Energy Method for Approximate Periodic Solution of Strongly Nonlinear Nonautonomous Systems

In this paper, based upon the concept of the conservation of average energy, a theorem for the periodic solution of strongly nonlinear nonautonomous systems is proved. By using this theorem not only the necessary and sufficient conditions for the existence and stability of the periodic solutions can be obtained but, at the same time, the approximate expressions for those periodic solutions can also be obtained.     

Local time-stepping procedures for the space-time conservation element and solution element method

A local time-stepping procedure for the space-time conservation element and solution element (CESE) method has been developed. This new procedure allows for variation of time-step size in both space and time, and can also be extended to become multi-dimensional solvers with structured/unstructured spatial grids. Moreover, it differs substantially in concept and methodology from the existing approaches. By taking full advantage of key concepts of the CESE method, in a simple and efficient manner it can enforce flux conservation across an interface separating grid zones of different time-step sizes. In particular, no correction pass is needed. Numerical experiments show that, for a variety of flow problems involving moving shock and flame discontinuities, accurate and robust numerical simulations can be achieved even with a reduction in time-step size on the order of 10 or higher for grids across a single interface.     

求解二维Euler方程的时一空守恒格式

将作者原来得出的一维时－空守恒格式推广到了二维情形,得到了二维Ｅｕｌｅｒ方程的时．空守恒格式,并用几个典型算例进行了检验计算,结果表明：得到的二维时一空守恒格式保留了一维格式所有的优点,格式简单,通用性强,对微波等间断具有很高的分辨率．     

Space conservation law in finite volume calculations of fluid flow

In the numerical solutions of fluid flow problems in moving co-ordinates, an additional conservation equation, namely the space conservation law, has to be solved simultaneously with the mass, momentum and energy conservation equations. In this paper a method of incorporating the space conservation law into a finite volume procedure is proposed and applied to a number of test cases. The results show that the method is efficient and produces accurate results for all grid velocities and time steps for which temporal accuracy suffices. It is also demonstrated, by analysis and test calculations, that not satisfying the space conservation law in a numerical solution procedure introduces errors in the form of artificial mass sources. These errors can be made negligible only by choosing a sufficiently small time step, which sometimes may be smaller than required by the temporal discretization accuracy.     

A PRESSURE-CORRECTION METHOD FOR THE SOLUTION OF INCOMPRESSIBLE VISCOUS FLOWS ON UNSTRUCTURED GRIDS

An unstructured grid, finite volume method is presented for the solution of two-dimensional viscous, incompressible flow. The method is based on the pressure-correction concept implemented on a semi-staggered grid. The computational procedure can handle cells of arbitrary shape, although solutions presented herein have been obtained only with meshes of triangular and quadrilateral cells. The discretization of the momentum equations is effected on dual cells surrounding the vertices of primary cells, while the pressure-correction equation applies to the primary-cell centroids and represents the conservation of mass across the primary cells. A special interpolation scheme s used to suppress pressure and velocity oscillations in cases where the semi-staggered arrangement does not ensure a sufficiently strong coupling between pressure and velocity to avoid such oscillations. Computational results presented for several viscous flows are shown to be in good agreement with analytical and experimental data reported in the open literature.     

An RVE-based multiscale modeling method for constitutive relations

This article reports an efficient method to characterize constitutive responses based on multiscale modeling for fluid flow in heterogeneous media based on the concept of representative volume element (RVE). Between different scales, it is considered as the basic principles for down-scaling information the conservation of velocity and of the strain rate tensor. Within this context, we formulate (i) the problem to be solved at the micro-scale, (ii) the up-scaling procedure which involves homogenization rules, and (iii) the generalized principle of multiscale virtual power. The complete theory for constitutive modeling is revisited and shown that when employing multiscale analysis among the suitable variational arguments we are able to obtain, in a straightforward manner, new constitutive behavior between kinematic motions and actions. Some examples of application of fluid flow in heterogeneous media with obstacles are presented to show the consequences of the proposed approach.     

二维时-空守恒格式及其在激波传播计算中的应用

将改进后的一维时－空守恒格式推广到了二维情形,得到了一个新的一般形式的二维Ｅｕｌｅｒ方程时－空守恒格式,并用该格式对几个具有复杂波系的流场进行了数值模拟。结果表明,该格式保留了一维格式通用性好、结构简单的优点,其计算结果精度高,对激波等间断具有很强的分辨率。     

On configurational compatibility and multiscale energy momentum tensors

In this work the continuum theory of defects has been revised through the development of kinematic defect potentials. These defect potentials and their corresponding variational principles provide a basis for constructing a new class of conservation laws associated with the compatibility conditions of continua. These conservation laws represent configurational compatibility conditions which are independent of the constitutive behavior of the continuum. They lead to the development of a new concept termed configurational compatibility, dual to the concept of configurational force. The contour integral of the corresponding conserved quantity is path-independent, if the domain encompassed by the integral is defect-free. It is shown that the Peach-Koehler force can be recovered as one of these invariant integrals. Based on the proposed defect potentials and their corresponding defect energies, two-field multiscale mixed variational principles can be employed to construct multiscale energy momentum tensors. An application is outlined in the form of a mode III elasto-plastic crack problem for which the new configurational quantities are calculated.     

On an extension of the d’Alembert solution to initial–boundary value problems in multi-layered,multi-material domains

This work introduces a method for the exact solution of initial–boundary value problems for linear, one-dimensional conservation laws in multi-layered, multi-material domains. The method is based on the geometry of the solutions of such conservation laws and represents an extension of the d’Alembert solution to initial–boundary value problems in multi-layered, multi-material domains.     

一般形式的二维时—空守恒格式

本文将经作者改进后的一维时－空守恒格式推广到了二维情形，得到了一个一般形式的二维Ｅｕｌｅｒ方程时－空守恒格式，该格式对各种不规则几何区域内的流动问题具有很强的适应性，同时它还保留了一维格式的优点。几个典型算例的计算结果表明，本文格式不仅精度高，通用性好，而且对激波等间断具有很高的分辨率。     

Symmetries of the hyperbolic shallow water equations and the Green–Naghdi model in Lagrangian coordinates

The observation that the hyperbolic shallow water equations and the Green–Naghdi equations in Lagrangian coordinates have the form of an Euler–Lagrange equation with a natural Lagrangian allows us to apply Noether's theorem for constructing conservation laws for these equations. In this study the complete group analysis of these equations is given: admitted Lie groups of point and contact transformations, classification of the point symmetries and all invariant solutions are studied. For the hyperbolic shallow water equations new conservation laws which have no analog in Eulerian coordinates are obtained. Using Noether's theorem a new conservation law of the Green–Naghdi equations is found. The dependence of solutions on the parameter is illustrated by self-similar solutions which are invariant solutions of both models.     

Theory of multiphase mixtures—A thermomechanical formulation

The paper presents a theory of mixtures with the nonzero interfacial area between the constituents of the mixture. The conservation laws are physically motivated by utilizing a volume averaging procedure and by the definition of a mapping transformation. It is shown that the theory constructed in this manner is consistent with the theory of mixtures with a vanishingly small interfacial area and that a second law of thermodynamics can be assigned for each phase of the mixture. The conservation laws are examined for invariance properties with the principle of the material frame indifference, and a particular constitutive assumption is discussed. Also presented in the paper are the conservation laws in the integral form and the jump conditions for the singular surfaces in the multiphase mixture.     

求解二维Euler方程的时一空守恒格式

将作者原来得出的一维时－空守恒格式推广到了二维情形,得到了二维Ｅｕｌｅｒ方程的时．空守恒格式,并用几个典型算例进行了检验计算,结果表明：得到的二维时一空守恒格式保留了一维格式所有的优点,格式简单,通用性强,对微波等间断具有很高的分辨率．     

Local and nonlocal conserved vectors of the system of two-dimensional generalized inviscid Burgers equations

The concept of non-linear self-adjointness for the construction of conservation laws has attracted a lot of interest in recent years. The most noteworthy aspect of it is the likelihood of explicitly constructing the conservation laws for any arbitrary systems of differential equations, in particular for those for which Noether׳s theorem is not applicable. In this study, we shall use both Noether׳s theorem and the non-linear self-adjoint method to construct local and nonlocal conserved vectors of the system of two-dimensional Burgers equations under consideration. The first integrals obtained not only give more credence to obtained results due to their generality with respect to any arbitrary functions of the velocity components but are also independent, nontrivial and infinitely many.     

RENEWAL OF BASIC LAWS AND PRINCIPLES FOR POLAR CONTINUUM THEORIES (Ⅹ)--MASTER BALANCE LAW

Through a comparison between the expressions of master balance laws and the conservation laws derived by Noether's theorem, a unified master balance law and six physically possible balance equations for micropolar continuum mechanics are naturally deduced. Among them, by extending the well-known conventional concept of energymomentum tensor, the rather general conservation laws and balance equations named after energy-momentum, energy-angular momentum and energy-energy are obtained. It is clear that the forms of the physical field quantities in the master balance law for the last three cases could not be assumed directly by perceiving through the intuition. Finally,some existing results are reduced immediately as special cases.     

RENEWAL OF BASIC LAWS AND PRINCIPLES FOR POLAR CONTINUUM THEORIES (Ⅹ) —MASTER BALANCE LAW

Through a comparison between the expressions of master balance laws and the conservation laws derived by Noether's theorem, a unified master balance law and six physically possible balance equations for micropolar continuum mechanics are naturally deduced. Among them, by extending the well-known conventional concept of energymomentum tensor, the rather general conservation laws and balance equations named after energy-momentum, energy-angular momentum and energy-energy are obtained. It is clear that the forms of the physical field quantities in the master balance law for the last three cases could not be assumed directly by perceiving through the intuition. Finally, some existing results are reduced immediately as special cases.     

Coupled solution of the species conservation equations using unstructured finite‐volume method

A coupled solver was developed to solve the species conservation equations on an unstructured mesh with implicit spatial as well as species‐to‐species coupling. First, the computational domain was decomposed into sub‐domains comprised of geometrically contiguous cells—a process similar to additive Schwarz decomposition. This was done using the binary spatial partitioning algorithm. Following this step, for each sub‐domain, the discretized equations were developed using the finite‐volume method, and solved using an iterative solver based on Krylov sub‐space iterations, that is, the pre‐conditioned generalized minimum residual solver. Overall (outer) iterations were then performed to treat explicitness at sub‐domain interfaces and nonlinearities in the governing equations. The solver is demonstrated for both two‐dimensional and three‐dimensional geometries for laminar methane–air flame calculations with 6 species and 2 reaction steps, and for catalytic methane–air combustion with 19 species and 24 reaction steps. It was found that the best performance is manifested for sub‐domain size of 2000 cells or more, the exact number depending on the problem at hand. The overall gain in computational efficiency was found to be a factor of 2–5 over the block (coupled) Gauss–Seidel procedure. All calculations were performed on a single processor machine. The largest calculations were performed for about 355 000 cells (4.6 million unknowns) and required 900 MB of peak runtime memory and 19 h of CPU on a single processor. Copyright © 2009 John Wiley & Sons, Ltd.     

Energy Method for Computing Periodic Solutions of Strongly Nonlinear Autonomous Systems with Multi-Degree-of-Freedom

In this paper with the use of conservation of average energy, a newmethod for computing the periodic solutions of strongly nonlinearautonomous systems with multi-degree-of-freedom is suggested. Thismethod cannot only decide the existence, but also give the approximateexpressions of the periodic solutions.     

High-Field Limit for the Vlasov-Poisson-Fokker-Planck System

This paper is concerned with the analysis of the stability of the Vlasov-Poisson-Fokker-Planck system with respect to the physical constants. If the scaled thermal mean free path converges to zero and the scaled thermal velocity remains constant, then a hyperbolic limit or equivalently a high-field limit equation is obtained for the mass density. The passage to the limit as well as the existence and uniqueness of solutions of the limit equation in L ¹, global or local in time, are analyzed according to the electrostatic or gravitational character of the field and to the space dimension. In the one-dimensional case a new concept of global solution is introduced. For the gravitational field this concept is shown to be equivalent to the concept of entropy solutions of hyperbolic systems of conservation laws. Accepted December 1, 2000?Published online April 23, 2001     

Contradiction between the C‐property and mass conservation in adaptive grid based shallow flow models: cause and solution

When performing shallow flow simulations on adaptive grids, the C‐property (i.e. conservation property) and the mass conservation may not be simultaneously preserved, that is, either C‐property or mass conservation is likely to be violated following grid refining or coarsening. The cause of such a contradiction is analyzed in detail in this work, which essentially links to the reconstruction of bed and flow information in those newly created cells during grid adaptation. An effective approach is subsequently proposed to resolve the contradiction by locally modifying the bed elevation in certain problematic cells when reconstructing flow information using linear interpolation as part of the grid adaptation procedure. Copyright © 2015 John Wiley & Sons, Ltd.