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NASA NTRS1978

Singularity computations

Direct descriptions of the structure of a singularity would describe the radial and angular distributions of the field quantities as explicitly as practicable along with some measure of the intensity of the singularity. This paper discusses such an approach based on recent development of numerical methods for elastoplastic flow. Attention is restricted to problems where one variable or set of variables is finite at the origin of the singularity but a second set is not.

Swedlow, J. L.↗

NASA NTRS1981

Interlaminar stress singularities at a straight free edge in composite laminates

A quasi-three-dimensional finite-element analysis was used to analyze the edge-stress problem in four-ply, composite laminates. The seven laminates that were considered belong to the laminate family where the outer ply angle is between 0 and 90 deg. Systematic convergence studies were made to explore the existence of stress singularities near the free edge. The present analysis appears to confirm the existence of stress singularities at the intersection of the interface and the free edge. The power of the stress singularity was the same for all seven laminates considered.

Raju, I. S.↗

NASA NTRS1983

The use of singular value gradients and optimization techniques to design robust controllers for multiloop systems

A method for designing robust feedback controllers for multiloop systems is presented. Robustness is characterized in terms of the minimum singular value of the system return difference matrix at the plant input. Analytical gradients of the singular values with respect to design variables in the controller are derived. A cumulative measure of the singular values and their gradients with respect to the design variables is used with a numerical optimization technique to increase the system's robustness. Both unconstrained and constrained optimization techniques are evaluated. Numerical results are presented for a two-input/two-output drone flight control system.

Newsom, J. R.↗

NASA NTRS1983

The use of singular value gradients and optimization techniques to design robust controllers for multiloop systems

Newsom, J. R.↗

NASA NTRS1986

Overcoming Robot-Arm Joint Singularities

Kinematic equations allow arm to pass smoothly through singular region. Report discusses mathematical singularities in equations of robotarm control. Operator commands robot arm to move in direction relative to its own axis system by specifying velocity in that direction. Velocity command then resolved into individual-joint rotational velocities in robot arm to effect motion. However, usual resolved-rate equations become singular when robot arm is straightened.

Barker, L. K.↗

NASA NTRS1985

A study of methods to predict and measure the transmission of sound through the walls of light aircraft. Integration of certain singular boundary element integrals for applications in linear acoustics

An alternative method for performing singular boundary element integrals for applications in linear acoustics is discussed. The method separates the integral of the characteristic solution into a singular and nonsingular part. The singular portion is integrated with a combination of analytic and numerical techniques while the nonsingular portion is integrated with standard Gaussian quadrature. The method may be generalized to many types of subparametric elements. The integrals over elements containing the root node are considered, and the characteristic solution for linear acoustic problems are examined. The method may be generalized to most characteristic solutions.

Zimmerle, D.↗

NASA NTRS1985

On the solution of integral equations with strong ly singular kernels

In this paper some useful formulas are developed to evaluate integrals having a singularity of the form (t-x) sup-m, m or = 1. Interpreting the integrals with strong singularities in Hadamard sense, the results are used to obtain approximate solutions of singular integral equations. A mixed boundary value problem from the theory of elasticity is considered as an example. Particularly for integral equations where the kernel contains, in addition to the dominant term (t,x) sup-m, terms which become unbounded at the end points, the present technique appears to be extremely effective to obtain rapidly converging numerical results.

Kaya, A. C.↗

NASA NTRS1986

On the solution of integral equations with strongly singular kernels

Some useful formulas are developed to evaluate integrals having a singularity of the form (t-x) sup-m ,m greater than or equal 1. Interpreting the integrals with strong singularities in Hadamard sense, the results are used to obtain approximate solutions of singular integral equations. A mixed boundary value problem from the theory of elasticity is considered as an example. Particularly for integral equations where the kernel contains, in addition to the dominant term (t-x) sup -m , terms which become unbounded at the end points, the present technique appears to be extremely effective to obtain rapidly converging numerical results.

Kaya, A. C.↗

NASA NTRS1986

Numerical solution of random singular integral equation appearing in crack problems

The solution of several elasticity problems, and particularly crack problems, can be reduced to the solution of one-dimensional singular integral equations with a Cauchy-type kernel or to a system of uncoupled singular integral equations. Here a method for the numerical solution of random singular integral equations of Cauchy type is presented. The solution technique involves a Chebyshev series approximation, the coefficients of which are the solutions of a system of random linear equations. This method is applied to the problem of periodic array of straight cracks inside an infinite isotropic elastic medium and subjected to a nonuniform pressure distribution along the crack edges. The statistical properties of the random solution are evaluated numerically, and the random solution is used to determine the values of the stress-intensity factors at the crack tips. The error, expressed as the difference between the mean of the random solution and the deterministic solution, is established. Values of stress-intensity factors at the crack tip for different random input functions are presented.

Sambandham, M.↗

NASA NTRS1987

On the solution of integral equations with strongly singular kernels

Some useful formulas are developed to evaluate integrals having a singularity of the form (t-x) sup-m, m greater than or equal 1. Interpreting the integrals with strong singularities in Hadamard sense, the results are used to obtain approximate solutions of singular integral equations. A mixed boundary value problem from the theory of elasticity is considered as an example. Particularly for integral equations where the kernel contains, in addition to the dominant term (t-x) sup-m, terms which become unbounded at the end points, the present technique appears to be extremely effective to obtain rapidly converging numerical results.

Kaya, A. C.↗

NASA NTRS1987

Cosmic ray-modified stellar winds. I - Solution topologies and singularities

In the present two-fluid hydrodynamical model for stellar wind flow modification due to its interaction with Galactic cosmic rays, these rays are coupled to the stellar wind by either hydromagnetic wave scattering or background flow irregularity propagation. The background flow is modified by the cosmic rays via their pressure gradient. The system of equations used possesses a line of singularities in (r, u, P sub c)-space, or a two-dimensional hypersurface of singularities in (r, u, P sub c, dP sub c/dr)-space, where r, u, and P sub c are respectively the radial distance from the star, the radial wind flow speed, and the cosmic ray pressure. The singular points may be nodes, foci, or saddle points.

Ko, C. M.↗

NASA NTRS1988

Singular perturbation and time scale approaches in discrete control systems

After considering a singularly perturbed discrete control system, a singular perturbation approach is used to obtain outer and correction subsystems. A time scale approach is then applied via block diagonalization transformations to decouple the system into slow and fast subsystems. To a zeroth-order approximation, the singular perturbation and time-scale approaches are found to yield equivalent results.

Naidu, D. S.↗

NASA NTRS1991

Nondimensional forms for singular perturbation analyses of aircraft energy climbs

This paper proposes a systematic approach for identifying the perturbation parameter in singular perturbation analysis of aircraft optimal guidance, and in particular considers a family of problems related to aircraft energy climbs. The approach, which is based on a nondimensionalization of the equations of motion, is used to evaluatae the appropriateness of forced singular perturbation formulations used in the past for transport and fighter aircraft, and to assess the applicability of energy state approximations and singular perturbation analysis for airbreathing transatmospheric vehicles with hypersonic cruise and orbital capabilities.

Calise, A. J.↗

NASA NTRS1992

Effective treatments of the singular line boundary problem for three dimensional grids

The axis singularity problem for 3D blunt body flows is examined. Two techniques of eliminating the axis-singularity difficulty are presented for utilization with finite-difference codes: (1) the 3D Navier-Stokes equations are reformulated utilizing a redefined Jacobian that is nonzero at the singular line, and (2) the 3D Navier-Stokes equations are solved using the Roe flux-difference splitting technique with a finite-volume based method for evaluation of the grid Jacobian and the metrics, and appropriate boundary conditions. Real-gas and viscous computations are also presented, demonstrating the generality of the two methods.

Palmer, Grant↗

NASA NTRS1992

Equilibrium stellar systems with spindle singularities

Equilibrium sequences of axisymmetric Newtonian clusters that tend toward singular states are constructed. The distribution functions are chosen to be of the form f = f(E, Jz). The numerical method then determines the density and gravitational potential self-consistently to satisfy Poisson's equation. For the prolate models, spindle singularities arise from the depletion of angular momentum near the symmetry axis. While the resulting density enhancement is confined to the region near the axis, the influence of the spindle extends much further out through its tidal gravitational field. Centrally condensed prolate clusters may contain strong-field regions even though the spindle mass is small and the mean cluster eccentricity is not extreme. While the calculations performed here are entirely Newtonian, the issue of singularities is an important topic in general relativity. Equilibrium solutions for relativistic star clusters can provide a testing ground for exploring this issue. The methods used in this paper for building nonspherical clusters can be extended to relativistic systems.

Shapiro, Stuart L.↗

NASA NTRS1992

Singularities of the Euler equation and hydrodynamic stability

Equations governing the motion of a specific class of singularities of the Euler equation in the extended complex spatial domain are derived. Under some assumptions, it is shown how this motion is dictated by the smooth part of the complex velocity at a singular point in the unphysical domain. These results are used to relate the motion of complex singularities to the stability of steady solutions of the Euler equation. A sufficient condition for instability is conjectured. Several examples are presented to demonstrate the efficacy of this sufficient condition which include the class of elliptical flows and the Kelvin-Stuart Cat's Eye.

Tanveer, S.↗

NASA NTRS1993

Singularities of the Euler equation and hydrodynamic stability

Tanveer, S.↗

NASA NTRS1996

Singularity Preserving Numerical Methods for Boundary Integral Equations

In the past twelve months (May 8, 1995 - May 8, 1996), under the cooperative agreement with Division of Multidisciplinary Optimization at NASA Langley, we have accomplished the following five projects: a note on the finite element method with singular basis functions; numerical quadrature for weakly singular integrals; superconvergence of degenerate kernel method; superconvergence of the iterated collocation method for Hammersteion equations; and singularity preserving Galerkin method for Hammerstein equations with logarithmic kernel. This final report consists of five papers describing these projects. Each project is preceeded by a brief abstract.

Kaneko, Hideaki↗