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All sourcesNASA NTRSDOE OSTI
NASA NTRS1990

Mathematical theory of a relaxed design problem in structural optimization

Various attempts have been made to construct a rigorous mathematical theory of optimization for size, shape, and topology (i.e. layout) of an elastic structure. If these are represented by a finite number of parametric functions, as Armand described, it is possible to construct an existence theory of the optimum design using compactness argument in a finite dimensional design space or a closed admissible set of a finite dimensional design space. However, if the admissible design set is a subset of non-reflexive Banach space such as L(sup infinity)(Omega), construction of the existence theory of the optimum design becomes suddenly difficult and requires to extend (i.e. generalize) the design problem to much more wider class of design that is compatible to mechanics of structures in the sense of variational principle. Starting from the study by Cheng and Olhoff, Lurie, Cherkaev, and Fedorov introduced a new concept of convergence of design variables in a generalized sense and construct the 'G-Closure' theory of an extended (relaxed) optimum design problem. A similar attempt, but independent in large extent, can also be found in Kohn and Strang in which the shape and topology optimization problem is relaxed to allow to use of perforated composites rather than restricting it to usual solid structures. An identical idea is also stated in Murat and Tartar using the notion of the homogenization theory. That is, introducing possibility of micro-scale perforation together with the theory of homogenization, the optimum design problem is relaxed to construct its mathematical theory. It is also noted that this type of relaxed design problem is perfectly matched to the variational principle in structural mechanics.

Kikuchi, Noboru↗
NASA NTRS1993

Relaxation schemes for Chebyshev spectral multigrid methods

Two relaxation schemes for Chebyshev spectral multigrid methods are presented for elliptic equations with Dirichlet boundary conditions. The first scheme is a pointwise-preconditioned Richardson relaxation scheme and the second is a line relaxation scheme. The line relaxation scheme provides an efficient and relatively simple approach for solving two-dimensional spectral equations. Numerical examples and comparisons with other methods are given.

Kang, Yimin↗
NASA NTRS2000

Modeling Step-Strain Relaxation and Cyclic Deformations of Elastomers

Data for step-strain relaxation and cyclic compressive deformations of highly viscous short elastomer cylinders are modeled using a large strain rubber viscoelastic constitutive theory with a rate-independent friction stress term added. In the tests, both small and large amplitude cyclic compressive strains, in the range of 1% to 10%, were superimposed on steady state compressed strains, in the range of 5% to 20%, for frequencies of 1 and 10 Hz. The elastomer cylinders were conditioned prior to each test to soften them. The constants in the viscoclastic-friction constitutive theory are determined by employing a nonlinear least-squares method to fit the analytical stresses for a Maxwell model, which includes friction, to measured relaxation stresses obtained from a 20% step-strain compression test. The simulation of the relaxation data with the nonlinear model is successful at compressive strains of 5%, 10%, 15%, and 20%. Simulations of hysteresis stresses for enforced cyclic compressive strains of 20% +/- 5% are made with the model calibrated by the relaxation data. The predicted hysteresis stresses are lower than the measured stresses.

Johnson, A.R.↗
NASA NTRS1994

Rotational Relaxation Analysis of Nitrogen in Low-Density Freejet Expansions

The phenomena of rotational relaxation of nitrogen has been examined by numerous investigators over many years. One of the experiments which has been performed examines nonequilibrium flow in low-density free jet expansions. Data have been taken in such flows using a variety of techniques, including time-of-flight methods and electron beam fluorescence spectroscopy. The direct flow properties measured in these different investigations, such as density and translational, rotational and vibrational temperatures generally show reasonable agreement. However, this kind of correlation from experiment to experiment tends to be lost when these data are analyzed to obtain rotational relaxation time or collision number. The goal of such data analyses is to generate a succinct model for rotational relaxation in nitrogen which is essential for the computation of nonequilibrium rarefied flows. The objective of the present work is to process a large body of experimental data in a consistent manner to yield relaxation model parameters of the greatest utility for flow computations.

Gochberg, Lawrence A.↗
NASA NTRS2001

Distributed Relaxation for Conservative Discretizations

A multigrid method is defined as having textbook multigrid efficiency (TME) if the solutions to the governing system of equations are attained in a computational work that is a small (less than 10) multiple of the operation count in one target-grid residual evaluation. The way to achieve this efficiency is the distributed relaxation approach. TME solvers employing distributed relaxation have already been demonstrated for nonconservative formulations of high-Reynolds-number viscous incompressible and subsonic compressible flow regimes. The purpose of this paper is to provide foundations for applications of distributed relaxation to conservative discretizations. A direct correspondence between the primitive variable interpolations for calculating fluxes in conservative finite-volume discretizations and stencils of the discretized derivatives in the nonconservative formulation has been established. Based on this correspondence, one can arrive at a conservative discretization which is very efficiently solved with a nonconservative relaxation scheme and this is demonstrated for conservative discretization of the quasi one-dimensional Euler equations. Formulations for both staggered and collocated grid arrangements are considered and extensions of the general procedure to multiple dimensions are discussed.

Diskin, Boris↗
NASA NTRS1997

Relaxation Revisited: A Fresh Look at Multigrid for Steady Flows

The year 1971 saw the publication of one of the landmark papers in computational aerodynamics, that of Murman and Cole. As with many seminal works, its significance lies not so much in the specific problem that it addressed| small disturbance, plane transonic flow - but in the identification of a general approach to the solution of a technically important and theoretically difficult problem. The key features of Murman and Cole's work were the use of type- dependent differencing to correctly account for the proper domain of dependence of a mixed elliptic/hyperbolic equation, and the introduction of line relaxation to solve the steady flow equation. All subsequent work in transonic potential flows was based on these concepts. Jameson extended Murman and Cole's ideas to the full potential equation with two important contributions. First, he introduced the rotated difference stencil, which generalized the Murman and Cole type-dependent difference operator to general coordinates. Second, he used the interpretation, introduced by Garabedian, of relaxation as an iteration in artificial time to construct stable relaxation schemes, generalizing the original line relaxation method of Reference. The decade of the 1970s saw an explosion of activity in the solution of transonic potential flows, which has been summarized in the review article of Caughey.

Roberts, Thomas W.↗
NASA NTRS2001

Determinants of diastolic myocardial tissue Doppler velocities: influences of relaxation and preload

Myocardial tissue Doppler echocardiography (TDE) has been proposed as a tool for the assessment of diastolic function. Controversy exists regarding whether TDE measurements are influenced by preload. In this study, left ventricular volume and high-fidelity pressures were obtained in eight closed-chest dogs during intermittent caval occlusion. The time constant of isovolumic ventricular relaxation (tau) was altered with varying doses of dobutamine and esmolol. Peak early diastolic myocardial (E(m)) and transmitral (E) velocities were measured before and after preload reduction. The relative effects of changes in preload and relaxation were determined for E(m) and compared with their effects on E. The following results were observed: caval occlusion significantly decreased E (DeltaE = 16.4 +/- 3.3 cm/s, 36.6 +/- 13.7%, P < 0.01) and E(m) (DeltaE(m) = 1. 3 +/- 0.4 cm/s, 32.5 +/- 26.1%, P < 0.01) under baseline conditions. However, preload reduction was similar for E under all lusitropic conditions (P = not significant), but these effects on E(m) decreased with worsening relaxation. At tau < 50 ms, changes in E(m) with preload reduction were significantly greater (DeltaE(m) = 2.8 +/- 0.6 cm/s) than at tau = 50-65 ms (DeltaE(m) = 1.2 +/- 0.2 cm/s) and at tau >65 ms (DeltaE(m) = 0.5 +/- 0.1 cm/s, P < 0.05). We concluded that TDE E(m) is preload dependent. However, this effect decreases with worsening relaxation.

NASA Discipline Cardiopulmonary↗
NASA NTRS2016

Separating the Influence of Environment from Stress Relaxation Effects on Dwell Fatigue Crack Growth

Seven different microstructural variations of LSHR were produced by controlling the cooling rate and the subsequent aging and thermal exposure heat treatments. Through cyclic fatigue crack growth testing performed both in air and vacuum, it was established that four out of the seven LSHR heat treatments evaluated, possessed similar intrinsic environmental resistance to cyclic crack growth. For these four heat treatments, it was further shown that the large differences in dwell crack growth behavior which still persisted, were related to their measured stress relaxation behavior. The apparent differences in their dwell crack growth resistance were attributed to the inability of the standard linear elastic fracture mechanics (LEFM) stress intensity parameter to account for visco-plastic behavior. Crack tip stress relaxation controls the magnitude of the remaining local tensile stresses which are directly related to the measured dwell crack growth rates. It was hypothesized that the environmentally weakened grain boundary crack tip regions fail during the dwells when their strength is exceeded by the remaining local crack tip tensile stresses. It was shown that the classical creep crack growth mechanisms such as grain boundary sliding did not contribute to crack growth, but the local visco-plastic behavior still plays a very significant role by determining the crack tip tensile stress field which controls the dwell crack growth behavior. To account for the influence of the visco-plastic behavior on the crack tip stress field, an empirical modification to the LEFM stress intensity parameter, Kmax, was developed by incorporating into the formulation the remaining stress level concept as measured by simple stress relaxation tests. The newly proposed parameter, Ksrf, did an excellent job in correlating the dwell crack growth rates for the four heat treatments which were shown to have similar intrinsic environmental cyclic fatigue crack growth resistance.

environmental damage↗
NASA NTRS1972

Study of excitation transfer in flowing helium afterglow pumped with a tuneable dye laser. 2: Measurement of the rate coefficient for the rotational relaxation of He2(3p (3 Pi g))

The rotational relaxation of the He2(3p(3pi g)) state is examined by optically pumping a flowing helium afterglow with a tuneable dye laser. The population of the J = 8 rotational state is enhanced by optically saturating the R sub 7 component of the transition connecting this state with the metastable He2(2s (3 Sigma u +)) molecular state. From the lifetime and yield of the Q sub 7 component, the rate coefficient for the rotational relaxation via the forbidden delta J = 1 channel is determined to be of the order of 2 x 10 to the minus 11th power cm3/sec. It is found that this represents about half of the total rate of rotational relaxation in this state.

Collins, C. B.↗
NASA NTRS1973

A general relaxation theory of simple liquids

A relatively simple relaxation theory to account for the behavior of liquids under dynamic conditions was proposed. The general dynamical equations are similar in form to the phenomenological relaxation equations used in theories of viscoelasticity, however, they differ in that all the coefficients of the present equations are expressed in terms of thermodynamic and molecular quantities. The theory is based on the concept that flow in a liquid distorts both the radial and the velocity distribution functions, and that relaxation equations describing the return of these functions to their isotropic distributions, characterizing a stationary liquid, can be written. The theory was applied to the problems of steady and oscillatory shear flows and to the propagation of longitudinal waves. In all cases classical results are predicted for strain rates, and an expression for the viscosity of a liquid, simular to the Macedo-Litovitz equation, is obtained.

Merilo, M.↗
NASA NTRS1973

Relaxation of the V = 4,5,6,7, Sigma g/+/ vibrational levels of carbon monoxide studied by laser absorption.

The vibrational relaxation of individual vibration rotation levels of carbon monoxide behind incident shock waves of carbon monoxide has been studied by the method of laser absorption. For the particular vibrational states (V = 4 to 7) and temperature range (2500 to 5500 K) studied, it is concluded that the characteristic relaxation times are in excellent agreement with those obtained via previous measurements of 'bulk' gas properties. Further, the data furnish strong corroboration of the idea that the individual levels are in Boltzmann vibrational equilibrium during the relaxation process.

Chackerian, C., Jr.↗
NASA NTRS1973

Relaxation time in disk galaxy simulations.

An initially stationary and stable axisymmetric disk of stars with a mass spectrum of stars is used to determine the collisional relaxation time. The relaxation time as determined from the rate of energy equipartition was found to be 560 rotation periods for stars with 10 times the mean star mass and was 1700 rotation periods for stars with 0.55 times the mean star mass. These times are in general agreement with theoretical predictions for the relaxation time of the two mass groups. The results show that the model used for the large-scale gravitational N-body calculations is indeed 'collisionless.'

Hohl, F.↗
NASA NTRS1974

Thermal relaxation of a two dimensional plasma in a dc magnetic field. Part 2: Numerical simulation

The thermal relaxation process for a spatially uniform two dimensional plasma in a uniform dc magnetic field is simulated numerically. Thermal relaxation times are defined in terms of the time necessary for the numerically computer Boltzman H-function to decrease through a given part of the distance to its minimum value. Dependence of relaxation time on two parameters is studied: number of particles per Debye square and ratio of gyrofrequency to plasma frequency.

Hsu, J. Y.↗
NASA NTRS1974

Relaxation of an unsteady turbulent boundary layer on a flat plate in an expansion tube

An analysis is presented for the relaxation of a turbulent boundary layer on a semi-infinite flat plate after passage of a shock wave and a trailing driver gas-driven gas interface. The problem has special application to expansion-tube flows. The flow-governing equations have been transformed into the Crocco variables, and a time-similar solution is presented in terms of the dimensionless distance-time variable alpha and the dimensionless velocity variable beta. An eddy-viscosity model, similar to that of time-steady boundary layers, is applied to the inner and outer regions of the boundary layer. A turbulent Prandtl number equal to the molecular Prandtl number is used to relate the turbulent heat flux to the eddy viscosity. The numerical results, obtained by using the Gauss-Seidel line-relaxation method, indicate that a fully turbulent boundary layer relaxes faster to the final steady-state values of heat transfer and skin friction than a laminar boundary layer. The results also give a fairly good estimate of the local skin friction and heat transfer for near steady-flow conditions.

Gurta, R. N.↗
NASA NTRS1973

Vibrational relaxation of carbon monoxide studied by two-wavelength infrared emission

Experimental results are presented for the vibrational relaxation of pure carbon monoxide behind incident shock waves over the temperature range 4000 to 6300 K. The data were obtained as infrared emission from the fundamental and overtone vibrational band systems (in some of the experiments the two-band systems were recorded simultaneously). The data are consistent with present theories for the vibrational relaxation of diatomic molecules and can be interpreted in terms of an initial Boltzmann vibrational distribution relaxing toward final equilibrium via a continuous sequence of intermediate Boltzmann distributions.

Chackerian, C., Jr.↗
NASA NTRS1973

Relaxation of the accelerating-gas boundary layer to the test-gas boundary layer on a flat plate in an expansion tube

An analytic investigation of the relaxation of the accelerating-gas boundary layer to the test-gas boundary layer over a flat plate mounted in an expansion tube has been conducted. In this treatment, nitrogen has been considered as the test gas and helium as the accelerating gas. The problem is analyzed in two conically similar limits: (1) when the time lag between the arrival of the shock and the interface at the leading edge of the plate is very large, and (2) when this time lag is negligible. The transient laminar boundary-layer equations of a perfect binary-gas mixture are taken as the flow governing equations. These coupled equations have been solved numerically by Gauss-Seidel line-relaxation method. The results predict the transient behavior as well as the time required for an all-helium accelerating-gas boundary layer to relax to an all-nitrogen boundary layer.

Gupta, R. N.↗
NASA NTRS1975

Stress relaxation and mechanical properties of RL-1973 and PD-200-16 silicone resin sponge materials

Stress relaxation tests were conducted by loading specimens in double-lap shear to a preselected strain level and monitoring the decay of stress with time. The stress relaxation response characteristics were measured over a temperature range of 100 to 300 K and four strain levels. It is concluded that only a slight amount of stress relaxation was observed, and the stiffness increased approximately two orders of magnitude over the range of temperatures.

Saylak, D.↗
NASA NTRS1977

Over-relaxation applied to the MacCormack finite-difference scheme

An over-relaxation procedure is applied to the MacCormack finite-difference scheme in order to reduce the computation time required to obtain a steady-state solution. The implementation of this acceleration procedure to an existing computer program using the regular MacCormack method is extremely simple and does not require additional storage. The over-relaxation procedure does not alter the steady-state solution, which is second-order accurate. The method is first applied to Burgers' equation. A stability condition and an expression for the increase in the rate of convergence are derived. The method is then applied to the calculation of the hypersonic viscous flow over a flat plate, using the complete Navier-Stokes equations, and the inviscid flow over a wedge. Reductions in computing time by factors of 3 and 1.5, respectively, are obtained by over-relaxation.

Desideri, J.-A.↗