Solving Large Deflection Bending Problem Of Rectangular Plate Based On Wavelet Integral Collocation Method

In the field of mechanics,nonlinear phenomena generally exist,which can be described as nonlinear initial-boundary value problems in mathematical form.However,due to the complexity of nonlinear problems,it is difficult for us to find analytical solutions at present,so numerical techniques are usually needed to solve current engineering problems.Although the existing numerical algorithms have achieved great success in this aspect,the existing research has not solved the nonlinear problem well.For example,when there are multiple spatial dimensions or high-order derivatives,they are generally not effectively solved.The existence of nonlinear problems makes the existing algorithms difficult to be effective,especially in the case of three couplings.Based on the current research status,this paper gives a high-precision solution method to the nonlinear problem of high-dimensional and high-order derivatives,and at the same time avoids the hourglass effect caused by finite element software simulation analysis when solving the bending problem of thin plate structure.Based on the one-dimensional wavelet method,this paper expands the multi-dimensional Coiflet wavelet integral approximation format,constructs a high-dimensional wavelet integral collocation method,and verifies the feasibility of the algorithm through numerical examples.The specific research content is divided into three parts and introduced as follows:(1)the compactly supported orthogonal Coiflet wavelet is introduced.Based on this,the multidimensional wavelet integral approximation format of the L~2 function on the bounded interval is obtained.The error accuracy of the approximation format is proved by Taylor polynomial interpolation.Afterwards,the jumping phenomenon existing at the boundary endpoints in the three-dimensional space was improved,and a more stable wavelet function integral form was obtained,and the numerical discrete format of the high-dimensional and high-order wavelet integral collocation method was given.(2)considering that the Poisson equation is often used to verify the pros and cons of a new algorithm,this paper uses extreme high-dimensional and high-order Poisson-like problems to verify the previously constructed the wavelet integral collocation method.We have analyzed the numerical accuracy of two-dimensional 4th to 8th order and three-dimensional 4th order Poisson-like equations respectively,and found that the solution accuracy of the method constructed in this paper does not depend on the spatial dimension and the highest derivative order.More importantly,it always maintains the same high accuracy as the direct approximation function.(3)for the large deflection bending problem of the rectangular thin plate in mechanic's structure analysis,the finite element algorithm will cause the hourglass effect because the order of the shape function is too low to describe the bending state.The wavelet method introduces high-order shape functions for interpolation,which can accurately express the bending state of the board,and the wavelet integral collocation method uses the idea of integration,which does not depend on the derivative and does not lose the accuracy of the solution.By loading the concentrated force in the center of the plate,we verify that the algorithm can completely avoid shear locking,and maintains the consistency with the theoretical analysis in terms of accuracy.