Non-periodic finite-element formulation of Kohn-Sham density functional theory

We present a real-space, non-periodic, finite-element formulation for Kohn-Sham density functional theory (KS-DFT). We transform the original variational problem into a local saddle-point problem, and show its well-posedness by proving the existence of minimizers. Further, we prove the convergence of finite-element approximations including numerical quadratures. Based on domain decomposition, we develop a parallel finite-element implementation of this formulation capable of performing both all-electron and pseudopotential calculations. We assess the accuracy of the formulation through selected test cases and demonstrate good agreement with the literature. We also evaluate the numerical performance of the implementation with regard to its scalability and convergence rates. We view this work as a step towards developing a method that can accurately study defects like vacancies, dislocations and crack tips using density functional theory (DFT) at reasonable computational cost by retaining electronic resolution where it is necessary and seamlessly coarse-graining far away.     

Non-periodic finite-element formulation of orbital-free density functional theory

We propose an approach to perform orbital-free density functional theory calculations in a non-periodic setting using the finite-element method. We consider this a step towards constructing a seamless multi-scale approach for studying defects like vacancies, dislocations and cracks that require quantum mechanical resolution at the core and are sensitive to long range continuum stresses. In this paper, we describe a local real-space variational formulation for orbital-free density functional theory, including the electrostatic terms and prove existence results. We prove the convergence of the finite-element approximation including numerical quadratures for our variational formulation. Finally, we demonstrate our method using examples.     

Scaling theory of adsorption-induced stresses in polymer brushes grafted onto compliant structures

The lateral forces exerted on a substrate by a layer of end-grafted polymer molecules are calculated on the basis of simple scaling arguments. The results are cast in terms of an equilibrium surface stress and an elastic constant, which describes the rate of change of the surface stress upon deformation of the substrate. This allows for straightforward integration of the present results into a continuum framework describing the response of a compliant structure, which facilitates device design and analysis. The results are illustrated with calculations for end-grafted poly(styrene) and poly(ethylene oxide), and the implications for building micromechanical devices based on adsorption-induced deformation are discussed.     

Stick-slip flow of high density polyethylene in a transparent slit die investigated by laser Doppler velocimetry

The oscillating flow instability of a molten linear high-density polyethylene is carefully studied using a single screw extruder equipped with a transparent slit die. Experiments are performed using laser Doppler velocimetry in order to obtain the local velocities field across the entire die width. At low flow rate, the extrusion is stable and steady state velocity profiles are obtained. During the instability, the velocity oscillates between two steady state limits, suggesting a periodic stick-slip transition mechanism. At high flow rate, the flow is mainly characterized by a pronounced wall slip. We show that wall slip occurs all along the die land. An investigation of the slip flow conditions shows that wall slip is not homogeneous in a cross section of the slit die, and that pure plug flow occurs only for very high flow rates. A numerical computation of the profile assuming wall slip boundary conditions is done to obtain the true local wall slip velocity. It confirms that slip velocities are of the same order of magnitude as those measured with a capillary rheometer.     

Combined electroosmotically and pressure driven flow of power-law fluids in a slit microchannel

Electroosmotic flow of power-law fluids in the presence of pressure gradient through a slit is analyzed. After numerically solving the Poisson–Boltzmann equation, the momentum equation with electroosmotic body force is solved through an iterative numerical procedure for both favorable and adverse pressure gradients. The results reveal that, in case of pressure assisted flow, shear-thinning fluids reach higher velocity magnitudes compared with shear-thickening fluids, whereas the opposite is true when an adverse pressure gradient is applied. The Poiseuille number is found to be an increasing function of the dimensionless Debye–Hückel parameter, the wall zeta potential, and the flow behavior index. Comparison between the exact and the results based on the Debye–Hückel linearization reveals that the simplified solution leads to large errors in evaluating the velocity profile for zeta potentials higher than 25 mV, except for shear-thickening fluids in the presence of favorable pressure gradient.     

Experimental velocity distribution measurements of high density polyethylene flowing into and within a slit

Using the Laser doppler technique we report experimental velocity profile measurements of molten polyethylene flowing into a slit die. Our experimental measurements are restricted to the centreline of the flow and three transverse sections within the slit. The results indicate that with the exception of a high flowrate centreline velocity overshoot, the normalised velocity profiles are relatively insensitive to both temperature, polymer grade and flowrate.We have also carried out an analysis and simulation to establish the effect on velocity measurements of both velocity gradients and solid boundaries within the probe volume of the intersecting laser beams used to measure the velocity profiles. Our results indicate that for our own experimental conditions we might expect to measure a finite velocity at the wall and that the presence of velocity gradients will not significantly effect the time dependence of the auto correlogram.     

Transient electro-osmotic and pressure driven flows of two-layer fluids through a slit microchannel

By method of the Laplace transform, this article presents semi-analytical solutions for transient electroosmotic and pressure-driven flows (EOF/PDF) of two-layer fluids between microparallel plates. The linearized Poisson-Boltzmann equation and the Cauchy momentum equation have been solved in this article. At the interface, the Maxwell stress is included as the boundary condition. By numerical computations of the inverse Laplace transform, the effects of dielectric constant ratio ε , density ratio ρ , pressure ratio p, viscosity ratio μ of layer II to layer I, interface zeta potential difference △ψ, interface charge density jump Q, the ratios of maximum electro-osmotic velocity to pressure velocity α , and the normalized pressure gradient B on transient velocity amplitude are presented.We find the velocity amplitude becomes large with the interface zeta potential difference and becomes small with the increase of the viscosity. The velocity will be large with the increases of dielectric constant ratio; the density ratio almost does not influence the EOF velocity. Larger interface charge density jump leads to a strong jump of velocity at the interface. Additionally, the effects of the thickness of fluid layers (h1 and h2 ) and pressure gradient on the velocity are also investigated.     

Diffraction of a sound wave by a slit

The diffraction of a sound wave by a slit in an unbounded plane is analyzed as an initial-boundary-value problem with a moving boundary for the two-dimensional wave equation. The initial-boundary-value problem is solved by the formation and inversion of Volterra integral equations. A solution is obtained in closed form in quadratures for an arbitrary angle of inclination of the incident wave front relative to the plane. The solution is presented in the form of recursion formulas, which take into account the influence of diffraction waves occurring in succession at the boundaries of the slit.     

Modeling wave-induced pore pressure and effective stress in a granular seabed

The response of a sandy seabed under wave loading is investigated on the basis of numerical modeling using a multi-scale approach. To that aim, the discrete element method is coupled to a finite volume method specially enhanced to describe compressible fluid flow. Both solid and fluid phase mechanics are upscaled from considerations established at the pore level. Model’s predictions are validated against poroelasticity theory and discussed in comparison with experiments where a sediment analog is subjected to wave action in a flume. Special emphasis is put on the mechanisms leading the seabed to liquefy under wave-induced pressure variation on its surface. Liquefaction is observed in both dilative and compactive regimes. It is shown that the instability can be triggered for a well-identified range of hydraulic conditions. Particularly, the results confirm that the gas content, together with the permeability of the medium are key parameters affecting the transmission of pressure inside the soil.     

Some primal problems of the axisymmetric theory of elasticity for space with a flat slit bounded by a circle

We consider the first and second primal problems of the axisymmetric theory of elasticity for space with a round slit and a mixed problem in which forces are specified on one side of the slit and displacements are specified on the other side. The problems reduce to conjugation problems for generalized analytic functions on rectilinear segments, whose solution is obtained in closed form. Institute of Applied Mechanics, Russian Academy of Sciences, Moscow 117334. Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 41, No. 1 pp. 144–151, January–February, 2000.     

Core temperature and density profile measurements in inertial confinement fusion implosions

We have measured time-integrated and time-gated electron temperature (T_e) and density (N_e) spatial profiles from indirect-drive implosions. In our experiments, we used a multiple-pinhole two-dimensional imaging spectrometer to obtain multispectral X-ray images of the imploded core. Quantitative comparisons between quasi-monochromatic images in different energy bands allowed T_e and N_e spatial profiles to be determined using two independent and validated techniques: a multi-objective search and reconstruction analysis, and an analytical analysis. We then compared the results to a simple one-dimensional (1D) mix-free hydrodynamics simulation in order to evaluate the ability of such a model to predict our experiments. Our data show spatial T_e profiles that are qualitatively consistent with the predictions of our 1D simulations, but we observe central cores that are 10–25% cooler and emit X-rays as late as 200 ps after peak compression. We infer time-gated spatial N_e profiles that are consistent with our 1D simulations near the times of peak compression, but we find significant disagreement between time-integrated data and 1D simulation predictions at large radii. Careful analysis of the time-gated and time-integrated T_e and N_e spatial profiles, together with streaked X-ray emission spectra from core and shell dopants, suggests mixing of shell material into the core is an important process that our 1D hydrodynamics simulations fail to capture, and comparison between image data and a simple analytical model suggests that 2–5 μm of the initial inner shell thickness mixes into the core during the time period of significant X-ray emission. This mix width is consistent with the predictions of a growth-factor analysis that treats instability growth seeded by capsule surface roughness, and points to the need to consider time-dependent mixing effects when interpreting T_e and N_e spatial profiles derived from multispectral X-ray image data, particularly at large radii where mixing effects will be most significant.     

Experimental and linear viscoelastic stress distribution measurements of high density polyethylene flowing into and within a slit

We report experimental measurements fo the centreline stress build up and relaxation as molten polyethelene flows into a slit die. The time-dependent extensional stress distribution is obtained using flow birefringence techniques and these observations complement the corresponding velocity field measurements already reported. Experimental measurements of the linear viscoelastic storage and loss modulus are obtained and, from these results, the polymers are characterized in terms of a modulus spectrum. Using this modulus spectrum together with a Maxwell-type constitutive equation and the experimental centreline kinematics, we find that it is possible to simulate successfully the experimentally observed stress distributions. Our results indicate that it is essential to include the polymers' broad spectrum of relaxation times when considering time dependent flow problems.     

侧限SHPB实验中软材料体压缩特性的测量方法

提出了利用侧限SHPB装置测量软材料体压缩特性的实验方法。该方法利用霍普金森压杆对试 样施加轴向压缩,同时依靠金属薄壁套筒约束试样的径向膨胀。通过套筒外壁的环向应变得出了试样径向应 力、径向应变,再结合压杆应变片信息,可以获得试样的流体压强-比容变化过程。利用该方法进行了高聚物 泡沫材料的动态压缩实验,证实了该方法对测量软材料的体积压缩特性是可行的。     

Cyclic density functional theory: A route to the first principles simulation of bending in nanostructures

We formulate and implement Cyclic Density Functional Theory (Cyclic DFT) — a self-consistent first principles simulation method for nanostructures with cyclic symmetries. Using arguments based on Group Representation Theory, we rigorously demonstrate that the Kohn-Sham eigenvalue problem for such systems can be reduced to a fundamental domain (or cyclic unit cell) augmented with cyclic-Bloch boundary conditions. Analogously, the equations of electrostatics appearing in Kohn-Sham theory can be reduced to the fundamental domain augmented with cyclic boundary conditions. By making use of this symmetry cell reduction, we show that the electronic ground-state energy and the Hellmann-Feynman forces on the atoms can be calculated using quantities defined over the fundamental domain. We develop a symmetry-adapted finite-difference discretization scheme to obtain a fully functional numerical realization of the proposed approach. We verify that our formulation and implementation of Cyclic DFT is both accurate and efficient through selected examples.The connection of cyclic symmetries with uniform bending deformations provides an elegant route to the ab-initio study of bending in nanostructures using Cyclic DFT. As a demonstration of this capability, we simulate the uniform bending of a silicene nanoribbon and obtain its energy-curvature relationship from first principles. A self-consistent ab-initio simulation of this nature is unprecedented and well outside the scope of any other systematic first principles method in existence. Our simulations reveal that the bending stiffness of the silicene nanoribbon is intermediate between that of graphene and molybdenum disulphide — a trend which can be ascribed to the variation in effective thickness of these materials. We describe several future avenues and applications of Cyclic DFT, including its extension to the study of non-uniform bending deformations and its possible use in the study of the nanoscale flexoelectric effect.     

Influential factors of loess liquefaction and pore pressure development

The liquefaction of loess under dynamic loading is studied experimentally with a dynamic triaxial test system. The effects of over-consolidation ratio (OCR), saturation degree and the frequency of dynamic loading upon loess liquefaction are investigated. The development of pore pressure within loess samples is also discussed. Based on the experimental results, the empirical relationship between pore pressure ratio and loading cycle number ratio is established for normal consolidated saturated loess.The project supported by the National Natural Science Foundation of China (50178005)     

Variable density flow in porous media. A study by means of pore level numerical simulations

This paper is devoted to the modelling of isothermal low Reynolds and Mach numbers transient compressible flow through porous media. Traditionally, this type of flow at the macroscopic level is described by the classical Darcy's law combined with a mass balance that includes the transient term. This model is called the ‘classic model’. The aim of this paper is to explore the validity of this classic model. To this end, the flow of an ideal gas is considered within two‐dimensional model porous media. The flow is due to the imposed pressure variations at the outlet of the fluid domain. At the microscopic level, the flow is computed by solving the full compressible Navier–Stokes equations in two dimensions. Special attention is given to the outlet boundary conditions. The analysis is based on the comparison between the macroscopic data, obtained on the one hand by spatially averaging the microscopic results, and on the other hand by solving the problem directly at the macroscopic level. Situations for which a good agreement is found between the two series of data and situations for which discrepancies are observed are exhibited. These various behaviours are discussed in terms of the various time scales controlling the flow and are explained by analysing the flow structure at pore level. Copyright © 1999 John Wiley & Sons, Ltd.