Engineering Papers⌕ Search

SEARCH · Engineering Papers

Results for “Nonlinear 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.

Quote a phrase for an exact phrase match. Source license links do not imply unrestricted reuse.

At least 847 records · Page 47
All sourcesNASA NTRSDOE OSTI
DOE OSTI2026

Reduced-action-integral approach for photon-photon interactions in vacuum

Electromagnetic waves propagating through vacuum can polarize virtual electron–positron pairs; this polarization, in turn, nonlinearly modifies their propagation. A semi-classical nonlinear wave equation describing the propagation is derived from the Euler–Heisenberg Lagrangian density, which captures vacuum polarization effects up to the one-loop level. In this article, we present a reduced-actionintegral approach that enables rapid modeling of nonlinear phenomena arising from the Euler– Heisenberg Lagrangian. Application of the variational principle to the reduced action provides equations of motion for familiar light-pulse parameters, such as spot size, phase, polarization, and phase-front curvature, without requiring full-field simulations. Three examples demonstrate the utility of the approach: phase modulation, birefringence, and frequency mixing.

72 PHYSICS OF ELEMENTARY PARTICLES AND FIELDS↗
DOE OSTI2021

Vacuum magnetic fields with exact quasisymmetry near a flux surface. Part 1. Solutions near an axisymmetric surface

While several results have pointed to the existence of exactly quasisymmetric fields on a surface (Garren & Boozer, Phys. Fluids B, vol. 3, 1991, pp. 2805–2821; 2822–2834; Plunk & Helander, J. Plasma Phys. , vol. 84, 2018, 905840205), we have obtained the first such solutions using a vacuum surface expansion formalism. We obtain a single nonlinear parabolic partial differential equation for a function $\eta$ such the field strength satisfies $B = B(\eta )$ . Closed-form solutions are obtained in cylindrical, slab and isodynamic geometries. Numerical solutions of the full nonlinear equations in general axisymmetric toroidal geometry are obtained, resulting in a class of quasihelical local vacuum equilibria near an axisymmetric surface. The analytic models provide additional insight into general features of the nonlinear solutions, such as localization of the surface perturbations on the inboard side. The local solutions thus obtained can be continued globally only for special initial surfaces.

Physics↗
NASA NTRS1994

Finite dimensional approximation of a class of constrained nonlinear optimal control problems

An abstract framework for the analysis and approximation of a class of nonlinear optimal control and optimization problems is constructed. Nonlinearities occur in both the objective functional and in the constraints. The framework includes an abstract nonlinear optimization problem posed on infinite dimensional spaces, and approximate problem posed on finite dimensional spaces, together with a number of hypotheses concerning the two problems. The framework is used to show that optimal solutions exist, to show that Lagrange multipliers may be used to enforce the constraints, to derive an optimality system from which optimal states and controls may be deduced, and to derive existence results and error estimates for solutions of the approximate problem. The abstract framework and the results derived from that framework are then applied to three concrete control or optimization problems and their approximation by finite element methods. The first involves the von Karman plate equations of nonlinear elasticity, the second, the Ginzburg-Landau equations of superconductivity, and the third, the Navier-Stokes equations for incompressible, viscous flows.

Gunzburger, Max D.↗
NASA NTRS1977

Stability analysis for laminar flow control, part 1

The basic equations for the stability analysis of flow over three dimensional swept wings are developed and numerical methods for their solution are surveyed. The equations for nonlinear stability analysis of three dimensional disturbances in compressible, three dimensional, nonparallel flows are given. Efficient and accurate numerical methods for the solution of the equations of stability theory were surveyed and analyzed.

Benney, D. J.↗
NASA NTRS1987

Covariance and the hierarchy of frame bundles

This is an essay on the general concept of covariance, and its connection with the structure of the nested set of higher frame bundles over a differentiable manifold. Examples of covariant geometric objects include not only linear tensor fields, densities and forms, but affinity fields, sectors and sector forms, higher order frame fields, etc., often having nonlinear transformation rules and Lie derivatives. The intrinsic, or invariant, sets of forms that arise on frame bundles satisfy the graded Cartan-Maurer structure equations of an infinite Lie algebra. Reduction of these gives invariant structure equations for Lie pseudogroups, and for G-structures of various orders. Some new results are introduced for prolongation of structure equations, and for treatment of Riemannian geometry with higher-order moving frames. The use of invariant form equations for nonlinear field physics is implicitly advocated.

Estabrook, Frank B.↗
NASA NTRS1990

A computational algorithm for spacecraft control and momentum management

Developments in the area of nonlinear control theory have shown how coordinate changes in the state and input spaces of a dynamical system can be used to transform certain nonlinear differential equations into equivalent linear equations. These techniques are applied to the control of a spacecraft equipped with momentum exchange devices. An optimal control problem is formulated that incorporates a nonlinear spacecraft model. An algorithm is developed for solving the optimization problem using feedback linearization to transform to an equivalent problem involving a linear dynamical constraint and a functional approximation technique to solve for the linear dynamics in terms of the control. The original problem is transformed into an unconstrained nonlinear quadratic program that yields an approximate solution to the original problem. Two examples are presented to illustrate the results.

Dzielski, John↗
NASA NTRS1997

Application of Least-Squares Adjustment Technique to Geometric Camera Calibration and Photogrammetric Flow Visualization

Flow visualization produces data in the form of two-dimensional images. If the optical components of a camera system are perfect, the transformation equations between the two-dimensional image and the three-dimensional object space are linear and easy to solve. However, real camera lenses introduce nonlinear distortions that affect the accuracy of transformation unless proper corrections are applied. An iterative least-squares adjustment algorithm is developed to solve the nonlinear transformation equations incorporated with distortion corrections. Experimental applications demonstrate that a relative precision on the order of 40,000 is achievable without tedious laboratory calibrations of the camera.

Chen, Fang-Jenq↗
NASA NTRS1974

Estimation of material properties of a nonlinearly elastic bar

A method of determining certain characteristic flexural rigidities and elastic properties of nonlinearly elastic materials is presented. An estimation method utilizing perturbation methods and a least squares fitting technique is used to solve the nonlinear differential equation derived from the moment curvature relation, subject to boundary values representing deflections of the bar at discrete points. Deflection data from numerical simulations of a nonlinearly elastic, prismatic bar are used to demonstrate the estimation method. Numerical experiments relating the accuracy of the identification to the number and accuracy of the boundary values are presented. Conclusions based on the numerical experiments are included.

Baker, B. E.↗
NASA NTRS1979

Classification and equivalence in estimation theory

A method is proposed for classifying estimation problems based on the Lie algebra generated by the operators which appear in the conditional density equation. A natural class of automorphisms of this algebra is examined and a systematic method of generating equivalent problems is developed. Finally, a new class of nonlinear filtering problems with essentially nonlinear filtering equations are presented.

Brockett, R. W.↗
NASA NTRS1992

A robust direct-integration method for rotorcraft maneuver and periodic response

The Newmark-Beta method and the Newton-Raphson iteration scheme are combined to develop a direct-integration method for evaluating the maneuver and periodic-response expressions for rotorcraft. The method requires the generation of Jacobians and includes higher derivatives in the formulation of the geometric stiffness matrix to enhance the convergence of the system. The method leads to effective convergence with nonlinear structural dynamics and aerodynamic terms. Singularities in the matrices can be addressed with the method as they arise from a Lagrange multiplier approach for coupling equations with nonlinear constraints. The method is also shown to be general enough to handle singularities from quasisteady control-system models. The method is shown to be more general and robust than the similar 2GCHAS method for analyzing rotorcraft dynamics.

Panda, Brahmananda↗
NASA NTRS1993

Numerical and experimental investigation of the bending response of thin-walled composite cylinders

A numerical and experimental investigation of the bending behavior of six eight-ply graphite-epoxy circular cylinders is presented. Bending is induced by applying a known end-rotation to each end of the cylinders, analogous to a beam in bending. The cylinders have a nominal radius of 6 inches, a length-to-radius ratio of 2 and 5, and a radius-to-thickness ratio of approximately 160. A (+/- 45/0/90)S quasi-isotropic layup and two orthotropic layups, (+/- 45/0 sub 2)S and (+/- 45/90 sub 2)S, are studied. A geometrically nonlinear special-purpose analysis, based on Donnell's nonlinear shell equations, is developed to study the prebuckling responses and gain insight into the effects of non-ideal boundary conditions and initial geometric imperfections. A geometrically nonlinear finite element analysis is utilized to compare with the prebuckling solutions of the special-purpose analysis and to study the buckling and post buckling responses of both geometrically perfect and imperfect cylinders. The imperfect cylinder geometries are represented by an analytical approximation of the measured shape imperfections. Extensive experimental data are obtained from quasi-static tests of the cylinders using a test fixture specifically designed for the present investigation. A description of the test fixture is included. The experimental data are compared to predictions for both perfect and imperfect cylinder geometries. Prebuckling results are presented in the form of displacement and strain profiles. Buckling end-rotations, moments, and strains are reported, and predicted mode shapes are presented. Observed and predicted moment vs. end-rotation relations, deflection patterns, and strain profiles are illustrated for the post buckling responses. It is found that a geometrically nonlinear boundary layer behavior characterizes the prebuckling responses. The boundary layer behavior is sensitive to laminate orthotropy, cylinder geometry, initial geometric imperfections, applied end-rotation, and non-ideal boundary conditions. Buckling end-rotations, strains, and moments are influenced by laminate orthotropy and initial geometric imperfections. Measured buckling results correlate well with predictions for the geometrically imperfect specimens. The postbuckling analyses predict equilibrium paths with a number of scallop-shaped branches that correspond to unique deflection patterns. The observed postbuckling deflection patterns and measured strain profiles show striking similarities to the predictions in some cases. Ultimate failure of the cylinders is attributed to an interlaminar shear failure mode along the nodal lines of the postbuckling deflection patterns.

Fuchs, J. P.↗
NASA NTRS1990

Nonlinear spatial evolution of externally excited instability waves in free shear layers

The effects of critical-layer nonlinearity on spatially growing instability waves on shear layers between parallel streams are considered. In the two-dimensional incompressible case, the flow in the critical layer is governed by a nonequilibrium, nonlinear vorticity equation. The initial exponential growth of the instability wave is converted into algebraic growth during the streamwise 'aging' of the critical layer into a quasi-equilibrium state. This leads to a next stage of evolution where the instability-wave growth is affected by both mean-flow-divergence and nonlinear effects and is eventually converted to decay. A uniformly valid composite formula for the instability wave amplitude, accounting for both nonparallel and nonlinear effects, is given and compared to experimental results. The inviscid solutions to these equations always end in a singularity at finite downstream distance.

Hultgren, L. S.↗
NASA NTRS1976

A block iterative finite element algorithm for numerical solution of the steady-state, compressible Navier-Stokes equations

An iterative method for numerically solving the time independent Navier-Stokes equations for viscous compressible flows is presented. The method is based upon partial application of the Gauss-Seidel principle in block form to the systems of nonlinear algebraic equations which arise in construction of finite element (Galerkin) models approximating solutions of fluid dynamic problems. The C deg-cubic element on triangles is employed for function approximation. Computational results for a free shear flow at Re = 1,000 indicate significant achievement of economy in iterative convergence rate over finite element and finite difference models which employ the customary time dependent equations and asymptotic time marching procedure to steady solution. Numerical results are in excellent agreement with those obtained for the same test problem employing time marching finite element and finite difference solution techniques.

Cooke, C. H.↗
NASA NTRS1990

Application of adjoint operators to neural learning

A technique for the efficient analytical computation of such parameters of the neural architecture as synaptic weights and neural gain is presented as a single solution of a set of adjoint equations. The learning model discussed concentrates on the adiabatic approximation only. A problem of interest is represented by a system of N coupled equations, and then adjoint operators are introduced. A neural network is formalized as an adaptive dynamical system whose temporal evolution is governed by a set of coupled nonlinear differential equations. An approach based on the minimization of a constrained neuromorphic energylike function is applied, and the complete learning dynamics are obtained as a result of the calculations.

Barhen, J.↗
DOE OSTI2021

Interference in nonlinear Compton scattering using a Schrödinger-equation approach

The interference between Compton scattering and nonlinear Compton scattering from a two-color field in the X-ray regime is theoretically examined for bound electrons. Here, the underlying phase shifts are analyzed using a perturbative approach in the incoming classical field. The perturbative approach is bench marked with a non-perturbative approach in the classical field. The interference for different combinations of linear polarization of the two fields is examined when the Compton and the nonlinear Compton scattered waves have the same wave vector and polarization. Only two cases exhibit interference. When there is interference, the calculations reveal an intrinsic phase difference between the Compton scattered wave function and the nonlinear Compton scattered wave function of either 0 or pi depending on the scattering angle.

74 ATOMIC AND MOLECULAR PHYSICS↗
NASA NTRS1977

A block iterative finite element algorithm for numerical solution of the steady-state, compressible Navier-Stokes equations

An iterative method for numerically solving the time independent Navier-Stokes equations for viscous compressible flows is presented. The method is based upon partial application of the Gauss-Seidel principle in block form to the systems of the nonlinear algebraic equations which arise in construction of finite element (Galerkin) models approximating solutions of fluid dynamic problems. The continuous cubic element on triangles is employed for function approximation. Computational results for a free shear flow at Re = 1000 indicate significant achievement of economy in iterative convergence rate over finite element and finite difference models which employ the customary time dependent equations and symptotic time marching procedure to steady solution. Numerical results are in excellent agreement with those obtained for the same test problem employing time marching finite element and finite difference solution techniques.

Cooke, C. H.↗
NASA NTRS1984

Analysis of lateral and torsional vibration characteristics of beams and shafts with end located rotational masses

Partial differential equations are derived for free lateral and torsional vibration of a uniform free-free beam with a rotational mass attached to each extremity. For appropriate boundary conditions, nonlinear algebraic equations are obtained using a symbolic manipulation computer program, the solutions of which enable the computation of the neutral frequencies and mode-shapes. The mode-shapes are linear combinations of trigonometric and hyperbolic sine and cosine functions. A computer program is written for the numerical solution of the algebraic equations mentioned above, which can compute the natural frequencies, mode-shapes, and node points for any given set of parameters, for any given number of modes.

Robertson, D. K.↗