Ancient Chinese algorithm: The Ying Buzu Shu (method of surplus and deficiency) vs Newton iteration method

Air exploratory discussion of an ancient Chinese algorithm, the Ying Buzu Shu, in about 2nd century BC, known as the rule of double false position in the West is given. In addition to pointing out that the rule of double false position is actually a translation version of the ancient Chinese algorithm, a comparison with well-known Newton iteration method is also made. If derivative is introduced, the ancient Chinese algorithm reduces to the Newton method. A modification of the ancient Chinese algorithm is also proposed, and some of applications to nonlinear oscillators are illustrated.

Complex potentials and singular integral equation for curve crack problem in antiplane elasticity

Four types of the fundamental complex potential in antiplane elasticity are introduced: (a) a point dislocation, (b) a concentrated force, (c) a dislocation doublet and (d) a concentrated force doublet. It is proven that if the axis of the concentrated force doublet is perpendicular to the direction of the dislocation doublet, the relevant complex potentials are equivalent. Using the obtained complex potentials, a singular integral equation for the curve crack problem is introduced. Some particular features of the obtained singular integral equation are discussed, and numerical solutions and examples are given.

Direct numerical test of the B-G-K model equation by the DSMC method

In the present paper the rarefied gas how caused by the sudden change of the wall temperature and the Rayleigh problem are simulated by the DSMC method which has been validated by experiments both in global flour field and velocity distribution function level. The comparison of the simulated results with the accurate numerical solutions of the B-G-K model equation shows that near equilibrium the BG-K equation with corrected collision frequency can give accurate result but as farther away from equilibrium the B-G-K equation is not accurate. This is for the first time that the error caused by the B-G-K model equation has been revealed.

A Study of Composite Beam with Shape Memory Alloy Arbitrarily Embedded under Thermal and Mechanical Loadings

The constitutive relations and kinematic assumptions on the composite beam with shape memory alloy (SMA) arbitrarily embedded are discussed and the results related to the different kinematic assumptions are compared. As the approach of mechanics of materials is to study the composite beam with the SMA layer embedded, the kinematic assumption is vital. In this paper, we systematically study the kinematic assumptions influence on the composite beam deflection and vibration characteristics. Based on the different kinematic assumptions, the equations of equilibrium/motion are different. Here three widely used kinematic assumptions are presented and the equations of equilibrium/motion are derived accordingly. As the three kinematic assumptions change from the simple to the complex one, the governing equations evolve from the linear to the nonlinear ones. For the nonlinear equations of equilibrium, the numerical solution is obtained by using Galerkin discretization method and Newton-Rhapson iteration method. The analysis on the numerical difficulty of using Galerkin method on the post-buckling analysis is presented. For the post-buckling analysis, finite element method is applied to avoid the difficulty due to the singularity occurred in Galerkin method. The natural frequencies of the composite beam with the nonlinear governing equation, which are obtained by directly linearizing the equations and locally linearizing the equations around each equilibrium, are compared. The influences of the SMA layer thickness and the shift from neutral axis on the deflection, buckling and post-buckling are also investigated. This paper presents a very general way to treat thermo-mechanical properties of the composite beam with SMA arbitrarily embedded. The governing equations for each kinematic assumption consist of a third order and a fourth order differential equation with a total of seven boundary conditions. Some previous studies on the SMA layer either ignore the thermal constraint effect or implicitly assume that the SMA is symmetrically embedded. The composite beam with the SMA layer asymmetrically embedded is studied here, in which symmetric embedding is a special case. Based on the different kinematic assumptions, the results are different depending on the deflection magnitude because of the nonlinear hardening effect due to the (large) deflection. And this difference is systematically compared for both the deflection and the natural frequencies. For simple kinematic assumption, the governing equations are linear and analytical solution is available. But as the deflection increases to the large magnitude, the simple kinematic assumption does not really reflect the structural deflection and the complex one must be used. During the systematic comparison of computational results due to the different kinematic assumptions, the application range of the simple kinematic assumption is also evaluated. Besides the equilibrium study of the composite laminate with SMA embedded, the buckling, post-buckling, free and forced vibrations of the composite beam with the different configurations are also studied and compared.

A lattice Boltzmann method for KDV equation

We prepose a 5-bit lattice Boltzmann model for KdV equation. Using Chapman-Enskog expansion and multiscale technique, we obtained high order moments of equilibrium distribution function, and the 3rd dispersion coefficient and 4th order viscosity. The parameters of this scheme can be determined by analysing the energy dissipation.

Modified Simplified Rate Equation Model of Flowing Chemical Oxygen-Iodine Laser and its Application

A modified simplified rate equation (RE) model of flowing chemical oxygen-iodine laser (COIL), which is adapted to both the condition of homogeneous broadening and inhomogeneous broadening being of importance and the condition of inhomogeneous broadening being predominant, is presented for performance analyses of a COIL. By using the Voigt profile function and the gain-equal-loss approximation, a gain expression has been deduced from the rate equations of upper and lower level laser species. This gain expression is adapted to the conditions of very low gas pressure up to quite high pressure and can deal with the condition of lasing frequency being not equal to the central one of spectral profile. The expressions of output power and extraction efficiency in a flowing COIL can be obtained by solving the coupling equations of the deduced gain expression and the energy equation which expresses the complete transformation of the energy stored in singlet delta state oxygen into laser energy. By using these expressions, the RotoCOIL experiment is simulated, and obtained results agree well with experiment data. Effects of various adjustable parameters on the performances of COIL are also presented.

A Continuation Method of Parameter Inversion for Non-Equilibrium Convection-Dispersion Equation

Based on the homotopy mapping, a globally convergent method of parameter inversion for non-equilibrium convection-dispersion equations (CDEs) is developed. Moreover, in order to further improve the computational efficiency of the algorithm, a properly smooth function, which is derived from the sigmoid function, is employed to update the homotopy parameter during iteration. Numerical results show the feature of global convergence and high performance of this method. In addition, even the measurement quantities are heavily contaminated by noises, and a good solution can be found.

经典轨道的封闭性和径向Schrodinger方程的因式分解

研究表明，保证经典轨道具有封闭性的Bertrand定理可以进一步推广，在适当的角动量睛，仍存在着非椭圆的闭合轨道。对于屏蔽Coulomb场，可获得广义Runge-Lenz矢量。这种轨道封闭性与径向Schrodinger方程因式分解相对应。

A lattice Boltzmann equation for waves

We propose a lattice Boltzmann model for the wave equation. Using a lattice Boltzmann equation and the Chapman-Enskog expansion, we get 1D and 2D wave equations with truncation error of order two. The numerical tests show the method can be used to simulate the wave motions.

Variational Approach to the Lane-Emden Equation

By the semi-inverse method, a variational principle is obtained for the Lane-Emden equation, which gives much numerical convenience when applying finite element methods or Ritz method.

A Variational Approach to the Burridge-Knopoff Equation

A variational principle is obtained for the Burridge-Knopoff model for earthquake faults, and this paper considers an analytic approach that does not require linearization or perturbation.

Variational Approach to the Thomas-Fermi Equation

By the semi-inverse method, a variational principle is obtained for the Thomas-Fermi equation, then the Ritz method is applied to solve an analytical solution, which is a much simpler and more efficient method.

Static Study of Cantilever Beam stiction under Electrostatic Force Influence

The model and analysis of the cantilever beam adhesion problem under the action of electrostatic force are given. Owing to the nonlinearity of electrostatic force, the analytical solution for this kind of problem is not available. In this paper, a systematic method of generating polynomials which are the exact beamsolutions of the loads with different distributions is provided. The polynomials are used to approximate the beam displacement due to electrostatic force. The equilibrium equation offers an answer to how the beam deforms but no information about the unstuck length. The derivative of the functional with respect to the unstuck length offers such information. But to compute the functional it is necessary to know the beam deformation. So the problem is iteratively solved until the results are converged. Galerkin and Newton-Raphson methods are used to solve this nonlinear problem. The effects of dielectric layer thickness and electrostatic voltage on the cantilever beamstiction are studied.The method provided in this paper exhibits good convergence. For the adhesion problem of cantilever beam without electrostatic voltage, the analytical solution is available and is also exactly matched by the computational results given by the method presented in this paper.