Recursion-transform method and potential formulae of the m×n cobweb and fan networks

In this paper, we made a new breakthrough, which proposes a new recursion–transform（RT） method with potential parameters to evaluate the nodal potential in arbitrary resistor networks. For the first time, we found the exact potential formulae of arbitrary m × n cobweb and fan networks by the RT method, and the potential formulae of infinite and semi-infinite networks are derived. As applications, a series of interesting corollaries of potential formulae are given by using the general formula, the equivalent resistance formula is deduced by using the potential formula, and we find a new trigonometric identity by comparing two equivalence results with different forms.

Superposition formulas of multi-solution to a reduced(3+1)-dimensional nonlinear evolution equation

We gave the localized solutions,the interaction solutions and the mixed solutions to a reduced(3+1)-dimensional nonlinear evolution equation.These solutions were characterized by superposition formulas of positive quadratic functions,the exponential and hyperbolic functions.According to the known lump solution in the outset,we obtained the superposition formulas of positive quadratic functions by plausible reasoning.Next,we constructed the interaction solutions between the localized solutions and the exponential function solutions with the similar theory.These two kinds of solutions contained superposition formulas of positive quadratic functions,which were turned into general ternary quadratic functions,the coefficients of which were all rational operation of vector inner product.Then we obtained linear superposition formulas of exponential and hyperbolic function solutions.Finally,for aforementioned various solutions,their dynamic properties were showed by choosing specific values for parameters.From concrete plots,we observed wave characteristics of three kinds of solutions.Especially,we could observe distinct generation and separation situations when the localized wave and the stripe wave interacted at different time points.

Bending analysis of magnetoelectroelastic nanoplates resting on Pasternak elastic foundation based on nonlocal theory

Based on the nonlocal theory and Mindlin plate theory,the governing equations(i.e.,a system of partial differential equations(PDEs)for bending problem)of magnetoelectroelastic(MEE)nanoplates resting on the Pasternak elastic foundation are first derived by the variational principle.The polynomial particular solutions corresponding to the established model are then obtained and further employed as basis functions with the method of particular solutions(MPS)to solve the governing equations numerically.It is confirmed that for the present bending model,the new solution strategy possesses more general applicability and superior flexibility in the selection of collocation points.The effects of different boundary conditions,applied loads,and geometrical shapes on the bending properties of MEE nanoplates are evaluated by using the developed method.Some important conclusions are drawn,which should be helpful for the design and applications of electromagnetic nanoplate structures.

BOUNDARY ELEMENT METHOD FOR MODEL ANALYSIS OF 2-D COMPOSITE STRUCTURE

The boundary element method is used for he modal analysis of freevibration of 2-D composite structures in this paper. Since theparticular solution method is used to treat the terms of body forces(inertial forces) in the equation of motion, only static fundamentalsolutions are needed in solving the problem. For an isotropiccantilever beam, the numerical results obtained by using the BEMpresented in this paper are in good agreement, with those of usingFEM or other BEM, but this BEM can also be used to analyze problemsfor anisotropic materials.

Numerical Simulation of Non-Linear Schrodinger Equations in Arbitrary Domain by the Localized Method of Approximate Particular Solution

The aim of this paper is to propose a fast meshless numerical scheme for the simulation of non-linear Schrodinger equations.In the proposed scheme,the implicit-Euler scheme is used for the temporal discretization and the localized method of approximate particular solution(LMAPS)is utilized for the spatial discretization.The multiple-scale technique is introduced to obtain the shape parameters of the multiquadric radial basis function for 2D problems and the Gaussian radial basis function for 3D problems.Six numerical examples are carried out to verify the accuracy and efficiency of the proposed scheme.Compared with well-known techniques,numerical results illustrate that the proposed scheme is of merits being easy-to-program,high accuracy,and rapid convergence even for long-term problems.These results also indicate that the proposed scheme has great potential in large scale problems and real-world applications.

The Recursive Formulation of Particular Solutions for Some Elliptic PDEs with Polynomial Source Functions

In this paper we develop an efficient meshless method for solving inhomogeneous elliptic partial differential equations.We first approximate the source function by Chebyshev polynomials.We then focus on how to find a polynomial particular solution when the source function is a polynomial.Through the principle of the method of undetermined coefficients and a proper arrangement of the terms for the polynomial particular solution to be determined,the coefficients of the particular solution satisfy a triangular system of linear algebraic equations.Explicit recursive formulas for the coefficients of the particular solutions are derived for different types of elliptic PDEs.The method is further incorporated into the method of fundamental solutions for solving inhomogeneous elliptic PDEs.Numerical results show that our approach is efficient and accurate.

BEM FOR MODAL ANALYSIS OF 3-D ANISOTROPIC STRUCTURES

The boundary element method for the modal analysis of freevibration for 3-D anisotropic structures using particular solutionshas been developed. The complete polynomials of order two are used toconstruct the particular solutions for general anisotropic materials.The numerical results for 3-D free vibra- tion analysis of anisotropic cantilever beam by the method presented is in goodagreement with the results us- ing the Ritz technique. Foranisotropic materials, the numerical results calculated form theproposed method are in good agreement with the results from MSC.NASTRAN.

BUCKLING AND POST-BUCKLING OF ANNULAR PLATES ON AN ELASTIC FOUNDATION

On the basis of von Karnmnequations, the axisymmetric buckling and post-buckling of annular plates on anelastic foundation is so wematically discussed byusing shooting

STRESS RATE INTEGRAL EQUATIONS OF ELASTOPLASTICITY

The stress rate integral equations of elastoplasticity are deduced based on Ref. [1] by consistent methods. The point at which the stresses and/or displacements are calculated can be in the body or on the boundary, and in the plastic region or elastic one. The existence of the principal value integral in the plastic region is demonstrated strictly, and the theoretical basis is presented for the paticular solution method by unit initial stress fields. In the present method, programming is easy and general, and the numerical results are excellent.

A Comparative Study on Polynomial Expansion Method and Polynomial Method of Particular Solutions

In this study,the polynomial expansion method(PEM)and the polynomial method of particular solutions(PMPS)are applied to solve a class of linear elliptic partial differential equations(PDEs)in two dimensions with constant coefficients.In the solution procedure,the sought solution is approximated by the Pascal polynomials and their particular solutions for the PEM and PMPS,respectively.The multiple-scale technique is applied to improve the conditioning of the resulted linear equations and the accuracy of numerical results for both of the PEM and PMPS.Some mathematical statements are provided to demonstrate the equivalence of the PEM and PMPS bases as they are both bases of a certain polynomial vector space.Then,some numerical experiments were conducted to validate the implementation of the PEM and PMPS.Numerical results demonstrated that the PEM is more accurate and well-conditioned than the PMPS and the multiple-scale technique is essential in these polynomial methods.

The Method of Fundamental Solutions for Solving Convection-Diffusion Equations with Variable Coefficients

A meshless method based on the method of fundamental solutions(MFS)is proposed to solve the time-dependent partial differential equations with variable coefficients.The proposed method combines the time discretization and the onestage MFS for spatial discretization.In contrast to the traditional two-stage process,the one-stage MFS approach is capable of solving a broad spectrum of partial differential equations.The numerical implementation is simple since both closed-form approximate particular solution and fundamental solution are easy to find than the traditional approach.The numerical results show that the one-stage approach is robust and stable.

A Revisit on the Derivation of the Particular Solution for the Differential Operator ∆^(2) ± λ^(2)

In this paper,we applied the polyharmonic splines as the basis functions to derive particular solutions for the differential operator ∆^(2) ± λ^(2).Similar to the derivation of fundamental solutions,it is non-trivial to derive particular solutions for higher order differential operators.In this paper,we provide a simple algebraic factorization approach to derive particular solutions for these types of differential operators in 2D and 3D.The main focus of this paper is its simplicity in the sense that minimal mathematical background is required for numerically solving higher order partial differential equations such as thin plate vibration.Three numerical examples in both 2D and 3D are given to validate particular solutions we derived.

Position optimization of particular solution sources for distributed source boundary point method by volume velocity-matching

Choosing particular solution source and its position have great influence on accu- racy of sound field prediction in distributed source boundary point method. An optimization method for determining the position of particular solution sources is proposed to get high accu- racy prediction result. In this method, tripole is chosen as the particular solution. The upper limit frequency of calculation is predicted by setting 1% volume velocity relative error limit using vibration velocity of structure surface. Then, the optimal position of particular solution sources, in which the relative error of volume velocity is minimum, is determined within the range of upper limit frequency by searching algorithm using volume velocity matching. The transfer matrix between pressure and surface volume velocity is constructed in the optimal position. After that, the sound radiation of structure is calculated by the matrix. The results of numerical simulation show that the calculation error is significantly reduced by the proposed method. When there are vibration velocity measurement errors, the calculation errors can be controlled within 5% by the method.

基于基准关联度与信息熵权法优选桂枝配方颗粒中挥发性成分的β-环糊精包合工艺

目的结合基准关联度与信息熵权法,采用正交试验对β-环糊精(β-Cyclodextrin,β-CD)与桂枝配方颗粒中挥发性成分的包合工艺进行优化。方法在单因素考察的基础上,选择β-CD与桂枝芳香水的投料比、包合温度、包合时间为主要影响因素,以包合物中桂皮醛的包合率、载药量及基准关联度为评价指标,采用信息熵权法计算各评价指标的权重系数,进行综合评价和优化正交试验包合工艺。通过薄层色谱法、紫外可见吸收光谱、傅里叶红外光谱、X射线衍射等方法对包合物进行表征。结果优选的最佳包合工艺为β-CD与芳香水的投料比3∶100(g·mL^(-1)),包合温度50℃,包合时间1 h。经验证,桂皮醛包合率为80.48%,载药量为8.63%,基准关联度为0.91。薄层、紫外、红外等表征实验表明芳香水中挥发性成分成功进入β-CD空腔,包合物制备成功。结论优选的包合工艺稳定可行,可为桂枝配方颗粒的制剂工艺研究提供参考。

Explicit Peck formula applied to ground displacement based on an elastic analytical solution for a shallow tunnel

Using the complex variable method,an elastic analytical solution of the ground displacement caused by a shallow circular tunneling is derived.Non-symmetric deformation relative to the horizontal center line of the tunnel cross-section is used as a boundary condition.A comparison between the proposed analytical method and the Finite Element Method is carried out to validate the rationality of the obtained analytical solution.Two parameters in the Peck formula,namely the maximum settlement of the ground surface center and the width coefficient of settlement curve,are fitted and determined.We propose a modified Peck formula by considering three input parameters,namely the tunnel depth,tunnel radius,and the tunnel gap.The influence of these three parameters on the modified Peck formula is analyzed.The applicability of the modified Peck formula is further investigated by reference to six engineering projects.The ground surface displacement obtained by the explicit Peck formula is in good agreement with the field data,and the maximum error is only 1.3 cm.The proposed formula can quickly and reasonably predict the ground surface settlement caused by tunnelling.

Efficient Algorithms for Approximating Particular Solutions of Elliptic Equations Using Chebyshev Polynomials

In this paper,we propose efficient algorithms for approximating particular solutions of second and fourth order elliptic equations.The approximation of the particular solution by a truncated series of Chebyshev polynomials and the satisfaction of the differential equation lead to upper triangular block systems,each block being an upper triangular system.These systems can be solved efficiently by standard techniques.Several numerical examples are presented for each case.

一类二阶常微分方程组的特解公式

采用待定系数法,给出了非齐次项为三角函数与指数函数乘积的三维二阶常系数线性微分方程组的特解公式,并通过算例验证了微分方程组的特解公式的正确性。

结构声辐射分析的全特解场边界元方法

提出一种新的高效、高精度的全特解场边界元方法。该方法通过一系列特解来建立边界积分方程，从而避免了其系数矩阵的直接计算。通过对算例及模拟齿轮箱辐射声场的边界元分析，证实了本文方法的有效性。

一类常系数微分方程组的通解

采用待定系数法,给出了一类非齐次项为二次多项式与指数函数之积的三维二阶常系数微分方程组的通解形式,并通过算例验证了特解公式的正确性。

一类二阶常微分方程组的通解

采用降阶和特征根(欧拉)方法,给出了一类三维二阶常系数微分方程组的通解公式,并通过算例与拉氏变换法进行了比较,说明了利用通解公式求解高阶微分方程组比采用其他方法求解更简捷。