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At least 703 records · Page 39
All sourcesNASA NTRSDOE OSTI
NASA NTRS1981

Spectral multigrid methods for elliptic equations

An alternative approach which employs multigrid concepts in the iterative solution of spectral equations was examined. Spectral multigrid methods are described for self adjoint elliptic equations with either periodic or Dirichlet boundary conditions. For realistic fluid calculations the relevant boundary conditions are periodic in at least one (angular) coordinate and Dirichlet (or Neumann) in the remaining coordinates. Spectral methods are always effective for flows in strictly rectangular geometries since corners generally introduce singularities into the solution. If the boundary is smooth, then mapping techniques are used to transform the problem into one with a combination of periodic and Dirichlet boundary conditions. It is suggested that spectral multigrid methods in these geometries can be devised by combining the techniques.

Zang, T. A.↗
NASA NTRS1983

A new consistent spatial differencing scheme for the transonic full-potential equation

A new spatial differencing scheme for the transonic full-potential equation in conservative form has been developed. This scheme guarantees zero truncation error on any curvilinear mesh for freestream flows in either two- or three-space dimensions. Solutions obtained with this new differencing scheme, away from freestream regions, exhibit greatly improved accuracy, especially for nonsmooth or singular meshes.

Flores, J.↗
NASA NTRS2015

Improved Discrete Ordinate Solutions in the Presence of an Anisotropically Reflecting Lower Boundary: Upgrades of the DISORT Computational Tool

A successor version 3 of DISORT (DISORT3) is presented with important upgrades that improve the accuracy, efficiency, and stability of the algorithm. Compared with version 2 (DISORT2 released in 2000) these upgrades include (a) a redesigned BRDF computation that improves both speed and accuracy, (b) a revised treatment of the single scattering correction, and (c) additional efficiency and stability upgrades for beam sources. In DISORT3 the BRDF computation is improved in the following three ways: (i) the Fourier decomposition is prepared "off-line", thus avoiding the repeated internal computations done in DISORT2; (ii) a large enough number of terms in the Fourier expansion of the BRDF is employed to guarantee accurate values of the expansion coefficients (default is 200 instead of 50 in DISORT2); (iii) in the post processing step the reflection of the direct attenuated beam from the lower boundary is included resulting in a more accurate single scattering correction. These improvements in the treatment of the BRDF have led to improved accuracy and a several-fold increase in speed. In addition, the stability of beam sources has been improved by removing a singularity occurring when the cosine of the incident beam angle is too close to the reciprocal of any of the eigenvalues. The efficiency for beam sources has been further improved from reducing by a factor of 2 (compared to DISORT2) the dimension of the linear system of equations that must be solved to obtain the particular solutions, and by replacing the LINPAK routines used in DISORT2 by LAPACK 3.5 in DISORT3. These beam source stability and efficiency upgrades bring enhanced stability and an additional 5-7% improvement in speed. Numerical results are provided to demonstrate and quantify the improvements in accuracy and efficiency of DISORT3 compared to DISORT2.

Lin, Z.↗
DOE OSTI2024

Dynamic gain and frequency comb formation in exceptional-point lasers

Abstract Exceptional points (EPs)—singularities in the parameter space of non-Hermitian systems where two nearby eigenmodes coalesce—feature unique properties with applications such as sensitivity enhancement and chiral emission. Existing realizations of EP lasers operate with static populations in the gain medium. By analyzing the full-wave Maxwell–Bloch equations, here we show that in a laser operating sufficiently close to an EP, the nonlinear gain will spontaneously induce a multi-spectral multi-modal instability above a pump threshold, which initiates an oscillating population inversion and generates a frequency comb. The efficiency of comb generation is enhanced by both the spectral degeneracy and the spatial coalescence of modes near an EP. Such an “EP comb” has a widely tunable repetition rate, self-starts without external modulators or a continuous-wave pump, and can be realized with an ultra-compact footprint. We develop an exact solution of the Maxwell–Bloch equations with an oscillating inversion, describing all spatiotemporal properties of the EP comb as a limit cycle. We numerically illustrate this phenomenon in a 5-μm-long gain-loss coupled AlGaAs cavity and adjust the EP comb repetition rate from 20 to 27 GHz. This work provides a rigorous spatiotemporal description of the rich laser behaviors that arise from the interplay between the non-Hermiticity, nonlinearity, and dynamics of a gain medium.

36 MATERIALS SCIENCE↗
NASA NTRS1971

The problem of an elastic stiffener bonded to a half plane.

The contact problem of an elastic stiffener bonded to an elastic half plane with different mechanical properties is considered. The governing integral equation is reduced to an infinite system of linear algebraic equations. It is shown that, depending on the value of a parameter which is a function of the elastic constants and the thickness of the stiffener, the system is either regular or quasi-regular. A complete numerical example is given for which the strength of the stress singularity and the contact stresses are tabulated.

Erdogan, F.↗
NASA NTRS1978

Stress intensity factors in a cracked infinite elastic wedge loaded by a rigid punch

A plane elastic wedge-shaped solid was split through the application of a rigid punch. It was assumed that the coefficient of friction on the the contact area was constant, and the problem had a plane of symmetry with respect to loading and geometry, with the crack in the plane of symmetry. The problem was formulated in terms of a system of integral equations with the contact stress and the derivative of the crack surface displacement as the unknown functions. The solution was obtained for an internal crack and for an edge crack. The results include primarily the stress intensity factors at the crack tips, and the measure of the stress singularity at the wedge apex, and at the end points of the contact area.

Erdogan, F.↗
NASA NTRS1980

A general panel method for the analysis and design of arbitrary configurations in incompressible flows

A method for solving the linear integral equations of incompressible potential flow in three dimensions is presented. Both analysis (Neumann) and design (Dirichlet) boundary conditions are treated in a unified approach to the general flow problem. The method is an influence coefficient scheme which employs source and doublet panels as boundary surfaces. Curved panels possessing singularity strengths, which vary as polynomials are used, and all influence coefficients are derived in closed form. These and other features combine to produce an efficient scheme which is not only versatile but eminently suited to the practical realities of a user-oriented environment. A wide variety of numerical results demonstrating the method is presented.

Johnson, F. T.↗
NASA NTRS1985

Finite-volume scheme for transonic potential flow about airfoils and bodies in an arbitrarily-shaped channel

A conservative finite-volume difference scheme is developed for the potential equation to solve transonic flow about airfoils and bodies in an arbitrary channel. The scheme employs a mesh which is a nearly-conformal 'O' mesh about the airfoil and nearly orthogonal at the channel walls. The mesh extends to infinity upstream and downstream, where the mapping is singular. Special procedures are required to treat the singularities at infinity, including computation of the metrics near those points. Channels with exit areas different from inlet areas are solved; a body with a sting mount is an example of such a case.

South, J. C., Jr.↗
NASA NTRS1986

Finite-volume scheme for transonic potential flow about airfoils and bodies in an arbitrarily shaped channel

A conservative finite-volume difference scheme is developed for the potential equation to solve transonic flow about airfoils and bodies in an arbitrarily shaped channel. The scheme employs a mesh which is a nearly conformal O mesh about the airfoil and nearly orthogonal at the channel walls. The mesh extends to infinity upstream and downstream, where the mapping is singular. Special procedures are required to treat the singularities at infinity, including computation of the metrics near those points. Channels with exit areas different from inlet areas are solved; a body with a sting mount is an example of such a case.

South, Jerry C., Jr.↗
NASA NTRS1993

Nonlinear vortex trail dynamics. II - Analytic solutions

Spatially periodic large amplitude solutions of the von Karman model are obtained in the neighborhood of singularities. These singularities correspond to vortex clusters in the physical plane. The quasi-periodic and unbounded solutions found analytically confirm earlier numerical work and show qualitative agreement with experimental observations of large-scale phenomena of vortex trails. Separatrices or heteroclinic orbits were explicitly found for an integrable approximate equation, which indicate that the von Karman model itself supports chaotic solutions.

Lim, Chjan C.↗
NASA NTRS1989

Crack-face displacements for embedded elliptic and semi-elliptical surface cracks

Analytical expressions for the crack-face displacements of an embedded elliptic crack in infinite solid subjected to arbitrary tractions are obtained. The tractions on the crack faces are assumed to be expressed in a polynomial form. These displacements expressions complete the exact solution of Vijayakumar and Atluri, and Nishioki and Atluri. For the special case of an embedded crack in an infinite solid subjected to uniform pressure loading, the present displacements agree with those by Green and Sneddon. The displacement equations derived were used with the finite-element alternating method (FEAM) for the analysis of a semi-elliptic surface crack in a finite solid subjected to remote tensile loading. The maximum opening displacements obtained with FEAM are compared to those with the finite-element method with singularity elements. The maximum crack opening displacements by the two methods showed good agreement.

Raju, I. S.↗
NASA NTRS1990

Crack-face displacements for an embedded elliptic crack in an infinite solid subjected to arbitrary tractions

Analytical expression for the crack-face displacements of an embedded elliptic crack in infinite solid subjected to arbitrary traction are obtained. The tractions on the crack faces are assumed to be expressed in a polynomial form. These displacements expressions complete the exact solution of Vijayakumar and Atluri, and Nishioki and Atluri. For the special case of an embedded crack in an infinite solid subjected to uniform pressure loading, the present displacements agree with those by Green and Sneddon. The displacement equations derived were used with the finite-element alternating method (FEAM) for the analysis of a semi-elliptic surface crack in a finite solid subjected to remote tensile loading. The maximum opening displacements obtained with FEAM are compared to those with the finite-element method with singularity elements. The maximum crack opening displacements by the two methods showed good agreement.

Raju, I. S.↗
NASA NTRS1990

Scattering of impact wave by a crack in composite plate

The surface responses due to impact load on an infinite uniaxial graphite/epoxy plate with and without delamination cracks are investigated both in time and frequency domain by using a hybrid method combining the finite element discretization of the near-field with boundary integral representation of the field outside a contour enclosing completely the crack. This combined method leads to a set of linear unsymmetric complex matrix equations which are solved to obtain the response in the frequency domain by biconjugate gradient method. The time domain response is then obtained by using an FFT. In order to capture the time-domain characteristics accurately, high order finite elements have been used. Also, both the six node singular elements and eight node transition elements are used around the crack tips to model the crack-tip singularity. It is shown that from the numerical results for surface responses both depth and length of this crack can be identified.

Ju, T. H.↗
NASA NTRS1994

Tensor-GMRES method for large sparse systems of nonlinear equations

This paper introduces a tensor-Krylov method, the tensor-GMRES method, for large sparse systems of nonlinear equations. This method is a coupling of tensor model formation and solution techniques for nonlinear equations with Krylov subspace projection techniques for unsymmetric systems of linear equations. Traditional tensor methods for nonlinear equations are based on a quadratic model of the nonlinear function, a standard linear model augmented by a simple second order term. These methods are shown to be significantly more efficient than standard methods both on nonsingular problems and on problems where the Jacobian matrix at the solution is singular. A major disadvantage of the traditional tensor methods is that the solution of the tensor model requires the factorization of the Jacobian matrix, which may not be suitable for problems where the Jacobian matrix is large and has a 'bad' sparsity structure for an efficient factorization. We overcome this difficulty by forming and solving the tensor model using an extension of a Newton-GMRES scheme. Like traditional tensor methods, we show that the new tensor method has significant computational advantages over the analogous Newton counterpart. Consistent with Krylov subspace based methods, the new tensor method does not depend on the factorization of the Jacobian matrix. As a matter of fact, the Jacobian matrix is never needed explicitly.

Feng, Dan↗
DOE OSTI2021

Solution Irregularity Remediation for Spatial Discretization Error Estimation for S N Transport Solutions

The discrete ordinates linear Boltzmann transport equation is typically solved in its spatially discretized form, incurring spatial discretization error. Quantification of this error for purposes such as adaptive mesh refinement or error analysis requires an a posteriori estimator, which utilizes the numerical solution to the spatially discretized equation to compute an estimate. Because the quality of the numerical solution informs the error estimate, irregularities, present in the true solution for any realistic problem configuration, tend to cause the largest deviation in the error estimate vis-a-vis the true error. In this paper, an analytical partial singular characteristic tracking (pSCT) procedure for reducing the estimator’s error is implemented within our novel residual source estimator for a zeroth-order discontinuous Galerkin scheme, at the additional cost of a single inner iteration. Here, a metric-based evaluation of the pSCT scheme versus the standard residual source estimator is performed over the parameter range of a Method of Manufactured Solutions test suite. The pSCT scheme generates near-ideal accuracy in the estimate in problems where the dominant source of the estimator’s error is the solution irregularity, namely, problems where the true solution is discontinuous and problems where the true solution’s first derivative is discontinuous and the scattering ratio is low. In problems where the scattering ratio is high and the true solution is discontinuous in the first derivative, the error in the scattering source, which is not converged by the pSCT scheme, is greater than the error incurred due to the irregularity. Ultimately, a pSCT scheme is judged to be useful for error estimation in problems where the computational cost of the scheme is justified. In the presence of many irregularities, such a scheme may be intractable for general use, but in benchmarks, as an analytical tool, or in problems that have nondissipative discontinuities, the scheme may prove invaluable.

73 NUCLEAR PHYSICS AND RADIATION PHYSICS↗
NASA NTRS1979

Stability of the nakedness of Weyl singularities

The stability of the nakedness of the Weyl singularities against matter perturbations is investigated. Consideration is given to the effects of infalling test matter on the convergence of outgoing null rays. It is shown that the additional convergence induced by infalling test matter does not blow up sufficiently fast to reconverge diverging outgoing rays, at least in the equator, and that the nakedness seems to be stable in this limited sense.

Haugan, M. P.↗
NASA NTRS1992

A numerical method for solving systems of linear ordinary differential equations with rapidly oscillating solutions

The present numerical method for accurate and efficient solution of systems of linear equations proceeds by numerically developing a set of basis solutions characterized by slowly varying dependent variables. The solutions thus obtained are shown to have a computational overhead largely independent of the small size of the scale length which characterizes the solutions; in many cases, the technique obviates series solutions near singular points, and its known sources of error can be easily controlled without a substantial increase in computational time.

Bernstein, Ira B.↗