Results for “Nonlinear approximation” — Page 73

NASA NTRS1991

Euler solutions to nonlinear acoustics of non-lifting rotor blades

For the first time a computational fluid dynamics (CFD) method is used to calculate directly the high-speed impulsive (HSI) noise of a non-lifting hovering rotor blade out to a distance of over three rotor radii. In order to accurately propagate the acoustic wave in a stable and efficient manner, an implicit upwind-biased Euler method is solved on a grid with points clustered along the line of propagation. A detailed validation of the code is performed for a rectangular rotor blade at tip Mach numbers ranging from 0.88 to 0.92. The agreement with experiment is excellent at both the sonic cylinder and at 2.18 rotor radii. The agreement at 3.09 rotor radii is still very good, showing improvements over the results from the best previous method. Grid sensitivity studies indicate that with special attention to the location of the boundaries a grid with approximately 60,000 points is adequate. This results in a computational time of approximately 40 minutes on a Cray-XMP. The practicality of the method to calculate HSI noise is demonstrated by expanding the scope of the investigation to examine the rectangular blade as well as a highly swept and tapered blade over a tip Mach number range of 0.80 to 0.95. Comparisons with experimental data are excellent and the advantages of planform modifications are clearly evident. New insight is gained into the mechanisms of nonlinear propagation and the minimum distance at which a valid comparison of different rotors can be made: approximately two rotor radii from the center of rotation.

Baeder, J. D.↗

NASA NTRS1991

Euler solutions to nonlinear acoustics of non-lifting hovering rotor blades

For the first time a computational fluid dynamics (CFD) method is used to calculate directly the high-speed impulsive (HSI) noise of a non-lifting hovering rotor blade out to a distance of over three rotor radii. In order to accurately propagate the acoustic wave in a stable and efficient manner, an implicit upwind-biased Euler method is solved on a grid with points clustered along the line of propagation. A detailed validation of the code is performed for a rectangular rotor blade at tip Mach numbers ranging from 0.88 to 0.92. The agreement with experiment is excellent at both the sonic cylinder and at 2.18 rotor radii. The agreement at 3.09 rotor radii is still very good, showing improvements over the results from the best previous method. Grid sensitivity studies indicate that with special attention to the location of the boundaries a grid with approximately 60,000 points is adequate. This results in a computational time of approximately 40 minutes on a Cray-XMP. The practicality of the method to calculate HSI noise is demonstrated by expanding the scope of the investigation to examine the rectangular blade as well as a highly swept and tapered blade over a tip Mach number range of 0.80 to 0.95. Comparisons with experimental data are excellent and the advantages of planform modifications are clearly evident. New insight is gained into the mechanisms of nonlinear propagation and the minimum distance at which a valid comparison of different rotors can be made: approximately two rotor radii from the center of rotation.

Baeder, J. D.↗

DOE OSTI2024

Nonexistence of integrable nonlinear magnetic fields with invariants quadratic in momenta

Nonlinear, completely integrable Hamiltonian systems that serve as blueprints for novel particle accelerators at the intensity frontier are promising avenues for research, as Fermilab’s Integrable Optics Test Accelerator (IOTA) example clearly illustrates. Here, we show that only very limited generalizations are possible when no approximations in the underlying Hamiltonian or Maxwell equations are allowed, as was the case for IOTA. Specifically, no such systems exist with invariants quadratic in the momenta, precluding straightforward generalization of the Courant-Snyder theory of linear integrable systems in beam physics. We also conjecture that no such systems exist with invariants of higher degree in the momenta. This leaves solenoidal magnetic fields, including their nonlinear fringe fields, as the only completely integrable static magnetic fields, albeit with invariants that are linear in the momenta. The difficulties come from enforcing Maxwell equations; without constraints, we show that there are many solutions. In particular, we discover a previously unknown large family of integrable Hamiltonians.

97 MATHEMATICS AND COMPUTING↗

NASA NTRS1988

Hybrid transfinite element modeling/analysis of nonlinear heat conduction problems involving phase change

The present paper describes the applicability of hybrid transfinite element modeling/analysis formulations for nonlinear heat conduction problems involving phase change. The methodology is based on application of transform approaches and classical Galerkin schemes with finite element formulations to maintain the modeling versatility and numerical features for computational analysis. In addition, in conjunction with the above, the effects due to latent heat are modeled using enthalpy formulations to enable a physically realistic approximation to be dealt computationally for materials exhibiting phase change within a narrow band of temperatures. Pertinent details of the approach and computational scheme adapted are described in technical detail. Numerical test cases of comparative nature are presented to demonstrate the applicability of the proposed formulations for numerical modeling/analysis of nonlinear heat conduction problems involving phase change.

Tamma, Kumar K.↗

DOE OSTI2021

Analysis of the SBP-SAT Stabilization for Finite Element Methods Part II: Entropy Stability

In the hyperbolic research community, there exists the strong belief that a continuous Galerkin scheme is notoriously unstable and additional stabilization terms have to be added to guarantee stability. In the first part of the series, the application of simultaneous approximation terms for linear problems is investigated where the boundary conditions are imposed weakly. By applying this technique, the authors demonstrate that a pure continuous Galerkin scheme is indeed linearly stable if the boundary conditions are imposed in the correct way. In this work, we extend this investigation to the nonlinear case and focus on entropy conservation. Here, by switching to entropy variables, we provide an estimation of the boundary operators also for nonlinear problems, that guarantee conservation. In numerical simulations, we verify our theoretical analysis.

97 MATHEMATICS AND COMPUTING↗

DOE OSTI2022

Operator inference for non-intrusive model reduction with quadratic manifolds

This paper proposes a novel approach for learning a data-driven quadratic manifold from high-dimensional data, then employing this quadratic manifold to derive efficient physics-based reduced-order models. The key ingredient of the approach is a polynomial mapping between high-dimensional states and a low-dimensional embedding. This mapping consists of two parts: a representation in a linear subspace (computed in this work using the proper orthogonal decomposition) and a quadratic component. The approach can be viewed as a form of data-driven closure modeling, since the quadratic component introduces directions into the approximation that lie in the orthogonal complement of the linear subspace, but without introducing any additional degrees of freedom to the low-dimensional representation. Combining the quadratic manifold approximation with the operator inference method for projection-based model reduction leads to a scalable non-intrusive approach for learning reduced-order models of dynamical systems. Further, applying the new approach to transport-dominated systems of partial differential equations illustrates the gains in efficiency that can be achieved over approximation in a linear subspace.

42 ENGINEERING↗

DOE OSTI2019

A hybrid point-particle force model that combines physical and data-driven approaches

Here this study improves upon the physics-based pairwise interaction extended point-particle (PIEP) model. The PIEP model leverages our physical understanding to predict fluid mediated interactions between solid particles. By considering the relative location of neighboring particles, the PIEP model is able to provide better predictions than existing drag models. While the current physical PIEP model is a powerful tool, its assumptions lead to increased error in flows with higher particle volume fractions. To reduce this error, a regression algorithm makes direct use of the results of direct numerical simulations (DNS) of an array of monodisperse solid particles subjected to uniform ambient flow at varying Reynolds numbers. The resulting statistical model and the physical PIEP model are superimposed to construct a hybrid, physics-based data-driven PIEP model. It must be noted that the performance of a pure data-driven approach without the model-form provided by the physical PIEP model is substantially inferior. The hybrid model's predictive capabilities are analyzed using additional DNS data that was not part of training the data-driven model. In every case tested, the hybrid models resulting from the regression were capable of (1) improving upon the physical PIEP model's prediction and (2) recovering underlying relevant physics from the DNS data. As the particle volume fraction increases, the physical PIEP model loses the ability to approximate the forces experienced by the particles, but the statistical model continues to produce accurate approximations.

97 MATHEMATICS AND COMPUTING↗

DOE OSTI2017

Parameter optimization for transitions between memory states in small arrays of Josephson junctions

Coupled arrays of Josephson junctions possess multiple stable zero voltage states. Such states can store information and consequently can be utilized for cryogenic memory applications. Basic memory operations can be implemented by sending a pulse to one of the junctions and studying transitions between the states. In order to be suitable for memory operations, such transitions between the states have to be fast and energy efficient. Here in this article we employed simulated annealing, a stochastic optimization algorithm, to study parameter optimization of array parameters which minimizes times and energies of transitions between specifically chosen states that can be utilized for memory operations (Read, Write, and Reset). Simulation results show that such transitions occur with access times on the order of 10–100 ps and access energies on the order of 10 -19 –5×10 -18 J. Numerical simulations are validated with approximate analytical results.

97 MATHEMATICS AND COMPUTING↗

DOE OSTI2018

Dynamical instability of the electric transport in superconductors

We develop a nonlinear theory of the electronic transport in superconductors in the framework of the time-dependent Ginzburg-Landau (TDGL) equation. We utilize self-consistent Gaussian approximation and reveal the conditions under which the current-voltage V(I) dependence (I-V characteristics) acquires an S-shape form leading to switching instabilities. We demonstrate that in two-dimensions the emergence of such an instability is a hallmark of the Berezinskii-Kosterlitz-Thou less (BKT) transition that we have detected by transport measurements of titanium nitride (TiN) films. Our theoretical findings compare favorably with our experimental results.

75 CONDENSED MATTER PHYSICS, SUPERCONDUCTIVITY AND↗

DOE OSTI2018

Solving high-dimensional partial differential equations using deep learning

Developing algorithms for solving high-dimensional partial differential equations (PDEs) has been an exceedingly difficult task for a long time, due to the notoriously difficult problem known as the “curse of dimensionality.” This paper introduces a deep learning-based approach that can handle general high-dimensional parabolic PDEs. To this end, the PDEs are reformulated using backward stochastic differential equations and the gradient of the unknown solution is approximated by neural networks, very much in the spirit of deep reinforcement learning with the gradient acting as the policy function. Numerical results on examples including the nonlinear Black–Scholes equation, the Hamilton–Jacobi–Bellman equation, and the Allen–Cahn equation suggest that the proposed algorithm is quite effective in high dimensions, in terms of both accuracy and cost. Furthermore, this opens up possibilities in economics, finance, operational research, and physics, by considering all participating agents, assets, resources, or particles together at the same time, instead of making ad hoc assumptions on their interrelationships.

97 MATHEMATICS AND COMPUTING↗

DOE OSTI2018

Flexible Foam Model.

Experiments were performed to characterize the mechanical response of a 15 pcf flexible polyurethane foam to large deformation at different strain rates and temperatures. Results from these experiments indicated that at room temperature, flexible polyurethane foams exhibit significant nonlinear elastic deformation and nearly return to their original undeformed shape when unloaded. However, when these foams are cooled to temperatures below their glass transition temperature of approximately -35 o C, they behave like rigid polyurethane foams and exhibit significant permanent deformation when compressed. Thus, a new model which captures this dramatic change in behavior with temperature was developed and implemented into SIERRA with the name Flex_Foam to describe the mechanical response of both flexible and rigid foams to large deformation at a variety of temperatures and strain rates. This report includes a description of recent experiments. Next, development of the Flex Foam model for flexible polyurethane and other flexible foams is described. Selection of material parameters are discussed and finite element simulations with the new Flex Foam model are compared with experimental results to show behavior that can be captured with this new model.

36 MATERIALS SCIENCE↗

NASA NTRS1971

Techniques for generation of control and guidance signals derived from optical fields, part 2

The development is reported of a high resolution technique for the detection and identification of landmarks from spacecraft optical fields. By making use of nonlinear regression analysis, a method is presented whereby a sequence of synthetic images produced by a digital computer can be automatically adjusted to provide a least squares approximation to a real image. The convergence of the method is demonstrated by means of a computer simulation for both elliptical and rectangular patterns. Statistical simulation studies with elliptical and rectangular patterns show that the computational techniques developed are able to at least match human pattern recognition capabilities, even in the presence of large amounts of noise. Unlike most pattern recognition techniques, this ability is unaffected by arbitrary pattern rotation, translation, and scale change. Further development of the basic approach may eventually allow a spacecraft or robot vehicle to be provided with an ability to very accurately determine its spatial relationship to arbitrary known objects within its optical field of view.

Hemami, H.↗

NASA NTRS1976

Indirect synthesis of multidegree-of-freedom transient systems

The indirect synthesis method is developed and shown to be capable of leading a near-optimal design of multidegree-of-freedom and multidesign-element transient nonlinear dynamical systems. The basis of the approach is to select the open design parameters such that the response of the portion of the system being designed approximates the limiting performances solution. The limiting performance problem can be formulated as one of linear programming by replacing all portions of the system subject to transient disturbances by control forces and supposing that the remaining portions are linear as are the overall kinematic constraints. One then selects the design parameters that respond most closely to the limiting performance solution, which can be achieved by unconstrained curve-fitting techniques.

Chen, Y. H.↗

NASA NTRS1978

Gaussian step-pressure loading of rigid viscoplastic plates

The response of a thin, rigid viscoplastic plate subjected to a spatially axisymmetric Gaussian step pressure impulse loading was studied analytically. A Gaussian pressure distribution in excess of the collapse load was applied to the plate, held constant for a length of time, and then suddenly removed. The plate deforms with monotonically increasing deflections until the dynamic energy is completely dissipated in plastic work. The simply supported plate of uniform thickness obeys the von Mises yield criterion and a generalized constitutive equation for rigid viscoplastic materials. For the small deflection bending response of the plate, the governing system of equations is essentially nonlinear. Transverse shear stress is neglected in the yield condition and rotary inertia in the equations of dynamic equilibrium. A proportional loading technique, known to give excellent approximations of the exact solution for the uniform load case, was used to linearize the problem and to obtain the analytical solutions in the form of eigenvalue expansions. The effects of load concentration, of an order of magnitude change in the viscosity of the plate material, and of load duration were examined while holding the total impulse constant.

Hayduk, R. J.↗

NASA NTRS1979

A comparison of linear acoustic theory with experimental noise data for a small-scale hovering rotor

Linear acoustic calculations based on full aerodynamic data as input are presented and compared with measured cases reported by Boxwell et al. (1978). The full aerodynamic data are obtained using three programs giving radial loading, chordwise loading, and chordwise position of transition. It is shown that in the theoretical results the most significant noise source mechanism is due to blade thickness. Thus the conclusions of Boxwell et al. as to the importance of nonlinearities around the blades are upheld. These conclusions concern the width, shape and the level of the acoustic pressure calculated from linear acoustic theory. Some of the approximations involved in the application of acoustic analogy using quadrupole sources are discussed. It is necessary that the near- and far-field problems of rotating blades be treated together as shown for the case of an oscillating sphere.

Farassat, F.↗

NASA NTRS1979

A search for the interplanetary quantity controlling the development of geomagnetic storms

An historical account is presented concerning the evolution of our present concept of geomagnetic storms. The present concept was formulated by Chapman (1927) in his magnetic data statistical studies of 'the initial rise' (now termed the initial phase) 'and subsequent larger decrease' (now termed the main phase) in H, followed by 'slow recovery'. The concept introduced by Alfven in 1940 of guiding center motions of a charged particle in a nonuniform magnetic field (ring currents) is also discussed. By 1963 it became quite certain that the ring current, namely a storm-time Van Allen belt, is formed in the magnetosphere during the storm's main phase. The search then began for the solar wind quantity controlling the development of the main phase. The author then gives a personal account of how our concept of geomagnetic storms has advanced and how new findings based on satellite and ground-based observations have made it possible to arrive at a first-approximation expression for the interplanetary quantity controlling the development of geomagnetic storms. Since a geomagnetic storm is a magnetic manifestation of a magnetospheric storm, which is a nonlinear superposition of intense magnetospheric substorms, the main emphasis is shifted toward the understanding of magnetospheric substorms in order to arrive at the parameters controlling the development of geomagnetic storms.

Akasofu, S.-I.↗

NASA NTRS1983

Implicit Total Variation Diminishing (TVD) schemes for steady-state calculations

The application of a new implicit unconditionally stable high resolution total variation diminishing (TVD) scheme to steady state calculations. It is a member of a one parameter family of explicit and implicit second order accurate schemes developed by Harten for the computation of weak solutions of hyperbolic conservation laws. This scheme is guaranteed not to generate spurious oscillations for a nonlinear scalar equation and a constant coefficient system. Numerical experiments show that this scheme not only has a rapid convergence rate, but also generates a highly resolved approximation to the steady state solution. A detailed implementation of the implicit scheme for the one and two dimensional compressible inviscid equations of gas dynamics is presented. Some numerical computations of one and two dimensional fluid flows containing shocks demonstrate the efficiency and accuracy of this new scheme.

Yee, H. C.↗

NASA NTRS1983

Implicit total variation diminishing (TVD) schemes for steady-state calculations

The application of a new implicit unconditionally stable high resolution total variation diminishing (TVD) scheme to steady state calculations. It is a member of a one parameter family of explicit and implicit second order accurate schemes developed by Harten for the computation of weak solutions of hyperbolic conservation laws. This scheme is guaranteed not to generate spurious oscillations for a nonlinear scalar equation and a constant coefficient system. Numerical experiments show that this scheme not only has a rapid convergence rate, but also generates a highly resolved approximation to the steady state solution. A detailed implementation of the implicit scheme for the one and two dimensional compressible inviscid equations of gas dynamics is presented. Some numerical computations of one and two dimensional fluid flows containing shocks demonstrate the efficiency and accuracy of this new scheme. Previously announced in STAR as N83-23085

Yee, H. C.↗