GPU Acceleration of Partial Differential Equation Solvers

Differential equations are often directly solvable by analytical means only in their one dimensional version. Partial differential equations are generally not solvable by analytical means in two and three dimensions, with the exception of few special cases. In all other cases, numerical approximation methods need to be utilized. One of the most popular methods is the finite element method. The main areas of focus, here, are the Poisson heat equation and the plate bending equation. The purpose of this paper is to provide a quick walkthrough of the various approaches that the authors followed in pursuit of creating optimal solvers, accelerated with the use of graphical processing units, and comparing them in terms of accuracy and time efficiency with existing or self-made non-accelerated solvers.

Volumetric Properties, Viscosities, and Isobaric Heat Capacities of Imidazolium Octanoate Protic Ionic Liquid in Molecular Solvents

Densities (F), viscosities (?), and isobaric heat molar capacities (Cp) of binary mixtures containing imidazolium octanoate, [Im][C7CO2], a protic ionic liquid (PIL), with four molecular solvents, water, acetonitrile, ethanol, and 1-octanol, are determined as a function of temperature from (298.15 to 323.15) K and within the whole composition range at atmospheric pressure. Excess molar volumes, VE, excess molar heat capacities, Cp E, and the deviation from additivity rules of viscosities, ??, of imidazolium octanoate solutions were then deduced from the experimental results, as well as apparent molar volumes, Vfi, and partial molar volumes, V j m,i. Results are discussed according to the nature of the interaction between the PIL and the molecules and the effect of temperature. The excess Gibbs energies of activation of viscous flow (?G*E) for these systems were then calculated at 298.15 K. The excess isobaric heat capacities, Cp E, of binary ([Im][C7CO2] + solvent) systems, depend also of the nature of the molecular solvent in mixture. The excess properties were then correlated, at each temperature, as a function of composition by a Redlich-Kister-type equation. Finally results have been discussed in terms of molecular interactions and molecular structures in these binary mixtures, and thermodynamic properties of investigated binary mixtures were then compared to literature values together to investigate the impact of the nature of the solvent on these reported properties.

Rheological and heat transfer behaviour of the ionic liquid, [C(4)mim][NTf2]

Systematic experiments have been carried out on the thermal and rheological behaviour of the ionic liquid, 1-butyl-3-methylimidazolium bis{(trifluoromethyl)sulfonyl} imide, [C(4)mim][NTf2], and, for the first time, on the forced convective heat transfer of an ionic liquid under the laminar flow conditions. The results show that the thermal conductivity of the ionic liquid is similar to 0.13 W m(-1) K-1, which is almost independent of temperature between 25 and 40 degrees C. Rheological measurements show that the [C(4)mim][NTf2] liquid is a Newtonian fluid with its shear viscosity decreasing with increasing temperature according to the exponential law over a temperature range of 20-90 degrees C. The convective heat transfer experiments demonstrate that the thermal entrance length of the ionic liquid is very large due to its high viscosity and low thermal conductivity. The convective heat transfer coefficient is observed to be much lower than that of distilled water under the same conditions. The convective heat transfer data are also found to fit well to the convectional Shah's equation under the conditions of this work. (C) 2007 Elsevier Inc. All rights reserved.

Generalized Langevin equation: An efficient approach to nonequilibrium molecular dynamics of open systems

The generalized Langevin equation (GLE) has been recently suggested to simulate the time evolution of classical solid and molecular systems when considering general nonequilibrium processes. In this approach, a part of the whole system (an open system), which interacts and exchanges energy with its dissipative environment, is studied. Because the GLE is derived by projecting out exactly the harmonic environment, the coupling to it is realistic, while the equations of motion are non-Markovian. Although the GLE formalism has already found promising applications, e. g., in nanotribology and as a powerful thermostat for equilibration in classical molecular dynamics simulations, efficient algorithms to solve the GLE for realistic memory kernels are highly nontrivial, especially if the memory kernels decay nonexponentially. This is due to the fact that one has to generate a colored noise and take account of the memory effects in a consistent manner. In this paper, we present a simple, yet efficient, algorithm for solving the GLE for practical memory kernels and we demonstrate its capability for the exactly solvable case of a harmonic oscillator coupled to a Debye bath.

Analysis of deformation behavior and workability of advanced 9Cr-Nb-V ferritic heat resistant steels

Hot compression tests were carried out on 9Cr–Nb–V heat resistant steels in the temperature range of 600–1200 °C and the strain rate range of 10−2–100 s−1 to study their deformation characteristics. The full recrystallization temperature and the carbon-free bainite phase transformation temperature were determined by the slope-change points in the curve of mean flow stress versus the inverse of temperature. The parameters of the constitutive equation for the experimental steels were calculated, including the stress exponent and the activation energy. The lower carbon content in steel would increase the fraction of precipitates by increasing the volume of dynamic strain-induced (DSIT) ferrite during deformation. The ln(εc) versus ln(Z) and the ln(σc) versus ln(Z) plots for both steels have similar trends. The efficiency of power dissipation maps with instability maps merged together show excellent workability from the strain of 0.05 to 0.6. The microstructure of the experimental steels was fully recrystallized upon deformation at low Z value owing to the dynamic recrystallization (DRX), and exhibited a necklace structure under the condition of 1050 °C/0.1 s−1 due to the suppression of the secondary flow of DRX. However, there were barely any DRX grains but elongated pancake grains under the condition of 1000 °C/1 s−1 because of the suppression of the metadynamic recrystallization (MDRX).

Constitutive modeling, microstructure evolution, and processing map for a nitride-strengthened heat-resistant steel

A constitutive equation was established to describe the deformation behavior of a nitride-strengthened (NS) steel through isothermal compression simulation test. All the parameters in the constitutive equation including the constant and the activation energy were precisely calculated for the NS steel. The result also showed that from the stress-strain curves, there existed two different linear relationships between critical stress and critical strain in the NS steel due to the augmentation of auxiliary softening effect of the dynamic strain-induced transformation. In the calculation of processing maps, with the change of Zener-Hollomon value, three domains of different levels of workability were found, namely excellent workability region with equiaxed-grain microstructure, good workability region with “stripe” microstructure, and the poor workability region with martensitic-ferritic blend microstructure. With the increase of strain, the poor workability region first expanded, then shrank to barely existing, but appeared again at the strain of 0.6.

Combined Conduction-Convection-Radiation Heat Transfer of Slip Flow inside a Micro-channel Filled with a Cellular Porous Material

Combined conduction–convection–radiation heat transfer is investigated numerically in a micro-channel filled with a saturated cellular porous medium, with the channel walls held at a constant heat flux. Invoking the velocity slip and temperature jump, the thermal behaviour of the porous–fluid system are studied by considering hydrodynamically fully developed flow and applying the Darcy–Brinkman flow model. One energy equation model based on the local thermal equilibrium condition is adopted to evaluate the temperature field within the porous medium. Combined conduction and radiation heat transfer is treated as an effective conduction process with a temperature-dependent effective thermal conductivity. Results are reported in terms of the average Nusselt number and dimensionless temperature distribution, as a function of velocity slip coefficient, temperature jump coefficient, porous medium shape parameter and radiation parameters. Results show that increasing the radiation parameter (Tr)(Tr) and the temperature jump coefficient flattens the dimensionless temperature profile. The Nusselt numbers are more sensitive to the variation in the temperature jump coefficient rather than to the velocity slip coefficient. Such that for high porous medium shape parameter, the Nusselt number is found to be independent of velocity slip. Furthermore, it is found that as the temperature jump coefficient increases, the Nusselt number decrease. In addition, for high temperature jump coefficients, the Nusselt number is found to be insensitive to the radiation parameters and porous medium shape parameter. It is also concluded that compared with the conventional macro-channels, wherein using a porous material enhances the rate of heat transfer (up to about 40 % compared to the clear channel), insertion of a porous material inside a micro-channel in slip regime does not effectively enhance the rate of heat transfer that is about 2 %.

Constant wall heat flux boundary conditions in micro-channels filled with porous material with internal heat generation under local thermal non-equilibrium conditions

Forced convection heat transfer in a micro-channel filled with a porous material saturated with rarefied gas with internal heat generation is studied analytically in this work. The study is performed by analysing the boundary conditions for constant wall heat flux under local thermal non-equilibrium (LTNE) conditions. Invoking the velocity slip and temperature jump, the thermal behaviour of the porous-fluid system is studied by considering thermally and hydrodynamically fully-developed conditions. The flow inside the porous material is modelled by the Darcy–Brinkman equation. Exact solutions are obtained for both the fluid and solid temperature distributions for two primary approaches models A and B using constant wall heat flux boundary conditions. The temperature distributions and Nusselt numbers for models A and B are compared, and the limiting cases resulting in the convergence or divergence of the two models are also discussed. The effects of pertinent parameters such as fluid to solid effective thermal conductivity ratio, Biot number, Darcy number, velocity slip and temperature jump coefficients, and fluid and solid internal heat generations are also discussed. The results indicate that the Nusselt number decreases with the increase of thermal conductivity ratio for both models. This contrasts results from previous studies which for model A reported that the Nusselt number increases with the increase of thermal conductivity ratio. The Biot number and thermal conductivity ratio are found to have substantial effects on the role of temperature jump coefficient in controlling the Nusselt number for models A and B. The Nusselt numbers calculated using model A change drastically with the variation of solid internal heat generation. In contrast, the Nusselt numbers obtained for model B show a weak dependency on the variation of internal heat generation. The velocity slip coefficient has no noticeable effect on the Nusselt numbers for both models. The difference between the Nusselt numbers calculated using the two models decreases with an increase of the temperature jump coefficient.