Noether's conservation laws of holonomic nonconservative dynamical systems in generalized mechanics

In the present paper, three kinds of forms for Noether's conservation laws of holonomic nonconservative dynamical systems in generalized mechanics are given. First Received Dec. 4, 1992     

Efficient high‐resolution relaxation schemes for hyperbolic systems of conservation laws

In this work, we present a total variation diminishing (TVD) scheme in the zero relaxation limit for nonlinear hyperbolic conservation law using flux limiters within the framework of a relaxation system that converts a nonlinear conservation law into a system of linear convection equations with nonlinear source terms. We construct a numerical flux for space discretization of the obtained relaxation system and modify the definition of the smoothness parameter depending on the direction of the flow so that the scheme obeys the physical property of hyperbolicity. The advantages of the proposed scheme are that it can give second‐order accuracy everywhere without introducing oscillations for 1‐D problems (at least with) smooth initial condition. Also, the proposed scheme is more efficient as it works for any non‐zero constant value of the flux limiter ? ? [0, 1], where other TVD schemes fail. The resulting scheme is shown to be TVD in the zero relaxation limit for 1‐D scalar equations. Bound for the limiter function is obtained. Numerical results support the theoretical results. Copyright © 2007 John Wiley & Sons, Ltd.     

An unconditionally stable,explicit Godunov scheme for systems of conservation laws

Common explicit, Godunov‐type schemes are subject to a stability constraint. The time‐line interpolation technique allows this constraint to be eliminated without having to make the scheme implicit or to linearize the equations. For 2×2 systems of conservation laws, a system of non‐linear equations has to be solved in the general case to determine the left and right states of the Riemann problems at the cell interfaces. However, if one cell in the domain is wide enough for the Courant number to be locally lower than unity, it is not necessary to solve a system anymore and the values at the next time step can be computed directly. The method is detailed for linear and non‐linear scalar advection, as well as for 2×2 systems of hyperbolic conservation laws. It is illustrated by an application to a simplified model for two‐phase flow in pipes, which is described using a 2×2 system of non‐linear hyperbolic equations. Copyright © 2002 John Wiley & Sons, Ltd.     

Group analysis of kinetic equations in a non-linear thermal transport problem

An application of modern group analysis to electron kinetic equations in non-linear thermal transport problem is discussed. The admitted symmetry group is calculated, and the optimal system of one and two-dimensional subalgebras is constructed. Representations of invariant solutions are presented.     

Shock wave solution of the Boltzmann kinetic equation in a 13-moment approximation

We consider the classic problem of a one-dimensional steady shock-wave solution of the Boltzmann kinetic equation utilizing a new type of 13-moment approximation proposed by Oguchi (1997). The model, unlike previous ones, expresses the collision term in an explicit function of the molecular velocity. This enables us to examine directly the nature of the singularity of the distribution function to this particular problem caused by the vanishing molecular velocity. We can thus obtain moment integrals directly because of its explicit expression. The principal value is utilized for the moment integral to cope with the singularity, and we can have five relations for five unknown functions to be determined with respect to the coordinate x. These relations can be reduced to a first-order differential equation that is solved to provide the familiar smooth monotonic transition from the upstream supersonic state to the subsonic downstream state. Computed values of shock thickness for various shock Mach numbers agree well with existing results obtained by different methods to the certain Mach number beyond which no solution exists.Received: 17 May 2002, Accepted: 1 May 2003, Published online: 15 August 2003PACS: 51.10. + y     

Application of a fourth-order relaxation scheme to hyperbolic systems of conservation laws

A fourth-order relaxation scheme is derived and applied to hyperbolic systems of conservation laws in one and two space dimensions. The scheme is based on a fourth-order central weighted essentially nonoscillatory (CWENO) reconstruction for one-dimensional cases, which is generalized to two-dimensional cases by the dimension-by-dimension approach. The large stability domain Runge-Kutta-type solver ROCK4 is used for time integration. The resulting method requires neither the use of Riemann solvers nor the computation of Jacobians and therefore it enjoys the main advantage of the relaxation schemes. The high accuracy and high-resolution properties of the present method are demonstrated in one- and two-dimensional numerical experiments. The project supported by the National Natural Science Foundation of China (60134010) The English text was polished by Yunming Chen.     

Multicomponent fluid flow analysis using a new set of conservation equations

In this work hydrodynamics of multicomponent ideal gas mixtures have been studied. Starting from the kinetic equations, the Eulerian approach is used to derive a new set of conservation equations for the multicomponent system where each component may have different velocity and kinetic temperature. The equations are based on the Grad's method of moment derived from the kinetic model in a relaxation time approximation (RTA). Based on this model which contains separate equation sets for each component of the system, a computer code has been developed for numerical computation of compressible flows of binary gas mixture in generalized curvilinear boundary conforming coordinates. Since these equations are similar to the Navier-Stokes equations for the single fluid systems, the same numerical methods are applied to these new equations. The Roe's numerical scheme is used to discretize the convective terms of governing fluid flow equations. The prepared algorithm and the computer code are capable of computing and presenting flow fields of each component of the system separately as well as the average flow field of the multicomponent gas system as a whole. Comparison of the present code results with those of a more common algorithm based on the mixture theory in a supersonic converging-diverging nozzle provides the validation of the present formulation. Afterwards, a more involved nozzle cooling problem with a binary ideal gas (helium-xenon) is chosen to compare the present results with those of the ordinary mixture theory. The present model provides the details of the flow fields of each component separately which is not available otherwise. It is also shown that the separate fluids treatment, such as the present study, is crucial when considering time scales on the order of (or shorter than) the intercollisions relaxation times.     

Characteristics, conservation laws, and symmetries of the kinetic equations of motion of bubbes in a fluid

Generalized characteristics and Riemann invariants that are preserved along the characteristics are found for a kinetic model of motion of bubbles in a fluid. Conditions that ensure the hyperbolicity of a set of equations of a bubbly flow are obtained. It is shown that the set of equations of motion has an infinite number of conservation laws. An infinite series of generalized symmetries admitted by the equations is constructed. Solutions that are invariant under the generalized symmetries of solution and describe the propagation of running and simple waves in a bubbly fluid are found. Lavrent’ev Institute of Hydrodynamics, Siberian Division, Russian Academy of Sciences, Novosibirsk 630090. Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 40, No., 2. pp. 86–100, March–April, 1999.     

An analysis of the relaxation oscillations of a nonlinear thermosyphon

The oscillatory behavior of an asymmetrically forced thermosyphon constituted by two connected vessels has been subjected to an asymptotically valid analysis using the vessel-volume ratio as expansion parameter. Due to the structure of the governing equations, the problem could not be dealt with using standard techniques; instead a phase-plane analysis was conducted. The analytically determined corrections to the previously established lowest-order discontinuous results proved to be useful even for comparatively large values of the expansion parameter. The relationship between these asymptotically valid corrections and the physics underlying the relaxation oscillation as well as the behavior of the system for strong thermal forcing is discussed. The study is concluded by an overview of some specific inconsistencies associated with the discontinuous lowest-order analysis and how these were alleviated by the asymptotically valid corrections.     

On 3-d strains in a thin coat of a plane domain

We formulate and solve the boundary value problem for a thin coat of a plane domain.     

Continuum approximation of the Peach-Koehler force on dislocations in a slip plane

We derive a continuum model for the Peach-Koehler force on dislocations in a slip plane. To represent the dislocations, we use the disregistry across the slip plane, whose gradient gives the density and direction of the dislocations. The continuum model is derived rigorously from the Peach-Koehler force on dislocations in a region that contains many dislocations. The resulting continuum model can be written as the variation of an elastic energy that consists of the contribution from the long-range elastic interaction of dislocations and a correction due to the line tension effect.     

An enhanced treatment of boundary conditions in implicit smoothed particle hydrodynamics简

In this work, an enhanced treatment of the solid boundaries is proposed for smoothed particle hydrodynamics with implicit time integration scheme （Implicit SPH）. Three types of virtual particles, i.e., boundary particles, image particles and mirror particles, are used to impose boundary conditions. Boundary particles are fixed on the solid boundary, and each boundary particle is associated with two fixed image particles inside the fluid domain and two fixed mirror particles outside the fluid domain. The image particles take the flow properties through fluid particles with moving least squares （MLS） interpolation and the properties of mirror particles can be obtained by the corresponding image particles. A repulsive force is also applied for boundary particles to prevent fluid particles from unphysical penetra- tion through solid boundaries. The new boundary treatment method has been validated with five numerical examples. All the numerical results show that Implicit SPH with this new boundary-treatment method can obtain accurate results for non-Newtonian fluids as well as Newtonian fluids, and this method is suitable for complex solid boundaries and can be easily extended to 3D problems.     

Numerical modeling of the acoustic wave dynamics on the basis of a kinetic approach

The results of the numerical modeling of a viscous compressible flow on a finite one-dimensional interval with periodic boundary conditions on the basis of a model kinetic equation are presented. The flow is excited by a small time-dependent force, periodic in space and time. It is shown that when the periodicity interval length L ≥ 10³ nonlinear steady oscillations with sharp temporal and spatial variations of the parameters arise. The system dynamics are investigated by means a pseudospectral method using the standard fast Fourier transformation procedure.     

Assessment of some iterative methods for non-symmetric linear systems arising in computational fluid dynamics

Various tests have been carried out in order to compare the performances of several methods used to solve the non-symmetric linear systems of equations arising from implicit discretizations of CFD problems, namely the scalar advection-diffusion equation and the compressible Euler equations. The iterative schemes under consideration belong to three families of algorithms: relaxation (Jacobi and Gauss-Seidel), gradient and Newton methods. Two gradient methods have been selected: a Krylov subspace iteration method (GMRES) and a non-symmetric extension of the conjugate gradient method (CGS). Finally, a quasi-Newton method has also been considered (Broyden). The aim of this paper is to provide indications of which appears to be the most adequate method according to the particular circumstances as well as to discuss the implementation aspects of each scheme.     

Semi-permeable crack analysis in a 2D magnetoelectroelastic medium based on the EMPS model

For a crack in a magnetoelectroelastic plane under the electrically and magnetically semi-permeable boundary condition, we derive the non-linear analytical solution of the strip electric–magnetic polarization saturation (EMPS) model. Using the extended dislocation theory and integral equation method, we obtain the electric and magnetic yielding zones, as well as the field intensity factor and local J-integral. Adapting an iterative method, numerical examples were performed to analyze the effect of different boundary conditions and the electric–magnetic saturated properties on the electric displacement and magnetic induction in the crack cavity, electric and magnetic yielding zones, stress intensity factor and local J-integral.     

The influence of normal flow boundary conditions on spurious modes in finite element solutions to the shallow water equations

Finite element solutions of the primitive equation (PE) form of the shallow water equations are notorious for the severe spurious 2Δx modes which appear. Wave equation (WE) solutions do not exhibit these numerical modes. In this paper we show that the severe spurious modes in PE solutions are strongly influenced by essential normal flow boundary conditions in the coupled continuity-momentum system of equations. This is demonstrated through numerical examples that avoid the use of essential normal flow boundary conditions either by specifying elevation values over the entire boundary or by implementing natural flow boundary conditions in the weak weighted residual form of the continuity equation. Results from a series of convergence tests show that PE solutions are of nearly the same quality as WE solutions when spurious modes are suppressed by alternative specification of the boundary conditions. Network intercomparisons indicate that varying nodal support does not excite spurious modes in a solution, although it does enhance the spurious modes introduced when an essential normal flow boundary condition is used. Dispersion analysis of discrete equations for interior and boundary nodes offers an explanation of the observed solution behaviour. For certain PE algorithms a mixed situation can arise where the boundary nodes exhibit a monotonic (noise-free) dispersion relationship and the interior nodes exhibit a folded (noisy) dispersion relationship. We have found that the mixed situation occurs when all boundary nodes are specified elevation nodes (which are enforced as essential conditions in the continuity equation) or when specified flow boundary nodes are treated as natural boundary conditions in the continuity equation. In either case the effect is to generate a solution that is essentially free of noise. Apparently, the monotonic dispersion behaviour at the boundaries suppresses the otherwise noisy behaviour caused by the folded dispersion relation on the interior.     

An analysis of exhaustion hardening in micron-scale plasticity

The rate-dependent behavior of micron-scale model planar crystals is investigated using the framework of mechanism-based discrete dislocation plasticity. Long-range interactions between dislocations are accounted for through elasticity. Mechanism-based constitutive rules are used to represent the short-range interactions between dislocations, including dislocation multiplication and dislocation escape at free surfaces. Emphasis is laid on circumstances where the deformed samples are not statistically homogeneous. The calculations show that dimensional constraints selectively set the operating dislocation mechanisms, thus giving rise to the phenomenon of exhaustion hardening whereby the applied strain rate is predominantly accommodated by elastic deformation. When conditions are met for this type of hardening to take place, the calculations reproduce some interesting qualitative features of plastic deformation in microcrystals, such as flow intermittency over coarse time-scales and large values of the flow stress with no significant accumulation of dislocation density. In addition, the applied strain rate is varied down to 0.1 s⁻¹ and is found to affect the rate of exhaustion hardening.     

One-dimensional dynamic equations of a piezoelectric semiconductor beam with a rectangular cross section and their application in static and dynamic characteristic analysis

Within the framework of continuum mechanics, the double power series expansion technique is proposed, and a series of reduced one-dimensional (1D) equations for a piezoelectric semiconductor beam are obtained. These derived equations are universal, in which extension, flexure, and shear deformations are all included, and can be degenerated to a number of special cases, e.g., extensional motion, coupled extensional and flexural motion with shear deformations, and elementary flexural motion without shear deformations. As a typical application, the extensional motion of a ZnO beam is analyzed sequentially. It is revealed that semi-conduction has a great effect on the performance of the piezoelectric semiconductor beam, including static deformations and dynamic behaviors. A larger initial carrier density will evidently lead to a lower resonant frequency and a smaller displacement response, which is a little similar to the dissipative effect. Both the derived approximate equations and the corresponding qualitative analysis are general and widely applicable, which can clearly interpret the inner physical mechanism of the semiconductor in the piezoelectrics and provide theoretical guidance for further experimental design.     

Optimal error bound in a Sobolev space of regularized approximation solutions for a sideways parabolic equation

The inverse heat conduction problem （IHCP） is a severely ill-posed problem in the sense that the solution （ if it exists） does not depend continuously on the data. But now the results on inverse heat conduction problem are mainly devoted to the standard inverse heat conduction problem. Some optimal error bounds in a Sobolev space of regularized approximation solutions for a sideways parabolic equation, i. e. , a non-standard inverse heat conduction problem with convection term which appears in some applied subject are given.     

Numerical analysis of longtime dynamic behavior in weakly damped forced KdV equation

ＩｎｔｒｏｄｕｃｔｉｏｎＯｎｅｏｆｔｈｅｍｏｓｔｉｍｐｏｒｔａｎｔａｎｄｉｎｔｅｒｅｓｔｉｎｇｐｒｏｂｌｅｍｓｉｎｔｈｅｆｉｅｌｄｏｆｎｏｎｌｉｎｅａｒｓｃｉｅｎｃｅｉｓｔｈｅｄｙｎａｍｉｃｓａｎａｌｙｓｉｓｏｆｓｐａｃｅ＿ｔｉｍｅｃｏｍｐｌｅｘｉｔｙ .Ｔｈｅｓｔｕｄｙｏｆｔｈｉｓｓｕｂｊｅｃｔｉｓｄｅｖｅｌｏｐｉｎｇａｌｏｎｇｔｗｏｄｉｒｅｃｔｉｏｎｓ.Ｏｎｏｎｅｈａｎｄ ,ｔｈｅｔｈｅｏｒｙｅｓｔａｂｌｉｓｈｅｄｂｙＴｅｍａｍ[1,2 ]ａｎｄｈｉｓｃｏ＿ｗｏｒｋｅｒｓｏｆｔｈｅｕｎｉｑｕｅｇｌｏｂ…