John M. Gary

Finite Element and Plate Theory Modeling of Acoustic Emission Waveforms

A comparison was made between two approaches to predict acoustic emission waveforms in thin plates. A normal mode solution method for Mindlin plate theory was used to predict the response of the flexural plate mode to a point source, step-function load, applied on the plate surface. The second approach used a dynamic finite element method to model the problem using equations of motion based on exact linear elasticity. Calculations were made using properties for both isotropic (aluminum) and anisotropic (unidirectional graphite/epoxy composite) materials. For simulations of anisotropic plates, propagation along multiple directions was evaluated. In general, agreement between the two theoretical approaches was good. Discrepancies in the waveforms at longer times were caused by differences in reflections from the lateral plate boundaries. These differences resulted from the fact that the two methods used different boundary conditions. At shorter times in the signals, before reflections, the slight discrepancies in the waveforms were attributed to limitations of Mindlin plate theory, which is an approximate plate theory. The advantages of the finite element method are that it used the exact linear elasticity solutions, and that it can be used to model real source conditions and complicated, finite specimen geometries as well as thick plates. These advantages come at a cost of increased computational difficulty, requiring lengthy calculations on workstations or supercomputers. The Mindlin plate theory solutions, meanwhile, can be quickly generated on personal computers. Specimens with finite geometry can also be modeled. However, only limited simple geometries such as circular or rectangular plates can easily be accommodated with the normal mode solution technique. Likewise, very limited source configurations can be modeled and plate theory is applicable only to thin plates.

Time-dependent modeling of erbium-doped waveguide lasers in lithium niobate pumped at 980 and 1480 nm

We have developed a rigorous phenomenological model for analyzing rare-earth doped waveguide lasers. The model is based on time-dependent laser rate equations for an arbitrary rare-earth-doped laser host with multiple energy levels. The rate equations are coupled with the laser signal and pump photon flux equations that have time-dependent boundary conditions. The formulation results in a large and stiff set of transcendental and coupled differential equations that are solved using finite difference discretization and the method of lines. Solutions for the laser signal power, pump power, and populations of ion energy levels as functions of space and time are obtained for waveguide lasers. We have used the model to predict the CW characteristics and Q-switched performance of waveguide lasers in lithium niobate pumped by a 980-nm source. Our analysis shows that hole burning can occur in erbium-doped lithium niobate lasers because of the intensity variation across guided transverse modes. We have predicted that Q-switch pulse peak powers can exceed 1 kW with pulsewidths less than 1 ns. Moreover, we have compared the CW and Q-switched performance of 980-nm pumped waveguide lasers and 1480 nm pumped waveguide lasers. An analysis of the effects of host- and fabrication-dependent parameters on CW 980-nm pumped lasers is included. These parameters include cooperative upconversion, excited state absorption, doping concentration, excess waveguide loss, cavity length, and mirror reflectance values. We demonstrate good quantitative agreement with waveguide laser experimental data obtained in our laboratory and with results from the literature.

CALCULATED REGENERATOR PERFORMANCE AT 4 K WITH HELIUM-4 AND HELIUM-3

The helium-4 working fluid in regenerative cryocoolers operating with the cold end near 4 K deviates considerably from an ideal gas. As a result, losses in the regenerator, given by the time-averaged enthalpy flux, are increased and are strong functions of the operating pressure and temperature. Helium-3, with its lower boiling point, behaves somewhat closer to an ideal gas in this low temperature range and can reduce the losses in 4 K regenerators. An analytical model is used to find the fluid properties that strongly influence the regenerator losses as well as the gross refrigeration power. The thermodynamic and transport properties of helium-3 were incorporated into the latest NIST regenerator numerical model, known as REGEN3.3, which was used to model regenerator performance with either helium-4 or helium-3. With this model we show how the use of helium-3 in place of helium-4 can improve the performance of 4 K regenerative cryocoolers. The effects of operating pressure, warm-end temperature, and frequency on regenerators with helium-4 and helium-3 are investigated and compared. The results are used to find optimum operating conditions. The frequency range investigated varies from 1 Hz to 30 Hz, with particular emphasis on higher frequencies.

Regenerator Behavior with Heat Input or Removal at Intermediate Temperatures

Regenerators with finite losses are capable of absorbing a limited amount of heat at intermediate temperatures along their length. This paper discusses a simple analytical model and a rigorous numerical model of regenerator behavior under the influence of heat input or heat removal at intermediate temperatures as well as the influence of a steady mass flow superimposed on the oscillating mass flow within the regenerator. The finite time-averaged enthalpy transport through the regenerator undergoes a discontinuity at the location of the heat input to satisfy the First Law of Thermodynamics. The discontinuous enthalpy flow leads to a discontinuous temperature gradient in the axial direction and to an increase in the regenerator loss that must be absorbed at the cold end. However, the increased loss is less than the heat input at the intermediate temperature, which allows the regenerator to provide a certain amount of cooling without the need for a separate expansion stage. This phenomenon is particularly useful for shield cooling and for precooling a gas continuously or at discrete regenerator locations prior to liquefaction at the cold end. For continuous precooling the total heat load can be reduced by as much as 23%.

Regenerator behavior at 4 K: Effect of volume and porosity

The low heat capacity of regenerator materials near 4 K and the real gas properties of helium at this temperature give rise to anomalous temperature profiles and behavior not seen at higher temperatures. This paper describes the behavior of regenerators calculated from the NIST computer code REGEN3.2 when the cold end is at 4 K. The results show that the regenerator loss is independent of the regenerator volume over a wide range and does not increase until the volume is decreased below some critical value, at which point the loss increases very rapidly. The transition occurs when the gas displacement amplitude at the warm end approaches the length of the regenerator. The model shows that reducing the porosity leads to a decreased regenerator loss at the plateau. The paper also describes the effect of the temperature at the warm end and the matrix heat capacity on the regenerator loss. The calculated temperature profiles agree with experimental measurements on regenerators in 4 K Gifford-McMahon refrigerators.

Successive overrelaxation, multigrid, and preconditioned conjugate gradients algorithms for solving a diffusion problem on a vector computer

The purpose of this paper is the treatment of three numerical algorithms [successive overrelaxation (SOR), multigrid (MG) and conjugate gradients preconditioned by a fast Poisson solver (CG)] for solving large but mildly behaved diffusion problems on a vector computer with memory- to-memory architecture. The problem is a symmetric nonnegative definite matrix equation arising from a cell-centered finite difference approximation of a 3-d diffusion equation with full Neumann boundary conditions.

Optimization Calculations for a 30 HZ, 4 K Regenerator with Helium-3 Working Fluid

The NIST numerical software, REGEN3.3, which incorporates both He‐4 and He‐3 properties, was used to calculate the losses and second law efficiencies of 4 K regenerators operating at 30 Hz. Operating parameters, such as average pressure, pressure ratio, and warm‐end temperature were varied to investigate the effect of non‐ideal gas properties. Regenerator parameters such as matrix material and shape, hydraulic diameter, and regenerator geometry were varied to investigate losses due to non‐ideal regenerator behavior. The results show that He‐3 can increase the regenerator efficiency by a factor of at least two compared to a regenerator optimized for He‐4. A layered regenerator of gadolinium oxysulfate (GOS) at the cold end and ErPr at the warm end is the best of many material combinations. A regenerator with parallel holes of about 20% porosity showed only slight improvement over one with packed spheres. The regenerator warm‐end temperature has little effect on its efficiency for temperatures below 35 K an...

Rigorous scalar modeling of Er- and Er/Yb-doped waveguide lasers

A rigorous scalar model for predicting the characteristics of rare-earth-doped waveguide lasers has been developed. The model consists of two nonhomogeneous wave equations: one for the forward-propagating laser signal power, the other for the backward-propagating laser signal. These equations are coupled with one forward-propagating, nonhomogeneous wave equation representing the pump signal. The three wave equations are coupled with the space dependent laser rate equations to form a system of time dependent differential equations. This large system of equations is solved, using appropriate initial and boundary conditions, by the method of lines using collocation for the spatial approximation. The solutions to this system yield data which predict the time and position-dependent laser signal power, pump power, and population densities in a waveguide laser cavity supporting an arbitrary guided mode. The assumptions made in this new model are that the transverse field maintains the same shape as a function of longitudinal position in the laser cavity and that the effects of spatial hole burning and standing waves are neglected. We have used this model to predict continuous wave and Q-switched laser performance for Er an Er/Yb-doped lasers. We have achieved favorable comparisons with actual laboratory operation of cw Yb/Er-co- doped waveguide lasers. Results from simulations of Er-doped and Yb/Er-doped Q-switched lasers are presented which show that high peak powers on the order of 500 W and 1 ns pulse widths can be achieved.

Regenerator performance in a Vuilleumier refrigerator compared with a third-order numerical model

A 3-stage Vuilleumier refrigerator was used to measure the performance of various third stage regenerators. The refrigerator operates between 2.5 and 5.0 Hz and, depending on the material used in the third stage regenerator, achieves temperatures of 8 to 20 K at the cold end of the third stage. This paper presents a comparison of regenerator performance for four regenerator materials: 229 μm diameter spheres of Pb+5%Sb, 229 μm diameter spheres of brass, 216 μm irregular-shaped GdRh powder, and a mixture of 229 μm and 762 μm diameter spheres of Pb+5%Sb. The experimental results are compared with a first-order model that neglects the void volume within the regenerator and with a third-order model that considers the effect of pressure oscillations in the regenerator void volume. Experimental results indicate that regenerator losses are dominated by the pressure oscillation in the void volume rather than the mass flow through the temperature gradient in the regenerator. These results are consistent with the third-order numerical model. This model shows that the heat capacity of the gas in the void space as well as the heat capacity of the matrix influences the regenerator performance.

A technique to evaluate benchmarks: a case study using the Livermore loops

This paper is devoted to an analysis of the data from the Livermore kernels benchmark. We will show that in the sense of least squares prediction the dimension of these data is rather small; a reduction of the data to dimen sion four has about the same predictive power as the original data. Two techniques are used that reduce the 72 kernel timings for each machine to a few scores by which the machine is characterized. The first is based on a principal component analysis, the second on a cluster analysis of the kernels. The validity of the reduction to lower dimension is checked by various means. The pos sible use of the Livermore data to predict the running time of larger codes is demonstrated.