MIKUSINSKI＇S OPERATORS SOLUTION OF THREE－ORDER LINEAR DIFFERENCE EQUATION WITH VARIABLE COEFFICIENTS

ＭＩＫＵＳＩＮＳＫＩ＇ＳＯＰＥＲＡＴＯＲＳＳＯＬＵＴＩＯＮＯＦＴＨＲＥＥ－ＯＲＤＥＲＬＩＮＥＡＲＤＩＦＦＥＲＥＮＣＥＥＱＵＡＴＩＯＮＷＩＴＨＶＡＲＩＡＢＬＥＣＯＥＦＦＩＣＩＥＮＴＳＺｈｏｕＺｈｉ－ｈｕ（周之虎）（ＡｎｈｕｉＡｒｃｈｉｔｅｃｔｕｒａｌＩ...     

whittaker’s Reduction Method for Poincare’s Dynamical Equations

ＷＨＩＴＴＡＫＥＲ＇ＳＲＥＤＵＣＴＩＯＮＭＥＴＨＯＤＦＯＲＰＯＩＮＣＡＲＥ＇ｓＤＹＮＡＭＩＣＡＬＥＱＵＡＴＩＯＮＳＱ．Ｋ．Ｇｈｏｒｉ（ＤｅｐａｒｔｍｅｎｔｏｆＭａｔｈｅｍａｔｉｃａｌＳｃｉｅｎｃｅｓ，ＫｉｎｇＦａｈｄＵｎｉｖｅｒｓｉｔｙｏｆＰｅｔｒｏ...     

A uniformly valid asymptotic solution of the Navier-Stokes equations

ＡＵＮＩＦＯＲＭＬＹＶＡＬＩＤＡＳＹＭＰＴＯＴＩＣＳＯＬＵＴＩＯＮＯＦＴＨＥＮＡＶＩＥＲ－ＳＴＯＫＥＳＥＱＵＡＴＩＯＮＳＱｉｎＳｈｅｎｇ－ｌｉ（秦圣立）ＺｈａｎｇＡｉ－ｓｈｕ（张爱淑）（Ｄｅｐｔ．ＯｆＰｈｙｓｉｃｓ,ＱｕｆｕＴｅａｃｈｅｒｓＵｎｉｖｅ...     

SPECTRAL METHOD IN TIME FOR KdV EQUATIONS

ＳＰＥＣＴＲＡＬＭＥＴＨＯＤＩＮＴＩＭＥＦＯＲＫｄＶＥＱＵＡＴＩＯＮＳＳＰＥＣＴＲＡＬＭＥＴＨＯＤＩＮＴＩＭＥＦＯＲＫｄＶＥＱＵＡＴＩＯＮＳ￥ＷｕＳｈｅｎｇｃｈａｎｇ（吴声昌）；ＬｉｕＸｉａｏｑｉｎｇ（刘小清）（ＲｅｃｅｉｖｅｄＦｅｂ．２２，１９９５...     

GEOMETRIC FRAMEWORK AND MINIMAL REALIZATIONS OF NONLINEAR SYSTEMS ON FIBRE BUNDLE

ＧＥＯＭＥＴＲＩＣＦＲＡＭＥＷＯＲＫＡＮＤＭＩＮＩＭＡＬＲＥＡＬＩＺＡＴＩＯＮＳＯＦＮＯＮＬＩＮＥＡＲＳＹＳＴＥＭＳＯＮＦＩＢＲＥＢＵＮＤＬＥＭｕＸｉａｏｗｕ（慕小武）（ＤｅｐａｒｔｍｅｎｔｏｆＭａｔｈｅｍａｔｉｃｓ，ＺｈｅｎｇｚｈｏｕＵｎｉｖｅｒｓ...     

EXISTENCE THEOREMS OF EXTREMAL SOLUTIONS FOR A CLASS OF NONLINEAR INTEGRO－DIFFERENTIAL EQUATIONS

ＥＸＩＳＴＥＮＣＥＴＨＥＯＲＥＭＳＯＦＥＸＴＲＥＭＡＬＳＯＬＵＴＩＯＮＳＦＯＲＡＣＬＡＳＳＯＦＮＯＮＬＩＮＥＡＲＩＮＴＥＧＲＯ－ＤＩＦＦＥＲＥＮＴＩＡＬＥＱＵＡＴＩＯＮＳＳｏｎｇＧｕａｎｇｘｉｎｇ（宋光兴）（ＲｅｃｅｉｖｅｄＭａｙ４，１９９５；Ｃｏｍ...     

CONSTRUCTION OF MODIFIED TAYLOR－GALERKIN FINITE ELEMENTS AND ITS APPLICATION IN COMPRESSIBLE FLOW COMPUTATION

ＣＯＮＳＴＲＵＣＴＩＯＮＯＦＭＯＤＩＦＩＥＤＴＡＹＬＯＲ－ＧＡＬＥＲＫＩＮＦＩＮＩＴＥＥＬＥＭＥＮＴＳＡＮＤＩＴＳＡＰＰＬＩＣＡＴＩＯＮＩＮＣＯＭＰＲＥＳＳＩＢＬＥＦＬＯＷＣＯＭＰＵＴＡＴＩＯＮＣＯＮＳＴＲＵＣＴＩＯＮＯＦＭＯＤＩＦＩＥＤＴＡＹＬＯＲ...     

CODIMENSION TWO BIFURCATIONS AND HOPF BIFURCATIONS OF AN IMPACTING VIBRATING

ＸｉｅＪｉａｎｈｕａ（谢建华）（ＲｅｃｅｉｖｅｄＯｃｔ．５，１９９４；ＣｏｍｍｕｎｉｃａｔｅｄｂｙＬｉＬｉ）ＣＯＤＩＭＥＮＳＩＯＮＴＷＯＢＩＦＵＲＣＡＴＩＯＮＳＡＮＤＨＯＰＦＢＩＦＵＲＣＡＴＩＯＮＳＯＦＡＮＩＭＰＡＣＴＩＮＧＶＩＢＲＡＴＩＮＧＳＹＳＴ...     

Similarity solutions of the super KdV equation

ＳＩＭＩＬＡＲＩＴＹＳＯＬＵＴＩＯＮＳＯＦＴＨＥＳＵＰＥＲＫｄＶＥＱＵＡＴＩＯＮＹｕＨｕｉｄａｎ（俞慧丹）Ｚｈａｎｇ．Ｊｉｅｆａｎｇ（张解放）（ｐｈｙｓｉｃｓＤｅｐａｒｔｍｅｎｔ,ＺｈｅｊｉａｎｇＮｏｒｍａｌＵｎｉｖｅｒｓｉｔｙ,Ｊｉｎｈｕａ,２３１...     

The similar solutions of nonlinear heat conduction equation

ＴＨＥＳＩＭＩＬＡＲＳＯＬＵＴＩＯＮＳＯＦＮＯＮＬＩＮＥＡＲＨＥＡＴＣＯＮＤＵＣＴＩＯＮＥＱＵＡＴＩＯＮＹｕａｎＹｉｗｕ（袁镒吾）（ＣｅｎｔｒａｌＳｏｕｔｈＵｎｉｖｅｒｓｉｔｙｏｆＴｅｃｈｎｏｌｏｇｙ,Ｃｈａｎｇｓｈａ４１００１２．Ｐ．Ｒ．Ｃｈｉｎａ...     

Testing the accuracy of the overlap criterion

Here we investigate the accuracy of the overlap criterion when applied to a simple near-integrable model in both its 2D and 3D versions. To this end, we consider, respectively, two and three quartic oscillators as the unperturbed system, and couple the degrees of freedom by a cubic, non-integrable perturbation. For both systems we compute the unperturbed resonances up to order O(ε²), and model each resonance by means of the pendulum approximation in order to estimate the theoretical critical value of the perturbation parameter for a global transition to chaos. We perform several surface of sections for the bi-dimensional case to derive an empirical value to be compared to our theoretical estimation. Although both values are of the same order of magnitude, there is a significant difference between them. For the 3D case a numerical estimate is attained that we observe matches quite well the critical value resulting from theoretical means. This confirms once again that calculating resonances up to O(ε²) suffices in order the overlap criterion to work out.     

Analytical procedure for determining the linear and nonlinear effective properties of the elastic composite cylinder

In this work we consider a cylindrical structure composed of a nonlinear core (inhomogeneity) surrounded by a different nonlinear shell (matrix). We elaborate a technique for determining its linear elastic moduli (second order elastic constants) and the nonlinear elastic moduli, which are called Landau coefficients (third order elastic constants). Firstly, we develop a nonlinear perturbation method which is able to turn the initial nonlinear elastic problem into a couple of linear problems. Then, we prove that only the solution of the first linear problem is necessary to calculate the linear and nonlinear effective properties of the heterogeneous structure. The following step consists in the exact solution of such a linear problem by means of the complex elastic potentials. As result we obtain the exact closed forms for the linear and nonlinear effective elastic moduli, which are valid for any volume fraction of the core embedded in the external shell.     

When and how do cracks propagate?

Crack propagation in an isotropic 2d brittle material is widely viewed as the interplay between two separate criteria. Griffith's cap on the energy release rate along the crack path decides when the crack propagates, while the Principle of Local Symmetry PLS decides how, that is, in which direction, that crack propagates. The PLS, which essentially predicts mode I propagation, cannot possibly hold in an anisotropic setting. Further it disagrees with its competitor, the principle of maximal energy release, according to which the direction of propagation should coincide with that of maximal energy release. Also, continuity of the time propagation is always implicitly assumed.In the spirit of the rapidly growing variational theory of fracture, we revisit crack path in the light of an often used tool in physics, i.e. energetic meta-stability of the current state among suitable competing crack states. In so doing, we do not need to appeal to either isotropy, or continuity in time. Here, we illustrate the impact of meta-stability in a 2d setting. In a 2d isotropic setting, it recovers the PLS for smooth crack paths. In the anisotropic case, it gives rise to a new criterion. But, of more immediate concern to the community, it also demonstrates that 2d crack kinking in an isotropic setting is incompatible with continuity in time of the propagation. Consequently, if viewing time continuity as non-negotiable, our work implies that the classical view of crack kinking along a single crack branch is not correct and that a change in crack direction necessarily involves more subtle geometries or evolutions.     

The issues of the uniqueness and the stability of the homogeneous response in uniaxial tests with gradient damage models

We consider a wide class of gradient damage models which are characterized by two constitutive functions after a normalization of the scalar damage parameter. The evolution problem is formulated following a variational approach based on the principles of irreversibility, stability and energy balance. Applied to a monotonically increasing traction test of a one-dimensional bar, we consider the homogeneous response where both the strain and the damage fields are uniform in space. In the case of a softening behavior, we show that the homogeneous state of the bar at a given time is stable provided that the length of the bar is less than a state dependent critical value and unstable otherwise. However, we also show that bifurcations can appear even if the homogeneous state is stable. All these results are obtained in a closed form. Finally, we propose a practical method to identify the two constitutive functions. This method is based on the measure of the homogeneous response in a situation where this response is stable without possibility of bifurcation, and on a procedure which gives the opportunity to detect its loss of stability. All the theoretical analyses are illustrated by examples.     

PRICE: primitive centred schemes for hyperbolic systems

We present first‐ and higher‐order non‐oscillatory primitive (PRI) centred (CE) numerical schemes for solving systems of hyperbolic partial differential equations written in primitive (or non‐conservative) form. Non‐conservative systems arise in a variety of fields of application and they are adopted in that form for numerical convenience, or more importantly, because they do not posses a known conservative form; in the latter case there is no option but to apply non‐conservative methods. In addition we have chosen a centred, as distinct from upwind, philosophy. This is because the systems we are ultimately interested in (e.g. mud flows, multiphase flows) are exceedingly complicated and the eigenstructure is difficult, or very costly or simply impossible to obtain. We derive six new basic schemes and then we study two ways of extending the most successful of these to produce second‐order non‐oscillatory methods. We have used the MUSCL‐Hancock and the ADER approaches. In the ADER approach we have used two ways of dealing with linear reconstructions so as to avoid spurious oscillations: the ADER TVD scheme and ADER with ENO reconstruction. Extensive numerical experiments suggest that all the schemes are very satisfactory, with the ADER/ENO scheme being perhaps the most promising, first for dealing with source terms and secondly, because higher‐order extensions (greater than two) are possible. Work currently in progress includes the application of some of these ideas to solve the mud flow equations. The schemes presented are generic and can be applied to any hyperbolic system in non‐conservative form and for which solutions include smooth parts, contact discontinuities and weak shocks. The advantage of the schemes presented over upwind‐based methods is simplicity and efficiency, and will be fully realized for hyperbolic systems in which the provision of upwind information is very costly or is not available. Copyright © 2003 John Wiley & Sons, Ltd.     

Computational and numerical analysis of a nonlinear mechanical system with bounded delay

Modern structures are increasingly resistant and complex. In many cases, such systems are modeled by numerical approximations methods, due to its complexities. In the study of vibration levels in the response of a system is important to consider issues like reliability and efficient design, since that such vibrations are undesirable phenomena that may cause damage, failure, and sometimes destruction of machines and structures. In this paper we investigated a modeling strategy of nonlinear system with damping, subject the time delayed. From models widely used in literature and with the help of numerical simulations a nonlinear damped system with two degree-of-freedom is analyzed. The system is constituted of a primary mass attached to the ground by a spring and damping with linear or nonlinear characteristics (primary system), and the secondary mass attached to the primary system by a spring and damping with linear or nonlinear characteristics (secondary system). It is well known that time delayed systems, due to its own nature, has singular behavior in its dynamics and that such singularities propagate over the time. Based on this, the main concerns of the present paper is to analyze the stability of a delayed system with two degree of freedom by means of the techniques development in [1] (Hu andWang, 2002). We also obtain the solution using the integration of equations of motions performing a Fourth Order Runge-Kutta Method. The behavior of a nonlinear main system with nonlinear secondary system will be investigated to many cases of resonances. In this case, various time delayed values are used to confirm its influence on the attenuation of vibrations, but, unfortunately, also the increase of nonlinearity (instable responses) of the system in question is observed.     

A non‐diffusive,divergence‐free,finite volume‐based double projection method on non‐staggered grids

Second‐order accurate projection methods for simulating time‐dependent incompressible flows on cell‐centred grids substantially belong to the class either of exact or approximate projections. In the exact method, the continuity constraint can be satisfied to machine‐accuracy but the divergence and Laplacian operators show a four‐dimension nullspace therefore spurious oscillating solutions can be introduced. In the approximate method, the continuity constraint is relaxed, the continuity equation being satisfied up to the magnitude of the local truncation error, but the compact Laplacian operator has only the constant mode. An original formulation for allowing the discrete continuity equation to be satisfied to machine‐accuracy, while using a finite volume based projection method, is illustrated. The procedure exploits the Helmholtz–Hodge decomposition theorem for deriving an additional velocity field that enforces the discrete continuity without altering the vorticity field. This is accomplished by solving a second elliptic field for a scalar field obtained by prescribing that its additional discrete gradients ensure discrete continuity based on the previously adopted linear interpolation of the velocity. The resulting numerical scheme is applied to several flow problems and is proved to be accurate, stable and efficient. This paper has to be considered as the companion of: 'F. M. Denaro, A 3D second‐order accurate projection‐based finite volume code on non‐staggered, non‐uniform structured grids with continuity preserving properties: application to buoyancy‐driven flows. IJNMF 2006; 52 (4):393–432. Now, we illustrate the details and the rigorous theoretical framework. Copyright © 2006 John Wiley & Sons, Ltd.     

Numerical detection and generation of solitary waves for a nonlinear wave equation

This paper presents an automatic algorithm for detecting and generating solitary waves of nonlinear wave equations. With this purpose, dynamic simulations are carried out, the solution of which evolves into a main pulse along with smaller dispersive tails. The solitary waves are detected automatically by the algorithm by checking that they have constant amplitude and are symmetric respect to its maximum value. Once the main wave has been detected, the algorithm cleans the dispersive tails for time enough so that the solitary wave is obtained with the required precision.In order to use our algorithm, we need a spatial discretization with local basis. The numerical experiments are carried out for the BBM equation discretized in space with cubic finite elements along with periodic boundary conditions. Moreover, a geometric integrator in time is used in order to obtain good approximations of the solitary waves.     

Numerical approximation of optimal control of unsteady flows using SQP and time decomposition

In this paper, we present numerical approximations of optimal control of unsteady flow problems using sequential quadratic programming method (SQP) and time domain decomposition. The SQP method is considered superior due to its fast convergence and its ability to take advantage of existing numerical techniques for fluid flow problems. It iteratively solves a sequence of linear quadratic optimal control problems converging to the solution of the non‐linear optimal control problem. The solution to the linear quadratic problem is characterized by the Karush–Kuhn–Tucker (KKT) optimality system which in the present context is a formidable system to solve. As a remedy various time domain decompositions, inexact SQP implementations and block iterative methods to solve the KKT systems are examined. Numerical results are presented showing the efficiency and feasibility of the algorithms. Copyright © 2004 John Wiley & Sons, Ltd.     

On the energy conservation during the active deformation in molecular dynamics simulations

In this paper, we examined the energy conservation for the current schemes of applying active deformation in molecular dynamics (MD) simulations. Specifically, two methods are examined. One is scaling the dimension of the simulation box and the atom positions via an affine transformation, suitable for the periodic system. The other is moving the rigid walls that interact with the atoms in the system, suitable for the non-periodic system. Based on the calculation of the external work and the internal energy change, we present that the atom velocities also need to be updated in the first deformation method; otherwise the energy conservation cannot be satisfied. The classic updating scheme is examined, in which any atom crossing the periodic boundary experiences a velocity delta that is equal to the velocity difference between the opposite boundaries. In addition, a new scheme which scales the velocities of all the atoms according to the strain increment is proposed, which is more efficient and realistic than the classic scheme. It is also demonstrated that the Virial stress instead of its interaction part is the correct stress definition that corresponds to Cauchy stress in the continuum mechanics.