Lagrangian Gas Dynamics in Two Dimensions and Lagrangian systems

We propose a new and canonical way of writing the equations of gas dynamics in Lagrangian coordinates in two dimensions as a weakly hyperbolic system of conservation laws. One part of the system is called the physical part and contains physical variables; the other part is the geometrical part. We show that the physical part is symmetrizable. We show that the weak hyperbolicity is due to shear contact discontinuities. Free divergence constraints play an important role in the system. We prove the L² stability of the physical part of the system. Based on this formulation, we derive a new conservative and entropy-consistent finite-volume numerical scheme. We prove the stability of the numerical scheme. Numerical results show the potential interest of this approach. Various examples (Born-Infeld, MHD, 3D lagrangian gas dynamics) can be written using the same abstract formalism.     

Complexification-averaging method via the averaged Lagrangian

In this paper, the complexification-averaging (CX-A) method for multi-DOF nonlinear vibratory systems is rederived in a new way based upon the averaged Lagrangian. The complex variables are introduced to represent the original displacements and velocities, and then the fast–slow decomposition of the complex variables is made. The time averaging of the Lagrangian over the fast variables is performed. Two different expressions for the kinetic energy are presented, and this results in two schemes for deriving the governing equations of the slow variables. For the scheme I, through the order analysis of the derivatives of the slow variables, it is shown that the second-order terms appeared in the averaged Lagrangian can be omitted, and thus a reduced averaged Lagrangian is obtained. Via the reduced averaged Lagrangian, the corresponding Lagrangian equations are derived. For the scheme II, through time averaging, the averaged Lagrangian is obtained, and then the corresponding equations for the slow variables can be obtained. Finally, two nonlinear vibratory systems with two-DOF and four-DOF, respectively, are given as examples to illustrate the new procedure for the CX-A method. The loci of nonlinear normal modes on the potential surface are studied in the first example, and the frequency-energy plot is investigated in the second example.     

Calculations of the transport properties within the PAW formalism

We implemented the calculation of the transport properties within the PAW formalism in the ABINIT code [1]. This feature allows the calculation of the electrical and optical properties, including the XANES spectrum, as well as the electronic contribution to the thermal conductivity. We present here the details of the implementation and results obtained for warm dense aluminum plasma.     

On the Lagrangian and instability of medium with defects

The article presents the Lagrangian of defects in the solids, equipped with bending and warp. The deformation of such elastic medium with defects is based on Riemann-Cartan geometry in three dimensional space. In the static theory for the media with dislocations and disclinations the possible choice of the geometric Lagrangian yield the equations of equilibrium. In this article, the assumed expression for the free energy leading is equal to a volume integral of the scalar function (the Lagrangian) that depends on metric and Ricci tensors only. In the linear elastic isotropic case the elastic potential is a quadratic function of the first and second invariants of strain and warp tensors with two Lame, two mixed and two bending constants. For the linear theory of homogeneous anisotropic elastic medium the elastic potential must be quadratic in warp and strain. The conditions of stability of media with defects are derived, such that the medium in its free state is stable. With the increasing strain the stability conditions could be violated. If the strain in material attains the critical value, the instability in form of emergence of new topological defects occurs. The medium undergoes the spontaneous symmetry breaking in form of emerging topological defects.     

Hamiltonian and Lagrangian theory of viscoelasticity

The viscoelastic relaxation modulus is a positive-definite function of time. This property alone allows the definition of a conserved energy which is a positive-definite quadratic functional of the stress and strain fields. Using the conserved energy concept a Hamiltonian and a Lagrangian functional are constructed for dynamic viscoelasticity. The Hamiltonian represents an elastic medium interacting with a continuum of oscillators. By allowing for multiphase displacement and introducing memory effects in the kinetic terms of the equations of motion a Hamiltonian is constructed for the visco-poroelasticity.

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关于地月系统的拉格朗日点

解释拉格朗日点的力学概念.计算地月系统的拉格朗日点.根据希尔曲线族的几何特征对拉格朗日点的稳定性给出直观的解释.     

Extended thermodynamics and the Jeffrey's constitutive equation

Extended irreversible thermodynamics provides an evolution equation for the viscous pressure tensor which reduces to the Jeffrey's constitutive equation in the long-wave limit. in contrast with Jeffrey's equation, the equation obtained in extended irreversible thermodynamics leads to finite speed of propagation for shear pulses. The nonlocal effects are included into the theory by allowing the entropy to depend on higher-order fluxes, instead of spatial gradients. The use of the former ones is clearly advantageous in the high-frequency domain.     

Modelling of dust lifting using the Lagrangian approach

The subject of this paper is dust lifting behind shock waves, a process that is important for the formation of explosive dust clouds in air. While Eulerian–Eulerian has been the standard numerical technique for such simulations, the Eulerian–Lagrangian technique has been used in this paper, making it possible to take into account more physical phenomena, such as particle–particle and particle–wall collisions. The results of the simulations are shown mainly graphically, as snapshots of particle positions at given times after the passing of the shock wave. The results show that the collisions, and the coefficient of restitution assumed for them, is important in determining the mobility and lifting of dust behind shock waves. The results also show that the idea of a horizontally travelling shock wave is an oversimplification: the strong pressure gradient at the surface results in a series of reflected waves generated at the surface and travelling into the gas phase.     

Effect of viscoplastic relations on the instability strain,shear band initiation strain,the strain corresponding to the minimum shear band spacing,and the band width in a thermoviscoplastic material

We study thermomechanical deformations of a steel block deformed in simple shear and model the thermoviscoplastic response of the material by four different relations. We use the perturbation method to analyze the stability of a homogeneous solution of the governing equations. The smallest value of the average strain for which the perturbed homogeneous solution becomes unstable is called the critical strain or the instability strain. For each one of the four viscoplastic relations, we investigate the dependence upon the nominal strain-rate of the critical strain, the shear band initiation strain, the shear band spacing and the band width. It is found that the qualitative responses predicted by the Wright–Batra, Johnson–Cook and the power law relations are similar but these differ from that predicted by the Bodner–Partom relation. The computed band width is found to depend upon the specimen height.     

The Smoluchowski equation for colloidal suspensions developed and analyzed through the GENERIC formalism

The Smoluchowski equation (SE) and the mechanical stress tensor for the over-damped dynamics of colloidal particles is derived directly at the pair distribution level starting from a thermodynamic basis using the general equations for equilibrium non-equilibrium reversible and irreversible coupling (GENERIC) formalism. Within the GENERIC formalism, the effect of the non-trivial convection due to hydrodynamic interactions is incorporated for the first time. The method generates a thermodynamically valid set of transport equations for the colloidal dispersion, thus properly identifying the extra stress due to the presence of the colloids. The derivation connects a formal entropy expansion to the many-body terms that arise in both the transport equation and the stress tensor, thus unifying their origin and providing a systematic path forward for improvement in the theory. The analysis identifies the thermodynamically valid stress expression, thus clarifying a long-standing problem in the literature that arises when separate derivations are performed for the transport equations and the stress tensor. The results of previous investigators are analyzed within this framework. Comparison with alternate methods of deriving the many-body Smoluchowski equation provide new insight into the nature of the many-body terms.     

The Stroh formalism and the reciprocity properties of reflection-transmission problems in crystal piezo-acoustics

The paper is concerned with the Helmholtz-Rayleigh reciprocity, which implies invariance of mode-into-mode transformation with respect to interchange of incident mode and reflected or transmitted mode. This concept is considered for a wide range of acoustic reflection-transmission problems in anisotropic piezoelectric media. Resorting to the ideas of the Stroh formalism and casting the wave solutions of a boundary problem into a self-orthogonal and complete set, we develop the common approach which allows us to prove the reciprocity properties in a similar fashion for reflection-transmission for various boundary conditions.     

The Anisotropic Lagrangian Averaged Euler and Navier-Stokes Equations

 The purpose of this paper is twofold. First, we give a derivation of the Lagrangian averaged Euler (LAE-α) and Navier-Stokes (LANS-α) equations. This theory involves a spatial scale α and the equations are designed to accurately capture the dynamics of the Euler and Navier-Stokes equations at length scales larger than α, while averaging the motion at scales smaller than α. The derivation involves an averaging procedure that combines ideas from both the material (Lagrangian) and spatial (Eulerian) viewpoints. This framework allows the use of a variant of G. I. Taylor's ``frozen turbulence' hypothesis as the foundation for the model equations; more precisely, the derivation is based on the strong physical assumption that fluctutations are frozen into the mean flow. In this article, we use this hypothesis to derive the averaged Lagrangian for the theory, and all the terms up to and including order α² are accounted for. The equations come in both an isotropic and anisotropic version. The anisotropic equations are a coupled system of PDEs (partial differential equations) for the mean velocity field and the Lagrangian covariance tensor. In earlier works by Foias, Holm & Titi [10], and ourselves [16], an analysis of the isotropic equations has been given. In the second part of this paper, we establish local in time well-posedness of the anisotropic LANS-α equations using quasilinear PDE type methods. (Accepted September 2, 2002) Published online November 26, 2002 Dedicated to Stuart Antman on the occasion of his 60th birthday Communicated by S. MüLLER