Lagrangian Approximation Research Articles

Multi-scale initial conditions for cosmological simulations

We discuss a new algorithm to generate multi-scale initial conditions with multiple levels of refinements for cosmological "zoom-in" simulations. The method uses an adaptive convolution of Gaussian white noise with a real space transfer function kernel together with an adaptive multi-grid Poisson solver to generate displacements and velocities following first (1LPT) or second order Lagrangian perturbation theory (2LPT). The new algorithm achieves RMS relative errors of order 10^(-4) for displacements and velocities in the refinement region and thus improves in terms of errors by about two orders of magnitude over previous approaches. In addition, errors are localized at coarse-fine boundaries and do not suffer from Fourier-space induced interference ringing. An optional hybrid multi-grid and Fast Fourier Transform (FFT) based scheme is introduced which has identical Fourier space behaviour as traditional approaches. Using a suite of re-simulations of a galaxy cluster halo our real space based approach is found to reproduce correlation functions, density profiles, key halo properties and subhalo abundances with per cent level accuracy. Finally, we generalize our approach for two-component baryon and dark-matter simulations and demonstrate that the power spectrum evolution is in excellent agreement with linear perturbation theory. For initial baryon density fields, it is suggested to use the local Lagrangian approximation in order to generate a density field for mesh based codes that is consistent with Lagrangian perturbation theory instead of the current practice of using the Eulerian linearly scaled densities.

Effect of internal mass in the simulation of a moving body by the immersed boundary method

We investigate the effect of internal mass in the simulation of a moving body by the immersed boundary method. In general, the force and the torque acting on the body are influenced by the internal mass, if they are obtained by the negative of the sum of body forces which are applied near the boundary in order to enforce the no-slip condition on the boundary. In this study, the following schemes for approximating the internal mass effect are introduced; no internal mass effect, rigid body approximation, and Lagrangian points approximation. By comparing these schemes through the simulations of a moving body, we examine the internal mass effect. The simulations of the flow around an oscillating circular cylinder and of the sedimentations of an elliptical cylinder and a sphere are performed by using an immersed boundary–lattice Boltzmann method, and it is found that the internal mass effect is significant to unsteady body motions for the Reynolds numbers over 10 and grows as the Reynolds number increases. We also find that for the angular motions of the body, the rigid body approximation causes errors for the rotational Reynolds numbers over 10.

Arbitrary Lagrangian Eulerian approximation with remeshing for Navier–Stokes equations

AbstractAn algorithm that allows remeshing in the arbitrary Lagrangian Eulerian (ALE) framework is presented. At every time step, we could triangulate the domain using either uniform size meshes or adapted meshes. We analyze the conditions when two time‐advancing algorithms based on the backward Euler scheme provide identical approximations. Numerical results are presented for Navier–Stokes equations on moving domain. For three academic tests presented in this paper, the uniform size mesh technique provides more accurate results than the classical ALE method, in particular when the domain is expanding particularly fast. Copyright © 2009 John Wiley & Sons, Ltd.

Extending the domain of validity of the Lagrangian approximation

We investigate convergence of Lagrangian Perturbation Theory (LPT) by analyzing the model problem of a spherical homogeneous top-hat in an Einstein-deSitter background cosmology. We derive the formal structure of the LPT series expansion, working to arbitrary order in the initial perturbation amplitude. The factors that regulate LPT convergence are identified by studying the exact, analytic solution expanded according to this formal structure. The key methodology is to complexify the exact solution, demonstrate that it is analytic and apply well-known convergence criteria for power series expansions of analytic functions.This analysis fully explains the previously reported observation that LPT fails to predict the evolution of an underdense, open region beyond a certain time. It also implies the existence of other examples, including overdense, closed regions, for which LPT predictions should also fail. We show that this is indeed the case by numerically computing the LPT expansion in these problematic cases. The formal limitations to the validity of LPT expansion are considerably more complicated than simply the first occurrence of orbit crossings as is often assumed. Evolution to a future time generically requires re-expanding the solution in overlapping domains that ultimately link the initial and final times, each domain subject to its own convergence criterion. We demonstrate that it is possible to handle all the problematic cases by taking multiple steps (LPT re-expansion). We characterize how the leading order numerical error for a solution generated by LPT re- expansion varies with the choice of Lagrangian order and of time step size. Convergence occurs when the Lagrangian order increases and/or the time step size decreases in a simple, well-defined manner. We develop a recipe for time step control for LPT re-expansion based on these results.

A new scheme for perturbation contribution in density functional theory and application to solvation force and critical fluctuations

To surpass a traditional mean field density functional approximation for a perturbation term of interparticle potential function in liquid state, a correlation term is introduced by using weighted density approximation to deal with the perturbation free energy beyond the mean field one. Consequently, a free energy density functional approximation is advanced by combining the mean field term and correlation term with a hard sphere term treated with a Lagrangian theorem-based density functional approximation in the present work. The present free energy density functional approximation is applied in the framework of classical density functional theory (DFT) to a hard core attractive Yukawa (HCAY) fluid subject to external fields; comparison of the resulted predictions for density profiles with available simulation data is favorable for the present DFT approach as a highly accurate predictive approach. Then, the DFT approach is employed to investigate influencing factors for solvation forces between two infinite planar surfaces immersed in an intervening solvent with the HCAY potential function. It is found that (i) critical fluctuations induce negative adsorptions and long-ranged solvation forces; (ii) for narrow slit, the effect of external potential range is kept down; instead strength of the external field contact potential plays dominating role; (iii) state point in the bulk phase diagram, where the most remarkable critical effects are displayed, is the one with a bulk density a little higher than the critical density; remnants of critical fluctuations remain close to the bulk gas-liquid coexistence curve.

A third-order asymptotic solution of nonlinear standing water waves in Lagrangian coordinates

Asymptotic solutions up to third-order which describe irrotational finite amplitude standing waves are derived in Lagrangian coordinates. The analytical Lagrangian solution that is uniformly valid for large times satisfies the irrotational condition and the pressure p = 0 at the free surface, which is in contrast with the Eulerian solution existing under a residual pressure at the free surface due to Taylor's series expansion. In the third-order Lagrangian approximation, the explicit parametric equation and the Lagrangian wave frequency of water particles could be obtained. In particular, the Lagrangian mean level of a particle motion that is a function of vertical label is found as a part of the solution which is different from that in an Eulerian description. The dynamic properties of nonlinear standing waves in water of a finite depth, including particle trajectory, surface profile and wave pressure are investigated. It is also shown that the Lagrangian solution is superior to an Eulerian solution of the same order for describing the wave shape and the kinematics above the mean water level.

A multiscale finite element space–time discretization method for transient transport phenomena using bubble functions

A space–time finite element discretization method for unsteady transport phenomena is formulated using a variational multiscale finite element scheme. The described discretization is based on the utilization of bubble function enriched finite elements. The scheme is applied to model unsteady diffusion and convection–diffusion equations. It is shown that any temporal and spatial instabilities can be removed by appropriate enrichment of linear Lagrangian finite element approximations in the context of the standard Galerkin method. The proposed scheme is compared with widely used θ time stepping method. Numerical results generated by the proposed scheme are validated via their comparison with the analytical solution of a bench mark problem.

Supersymmetric QCD corrections to associated Higgs boson-bottom quark production

The associated production of a Higgs boson with a b quark is a discovery mode for a minimal supersymmetric standard model (MSSM) Higgs boson at large tan{beta}. We compute the SUSY QCD corrections from gluino and squark loops to this process and combine them with the O({alpha}{sub s}{sup 2}) next-to-leading order (NLO) QCD corrections to obtain reliable predictions for the rate. Finally, we compare our results with an effective Lagrangian approximation which includes only the low energy corrections from squark and gluino loops to the bb Higgs vertices.

On the Lagrangian description of steady surface gravity waves

This paper concerns the mathematical formulation of two-dimensional steady surface gravity waves in a Lagrangian description of motion. It is demonstrated first that classical second-order Lagrangian Stokes-like approximations do not exactly represent a steady wave motion in the presence of net mass transport (Stokes drift). A general mathematically correct formulation is then derived. This derivation leads naturally to a Lagrangian Stokes-like perturbation scheme that is uniformly valid for all time – in other words, without secular terms. This scheme is illustrated, both for irrotational waves, with seventh-order and third-order approximations in deep water and finite depth, respectively, and for rotational waves with a third-order approximation of the Gerstner-like wave on finite depth. It is also shown that the Lagrangian approximations are more accurate than their Eulerian counterparts of the same order.

A stochastic particle-mesh scheme for uncertainty propagation in vortical flows

A new mesh-particle scheme is constructed for uncertainty propagation in vortical flow. The scheme is based on the incorporation of polynomial chaos (PC) expansions into a Lagrangian particle approximation of the Navier–Stokes equations. The main idea of the method is to use a unique set of particles to transport the stochastic modes of the solution. The particles are transported by the mean velocity field, while their stochastic strengths are updated to account for diffusive and convective effects induced by the coupling between stochastic modes. An integral treatment is used for the evaluation of the coupled stochastic terms, following the framework of the particle strength exchange (PSE) methods, which yields a conservative algorithm. It is also shown that it is possible to apply solution algorithms used in deterministic setting, including particle-mesh techniques and particle remeshing. Thus, the method combines the advantages of particles discretizations with the efficiency of PC representations. Validation of the method on uncertain diffusion and convection problems is first performed. An example is then presented of natural convection of a hot patch of fluid in infinite domain, and the computations are used to illustrate the effectiveness of the approach for both large number of particles and high-order PC expansions.

Effect of foundation embedment on seismic behavior of elevated tanks considering fluid–structure-soil interaction

This paper investigates the effects of foundation embedment on the seismic behavior of fluid-elevated tank-foundation–soil system with a structural frame supporting the fluid containing tank. Six different soil types defined in the well-known seismic codes were considered. Both the sloshing effects of the fluid and soil-structure interaction of the elevated tanks located on these six different soils were included in the analyses. Fluid-elevated tank-foundation–soil systems were modeled with the finite element (FE) technique. The fluid-structure interaction was taken into account using Lagrangian fluid FE approximation implemented in the general purpose structural analysis computer program, ANSYS. FE model with viscous boundary was used to include elevated tank-foundation–soil interaction effects. The models were analyzed for the foundations with and without embedment. It was found that the tank roof displacements were affected significantly by the embedment in soft soil, however, this effect was smaller for stiff soil types. Except for soft soil types, embedment did not affect the other response parameters, such as sloshing displacement, of the systems considered in this study.

Chaos and order in models of black hole pairs

Chaos in the orbits of black hole pairs has by now been confirmed by several independent groups. While the chaotic behavior of binary black hole orbits is no longer argued, it remains difficult to quantify the importance of chaos to the evolutionary dynamics of a pair of comparable mass black holes. None of our existing approximations are robust enough to offer convincing quantitative conclusions in the most highly nonlinear regime. It is intriguing to note that, in three different approximations to a black hole pair built of a spinning black hole and a nonspinning companion, two approximations exhibit chaos and one approximation does not. The fully relativistic scenario of a spinning test mass around a Schwarzschild black hole shows chaos, as does the post-Newtonian Lagrangian approximation. However, the approximately equivalent post-Newtonian Hamiltonian approximation does not show chaos when only one body spins. It is well known in dynamical systems theory that one system can be regular while an approximately related system is chaotic, so there is no formal conflict. However, the physical question remains: Is there chaos for comparable mass binaries when only one object spins? We are unable to answer this question given the poor convergence of the post-Newtonian approximation to the fully relativistic system. A resolution awaits better approximations that can be trusted in the highly nonlinear regime.

Adhesive gravitational clustering

The notion of `adhesion' has been advanced for the phenomenon of stabilization of large-scale structure emerging from gravitational instability of a cold medium. Recently, the physical origin of adhesion has been identified: a systematic derivation of the equations of motion for the density and the velocity fields leads naturally to the key equation of the `adhesion approximation' - however, under a set of strongly simplifying assumptions. In this work, we provide an evaluation of the current status of adhesive gravitational clustering and a clear explanation of the assumptions involved. Furthermore, we propose systematic generalizations with the aim to relax some of the simplifying assumptions. We start from the general Newtonian evolution equations for self-gravitating particles on an expanding Friedmann background and recover the popular `dust model' (pressureless fluid), which breaks down after the formation of density singularities; then we investigate, in a unified framework, two other models which, under the restrictions referred to above, lead to the `adhesion approximation'. We apply the Eulerian and Lagrangian perturbative expansions to these new models and, finally, we discuss some non-perturbative results that may serve as starting points for workable approximations of non-linear structure formation in the multi-stream regime. In particular, we propose a new approximation that includes, in limiting cases, the standard `adhesion model' and the Eulerian as well as Lagrangian first-order approximations.

Local Solvent Density Augmentation around a Solute in Supercritical Solvent Bath: 1. A Mechanism Explanation and a New Phenomenon

A recently proposed partitioned density functional (DF) approximation (Phys. Rev. E 2003, 68, 061201) and an adjustable parameter-free version of a Lagrangian theorem-based DF approximation (LTDFA: Phys. Lett. A 2003, 319, 279) are combined to propose a DF approximation for nonuniform Lennard-Jones (LJ) fluid. Predictions of the present DF approximation for local LJ solvent density inhomogeneity around a large LJ solute particle or hard core Yukawa particle are in good agreement with existing simulation data. An extensive investigation about the effect of solvent bath temperature, solvent-solute interaction range, solvent-solute interaction magnitude, and solute size on the local solvent density inhomogeneity is carried out with the present DF approximation. It is found that a plateau of solvent accumulation number as a function of solvent bath bulk density is due to a coupling between the solvent-solute interaction and solvent correlation whose mathematical expression is a convolution integral appearing in the density profile equation of the DF theory formalism. The coupling becomes stronger as the increasing of the whole solvent-solute interaction strength, solute size relative to solvent size, and the closeness to the critical density and temperature of the solvent bath. When the attractive solvent-solute interaction becomes large enough and the bulk state moves close enough to the critical temperature of the solvent bath, the maximum solvent accumulation number as a function of solvent bath bulk density appears near the solvent bath critical density; the appearance of this maximum is in contrast with a conclusion drawn by a previous investigation based on an inhomogeneous version of Ornstein-Zernike integral equation carried out only for a smaller parameter space than that in the present paper. Advantage of the DFT approach over the integral equation is discussed.

Sedimentation Equilibrium of Colloidal Suspensions in a Planar Pore Based on Density Functional Theory and the Hard-Core Attractive Yukawa Model

The sedimentation equilibrium of colloidal suspensions modeled by hard-core attractive Yukawa (HCAY) fluids in a planar pore is studied. The density profile of the HCAY fluid in a gravitational field and its distribution between the pore and uniform phases are investigated by a density functional theory (DFT) approach, which results from employing a recently proposed parameter-free version of the Lagrangian theorem-based density functional approximation (Zhou, S. Phys. Lett. A 2003, 319, 279) for hard-sphere fluids to the hard-core part of the HCAY fluid, and the second-order functional perturbation expansion approximation to the tail part as was done in a recent partitioned density functional approximation (Zhou, S. Phys. Rev. E: Stat. Phys., Plasmas, Fluids, Relat. Interdiscip. Top. 2003, 68, 061201). The resultant DFT approach is, thus, the first adjustable parameter-free DFT for HCAY fluids. The validity of the present DFT for HCAY fluids of reduced range parameter z(red) = 1.8 under various external potentials is established in the first of the papers cited previously. The present DFT for HCAY fluids can predict the radial distribution function for the bulk HCAY fluid accurately in the colloidal limit (large value of z(red)), and in the hard-sphere limit, its prediction for the density profile of the hard-sphere fluid in a gravitational field is in very good agreement with the existing simulation data. The dependence of the density profile and distribution coefficient on the magnitude of the interparticle attraction, gravitational field, and degree of confinement is investigated in detail by the present DFT approach. Intuitive and qualitative analyses are also compared with the quantitative DFT calculational results.

Further investigation about Lagrangian theorem-based density functional approximation: test by non-uniform polymer melt

A Lagrangian theorem-based density functional approximation [S. Zhou, New J. Phys. 4 (2002) 36] for hard sphere fluid is employed to describe non-uniform polymer melt in the framework of density functional theory. A required bulk second order direct correlation function (DCF) within the whole density range is obtained by solving the polymer-RISM integral equation, the associated adjustable parameter is specified by a hard wall sum rule, and is found to be a negative value when the bulk density is low and the number of chain segment is large. However, the mathematically meaningless value can be physically meaningful by the observation that the present recipe can produce out density profile in very good agreement with simulation data not only at the contact region, but also at the region far away from the surface, and that the predicted global quantities such as surface excess and surface tension are also in good agreement with the simulation data. It is considered that the LTDFA has a property of self correction, which enables the LTDFA-based DFA for non-uniform polymer melt performs quite well even with a not very accurate second order DCF as input. Potential applications of the self correction peculiarity are discussed.

Statistical Mechanics of Turbulent Flows

Statistical Mechanics of Turbulent Flows

On uniform convergence rates for Eulerian and Lagrangian finite element approximations of convection-dominated diffusion problems

Standard error estimates in the literature for finite element approximations of nonstationary convection-diffusion problems depend either reciprocally on the diffusion parameter e or on higher order norms of the solution. Therefore, the estimates generally become worthless in the convection-dominated case with 0 < e ≪ 1. In this work we develop a rigorous e-uniform convergence theory for finite element discretizations of convection-dominated diffusion problems in Eulerian and Lagrangian coordinates. Here, the constants that arise in the error estimates depend on norms of the data and not of the solution and remain bounded in the hyperbolic limit e → 0. In particular in the Lagrangian case this requires modifications to standard finite element error analyses. In the Eulerian formulation e-uniform convergence of order one half is proven whereas in the Lagrangian framework e-uniform convergence of optimal order is established. The estimates are based on e-uniform a priori estimates for the solution of the continuous problems which are derived first.

Application of Lagrangian theorem-based density-functional approximation free of adjustable parameters to nonhard-sphere fluid

A recently proposed parameter free version of a Lagrangian theorem-based density functional approximation (LTDFA) [S. Zhou, Phys. Lett. A 319, 279 (2003)] for hard-sphere fluid is applied to hard-core attractive Yukawa model fluid by dividing bulk second-order direct correlation function (DCF) of fluid under consideration into hard-core part and tail part. The former is treated by the parameter free version of the LTDFA, while the tail part is treated by second-order functional perturbation expansion approximation as done in a recent partitioned DFA [S. Zhou, Phys. Rev. E 68, 061201 (2003)]. Two versions of mean spherical approximation (MSA) for the bulk second-order DCF are employed as input, one is the less accurate plain MSA whose tail part of the second-order DCF is strictly independent of a density argument, the other is the more accurate inverse temperature expansion version of the MSA whose tail part is not strictly independent of the density argument. Calculational results indicate that prediction based on the plain MSA is far more accurate than that based on the inverse temperature expansion version of the MSA. The reason is considered to be that the partitioned DFA requires that the tail part is highly or completely independent of the density argument, the plain MSA, by assuming that the tail part is exactly the potential itself, embodies all of the nonlinearities into the hard-core part which can be treated satisfactorily by the parameter free version of the LTDFA. The present investigation results in a universal method for constructing DFA for nonuniform any nonhard-sphere interaction potential fluids.

Quark Condensate in the Strange Matter**The project supported in part by Key Research Plan of Theoretical Physics and Cross Science under Grant No. 90103018 and National Natural Science Foundation of China under Grant No. 19975062

In a nonlinear chiral SU(3) framework, we investigate the quark condensate in the strange matter including and Λ, making use of chiral symmetry spontaneous breaking Lagrangian and mean-field approximation. The results show that the chiral symmetry is restored partially when the strange matter density increases and that plays a very important role in the strange matter which may approach the constituents of the neutron stars. In addition, we can find that the strange matter density where the π-condensate emerges leads to the ratio of the nucleon number to baryon number.