials is developed and interactions of dislocations among all slip modes are taken into account. A mechanism for dislocation density evolution during twinning and phase transformation is proposed. The shock loading along the [0001] and [10$\bar{1}1$ directions of single crystal high purity Ti is investigated computationally. Very good correspondence between simulation and experiment is obtained, which includes pole figures, volume fractions of components, free surface velocity, peak pressure, and phase transformation pressure. Multiple experimental phenomena are interpreted based upon the progression of dislocation slip, deformation twinning, and phase transformation. In compression with the [10$\bar{1}1$ crystal, a higher volume fraction in the primary twins but a lower secondary twin volume fraction in the [0001] crystal in experiment was observed. Finally, the higher propensity for phase transformation occurs in the [0001] crystal is reproduced. In addition to material texture, distributions of temperature, stresses, and plastic strains dependent on the impact loading directions are revealed.

36 MATERIALS SCIENCE↗

DOE OSTI2012

First-principles studies on linear and nonlinear optical effects in Ln{sub 4}GaSbS{sub 9} (Ln=Ce-Nd, Sm, Gd-Tm, Lu)

Density functional theory (DFT) calculations have been performed on electronic structures of 11 middle-infrared (mid-IR) nonlinear optical quaternary sulfides: Ln{sub 4}GaSbS{sub 9} (Ln=Ce-Nd, Sm, Gd-Tm, Lu). Our results show that Ln{sub 4}GaSbS{sub 9} are indirect gap semiconductors with a slight band gap increase from Ce to Lu. Their linear optical properties, including refractive index, absorption coefficient and energy loss function, as well as the nonlinear optical coefficients including static d{sub 31}, d{sub 32}, d{sub 33} and dynamic d{sub 32}, are calculated. More importantly, the strong SHG response of Ln{sub 4}GaSbS{sub 9} can be attributed to the electronic transitions from S 3p states in valence bands (VB) to Sb-S and Ln-S antibonding states (CB). - Graphical Abstract: Density functional theory (DFT) calculations on Ln{sub 4}GaSbS{sub 9} reveal their linear optical properties, including refractive index, adsorption coefficient and energy loss function, the nonlinear optical coefficients including static d{sub 31}, d{sub 32}, d{sub 33} and dynamic d{sub 32}, as well as the origins of the strong SHG response. Highlights: Black-Right-Pointing-Pointer First-principles studies on a newly discovered NLO active family Ln{sub 4}GaSbS{sub 9}. Black-Right-Pointing-Pointer Reports static SHG coefficients (d{sub 31}, d{sub 32}, d{sub 33}) and dynamic d{sub 32}. Black-Right-Pointing-Pointer Reveals the origin of the strong SHG response.

37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CH↗

DOE OSTI2012

Crystal structure, optical and thermal studies of a new organic nonlinear optical material: L-Histidinium maleate 1.5-hydrate

Highlights: ► L-Histidinium maleate 1.5-hydrate, a new organic crystal has been grown for the first time. ► The crystal structure is reported for the first time (CCDC 845975). ► The crystal belongs to monoclinic system with space group P2{sub 1}, Z = 4, a = 11.4656(7) Å, b = 8.0530(5) Å, c = 14.9705(9) Å and β = 101.657(2)°. ► The optical absorption study substantiates the complete transparency of the crystal. ► Kurtz powder SHG test confirms the nonlinear property of the crystal. -- Abstract: A new organic nonlinear optical material L-histidinium maleate 1.5-hydrate (LHM) with the molecular formula C{sub 10}H{sub 16}N{sub 3}O{sub 7.5} has been successfully synthesized from aqueous solution by slow solvent evaporation method. The structural characterization of the grown crystal was carried out by single crystal X-ray diffraction at 293(2) K. In the crystal, molecules are linked through inter and intramolecular N-H⋯O and O-H⋯O hydrogen bonds, generate edge fused ring motif. The hydrogen bonded motifs are linked to each other to form a three dimensional network. The FT-IR spectroscopy was used to identify the functional groups of the synthesized compound. The optical behavior of the grown crystal was examined by UV–visible spectral analysis, which shows that the optical absorption is almost negligible in the wavelength range 280–1300 nm. The nonlinear optical property was confirmed by the powder technique of Kurtz and Perry. The thermal behavior of the grown crystal was analyzed by thermogravimetric analysis.

36 MATERIALS SCIENCE↗

DOE OSTI2012

Preparation and nonlinear optical properties of indium nanocrystals in sodium borosilicate glass by the sol–gel route

Graphical abstract: The sodium borosilicate glass doped with indium nanocrystals have been successfully prepared by sol–gel methods. And the indium nanocrystals in tetragonal crystal system have formed uniformly in the glass, and the average diameter of indium nanocrystals is about 30 nm. The third-order optical nonlinear refractive index γ, absorption coefficient β, and susceptibility χ{sup (3)} of the glass are determined to be −4.77 × 10{sup −16} m{sup 2}/W, 2.67 × 10{sup −9} m/W, and 2.81 × 10{sup −10} esu, respectively. Highlights: ► Indium nanocrystals embedded in glass matrix have been prepared by sol–gel route. ► The crystal structure and composition are investigated by XRD and XPS. ► Size and distribution of indium nanocrystals is determined by TEM. ► The third-order optical nonlinearity is investigated by using Z-scan technique. -- Abstract: The sodium borosilicate glass doped with indium nanocrystals have been successfully prepared by sol–gel route. The thermal stability behavior of the stiff gel is investigated by thermogravimetric (TG) and differential thermal (DTA) analysis. The crystal structure of the glass is characterized by X-ray powder diffraction (XRD). Particle composition is determined by X-ray photoelectron spectroscopy (XPS). Size and distribution of the nanocrystals are characterized by transmission electron microscopy (TEM) as well as high-resolution transmission electron microscopy (HRTEM). Results show that the indium nanocrystals in tetragonal crystal structure have formed in glass, and the average diameter is about 30 nm. Further, the glass is measured by Z-scan technique to investigate the nonlinear optical (NLO) properties. The third-order NLO coefficient χ{sup (3)} of the glass is determined to be 2.81 × 10{sup −10} esu. The glass with large third-order NLO coefficient is promising materials for applications in optical devices.

36 MATERIALS SCIENCE↗

DOE OSTI2015

One-pot synthesis of silica-hybridized Ag{sub 2}S–CuS nanocomposites with tunable nonlinear optical properties

Highlights: • Silica modified QDs of CuS and Ag{sub 2}S is developed at room temperature. • Formation of Ag{sub 2}S/CuS nanocomposites is confirmed from XRD and FFT of HRTEM images. • The concentration dependent growth of silica modified QDs is discussed. • Nonlinear absorption observed in ns excitations is dominated by SA and ESA. • Tuning of optical limiting efficiency is achieved with relative Ag{sub 2}S content. - Abstract: In the present work we report a simple, facile route developed for preparing silica hybridized copper sulfide and silver sulfide quantum dots at room temperature. By adjusting the concentration of the precursors, Ag{sub 2}S can form Ag{sub 2}S–CuS nanocomposites which are self regulated in one pot. Their crystalline, structural and optical properties have been investigated in detail, and the optical limiting nature is studied from fluence-dependent transmittance measurements employing short (5 ns) laser pulses at 532 nm. Ag{sub 2}S nanoparticles are found to have large third order nonlinear optical coefficients with a relatively lower optical limiting threshold of 1.7 J cm{sup −2}, while the nonlinearity of the nanocomposites is found to lie in between that of Ag{sub 2}S and CuS nanoparticles. These results suggest pathways for designing good quality optical limiters with tunable optical limiting efficiencies by varying the constituent nanocrystal compositions.

77 NANOSCIENCE AND NANOTECHNOLOGY↗

DOE OSTI2025

Bayesian learning with Gaussian processes for low-dimensional representations of time-dependent nonlinear systems

This work presents a data-driven method for learning low-dimensional time-dependent physics-based surrogate models whose predictions are endowed with uncertainty estimates. We use the operator inference approach to model reduction that poses the problem of learning low-dimensional model terms as a regression of state space data and corresponding time derivatives by minimizing the residual of reduced system equations. Standard operator inference models perform well with accurate training data that are dense in time, but producing stable and accurate models when the state data are noisy and/or sparse in time remains a challenge. Another challenge is the lack of uncertainty estimation for the predictions from the operator inference models. Our approach addresses these challenges by incorporating Gaussian process surrogates into the operator inference framework to (1) probabilistically describe uncertainties in the state predictions and (2) procure analytical time derivative estimates with quantified uncertainties. The formulation leads to a generalized least-squares regression and, ultimately, reduced-order models that are described probabilistically with a closed-form expression for the posterior distribution of the operators. The resulting probabilistic surrogate model propagates uncertainties from the observed state data to reduced-order predictions. Furthermore, we demonstrate the method is effective for constructing low-dimensional models of two nonlinear partial differential equations representing a compressible flow and a nonlinear diffusion–reaction process, as well as for estimating the parameters of a low-dimensional system of nonlinear ordinary differential equations representing compartmental models in epidemiology.

Data-driven model reduction↗

DOE OSTI2019

Correcting nonlinearity and mass-bias in measurements by ICP quadrupole mass spectrometry

A two-parameter modeling formula is proposed for quadrupole ICP-MS (ICP-QMS) data, which corrects for nonlinearity with a “dead-time” function and makes subsequent weighted mass-bias corrections to normalize isotope ratio data to pertinent isotopic reference materials. The methodology improves the short-term and long-term precision as well as accuracy of the measurements. Applying the model to isotopes spanning a wide mass range revealed that values for the parameters describing nonlinearity and mass bias do not conform to expectations based on conventional interpretation. Within a single analysis run for a specific element, nonlinearity and mass-bias corrections can be evaluated to very tight tolerances and the model fits the data to within a few parts per thousand. However, in a different run or with a different element, the values of the parameters might change. A comprehensive strategy for conducting measurements of isotope ratios or trace-element concentrations is described, which takes advantage of the model’s strengths and accommodates the observed run-to-run variations. in conclusion, supporting data is presented to demonstrate improved precision and accuracy with the widely available ICP-QMS system.

73 NUCLEAR PHYSICS AND RADIATION PHYSICS↗