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Results for “nonlinear partial differential equations”

Search indexed NASA NTRS and DOE OSTI research on propulsion, heat transfer, battery materials and energy systems. Follow report and document links to the original sources.

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At least 235 records · Page 13
All sourcesNASA NTRSDOE OSTI
NASA NTRS1991

Spillover, nonlinearity, and flexible structures

Many systems whose evolution in time is governed by Partial Differential Equations (PDEs) are linearized around a known equilibrium before Computer Aided Control Engineering (CACE) is considered. In this case, there are infinitely many independent vibrational modes, and it is intuitively evident on physical grounds that infinitely many actuators would be needed in order to control all modes. A more precise, general formulation of this grave difficulty (spillover problem) is due to A.V. Balakrishnan. A possible route to circumvention of this difficulty lies in leaving the PDE in its original nonlinear form, and adding the essentially finite dimensional control action prior to linearization. One possibly applicable technique is the Liapunov Schmidt rigorous reduction of singular infinite dimensional implicit function problems to finite dimensional implicit function problems. Omitting details of Banach space rigor, the formalities of this approach are given.

Bass, Robert W.↗
NASA NTRS1992

Linear and nonlinear stability of the Blasius boundary layer

Two new techniques for the study of the linear and nonlinear instability in growing boundary layers are presented. The first technique employs partial differential equations of parabolic type exploiting the slow change of the mean flow, disturbance velocity profiles, wavelengths, and growth rates in the streamwise direction. The second technique solves the Navier-Stokes equation for spatially evolving disturbances using buffer zones adjacent to the inflow and outflow boundaries. Results of both techniques are in excellent agreement. The linear and nonlinear development of Tollmien-Schlichting (TS) waves in the Blasius boundary layer is investigated with both techniques and with a local procedure based on a system of ordinary differential equations. The results are compared with previous work and the effects of non-parallelism and nonlinearity are clarified. The effect of nonparallelism is confirmed to be weak and, consequently, not responsible for the discrepancies between measurements and theoretical results for parallel flow.

Bertolotti, F. P.↗
NASA NTRS1971

Similarity analysis of differential equations by Lie group.

Methods for transforming partial differential equations into forms more suitable for analysis and solution are investigated. The idea of Lie's infinitesimal contact transformation group is introduced to develop a systematic method which involves mostly algebraic manipulations. A thorough presentation of the application of this general method to the problem of similarity analysis in a broader sense - namely, the similarity between partial and ordinary differential equations, boundary value and initial value problems, and nonlinear and linear equations - is given with new and very general methods evolved for deriving the possible groups of transformations.

Na, T. Y.↗
NASA NTRS1992

Fast neural solution of a nonlinear wave equation

A neural algorithm for rapidly simulating a certain class of nonlinear wave phenomena using analog VLSI neural hardware is presented and applied to the Korteweg-de Vries partial differential equation. The corresponding neural architecture is obtained from a pseudospectral representation of the spatial dependence, along with a leap-frog scheme for the temporal evolution. Numerical simulations demonstrated the robustness of the proposed approach.

Toomarian, Nikzad↗
NASA NTRS1993

On nonlinear Tollmien-Schlichting/vortex interaction in three-dimensional boundary layers

The instability of an incompressible three-dimensional boundary layer (that is, one with cross-flow) is considered theoretically and computationally in the context of vortex/wave interactions. Specifically the work centers on two low amplitude, lower-branch Tollmien-Schlichting waves which mutually interact to induce a weak longitudinal vortex flow; the vortex motion, in turn, gives rise to significant wave-modulation via wall-shear forcing. The characteristic Reynolds number is taken as a large parameter and, as a consequence, the waves' and the vortex motion are governed primarily by triple-deck theory. The nonlinear interaction is captured by a viscous partial-differential system for the vortex coupled with a pair of amplitude equations for each wave pressure. Three distinct possibilities were found to emerge for the nonlinear behavior of the flow solution downstream - an algebraic finite-distance singularity, far downstream saturation or far-downstream wave-decay (leaving pure vortex flow) - depending on the input conditions, the wave angles, and the size of the cross-flow.

Davis, Dominic A. R.↗
DOE OSTI2020

Gradient-based constrained optimization using a database of linear reduced-order models

A methodology grounded in model reduction is presented for accelerating the gradient-based solution of a family of linear or nonlinear constrained optimization problems where the constraints include at least one linear Partial Differential Equation (PDE). A key component of this methodology is the construction, during an offline phase, of a database of pointwise, linear, Projection-based Reduced-Order Models (PROM)s associated with a design parameter space and the linear PDE(s). A parameter sampling procedure based on an appropriate saturation assumption is proposed to maximize the efficiency of such a database of PROMs. A real-time method is also presented for interpolating at any queried but unsampled parameter vector in the design parameter space the relevant sensitivities of a PROM. The practical feasibility, computational advantages, and performance of the proposed methodology are demonstrated for several realistic, nonlinear, aerodynamic shape optimization problems governed by linear aeroelastic constraints.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2022

Development and Analysis of Optimal Multilevel Solvers on Advanced Computers. Final Report

Constrained optimization in the context of time dependent, partial differential equations (PDE) leads to a symmetric, block-tridiagonal system of nonlinear equations that must be solved repeatedly in an iterative solution strategy. The blocks represent spatial discretization, while the connection between the blocks represents a forward and backward integration in time. The focus of this project is to apply a parallel-in-time (PiT) solution technique to the large block-triangular system.

97 MATHEMATICS AND COMPUTING↗
NASA NTRS1979

Formulation of coupled rotor/fuselage equations of motion

The governing equations of motion of a helicopter rotor coupled to a rigid body fuselage are derived. A consistent formulation is used to derive nonlinear periodic coefficient equations of motion which can be used to study coupled rotor/fuselage dynamics in forward flight. The methodology of rotor/fuselage coupling is clearly described and the importance of an ordering scheme in deriving consistent nonlinear equations of motion is reviewed. The final equations which are presented in partial differential form can be used to model coupled rotor/fuselage aeroelastic response or stability problems.

Warmbrodt, W.↗
NASA NTRS1983

Singular Perturbations in Systems and Control

The application of singular perturbation theory to problems in systems and control is addressed. The types of systems considered include linear and nonlinear, discrete and continuous, and stochastic and deterministic. Both ordinary and partial differential equations are covered.

Ardema, M. D.↗
NASA NTRS1986

Concurrent processing in nonlinear structural stability

A concurrent processing algorithm is developed for materially nonlinear stability analysis of imperfect columns with biaxial partial rotational end restraints. The algorithm for solving the governing nonlinear ordinary differential equations is implemented on a multiprocessor computer called the 'Finite Element Machine', developed at the NASA Langley Research Center. Numerical results are obtained on up to nine concurrent processors. A substantial computational gain is achieved in using the parallel processing approach.

Darbhamulla, S. P.↗
NASA NTRS2010

Analysis of Instabilities in Non-Axisymmetric Hypersonic Boundary Layers Over Cones

Hypersonic flows over circular cones constitute one of the most important generic configurations for fundamental aerodynamic and aerothermodynamic studies. In this paper, numerical computations are carried out for Mach 6 flows over a 7-degree half-angle cone with two different flow incidence angles and a compression cone with a large concave curvature. Instability wave and transition-related flow physics are investigated using a series of advanced stability methods ranging from conventional linear stability theory (LST) and a higher-fidelity linear and nonlinear parabolized stability equations (PSE), to the 2D eigenvalue analysis based on partial differential equations. Computed N factor distribution pertinent to various instability mechanisms over the cone surface provides initial assessments of possible transition fronts and a guide to corresponding disturbance characteristics such as frequency and azimuthal wave numbers. It is also shown that strong secondary instability that eventually leads to transition to turbulence can be simulated very efficiently using a combination of advanced stability methods described above.

Li, Fei↗
NASA NTRS1994

A note on the accuracy of spectral method applied to nonlinear conservation laws

Fourier spectral method can achieve exponential accuracy both on the approximation level and for solving partial differential equations if the solutions are analytic. For a linear partial differential equation with a discontinuous solution, Fourier spectral method produces poor point-wise accuracy without post-processing, but still maintains exponential accuracy for all moments against analytic functions. In this note we assess the accuracy of Fourier spectral method applied to nonlinear conservation laws through a numerical case study. We find that the moments with respect to analytic functions are no longer very accurate. However the numerical solution does contain accurate information which can be extracted by a post-processing based on Gegenbauer polynomials.

Shu, Chi-Wang↗
NASA NTRS1983

Linear approximations of nonlinear systems

A method for designing an automatic flight controller for short and vertical takeoff aircraft is presently being developed at NASA Ames Research Center. This technique involves transformations of nonlinear systems to controllable linear systems and takes into account the nonlinearities of the aircraft. In general, the transformations cannot always be given in closed form. Using partial differential equations, an approximate linear system, called the modified tangent model, was recently introduced. A linear transformation of this tangent model to Brunovsky canonical form can be constructed, and from this an approximation of an exact transformation for the nonlinear system can be found. It is shown that a canonical expansion in Lie brackets about the point x(0) yields the same modified tangent model.

Hunt, L. R.↗
DOE OSTI2025

McCormick envelopes in mixed-integer PDE-constrained optimization

McCormick envelopes are a standard tool for deriving convex relaxations of optimization problems that involve polynomial terms. Such McCormick relaxations provide lower bounds, for example, in branch-and-bound procedures for mixed-integer nonlinear programs but have not gained much attention in PDE-constrained optimization so far. This lack of attention may be due to the distributed nature of such problems, which on the one hand leads to infinitely many linear constraints (generally state constraints that may be difficult to handle) in addition to the state equation for a pointwise formulation of the McCormick envelopes and renders bound-tightening procedures that successively improve the resulting convex relaxations computationally intractable. We analyze McCormick envelopes for a model problem class that is governed by a semilinear PDE involving a bilinearity and integrality constraints. We approximate the nonlinearity and in turn the McCormick envelopes by averaging the involved terms over the cells of a partition of the computational domain on which the PDE is defined. This yields convex relaxations that underestimate the original problem up to an a priori error estimate that depends on the mesh size of the discretization. These approximate McCormick relaxations can be improved by means of an optimization-based bound-tightening procedure. We show that their minimizers converge to minimizers to a limit problem with a pointwise formulation of the McCormick envelopes when driving the mesh size to zero. We provide a computational example, for which we certify all of our imposed assumptions. The results point to both the potential of the methodology and the gaps in the research that need to be closed. Our methodology provides a framework first for obtaining pointwise underestimators for nonconvexities and second for approximating them with finitely many linear inequalities in an infinite-dimensional setting.

Approximations and Expansions↗
NASA NTRS1994

Influence of cross-flow on nonlinear Tollmien-Schlichting/vortex interaction

The transition of an incompressible three-dimensional boundary layer with strong cross-flow is considered theoretically and computationally in the context of vortex/wave interactions. Specifically the work centers on two lower-branch Tollmien-Schlichting waves which mutually interact nonlinearly to induce a longitudinal vortex flow. The vortex motion in turn gives rise to significant wave modulation via wall-shear forcing. The characteristic Reynolds number is large and, as a consequence, the waves' and the vortex motion are governed primarily by triple deck theory. The nonlinear interaction is captured by a viscous partial-differential system for the vortex coupled with a pair of amplitude equations for each wave pressure. Following analysis and computation over a wide range of parameters, three distinct responses are found to emerge in the nonlinear behavior of the flow solution downstream: an algebraic finite-distance singularity, far-downstream saturation or far-downstream wave decay leaving pure vortex flow. These depend on the input conditions, the wave angles and the size of the cross flow.

Davis, D. A. R.↗
NASA NTRS1983

Linear approximations of nonlinear systems

The development of a method for designing an automatic flight controller for short and vertical take off aircraft is discussed. This technique involves transformations of nonlinear systems to controllable linear systems and takes into account the nonlinearities of the aircraft. In general, the transformations cannot always be given in closed form. Using partial differential equations, an approximate linear system called the modified tangent model was introduced. A linear transformation of this tangent model to Brunovsky canonical form can be constructed, and from this the linear part (about a state space point x sub 0) of an exact transformation for the nonlinear system can be found. It is shown that a canonical expansion in Lie brackets about the point x sub 0 yields the same modified tangent model.

Hunt, L. R.↗
DOE OSTI2023

Development of the Senseiver for efficient field reconstruction from sparse observations

Abstract The reconstruction of complex time-evolving fields from sensor observations is a grand challenge. Frequently, sensors have extremely sparse coverage and low-resource computing capacity for measuring highly nonlinear phenomena. While numerical simulations can model some of these phenomena using partial differential equations, the reconstruction problem is ill-posed. Data-driven-strategies provide crucial disambiguation, but these suffer in cases with small amounts of data, and struggle to handle large domains. Here we present the Senseiver, an attention-based framework that excels in reconstructing complex spatial fields from few observations with low overhead. The Senseiver reconstructs n -dimensional fields by encoding arbitrarily sized sparse sets of inputs into a latent space using cross-attention, producing uniform-sized outputs regardless of the number of observations. This allows efficient inference by decoding only a sparse set of output observations, while a dense set of observations is needed to train. This framework enables training of data with complex boundary conditions and extremely large fine-scale simulations. We build on the Perceiver IO by enabling training models with fewer parameters, which facilitates field deployment, and a training framework that allows a flexible number of sensors as input, which is critical for real-world applications. We show that the Senseiver advances the state-of-the-art of field reconstruction in many applications.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2024

Paired autoencoders for likelihood-free estimation in inverse problems

Abstract We consider the solution of nonlinear inverse problems where the forward problem is a discretization of a partial differential equation. Such problems are notoriously difficult to solve in practice and require minimizing a combination of a data-fit term and a regularization term. The main computational bottleneck of typical algorithms is the direct estimation of the data misfit. Therefore, likelihood-free approaches have become appealing alternatives. Nonetheless, difficulties in generalization and limitations in accuracy have hindered their broader utility and applicability. In this work, we use a paired autoencoder framework as a likelihood-free estimator (LFE) for inverse problems. We show that the use of such an architecture allows us to construct a solution efficiently and to overcome some known open problems when using LFEs. In particular, our framework can assess the quality of the solution and improve on it if needed. We demonstrate the viability of our approach using examples from full waveform inversion and inverse electromagnetic imaging.

Chung, Matthias (ORCID:0000000178224539)↗