Engineering Papers⌕ Search

SEARCH · Engineering Papers

Results for “Galerkin method”

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 181 records · Page 10
All sourcesNASA NTRSDOE OSTI
NASA NTRS1986

Influence of third-degree geometric nonlinearities on the vibration and stability of pretwisted, preconed, rotating blades

The governing coupled flapwise bending, edgewise bending, and torsional equations are derived including third-degree geometric nonlinear elastic terms by making use of the geometric nonlinear theory of elasticity in which the elongations and shears are negligible compared to unity. These equations are specialized for blades of doubly symmetric cross section with linear variation of pretwist over the blade length. The nonlinear steady state equations and the linearized perturbation equations are solved by using the Galerkin method, and by utilizing the nonrotating normal modes for the shape functions. Parametric results obtained for various cases of rotating blades from the present theoretical formulation are compared to those produced from the finite element code MSC/NASTRAN, and also to those produced from an in-house experimental test rig. It is shown that the spurious instabilities, observed for thin, rotating blades when second degree geometric nonlinearities are used, can be eliminated by including the third-degree elastic nonlinear terms. Furthermore, inclusion of third degree terms improves the correlation between the theory and experiment.

Subrahmanyam, K. B.↗
NASA NTRS1985

Nonlinear effects in the coupled response of tiles bonded to a plate

The coupled response to a large impulse on tiles bonded to a finite plate is studied. The analysis includes geometric nonlinearity born from boundary restraint in the plane, which stiffens the plate transversely. It also includes material nonlinearities born from plasticity of the plate's material and from properties of a polymer bond including memory and dissipation. The equations of motion are solved by the Galerkin method using linearized eigenfunctions of the system as trial functions. A strip of plate which is one tile wide is modeled.

El-Raheb, M.↗
NASA NTRS1985

Topics in spectral methods

After detailing the construction of spectral approximations to time-dependent mixed initial boundary value problems, a study is conducted of differential equations of the form 'partial derivative of u/partial derivative of t = Lu + f', where for each t, u(t) belongs to a Hilbert space such that u satisfies homogeneous boundary conditions. For the sake of simplicity, it is assumed that L is an unbounded, time-independent linear operator. Attention is given to Fourier methods of both Galerkin and pseudospectral method types, the Galerkin method, the pseudospectral Chebyshev and Legendre methods, the error equation, hyperbolic partial differentiation equations, and time discretization and iterative methods.

Gottlieb, D.↗
NASA NTRS1986

Generalized INF-SUP condition for Chebyshev approximation of the Navier-Stokes equations

An abstract mixed problem and its approximation are studied; both are well-posed if and only if several inf-sup conditions are satisfied. These results are applied to a spectral Galerkin method for the Stokes problem in a square, when it is formulated in Chebyshev weighted Sobolev spaces. Finally, a collocation method for the Navier-Stokes equations at Chebyshev nodes is analyzed.

Bernardi, Christine↗
NASA NTRS1987

Nonlinear vibration and stability of rotating, pretwisted, preconed blades including Coriolis effects

The coupled bending-bending-torsional equations of dynamic motion of rotating, linearly pretwisted blades are derived including large precone, second degree geometric nonlinearities and Coriolis effects. The equations are solved by the Galerkin method and a linear perturbation technique. Accuracy of the present method is verified by conparisons of predicted frequencies and steady state deflections with those from MSC/NASTRAN and from experiments. Parametric results are generated to establish where inclusion of only the second degree geometric nonlinearities is adequate. The nonlinear terms causing torsional divergence in thin blades are identified. The effects of Coriolis terms and several other structurally nonlinear terms are studied, and their relative importance is examined.

Subrahmanyam, K. B.↗
NASA NTRS1987

Influence of third-degree geometric nonlinearities on the vibration and stability of pretwisted, preconed, rotating blades

The governing coupled flapwise bending, edgewise bending, and torsional equations are derived including third-degree goemetric nonlinear elastic terms by making use of the geometric nonlinear theory of elasticity in which the elongations and shears are negligible compared to unity. These equations are specialized for blades of doubly symmetric cross section with linear variation of Pretwist over the blade length. The nonlinear steady state equations and the linearized perturbation equations are solved by using the Galerkin method, and by utilizing the nonrotating normal modes for the shape functions. Parametric results obtained for various cases of rotating blades from the present theoretical formulation are compared to those produced from the finite element code MSC/NASTRAN, and also to those produced from an in-house experimental test rig. It is shown that the spurious instabilities, observed for thin, rotating blades when second degree geometric nonlinearities are used, can be eliminated by including the third-degree elastic nonlinear terms. Furthermore, inclusion of third degree terms improves the correlation between the theory and experiment.

Subrahmanyam, K. B.↗
NASA NTRS1987

Dynamics of runaway tails with time-dependent sub-Dreicer dc fields in magnetized plasmas

The evolution of runaway tails driven by sub-Dreicer time-dependent dc fields in a magnetized plasma are studied numerically using a quasi-linear code based on the Ritz-Galerkin method and finite elements. It is found that the runaway tail maintained a negative slope during the dc field increase. Depending on the values of the dc electric field at t = 0 and the electron gyrofrequency to the plasma frequency ratio the runaway tail became unstable to the anomalous Doppler resonance or remained stable before the saturation of the dc field at some maximum value. The systems that remained stable during this stage became unstable to the anomalous Doppler or the Cerenkov resonances when the dc field was kept at the saturation level or decreased. Once the instability is triggered, the runaway tail is isotropized.

Moghaddam-Taaheri, E.↗
NASA NTRS1989

Rigorous analysis of a circular patch antenna excited by a microstrip transmission line

Boundary conditions are enforced on a portion of the microstrip feed line as well as the patch antenna. The integral equation for the unknown currents on the antenna and feed is solved by applying the Galerkin method of moments in the Fourier transform domain. The validity of the solution is tested by comparison of computed results with experimental data. The theoretical treatment proves to be applicable to the most common feeding arrangements, namely, the direct edge-feed and proximity coupling excitation. In the latter case, two-layer substrates having distinct dielectric constants are studied. The purpose of the study is to deduce, for a given overall substrate thickness, the smallest line-ground plane separation for which a match of the radiator to the feed line is still possible. The advantages of such a configuration are discussed.

Davidovitz, Marat↗
NASA NTRS1990

Acceleration of runaway electrons in solar flares

The dc electric field acceleration of electrons out of a thermal plasma and the evolution of the runaway tail are studied numerically, using a relativistic quasi-linear code based on the Ritz-Galerkin method and finite elements. A small field-aligned electric field is turned on at a certain time. The resulting distribution function from the runaway process is used to calculate the synchrotron emission during the evolution of the runaway tail. It is found that, during the runaway tail formation, which lasts a few tens of seconds for typical solar flare conditions, the synchrotron emission level is low, almost ot the same order as the emission from the thermal plasma, at the high-frequency end of the spectrum. However, the emission is enhanced explosively in a few microseconds by several orders of magnitude at the time the runaway tail stops growing along the magnetic field and tends toward isotropy due to the pitch-angle scattering of the fast particles. Results indicate that, in order to account for the observed synchrotron emission spectrum of a typical solar flare, the electric field acceleration phase must be accompanied or preceded by a heating phase which yields an enhanced electron temperature of about 2-15 keV in the flare region if the electric field is 0.1-0.2 times the Dreicer field and cyclotron-to-plasma frequency ratios are of order 1-2.

Moghaddam-Taaheri, E.↗
NASA NTRS1990

Finite-element schemes for extended integrations of atmospheric models

Of the two finite-element models presently used to investigate the effect of the conservation of integral invariants, by means of finite-element discretization schemes of the shallow-water equations, in order to serve as a paradigm of long-term atmospheric-model integrations, the first employs rectangular elements and conserves total energy, while the second uses triangular elements and a high-accuracy two-stage Numerov-Galerkin method. Attention is given to critical times for numerical nonlinear instability, as well as to the determination of the critical degree of dissipation entailed by the achievement of stable long-term integrations. Relative computational efficiency and accuracy comparisons are presented for the two finite-element schemes.

Steppeler, J.↗
NASA NTRS1991

Numerical recovery of material parameters in Euler-Bernoulli beam models

A fully Sinc-Galerkin method for recovering the spatially varying stiffness parameter in fourth-order time-dependence problems with fixed and cantilever boundary conditions is presented. The forward problems are discretized with a sinc basis in both the spatial and temporal domains. This yields an approximation solution which converges exponentially and is valid on the infinite time interval. When the forward methods are applied to parameter recovery problems, the resulting inverse problems are ill-posed. Tikhonov regularization is applied and the resulting minimization problems are solved via a quasi-Newton/trust region algorithm. The L-curve method is used to determine an appropriate value of the regularization parameter. Numerical results which highlight the method are given for problems with both fixed and cantilever boundary conditions.

Smith, R. C.↗
NASA NTRS1991

Sinc-Galerkin estimation of diffusivity in parabolic problems

A fully Sinc-Galerkin method for the numerical recovery of spatially varying diffusion coefficients in linear partial differential equations is presented. Because the parameter recovery problems are inherently ill-posed, an output error criterion in conjunction with Tikhonov regularization is used to formulate them as infinite-dimensional minimization problems. The forward problems are discretized with a sinc basis in both the spatial and temporal domains thus yielding an approximate solution which displays an exponential convergence rate and is valid on the infinite time interval. The minimization problems are then solved via a quasi-Newton/trust region algorithm. The L-curve technique for determining an approximate value of the regularization parameter is briefly discussed, and numerical examples are given which show the applicability of the method both for problems with noise-free data as well as for those whose data contains white noise.

Smith, Ralph C.↗
NASA NTRS1992

Effect of surface tension on the onset of convection in a double-diffusive layer

The effect of surface tension on the stability of a double-diffusive layer is considered using linear stability analysis. The surface tension is assumed to vary linearly with temperature and solute concentration. The eigenvalue problem is solved by the Galerkin method. Results show that the predicted stability boundary based on Marangoni effects alone is completely altered in the presence of buoyancy effects induced by low gravity levels (about 10 exp -5 g). At reduced gravity levels, salt-finger instability may onset in the overstable mode due to the stabilizing effect of surface tension. Fluid properties in terms of the Prandtl and the Lewis numbers have a profound effect on the stability conditions; opposite stability characteristics are found in salt solutions and in molten metals.

Chen, C. F.↗
NASA NTRS1993

Double-diffusive fingering convection in a porous medium

The characteristics of nonlinear two-dimensional horizontally periodic double-diffusive fingering convection in a saturated porous medium is investigated, using the Darcy equation including Brinkman and Forchheimer terms for the momentum equation. To solve the equations and the corresponding initial and boundary conditions, a Galerkin method is applied in the horizontal direction, and a hybrid finite difference method is used in the vertical direction. The developed computer code was used to compute the thermal convection case, and the results were found to be in good agreement with existing results.

Chen, Falin↗
NASA NTRS1993

Simulation of tail buffet using delta wing-vertical tail configuration

Computational simulation of the vertical tail buffet problem is accomplished using a delta wing-vertical tail configuration. Flow conditions are selected such that the wing primary-vortex cores experience vortex breakdown and the resulting flow interacts with the vertical tail. This multidisciplinary problem is solved successively using three sets of equations for the fluid flow, aeroelastic deflections and grid displacements. For the fluid dynamics part, the unsteady, compressible, full Navier-Stokes equations are solved accurately in time using an implicit, upwind, flux-difference splitting, finite-volume scheme. For the aeroelastic part, the aeroelastic equation for bending vibrations is solved accurately in time using the Galerkin method and the four-stage Runge-Kutta scheme. The grid for the fluid dynamics computations is updated every few time steps using a third set of interpolation equations. The computational application includes a delta wing of aspect ratio 1 and a rectangular vertical tail of aspect ratio 2, which is placed at 0.5 root-chord length downstream of the wing trailing edge. The wing angle of attack is 35 deg and the flow Mach number and Reynolds number are 0.4 and 10,000, respectively.

Kandil, Osama A.↗
NASA NTRS1950

Critical stress of ring-stiffened cylinders in torsion

A chart in terms of nondimensional parameters is presented for the theoretical critical stress in torsion of simply supported cylinders stiffened by identical equally spaced rings of zero torsional stiffness. The results are obtained by solving the equation of equilibrium by means of the Galerkin method. Comparison of the theoretical results with experimental results indicates that ring-stiffened cylinders buckle, on the average, at a buckling stress about 15 percent below the theoretical buckling stress. (author)

Stein, Manuel↗
NASA NTRS1993

Wavelet multiresolution analyses adapted for the fast solution of boundary value ordinary differential equations

We present ideas on how to use wavelets in the solution of boundary value ordinary differential equations. Rather than using classical wavelets, we adapt their construction so that they become (bi)orthogonal with respect to the inner product defined by the operator. The stiffness matrix in a Galerkin method then becomes diagonal and can thus be trivially inverted. We show how one can construct an O(N) algorithm for various constant and variable coefficient operators.

Jawerth, Bjoern↗
NASA NTRS1995

Scattering Of Nonplanar Acoustic Waves

Report presents theoretical study of scattering of nonplanar acoustic waves by rigid bodies. Study performed as part of effort to develop means of predicting scattering, from aircraft fuselages, of noise made by rotating blades. Basic approach was to model acoustic scattering by use of boundary integral equation to solve equation by the Galerkin method.

Gillman, Judith M.↗