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At least 19 records
All sourcesNASA NTRSDOE OSTI
DOE OSTI2024

The development of Gibbs's dyadic and implications for the gradient of a vector field

In this paper, we review the history of the dyadic as developed by Gibbs. This mathematical construct appeared in the second part of Gibbs's pamphlet on vector analysis (published in 1884), and it represented the first known development of a Cartesian theory of tensors. Gibbs made extensive use of the dyadic to express his theory of linear vector functions, that is, functions that acted on vectors and mapped them to new vectors. The dyadic proved to be a capable vehicle in Gibbs's hands, and his theory for dyadics (which we would now call second-order Cartesian tensors) was relatively advanced. The theory detailed notions such as the decomposition of vectors and conditions under which a tensor would have an inverse. While Gibbs's theory for linear operators expressed by dyadics was robust, it did not seem to garner the attention that the more conventional vector analysis (published in the first half of his pamphlet in 1881) did. Perhaps in part because of the general unfamiliarity with the dyadic, two distinct and conflicting definitions of the gradient of a vector field have arisen in the literature. The details of these differences in notation, possible reasons for the difference, and a potential resolution are proposed.

97 MATHEMATICS AND COMPUTING↗
NASA NTRS1974

The dyadic diffraction coefficient for a curved edge

A compact dyadic diffraction coefficient for electromagnetic waves obliquely incident on a curved edge formed by perfectly conducting curved or plane surfaces is obtained. This diffraction coefficent remains valid in the transition regions adjacent to shadow and reflection boundaries, where the diffraction coefficients of Keller's original theory fail. The method is on Keller's method of the canonical problem, which in this case is the perfectly conducting wedge illuminated by plane, cylindrical, conical, and spherical waves. When the proper ray fixed coordinate system is introduced, the dyadic diffraction coefficient for the wedge is found to be the sum of only two dyads, and it is shown that this is also true for the dyadic diffraction coefficients of higher order edges. One dyad contains the acoustic soft diffraction coefficient; the other dyad contains the acoustic hard diffraction coefficient. The expressions for the acoustic wedge diffraction coefficients contain Fresnel integrals, which ensure that the total field is continuous at shadow and reflection boundaries. The diffraction coefficients have the same form for the different types of edge illumination; only the arguments of the Fresnel integrals are different. Since diffraction is a local phenomenon, and locally the curved edge structure is wedge shaped, this result is readily extended to the curved edge.

Kouyoumjian, R. G.↗
NASA NTRS1997

Analysis of Discontinuities in a Rectangular Waveguide Using Dyadic Green's Function Approach in Conjunction with Method of Moments

The dyadic Green's function for an electric current source placed in a rectangular waveguide is derived using a magnetic vector potential approach. A complete solution for the electric and magnetic fields including the source location is obtained by simple differentiation of the vector potential around the source location. The simple differentiation approach which gives electric and magnetic fields identical to an earlier derivation is overlooked by the earlier workers in the derivation of the dyadic Green's function particularly around the source location. Numerical results obtained using the Green's function approach are compared with the results obtained using the Finite Element Method (FEM).

Deshpande, M. D.↗
NASA NTRS1974

A vector-dyadic development of the equations of motion for N-coupled rigid bodies and point masses

The equations of motion are derived, in vector-dyadic format, for a topological tree of coupled rigid bodies, point masses, and symmetrical momentum wheels. These equations were programmed, and form the basis for the general-purpose digital computer program N-BOD. A complete derivation of the equations of motion is included along with a description of the methods used for kinematics, constraint elimination, and for the inclusion of nongyroscope forces and torques acting external or internal to the system.

Frisch, H. P.↗
NASA NTRS1981

Mean dyadic Green's function for a two layer random medium

The mean dyadic Green's function for a two-layer random medium with arbitrary three-dimensional correlation functions has been obtained with the zeroth-order solution to the Dyson equation by applying the nonlinear approximation. The propagation of the coherent wave in the random medium is similar to that in an anisotropic medium with different propagation constants for the characteristic transverse electric and transverse magnetic polarizations. In the limit of a laminar structure, two propagation constants for each polarization are found to exist.

Zuniga, M. A.↗
NASA NTRS1983

Dyadic Green's functions for layered anisotropic medium

The dyadic Green's functions (DGF) for unbounded and layered anisotropic media have been obtained. The anisotropic medium is assumed to be tilted uniaxial. With the availability of the DGF's, many problems involving radiation and scattering of electromagnetic waves can readily be solved.

Lee, J. K.↗
NASA NTRS1989

One- and two-dimensional dyadic Green's functions in chiral media

The one-dimensional and two-dimensional dyadic Green functions are determined for an unbounded lossless reciprocal chiral medium which is electromagnetically described by a set of symmetric constitutive relations. It is shown that, as in the three-dimensional case, the medium supports two eigenmodes of propagation with two different wavenumbers, corresponding to the right- and left-circularly polarized waves, respectively. The eigenmode amplitudes a and b are similar to those of the three-dimensional case.

Engheta, Nader↗
NASA NTRS1975

A vector-dyadic development of the equations of motion for N-coupled flexible bodies and point masses

The equations of motion for a system of coupled flexible bodies, rigid bodies, point masses, and symmetric wheels were derived. The equations were cast into a partitioned matrix form in which certain partitions became nontrivial when the effects of flexibility were treated. The equations are shown to contract to the coupled rigid body equations or expand to the coupled flexible body equations all within the same basic framework. Furthermore, the coefficient matrix always has the computationally desirable property of symmetry. Making use of the derived equations, a comparison was made between the equations which described a flexible body model and those which described a rigid body model of the same elastic appendage attached to an arbitrary coupled body system. From the comparison, equivalence relations were developed which defined how the two modeling approaches described identical dynamic effects.

Frisch, H. P.↗
NASA NTRS1984

Issues in Developing a Normative Descriptive Model for Dyadic Decision Making

Most research in modelling human information processing and decision making has been devoted to the case of the single human operator. In the present effort, concepts from the fields of organizational behavior, engineering psychology, team theory and mathematical modelling are merged in an attempt to consider first the case of two cooperating decisionmakers (the Dyad) in a multi-task environment. Rooted in the well-known Dynamic Decision Model (DDM), the normative descriptive approach brings basic cognitive and psychophysical characteristics inherent to human behavior into a team theoretic analytic framework. An experimental paradigm, involving teams in dynamic decision making tasks, is designed to produce the data with which to build the theoretical model.

Serfaty, D.↗
NASA NTRS1993

The problem of exact interior solutions for rotating rigid bodies in general relativity

The (3 + 1) dyadic formalism for timelike congruences is applied to derive interior solutions for stationary, axisymmetric, rigidly rotating bodies. In this approach the mathematics is formulated in terms of three-space-covariant, first-order, vector-dyadic, differential equations for a and Omega, the acceleration and angular velocity three-vectors of the rigid body; for T, the stress dyadic of the matter; and for A and B, the 'electric' and 'magnetic' Weyl curvature dyadics which describe the gravitational field. It is shown how an appropriate ansatz for the forms of these dyadics can be used to discover exact rotating interior solutions such as the perfect fluid solution first published in 1968. By incorporating anisotropic stresses, a generalization is found of that previous solution and, in addition, a very simple new solution that can only exist in toroidal configurations.

Wahlquist, H. D.↗
NASA NTRS1974

A uniform geometrical theory of diffraction for an edge in a perfectly conducting surface

A compact dyadic diffraction coefficient for electromagnetic waves obliquely incident on a curved edge formed by perfectly conducting curved or plane surfaces is obtained. This diffraction coefficient remains valid in the transition regions adjacent to shadow and reflection boundaries, where the diffraction coefficients of Keller's original theory fail. Our method is based on Keller's method of the canonical problem, which in this case is the perfectly conducting wedge illuminated by plane, cylindrical, conical, and spherical waves. When the proper ray-fixed coordinate system is introduced, the dyadic diffraction coefficient for the wedge is found to be the sum of only two dyads, and it is shown that this is also true for the dyadic diffraction coefficients of higher order edges. One dyad contains the acoustic soft diffraction coefficient; the other dyad contains the acoustic hard diffraction coefficient. The expressions for the acoustic wedge diffraction coefficients contain Fresnel integrals, which ensure that the total field is continuous at shadow and reflection boundaries.

Kouyoumjian, R. G.↗
NASA NTRS2018

Overview of Methods for Deriving the Radiative Transfer Theory from the Maxwell Equations. I: Approach Based on the Far-Field Foldy Equations

In this paper, we revisit, with further enhancements and clarifications, the self-consistent first-principles approach developed previously for deriving the vector radiative transfer theory for a discrete random medium with a sparse concentration of particles. We specifically consider the case of a plane-parallel particulate layer embedded in an otherwise homogeneous unbounded medium. The solution method is based on the far-field Foldy equations, an order-of-scattering expansion for the total field derived under the Twersky approximation, the computation of the coherent field by assuming that the positions of the particles are uncorrelated, and the ladder approximation for the coherency dyadic. The latter yields an integral equation for the diffuse specific coherency dyadic, defined through an angular spectrum repre- sentation for the coherency dyadic, which in turn, gives the vector radiative transfer equation for the diffuse specific intensity column vector. We analyze specifically the computation of the coherent field for inhomogeneous particulate media and multiple species of particles, the continuous extension of the far-field representation to the near field, the Foldy approximation, and the Foldy integral equation for the coherent field. Finally, we discuss the transition from the vector to the scalar radiative transfer equation.

Doicu, Adrian↗
NASA NTRSDate not supplied

Attitude Dynamics of On-orbit Refueling Configurations

On-orbit refueling is a key enabling technology that will allow a significant increase in the amount of payload mass delivered beyond low-Earth orbit. Despite the potential benefits, there are numerous concerns regarding the operability and scalability of this critical technology. This paper explores the attitude dynamics of two docked spacecraft performing propellant transfer. A vector-dyadic equation is derived to account for the change over time in the mass distribution and the position of the mass center of the stack and moving mass terms omitted in previous literature. Subsequently, the result is applied to two SpaceX-inspired refueling configurations to assess the relative magnitudes of the various terms in the vector-dyadic equation in comparison to the gravity gradient torque.

vector-dyadic equation↗
NASA NTRS1974

Investigation of rank 2 and higher output feedback for pole placement

A common feature of several pole placement techniques is discussed and the use of a dyadic feedback matrix is presented. The limitation of this design is examined and a design involving output feedback matrices of Rank greater than one is developed as a logical extension of the dyadic feedback design. An example is presented to illustrate the design procedure.

Sridhar, B.↗