Study of a collocated Lagrange‐remap scheme for multi‐material flows adapted to HPC

This paper describes a collocated numerical scheme for multi‐material compressible Euler equations, which attempts to suit to parallel computing constraints. Its main features are conservativity of mass, momentum, total energy and entropy production, and second order in time and space. In the context of a Eulerian Lagrange‐remap scheme on planar geometry and for rectangular meshes, we propose and compare remapping schemes using a finite volume framework. We consider directional splitting or fully multi‐dimensional remaps, and we focus on a definition of the so‐called corner fluxes. We also address the issue of the internal energy behavior when using a conservative total energy remap. It can be perturbed by the duality between kinetic energy obtained through the conservative momentum remap or implicitly through the total energy remap. Therefore, we propose a kinetic energy flux that improves the internal energy remap results in this context. Copyright © 2016 John Wiley & Sons, Ltd.     

Conservation laws in non-linear dissipative systems

In Hamiltonian theory, Noether's theorem commonly is used to show the conservation of linear momentum and energy as a consequence of symmetry properties. The possibility of enclosing Hamiltonian theory in a wider context by use of Gibbs-Falkian thermodynamical methods, offering the opportunity to cover mechanical and thermodynamical systems with the same mathematical tools, is considered. Consequently it is shown how Noether's identity can be extended for dissipative systems which are appropriate to describe real life phenomena. By use of the principle of least action an extended version of Noether's theorem is calculated, from which the conservation of linear momentum and total energy can be derived. Additionally, the condition of absolute invariance is shown to be too restrictive for physical applications.     

Sharp-crack limit of a phase-field model for brittle fracture

A matched asymptotic analysis is used to establish the correspondence between an appropriately scaled version of the governing equations of a phase-field model for fracture and the equations of the two-dimensional sharp-crack theory of Gurtin and Podio-Guidugli (1996) that arise on assuming that the bulk constitutive behavior is nonlinearly elastic, requiring that surface energy provides the only factor limiting crack propagation, and assuming that the fracture kinetics are isotropic. Consistent with the prominence of the configurational momentum balance at the crack tip in the latter theory, the approach capitalizes on the configurational momentum balance that arises naturally in the context of the phase-field model. The model developed and utilized here incorporates irreversibility of the phase-field evolution. This is achieved by introducing a suitable constraint and by carefully heeding the influence of that constraint on the kinetics underlying microstructural changes associated with fracture. The analysis is predicated on the assumption that the phase-field variable takes values in the closed interval between zero and unity.     

Shape sensitivity analysis and the energy momentum tensor for the kinematic and static models of torsion

The aim of this paper is to bridge shape sensitivity analysis and configurational mechanics by means of a widespread use of the shape derivative concept. This technique will be applied as a systematic procedure to obtain the Eshelby’s energy momentum tensor associated to the problem under consideration. In order to highlight special features of this procedure and without loss of generality, we focus our attention in the application of shape sensitivity analysis to the problem of twisted straight bars within the framework of linear elasticity.Kinematic and static variational formulations as well as the direct method of sensitivity analysis are used to perform shape derivatives of both models. Integral expressions of first and second order shape derivatives of the total potential energy and the complementary potential energy with respect to an arbitrary transverse cross-section shape change, are achieved. These integral expressions put in evidence the relationship between shape sensitivity analysis and the first and second order Eshelby’s energy momentum tensors. Also, the null divergence property of these tensors is easily proved by comparing, in each case, the domain and boundary integral shape derivative arrived at. Finally, an example with a known exact solution, corresponding to an elastic bar with elliptical transverse cross-section submitted to twist, is presented in order to illustrate the usefulness of these tensors to compute the corresponding shape derivatives.     

Stability of a Crank–Nicolson pressure correction scheme based on staggered discretizations

In the context of LES of turbulent flows, the control of kinetic energy seems to be an essential requirement for a numerical scheme. Designing such an algorithm, that is, as less dissipative as possible while being simple, for the resolution of variable density Navier–Stokes equations is the aim of the present work. The developed numerical scheme, based on a pressure correction technique, uses a Crank–Nicolson time discretization and a staggered space discretization relying on the Rannacher–Turek finite element. For the inertia term in the momentum balance equation, we propose a finite volume discretization, for which we derive a discrete analogue of the continuous kinetic energy local conservation identity. Contrary to what was obtained for the backward Euler discretization, the dissipation defect term associated to the Crank–Nicolson scheme is second order in time. This behavior is evidenced by numerical simulations. Copyright © 2013 John Wiley & Sons, Ltd.     

Continuum thermodynamic formulation of models for electromagnetic thermoinelastic solids with application in electromagnetic metal forming

The purpose of this work is the formulation and application of a continuum thermodynamic approach to the phenomenological modeling of a class of engineering materials which can be dynamically formed using strong magnetic fields. This is carried out in the framework of a thermodynamic, internal-variable-based formulation in which the deformation, temperature and magnetic fields are in general coupled. This coupling takes the form of the Lorentz force as an additional supply of momentum, and the electromotive power as an additional supply of energy, in the material. In the current approach, the basic thermomechanical field relations for mass, momentum and moment of momentum are obtained from the total energy balance via invariance, and completed by Maxwells field equations. The constitutive formulation is based on the exploitation of the Müller-Liu entropy principle, here for the case of isotropic thermoelastic, viscoplastic material behaviour. The resulting reduced constitutive and field relations and restrictions are then applied to the modeling and simulation of high-speed electromagnetic forming of metal tubes and sheet metal. In this context, scaling arguments show that, over the relevant length- and timescales of engineering interest, the evolution of the magnetic field is diffusive in nature, and thermal conduction is negligible. Comparison of the simulation and experimental results for the final sheet metal form shows very good agreement.Received: 16 March 2004, Accepted: 6 May 2004, Published online: 17 September 2004PACS: 46.05. + b, 46.25.Hf, 46.35 + z Correspondence to: B. Svendsen     

RENEWAL OF BASIC LAWS AND PRINCIPLES FOR POLAR CONTINUUM THEORIES (Ⅵ)—CONSERVATION LAWS OF MASS AND INERTIA

IntroductionThisworkisadirectcontinuationandasupplementofRefs .[1～8] .InRefs.[1～8]thecoupledbalancelawsandequationsofmomentum ,angularmomentumandenergyaswellasthenewHamiltonprinciple,principleofvirtualpowerandNoethertheoremhavebeenpresented .However,thecoupledconservationlawsofmassandinertiahavenotbeenreestablishedyet.Thepurposeofthispaperistoreestablishtheconservationlawsandequationsofmassandinertiaandtocombinethemwiththecoupledbalancelawsandequationsofmomentum ,angularmomentum ,energyand…     

Experimental and numerical analysis of thermo-magnetic convection in a vertical annular enclosure

In the present paper an experimental and numerical analysis of a thermo-magnetic convective flow of paramagnetic fluid in an annular enclosure with a round rod core and a cylindrical outer wall is presented. It is complemented by an experimental analysis of natural convection depending on the inclination angle to show the stability of the present configuration. Convection in an annulus between two vertical co-axial cylinders resulting from gravitational and magnetic environments has been investigated. A strong magnetic field can be an alternative to heat transfer enhancement. The effect of the magnetic field on the convection of the paramagnetic fluid in the annular vessel in various positions was compared. The numerical analysis was done based on the continuity, momentum and energy equations. A term related to the magnetic buoyancy force was added to the momentum equation. The distributions of Nusselt number present minima in two positions of the enclosure, which depends on the reciprocal relationship between the gravitational and magnetising forces.     

Topology of the Integral Manifolds of the N-Body Problem with Positive Energy

The integral manifolds of the N-body problem are the level sets of energy and angular momentum. For positive energy and non-zero angular momentum, all level sets are diffeomorphic to a non-zero level set of angular momentum on the unit tangent bundle of the configuration space. The one complication that arises in attempting to describe this level set explicitly is the degeneracy at the syzygies of the equations that define angular momentum. In this work, we analyze the behavior of the angular momentum near syzygies, and show how to construct local coordinates near the syzygies. In particular, we show that the projection of the integral manifold onto the configuration space _c is a homotopy equivalence, and use this to compute the homology of the integral manifolds.     

On the principle of minimal entropy production for Navier-Stokes-Fourier fluids

In this paper we discuss the principle of minimal entropy production, proposed by Prigogine [1], which affirms that the global entropy production approaches a minimum as a process becomes stationary. We point out in two particular cases that this principle produces field equations that do not agree with the equations of balance of mass, momentum and energy. The processes considered are: • heat conduction in a fluid at rest • shear flow and heat conduction in an incompressible fluid. Now is the appropriate time to review Prigogine's principle, since in recent years a new, and different principle of minimal entropy production has been proposed. This is the “minimax principle” postulated by Struchtrup & Weiss [2]. Within the context of extended thermodynamics this new principle shows great promise. Received April 1, 1999     

Bilans d'energie et de quantité de mouvement pour un continu deformable,polarisable et magnétisable

In this paper, it is shown that the different electromagnetic energy-momentum tensors proposed by various authors for a continuum interacting with an electromagnetic field all lead to the same equations of balance for energy and momentum provided the definitions of stress and internal energy are suitably related. These various tensors come out from different partitions of the total energy-momentum tensor. From a particular partition, we derive an expression of the balance of energy suitable for application to continuum thermodynamics. In the classical approximation, the corresponding equation of balance of momentum gives rise to an expression for the electromagnetic force in a polarizable and magnetizable continuum.     

On canonical equations of continuum thermomechanics

It is shown that the canonical balance of momentum of continuum mechanics can be formulated in a general way, but not independently of the usual balance of linear momentum, even in the absence of specified constitutive equations. A parallel construct is made of necessity for the accompanying time-like canonical energy equation. On specifying the energy, previous particular cases can be deduced including pure elasticity, inhomogeneous thermoelasticity of conductors, and the case of dissipative solid-like materials described by means of a diffusive internal variable (such as in damage or weakly non-local plasticity). A redefinition of the entropy flux is necessarily accompanied by a redefinition of the Eshelby stress tensor.     

格子Boltzmann模型的改进与流体力学方程

流体的流动可以看成是分子以上水平的粒子基本运动组合而成,任何一个粒子系统的Hamiltonian都是由动能和势能这两部分所组成.借助于Hamiltonian建立了微观粒子和宏观流体之间的能量守恒准则,发展了一个适合于热流场数值模拟的格子Boltzmann模型.从该模型可以还原出宏观的流体力学方程,所得动量方程的黏性输运项除了具有Navier-Stokes黏性力的特征外还与非定常的、非线性的动量通量和非定常的内能相关.用该模型对Benard热对流进行了数值模拟,很好地再现了Benard cell,并且克服了热格子Boltzmann模型数值稳定性差的不足.     

RENEWAL OF BASIC LAWS AND PRINCIPLES FOR POLAR CONTINUUM THEORIES (Ⅵ)—CONSERVATION LAWS OF MASS AND INERTIA

The purpose is to reestablish the coupled conservation laws, the local conservation equations and the jump conditions of mass and inertia for polar continuum theories. In this connection the new material derivatives of the deformation gradient, the line element, the surface element and the volume element were derived and the generalized Reynolds transport theorem was presented. Combining, these conservation laws of mass and inertia with the balance laws of momentum, angular momentum and energy derived in our previous papers of this series, a rather complete system of coupled basic laws and principles for polar continuum theories is constituted on the whole. From this system the coupled nonlocal balance equations of mass, inertia, momentum, angular momentum and energy may be obtained by the usual localization. Contributed by DAI Tian-min, Original Member of Editorial Committee, AMM Foundation items: the National Natural Science Foundation of China (10072024); the Research Foundation of Liaoning Education Committee (990111001) Biography: DAI Tian-min (1931≈)     

Hydrodynamic Limit for a Hamiltonian System with Boundary Conditions and Conservative Noise

We study the hyperbolic scaling limit for a chain of N coupled anharmonic oscillators. The chain is attached to a point on the left and there is a force (tension) τ acting on the right. In order to provide good ergodic properties to the system, we perturb the Hamiltonian dynamics with random local exchanges of velocities between the particles, so that momentum and energy are locally conserved. We prove that in the macroscopic limit the distributions of the elongation, momentum and energy converge to the solution of the Euler system of equations in the smooth regime.     

Hydraulic jumps in two-layer flow with a free surface

This paper presents a closure relation which describes hydraulic jumps in two-layer flows with a free surface over a flat bottom. This relation is derived from the momentum equations for each layer, which, subject to the condition of conservation of the total momentum and mass of each layer, become conservative in a sense. It is shown that use of this relation provides a reduction in the total energy at the jump.     

GENERALIZED THEORY OF NONLINEAR AND UNSTEADY MECHANICS AND APPLICATIONS TO PARTICLE PHYSICS

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Approximate Analytical Solutions for Flow of a Third-Grade Fluid Through a Parallel-Plate Channel Filled with a Porous Medium

The flow of a non-Newtonian fluid through a porous media in between two parallel plates at different temperatures is considered. The governing momentum equation of third-grade fluid with modified Darcy’s law and energy equation have been derived. Approximate analytical solutions of momentum and energy equations are obtained by using perturbation techniques. Constant viscosity, Reynold’s model viscosity, and Vogel’s model viscosity cases are treated separately. The criteria for validity of approximate solutions are derived. A numerical residual error analysis is performed for the solutions. Within the validity range, analytical and numerical solutions are in good agreement.     

Relativistic equations of balance in continuum mechanics

In this paper we formulate the global expression of balance of energy starting from the relativistic definition of force, mechanical work, heat supply, and kinetic energy. We then prove that invariance under Lorentz transformation leads to the local expression for balance of momentum and to the equation of decomposition of the energy-momentum tensor.Previously, the principles of balance of momentum and balance of energy have been introduced as independent principles.     

壁面加热湍流相干结构的子波分析实验研究

通过在平板湍流边界层沿流向固壁表面平行放置若干条通电加热的金属细丝,在平板表面形成沿展向周期性分布的温度场,利用该温度场引起的空气热对流,在湍流边界层近壁区域产生一组沿湍流边界层展向周期分布的大尺度流向涡结构,改变了平板湍流边界层中不同尺度结构及其能量分布。采用对壁湍流多尺度结构的子波分析表明,在湍流边界层近壁区域产生规则的流向涡结构将壁湍流各种尺度湍涡结构不规则的脉动有序地组织起来,抑制了壁湍流各种尺度湍涡结构脉动,特别抑制了能量最大尺度湍涡结构的脉动,减小由于湍流脉动引起的在湍流边界层法向和展向的动量和能量损耗,从而减小了湍流的阻力。