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.

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.

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.

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.

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.

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.

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.

A Boundary Meshless Method for Solving Heat Transfer Problems Using the Fourier Transform

Fourier transform is applied to remove the time-dependent variable in the diffusion equation.Under non-harmonic initial conditions this gives rise to a non-homogeneous Helmholtz equation,which is solved by the method of fundamental solutions and the method of particular solutions.The particular solution of Helmholtz equation is available as shown in[4,15].The approximate solution in frequency domain is then inverted numerically using the inverse Fourier transform algorithm.Complex frequencies are used in order to avoid aliasing phenomena and to allow the computation of the static response.Two numerical examples are given to illustrate the effectiveness of the proposed approach for solving 2-D diffusion equations.

Using Gaussian Eigenfunctions to Solve Boundary Value Problems

Kernel-basedmethods are popular in computer graphics,machine learning,and statistics,among other fields;because they do not require meshing of the domain under consideration,higher dimensions and complicated domains can be managed with reasonable effort.Traditionally,the high order of accuracy associated with these methods has been tempered by ill-conditioning,which ariseswhen highly smooth kernels are used to conduct the approximation.Recent advances in representing Gaussians using eigenfunctions have proven successful at avoiding this destabilization in scattered data approximation problems.This paper will extend these techniques to the solution of boundary value problems using collocation.The method of particular solutions will also be considered for elliptic problems,using Gaussian eigenfunctions to stably produce an approximate particular solution.