Novel implementation of homogenization method to predict effective properties of periodic materials

Representative volume element （RVE） method and asymptotic homogenization （AH） method are two widely used methods in predicting effective properties of pe- riodic materials. This paper develops a novel implementa- tion of the AH method, which has rigorous mathematical foundation of the AH method, and also simplicity as the RVE method. This implementation can be easily realized using commercial software as a black box, and can use all kinds of elements available in commercial software to model unit cells with rather complicated microstructures, so the model may remain a fairly small scale. Several examples were car- fled out to demonstrate the simplicity and effectiveness of the new implementation.

Two-scale finite element method for piezoelectric problem in periodic structure

The prediction of the mechanical and electric properties of piezoelectric fibre composites has become an active research area in recent years. By means of introducing a boundary layer problem, some new kinds of two-scale finite element methods for solutions to the electric potential and the displacement for composite material in periodic struc- ture under the coupled piezoelectricity are derived. The coupled two-scale relation of the electric potential and the displacement is set up, and some finite element approximate estimates and numerical examples which show the effectiveness of the method are presented.

Construction of doubly-periodic solutions to nonlinear partial differential equations using improved Jacobi elliptic function expansion method and symbolic computation

Some doubly-periodic solutions of the Zakharov-Kuznetsov equation are presented. Our approach is to introduce an auxiliary ordinary differential equation and use its Jacobi elliptic function solutions to construct doubly-periodic solutions of the Zakharov-Kuznetsov equation, which has been derived by Gottwald as a two-dimensional model for nonlinear Rossby waves. When the modulus k →1, these solutions reduce to the solitary wave solutions of the equation.

Extended F-Expansion Method and Periodic Wave Solutions for Klein-Gordon-SchrSdinger Equations

我们在场为在数学物理发现非线性的进化方程的周期的波浪答案的一个扩大 F 扩大方法。由使用扩大 F 扩大方法，各种各样的 Jacobi 椭圆形的功能为 theKlein-Gordon-Schroedinger 方程表示的许多周期的波浪答案被获得。在限制盒子中，方程的独居的波浪解决方案和三角法的函数解决方案也被获得。

Re-study on Recurrence Period of Stokes Wave Train with High Order Spectral Method

Owing to the Benjamin-Feir instability, the Stokes wave train experiences a modulation-demodulation process, and presents a recurrence characteristics. Stiassnie and Shemer researched the unstable evolution process and provided a theoretical formulation for the recurrence period in 1985 on the basis of the nonlinear cubic Schrodinger equation （NLS）. However, NLS has limitations on the narrow band and the weak nonlinearity. The recurrence period is re-investigated in this paper by using a highly efficient High Order Spectral （HOS） method, which can be applied for the direct phase- resolved simulation of the nonlinear wave train evolution. It is found that the Stiassnie and Shemer＇s formula should be modified in the cases with most unstable initial conditions, which is important for such topics as the generation mechanisms of freak waves. A new recurrence period formula is presented and some new evolution characteristics of the Stokes wave train are also discussed in details.

Finding Periodic Solutions of Ordinary Differential Equations by the Homotopy Method

As an important aspect of applications, it is discussed how to find periodic solutions for ordinary differential equations. By using the homotopy method, a global method for finding those solutions is proposed.

Doubly Periodic Wave Solutions of Jaulent-Miodek Equations Using Variational Iteration Method Combined with Jacobian-function Method

变化重复方法的优点之一是起始的猜测的免费选择。在这篇论文，我们使用 Jacobian 功能方法的基本想法与一些未知参数构造概括试用功能。Jaulent Miodek 方程被用来说明这个方法的有效性和便利，一些新明确的准确旅行波浪答案被获得了，它包括钟类型 soliton 答案，纽结类型 soliton 答案，独居的波浪答案，和二倍地周期的波浪答案。

Synthesis and Optimization of Almost Periodic Antennas Using Floquet Modal Analysis and MoM-GEC Method

This paper presented a new Floquet analysis used to calculate the radiation for 1-D and 2-D coupled periodic antenna systems. In this way, an accurate evaluation of mutual coupling can be proven by using a new mutual interaction expression that was based on Fourier analysis. Then, this work indicated how Floquet analysis can be used to study a finite array with uniform amplitude and linear phase distribution in both x and y directions. To modelize the proposed structures, two formulations were given in a spectral and spatial domain, where the Moment (MoM) method combined with a generalized equivalent circuit (GEC) method was applied. Radiation pattern of coupled periodic antenna was shown by varying many parameters, such as frequencies, distance and Floquet states. The 3-D radiation beam of the coupled antenna array was analyzed and compared in several steering angles θs and coupling values dx. The simulation of this structure demonstrated that directivity decreased at higher coupling values. The secondary lobs in the antenna radiation pattern affected the main lobe gain by energy dispersal and considerable increasing of side lobe level (SLL) may be achieved. Therefore, the sweeping of the radiation beam in several steering directions affected the electromagnetic performance of the antenna system: the directivity at the steering angle θs = π⁄3 was more damaged and had 19.99 dB while the second at θs = 0 had about 35.11 dB. This parametric study of coupled structure used to concept smart periodic antenna with sweeping radiation beam.

Improving Wilson-θ and Newmark-β Methods for Quasi-Periodic Solutions of Nonlinear Dynamical Systems

Quasi-periodic responses can appear in a wide variety of nonlinear dynamical systems. To the best of our knowledge, it has been a tough job for years to solve quasi-periodic solutions, even by numerical algorithms. Here in this paper, we will present effective and accurate algorithms for quasi-periodic solutions by improving Wilson-θ and Newmark-β methods, respectively. In both the two methods, routinely, the considered equations are rearranged in the form of incremental equilibrium equations with the coefficient matrixes being updated in each time step. In this study, the two methods are improved via a predictor-corrector algorithm without updating the coefficient matrixes, in which the predicted solution at one time point can be corrected to the true one at the next. Numerical examples show that, both the improved Wilson-θ and Newmark-β methods can provide much more accurate quasi-periodic solutions with a smaller amount of computational resources. With a simple way to adjust the convergence of the iterations, the improved methods can even solve some quasi-periodic systems effectively, for which the original methods cease to be valid.

Finding Periodic Solutions of High Order Duffing Equations via Homotopy Method

This paper presents a detailed analysis of finding the periodic solutions for the high order Duffing equation x^（2n） ＋ g（x） = e（t） （n ≥ 1）. Firstly, we give a constructive proof for the existence of periodic solutions via the homotopy method. Then we establish an efficient and global convergence method to find periodic solutions numerically.

A two-scale method for identifying mechanical parameters of composite materials with periodic configuration

In this paper, a two-scale method （TSM） is presented for identifying the mechanics parameters such as stiffness and strength of composite materials with small periodic configuration. Firstly, a formulation is briefly given for two-scale analysis （TSA） of the composite materials. And then a two-scale computation formulation of strains and stresses is developed by displacement solution with orthotropic material coefficients for three kinds of such composites structures, i.e., the tension column with a square cross section, the bending cantilever with a rectangular cross section and the torsion column with a circle cross section. The strength formulas for the three kinds of structures are derived and the TSM procedure is discussed. Finally the numerical results of stiffness and strength are presented and compared with experimental data. It shows that the TSM method in this paper is feasible and valid for predicting both the stiffness and the strength of the composite materials with periodic configuration.

A New Kind of Iteration Method for Finding Approximate Periodic Solutions to Ordinary Diferential Equations

In this paper, a new kind of iteration technique for solving nonlinear ordinary differential equations is described and used to give approximate periodic solutions for some well-known nonlinear problems. The most interesting features of the proposed methods are its extreme simplicity and concise forms of iteration formula for a wide range of nonlinear problems.

Study of the emissivity of rough surfaces periodic using the method of coupled waves analysis (CWA) compared with method of geometrical optics approximation (GOA)

We present in this paper a numerical study of the validity limit of the geometrical optics approximation compared with a differential method which is established according to rigorous formalisms based on the electromagnetic theory. The precedent studies show that this method is adapted to the study of diffraction by periodic rough surfaces. We determine by two methods the emissivity of gold and tungsten for surfaces with a rectangular or sinusoidal profile, for a wavelength equal to 0.55 microns. The monochromatic directional emissivity of these surfaces clearly depends on the angle of incidence, the surface profile, height, period and the nature of the material. We perform our calculations by a method of coupled wave analysis (CWA) and a geometric optics method (GOA). The latter method is theoretically valid only when the dimensions of the cavities are very large compared to the wavelength, while the CWA is theoretically correct whatever these dimensions. The main purpose of this work is to investigate the validity limit of GOA compared with CWA. The obtained results for a fixed height of the grating, allowed us to delimit the validity domain of the optic geometrical approximation for the treated cases. Finally, the agreement between the emissivity calculated by the differential method and that given on the basis of the homogenization theory is satisfactory when the period is much smaller than the wavelength.

Exponentially-Fitted 2-Step Simpson’s Method for Oscillatory/Periodic Problems

Following a six-step flow chart, exponentially-fitted variant of the 2-step Simpson’s method suitable for solving ordinary differential equations with periodic/oscillatory behaviour is constructed. The qualitative properties of the constructed methods are also investigated. Numerical experiments on standard problems confirming the theoretical expectations regarding the constructed methods compared with other existing standard methods are also presented. Our results unify and improve the existing classical 2-step Simpson’s method.

Construction of solitonary and periodic solutions to some nonlinear equations using EXP-function method

This paper applies the EXP-function method to find exact solutions of various nonlinear equations. Tzitzeica- Dodd-Bullough (TDB) equation was selected to illustrate the effectiveness and convenience of the suggested method. More generalized solitonary solutions with free parameters were obtained by suitable choice of the free parameters, and also the obtained solitonary solutions can be converted into periodic solutions.

Existence of Periodic Solutions for Odd Order Ordinary Differential Equations via the Homotopy Method

This paper deals with the problems of finding periodic solutions for the third order ordinary differential equations of the form (1) where T is a fixed positive number and f satisfies some additional conditions which will be stated later.The periodicity problem has been one of main topics in the qualitative theory of ordinary

An asymptotic solving method for the periodic solution of a class of disturbed nonlinear evolution equation

A class of disturbed evolution equation is considered using a simple and valid technique. We first introduce the periodic traveling-wave solution of a corresponding typical evolution equation. Then the approximate solution for an original disturbed evolution equation is obtained using the asymptotic method. We point out that the series of approximate solution is convergent and the accuracy of the asymptotic solution is studied using the fixed point theorem for the functional analysis.

ANALYSIS OF OPEN MICROSTRIP STRUCTURES BY USING DIAKOPTIC METHOD OF LINES COMBINED WITH PERIODIC BOUNDARY CONDITIONS

This paper presents the analysis of open microstrip structures by using diakoptic method of lines (ML) combined with periodic boundary conditions (PBC). The parameters of microstrip patch are obtained from patch current excited by plane wave. Impedance matrix elements are computed by using fast Fourier transform(FFT), and reduced equation is solved by using diakoptic technique. Consequently, the computing time is reduced significantly. The convergence property of simulating open structure by using PBC and the comparison of the computer time between using PBC and usual absorbing boundary condition (ABC) show the validity of the method proposed in this paper. Finally, the resonant frequency of a microstrip patch is computed. The numerical results obtained are in good agreement with those published.

Hybrid PMM-MoM Method for the Analysis of Finite Periodic Structures

In this paper, a hybrid method (hybrid PMM-MoM method) is presented for the effective and accurate analysis of finite periodic structures. This method divides a finite periodic structure into two parts. The inner part of an approximate infinite periodic structure is analyzed by periodic method of moment (PMM);the outer part is then analyzed by method of moments (MoM). For the finite periodic structures, the accuracy of the new method is much better than that of the pure PMM, and is almost the same as that of pure MoM. Because pure PMM uses the periodic boundary conditions, it takes much less memory resources and computation time. For hybrid PMM-MoM method, because the inner part is calculated by PMM, the calculation work concentrates on the outer part. Consequently, compared with the exact MoM, the new method saves much more memory resources and computation time, which provides a drastic reduction of unknowns.

Existence and Uniqueness of 2π-Periodic Solution About Duffing Equation and Its Numerical Method

A new provement of the existence and uniqueness about periodic boundary value Duffing equation is established by using global inverse function theorem. An algorithm for solving differential equation that has a large convergence domain is given. Finally, a numerical example is given.