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All sourcesNASA NTRSDOE OSTI
NASA NTRS1988

An exact solution to the relativistic equation of motion of a charged particle driven by a linearly polarized electromagnetic wave

An exact analytic solution is found for a basic electromagnetic wave-charged particle interaction by solving the nonlinear equations of motion. The particle position, velocity, and corresponding time are found to be explicit functions of the total phase of the wave. Particle position and velocity are thus implicit functions of time. Applications include describing the motion of a free electron driven by an intense laser beam..

Shebalin, John V.↗
DOE OSTI2012

Modified semi-classical methods for nonlinear quantum oscillations problems

We develop a modified semi-classical approach to the approximate solution of Schroedinger's equation for certain nonlinear quantum oscillations problems. In our approach, at lowest order, the Hamilton-Jacobi equation of the conventional semi-classical formalism is replaced by an inverted-potential-vanishing-energy variant thereof. With suitable smoothness, convexity and coercivity properties imposed on its potential energy function, we prove, using methods drawn from the calculus of variations together with the (Banach space) implicit function theorem, the existence of a global, smooth 'fundamental solution' to this equation. Higher order quantum corrections thereto, for both ground and excited states, can then be computed through the integration of associated systems of linear transport equations, derived from Schroedinger's equation, and formal expansions for the corresponding energy eigenvalues obtained therefrom by imposing the natural demand for smoothness on the (successively computed) quantum corrections to the eigenfunctions. For the special case of linear oscillators our expansions naturally truncate, reproducing the well-known exact solutions for the energy eigenfunctions and eigenvalues. As an explicit application of our methods to computable nonlinear problems, we calculate a number of terms in the corresponding expansions for the one-dimensional anharmonic oscillators of quartic, sectic, octic, and dectic types and compare the results obtained with those of conventional Rayleigh/Schroedinger perturbation theory. To the orders considered (and, conjecturally, to all orders) our eigenvalue expansions agree with those of Rayleigh/Schroedinger theory whereas our wave functions more accurately capture the more-rapid-than-gaussian decay known to hold for the exact solutions to these problems. For the quartic oscillator in particular our results strongly suggest that both the ground state energy eigenvalue expansion and its associated wave function expansion are Borel summable to yield natural candidates for the actual exact ground state solution and its energy. Our techniques for proving the existence of the crucial 'fundamental solution' to the relevant (inverted-potential-vanishing-energy) Hamilton-Jacobi equation have the important property of admitting interesting infinite dimensional generalizations. In a project paralleling the present one we shall show how this basic construction can be carried out for the Yang-Mills equations in Minkowski spacetime.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗
DOE OSTI2021

Learning viscoelasticity models from indirect data using deep neural networks

In this study, we propose a novel approach to model viscoelasticity materials, where rate-dependent and non-linear constitutive relationships are approximated with deep neural networks. We assume that inputs and outputs of the neural networks are not directly observable, and therefore common training techniques with input–output pairs for the neural networks are inapplicable. To that end, we develop a novel computational approach to both calibrate parametric and learn neural-network-based constitutive relations of viscoelasticity materials from indirect displacement data in the context of multiple-physics systems. We show that limited displacement data holds sufficient information to quantify the viscoelasticity behavior. We formulate the inverse computation – modeling viscoelasticity properties from observed displacement data – as a PDE-constrained optimization problem and minimize the error functional using a gradient-based optimization method. The gradients are computed by a combination of automatic differentiation and implicit function differentiation rules. The effectiveness of our method is demonstrated through numerous benchmark problems in geomechanics and porous media transport.

97 MATHEMATICS AND COMPUTING↗
NASA NTRS1972

Application of Contraction Mappings to the Control of Nonlinear Systems

The theoretical and applied aspects of successive approximation techniques are considered for the determination of controls for nonlinear dynamical systems. Particular emphasis is placed upon the methods of contraction mappings and modified contraction mappings. It is shown that application of the Pontryagin principle to the optimal nonlinear regulator problem results in necessary conditions for optimality in the form of a two point boundary value problem (TPBVP). The TPBVP is represented by an operator equation and functional analytic results on the iterative solution of operator equations are applied. The general convergence theorems are translated and applied to those operators arising from the optimal regulation of nonlinear systems. It is shown that simply structured matrices and similarity transformations may be used to facilitate the calculation of the matrix Green functions and the evaluation of the convergence criteria. A controllability theory based on the integral representation of TPBVP's, the implicit function theorem, and contraction mappings is developed for nonlinear dynamical systems. Contraction mappings are theoretically and practically applied to a nonlinear control problem with bounded input control and the Lipschitz norm is used to prove convergence for the nondifferentiable operator. A dynamic model representing community drug usage is developed and the contraction mappings method is used to study the optimal regulation of the nonlinear system.

Killingsworth, W. R., Jr.↗
NASA NTRS1972

Liapunov functions for non-linear difference equation stability analysis.

Liapunov functions to determine the stability of non-linear autonomous difference equations can be developed through the use of auxiliary exact difference equations. For this purpose definitions are introduced for the gradient of an implicit function of a discrete variable, a principal sum, a definite sum and an exact difference equation, and a theorem for exactness of a difference form is proved. Examples illustrate the procedure.

Park, K. E.↗
NASA NTRS1984

The ADS general-purpose optimization program

The mathematical statement of the general nonlinear optimization problem is given as follows: find the vector of design variables, X, that will minimize f(X) subject to G sub J (x) + or - 0 j=1,m H sub K hk(X) = 0 k=1,l X Lower I approx less than X sub I approx. less than X U over I i = 1,N. The vector of design variables, X, includes all those variables which may be changed by the ADS program in order to arrive at the optimum design. The objective function F(X) to be minimized may be weight, cost or some other performance measure. If the objective is to be maximized, this is accomplished by minimizing -F(X). The inequality constraints include limits on stress, deformation, aeroelastic response or controllability, as examples, and may be nonlinear implicit functions of the design variables, X. The equality constraints h sub k(X) represent conditions that must be satisfied precisely for the design to be acceptable. Equality constraints are not fully operational in version 1.0 of the ADS program, although they are available in the Augmented Lagrange Multiplier method. The side constraints given by the last equation are used to directly limit the region of search for the optimum. The ADS program will never consider a design which is not within these limits.

Vanderplaats, G. N.↗
NASA NTRS1991

An eddy viscosity calculation method for a turbulent duct flow

The mean velocity profile across a fully developed turbulent duct flow is obtained from an eddy viscosity relation combined with an empirical outer region wake function. Results are in good agreement with experiments and with direct numerical simulations in the same flow at two Reynolds numbers. In particular, the near-wall trend of the Reynolds shear stress and its variation with Reynolds number are similar to those of the simulations. The eddy viscosity method is more accurate than previous mixing length or implicit function methods.

Antonia, R. A.↗
NASA NTRS1974

Optimal control problems with mixed control-phase variable equality and inequality constraints

In this paper, necessary conditions are obtained for optimal control problems containing equality constraints defined in terms of functions of the control and phase variables. The control system is assumed to be characterized by an ordinary differential equation, and more conventional constraints, including phase inequality constraints, are also assumed to be present. Because the first-mentioned equality constraint must be satisfied for all t (the independent variable of the differential equation) belonging to an arbitrary (prescribed) measurable set, this problem gives rise to infinite-dimensional equality constraints. To obtain the necessary conditions, which are in the form of a maximum principle, an implicit-function-type theorem in Banach spaces is derived.

Makowski, K.↗
DOE OSTI2018

Two patterns of PT-symmetry breakdown in a non-numerical six-state simulation

Highlights: • Hot topic (PT symmetry breakdown in quantum theory) studied. • Next to elementary toy model considered. • Characteristic features (like quantum phase transitions) sampled. • Surprisingly, the results are non-numerical. • The method (inverse-function treatment) is brand new. - Abstract: Three-parametric family of non-Hermitian but PT-symmetric six-by-six matrix Hamiltonians H{sup (6)}(x,y,z) is considered. The PT-symmetry remains spontaneously unbroken (i.e., the spectrum of the bound-state energies remains real so that the unitary-evolution stability of the quantum system in question is shown guaranteed) in a non-empty domain D{sup (physical)} of parameters x,y,z. The construction of the exceptional-point (EP) boundary ∂D{sup (physical)} of the physical domain is preformed using an innovative non-numerical implicit-function-construction strategy. The topology of the resulting EP boundary of the spontaneous PT-symmetry breakdown (i.e., of the physical “horizon of stability”) is shown similar to its much more elementary N=4 predecessor. Again, it is shown to consist of two components, viz., of the region of the quantum phase transitions of the first kind (during which at least some of the energies become complex) and of the quantum phase transitions of the second kind (during which some of the level pairs only cross but remain real).

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗
DOE OSTI2022

$\mathrm{IH}$-$\mathrm{GAN}$: A conditional generative model for implicit surface-based inverse design of cellular structures

Variable-density cellular structures can overcome connectivity and manufacturability issues of topologically optimized structures, particularly those represented as discrete density maps. However, the optimization of such cellular structures is challenging due to the multiscale design problem. Past work addressing this problem generally either only optimizes the volume fraction of single-type unit cells but ignoring the effects of unit cell geometry on properties, or considers the geometry–property relation but builds this relation via heuristics. In contrast, we propose a simple yet more principled way to accurately model the property to geometry mapping using a conditional deep generative model, named Inverse Homogenization Generative Adversarial Network (IH-GAN). It learns the conditional distribution of unit cell geometries given properties and can realize the one-to-many mapping from properties to geometries. Here we further reduce the complexity of IH-GAN by using the implicit function parameterization to represent unit cell geometries. Results show that our method can 1) generate various unit cells that satisfy given material properties with high accuracy (R 2 -scores between target properties and properties of generated unit cells >98%) and 2) improve the optimized structural performance over the conventional variable-density single-type structure. In the minimum compliance example, our IH-GAN generated structure achieves a 79.7% reduction in concentrated stress and an extra 3.03% reduction in displacement. In the target deformation examples, our IH-GAN generated structure reduces the target matching error by 86.4% and 79.6% for two test cases, respectively. We also demonstrated that the connectivity issue for multi-type unit cells can be solved by transition layer blending.

42 ENGINEERING↗
DOE OSTI2019

Bootstrap-based confidence interval estimation for thermal security region of bulk power grid

Thermal security region (TSR) is a powerful tool for monitoring and controlling the thermal security of bulk power grid with high penetration of renewable energy. One of the major challenges of TSR in practical application is to obtain the exact analytical expression of the TSR boundary (TSRB) because TSRB is a nonlinear hypersurface described by an implicit function. For this reason, the TSRB is usually approximated by hyperplane instead of getting the exact analytical expression. Traditionally, the coefficients of the hyperplane are estimated with a set of original TSRB points using least square estimation (LSE). However, LSE is a point estimation method, which is incapable of evaluating the quality of the estimation. In addition, the accuracy and computational efficiency of the hyperplane approximation are influenced by the number of TSRB points significantly. Here, a bootstrap based confidence interval estimation is proposed to estimate not only the coefficients of TSRB approximation hyperplane, but also the standard deviations and confidence intervals of the coefficients for evaluating the quality and reliability of the approximation results. First, empirical distribution functions (EDFs) of the coefficients of TSRB approximation hyperplane are approximated from a set of original TSRB points by using residual resampling bootstrap method. Then, the EDFs of the coefficients are employed to estimate the coefficients of the approximation hyperplane. Meanwhile, the standard deviations and confidence intervals of the estimated coefficients of the hyperplanes are also calculated for evaluating the quality and reliability of the approximation. The proposed approach is tested on the China Southern Power Grid (CSG). Results of simulations validate the accuracy and efficiency of the proposed approach in approximating the TSRB.

32 ENERGY CONSERVATION, CONSUMPTION, AND UTILIZATI↗
DOE OSTI2023

A high order Cartesian grid, finite volume method for elliptic interface problems

We present a higher-order finite volume method for solving elliptic PDEs with jump conditions on interfaces embedded in a 2D Cartesian grid. Second, fourth, and sixth order accuracy is demonstrated on a variety of tests including problems with high-contrast and spatially varying coefficients, large discontinuities in the source term, and complex interface geometries. We include a generalized truncation error analysis based on cell-centered Taylor series expansions, which then define stencils in terms of local discrete solution data and geometric information. In the process, we develop a simple method based on Green's theorem for computing exact geometric moments directly from an implicit function definition of the embedded interface. This approach produces stencils with a simple bilinear representation, where spatially-varying coefficients and jump conditions can be easily included and finite volume conservation can be enforced.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2014

Coherent states formulation of polymer field theory

We introduce a stable and efficient complex Langevin (CL) scheme to enable the first direct numerical simulations of the coherent-states (CS) formulation of polymer field theory. In contrast with Edwards’ well-known auxiliary-field (AF) framework, the CS formulation does not contain an embedded nonlinear, non-local, implicit functional of the auxiliary fields, and the action of the field theory has a fully explicit, semi-local, and finite-order polynomial character. In the context of a polymer solution model, we demonstrate that the new CS-CL dynamical scheme for sampling fluctuations in the space of coherent states yields results in good agreement with now-standard AF-CL simulations. The formalism is potentially applicable to a broad range of polymer architectures and may facilitate systematic generation of trial actions for use in coarse-graining and numerical renormalization-group studies.

37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CH↗
DOE OSTI2016

One-baryon spectrum and analytical properties of one-baryon dispersion curves in 3 + 1 dimensional strongly coupled lattice QCD with three flavors

Considering a 3 + 1 dimensional lattice quantum chromodynamics (QCD) model defined with the improved Wilson action, three flavors, and 4 × 4 Dirac spin matrices, in the strong coupling regime, we reanalyze the question of the existence of the eightfold way baryons and complete our previous work where the existence of isospin octet baryons was rigorously solved. Here, we show the existence of isospin decuplet baryons which are associated with isolated dispersion curves in the subspace of the underlying quantum mechanical Hilbert space with vectors constructed with an odd number of fermion and antifermion basic quark and antiquark fields. Moreover, smoothness properties for these curves are obtained. The present work deals with a case for which the traditional method to solve the implicit equation for the dispersion curves, based on the use of the analytic implicit function theorem, cannot be applied. We do not have only one but two solutions for each one-baryon decuplet sector with fixed spin third component. Instead, we apply the Weierstrass preparation theorem, which also provides a general method for the general degenerate case. This work is completed by analyzing a spectral representation for the two-baryon correlations and providing the leading behaviors of the field strength normalization and the mass of the spectral contributions with more than one-particle. These are needed results for a rigorous analysis of the two-baryon and meson-baryon particle spectra.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗
DOE OSTI2014

Definition of the metric on the space clos{sub ∅}(X) of closed subsets of a metric space X and properties of mappings with values in clos{sub ∅}(R{sup n})

The paper is concerned with the extension of tests for superpositional measurability, Filippov's implicit function lemma and the Scorza Dragoni property to set-valued (and, as a corollary, to single-valued) mappings that fail to satisfy the Carathéodory conditions (the upper Carathéodory conditions) and are not continuous (upper semicontinuous) in the phase variable. The corresponding results depend on the introduction of the space clos{sub ∅}(X) of all closed subsets (including the empty set) of an arbitrary metric space X; a metric on clos{sub ∅}(X) is proposed; the space clos{sub ∅}(X) is shown to be complete whenever the original space X is; a criterion for convergence of a sequence is put forward; mappings with values in clos{sub ∅}(X) are studied. Some results on set-valued mappings satisfying the Carathéodory conditions and having compact values in R{sup n} are shown to hold for mappings with values in clos{sub ∅}(R{sup n}), measurable in the first argument, and continuous in the proposed metric in the second argument. Bibliography: 22 titles.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2014

WE-E-18C-01: Multi-Energy CT: Current Status and Recent Innovations

Conventional computed tomography (CT) uses a single polychromatic x-ray spectrum and energy integrating detectors, and produces images whose contrast depends on the effective attenuation coefficient of the broad spectrum beam. This can introduce errors from beam hardening and does not produce the optimal contrast-to-noise ratio. In addition, multiple materials can have the same effective attenuation coefficient, causing different materials to be indistinguishable in conventional CT images. If transmission measurements at two or more energies are obtained, even with polychromatic beams, more specific information about the object can be obtained. If the object does not contain materials with k-edges in the spectrum, the x-ray attenuation can be well-approximated by a linear combination of two processes (photoelectric absorption and Compton scattering) or, equivalently, two basis materials. For such cases, two spectral measurements suffice, although additional measurements can provide higher precision. If K-edge materials are present, additional spectral measurements can allow these materials to be isolated. Current commercial implementations use varied approaches, including two sources operating a different kVp, one source whose kVp is rapidly switched in a single scan, and a dual layer detector that can provide spectral information in every reading. Processing of the spectral information can be performed in the raw data domain or in the image domain. The process of calculating the amount of the two basis functions implicitly corrects for beam hardening and therefore can lead to improvements in quantitative accuracy. Information can be extracted to provide material specific information beyond that of conventional CT. This additional information has been shown to be important in several clinical applications, and can also lead to more efficient clinical protocols. Recent innovations in x-ray sources, detectors, and systems have made multi-energy CT much more practical and improved its performance. In addition, this is a very active area of research and further improvements are expected through further technological improvements. Learning Objectives: Basic principles of multi-energy CT Current implementations of mutli-energy CT Data and image analysis methods in multi-energy CT Current clinical applications of dual energy CT5. recent innovations and anticipated advances in multi-energy CT.

62 RADIOLOGY AND NUCLEAR MEDICINE↗
DOE OSTI2015

SU-E-J-130: Automating Liver Segmentation Via Combined Global and Local Optimization

Purpose: The aim of this work is to develop a robust algorithm for accurate segmentation of liver with special attention paid to the problems with fuzzy edges and tumor. Methods: 200 CT images were collected from radiotherapy treatment planning system. 150 datasets are selected as the panel data for shape dictionary and parameters estimation. The remaining 50 datasets were used as test images. In our study liver segmentation was formulated as optimization process of implicit function. The liver region was optimized via local and global optimization during iterations. Our method consists five steps: 1)The livers from the panel data were segmented manually by physicians, and then We estimated the parameters of GMM (Gaussian mixture model) and MRF (Markov random field). Shape dictionary was built by utilizing the 3D liver shapes. 2)The outlines of chest and abdomen were located according to rib structure in the input images, and the liver region was initialized based on GMM. 3)The liver shape for each 2D slice was adjusted using MRF within the neighborhood of liver edge for local optimization. 4)The 3D liver shape was corrected by employing SSR (sparse shape representation) based on liver shape dictionary for global optimization. Furthermore, H-PSO(Hybrid Particle Swarm Optimization) was employed to solve the SSR equation. 5)The corrected 3D liver was divided into 2D slices as input data of the third step. The iteration was repeated within the local optimization and global optimization until it satisfied the suspension conditions (maximum iterations and changing rate). Results: The experiments indicated that our method performed well even for the CT images with fuzzy edge and tumors. Comparing with physician delineated results, the segmentation accuracy with the 50 test datasets (VOE, volume overlap percentage) was on average 91%–95%. Conclusion: The proposed automatic segmentation method provides a sensible technique for segmentation of CT images. This work is supported by NIH/NIBIB (1R01-EB016777), National Natural Science Foundation of China (No.61471226 and No.61201441), Research funding from Shandong Province (No.BS2012DX038 and No.J12LN23), and Research funding from Jinan City (No.201401221 and No.20120109)

60 APPLIED LIFE SCIENCES↗
DOE OSTI2019

ECP Milestone Memo WBS 2.3.4.13 ECP/VTK-m FY19Q1 [MS-19/01-03] ZFP / Release / Clip STDA05-17

The STDA05-17 milestone comprises the following 3 deliverables. VTK-m Release 2 We will provide a release of VTK-m software and associated documentation. The source code repository will be tagged at a stable state, and, at a minimum, tarball captures of the source code will be made available from the web site. A version of the VTK-m User's Guide documenting this release will also be made available. Productionize zfp compression The "ZFP: Compressed Floating-Point Arrays" project (WBS 1.3.4.13) is creating an implementation of ZFP compression in VTK-m. Their implementation will be focused on operating in CUDA. The VTK-m project will assist by generalizing the implementation to other devices (such as multi-core CPUs). We will also assist in productionizing the code such that it can be used by external projects and products. Clip Clip operations intersect meshes with implicit functions. It is the foundation of spatial subsetting algorithms, such as "box," and the foundation of data-based subsetting, such as "isovolume." The algorithm requires considering thousands of possible cases, and is thus quite difficult to implement. This milestone will implement clipping to be sufficient for Visit's and ParaView's needs.

97 MATHEMATICS AND COMPUTING↗