A Meshless Method For Solving The Bending Problem Of Heterogeneous Thin Plates

In this paper,a meshless method for the bending problem of elastic thin plates is presented,which is suitable for solving the bending problem of thin plates under arbitrary load forms and different boundary conditions.At present,only a few kinds of elastic thin plate deflection can be obtained in simple form,and all of them focus on the solution of axisymmetric and relatively simple cases.On the contrary,for the case of complex and asymmetric loads,the approximate solution or numerical simulation method is mainly adopted,and the form of series is often used.As a widely used numerical method,finite element method(FEM)requires global meshing and has relatively low computational efficiency.On this basis,a meshless method is proposed,in which the unit concentrated force acting vertically on the plate plane is regarded as the point source to obtain the special solution of the fourth order partial differential equation of the bending of the corresponding thin plate.Combined with the general solution in the form of Fourier series,a meshless method for the bending of the elastic thin plate is obtained.In this paper,a meshless method,which satisfies the governing differential equations of plates,can be used to solve the undetermined coefficients by using the boundary conditions,the least square method and the collocation method.Considering the common boundary conditions,the two-dimensional Dirac function is first introduced to transform the fourth order partial differential equation of thin plate bending,and the basic solution of the two-dimensional Laplace equation is obtained.By contrast,the particular solution of the partial differential equation can be obtained.Secondly,the solution of the homogeneous equation is expanded by Fourier series into the homogeneous equation,and the form is transformed into the form of Euler equation.After variable substitution,the general solution form can be expressed,and the general solution of the problem can be synthesized with the special solution group.For example,under the action of the concentrated force,the boundary condition of the fixed edge of the homogeneous circular plate is that the deflection is zero and the rotation Angle is zero.When the general solution of deflection is substituted into the boundary value conditions and the linear equations are formed,it is easy to obtain the constant coefficient term,thus obtaining the general solution of deflection,and the internal force variation and deflection variation characteristics in the plate can be obtained by taking points in the plate.This meshless method is used to calculate the bending problems of several kinds of plates,including single homogeneous thin plate,heterogeneous thin plate and continuous multi-span thin plate.In the solution process,there is no need for meshing,and the series converges quickly,and the calculation is small,but the calculation results can meet the requirements of any precision.