An Adaptive Wavelet Method With Closed Property For Solving Strong Nonlinear Systems And Its Application To Mechanics

The high-precision quantitative solution technology of nonlinear systems,especially strong nonlinear systems,is a common scientific problem in nonlinear scientific research.And it is also one of the key problems to be solved urgently.Most of the existing numerical methods are difficult to establish a unified quantitative solution format for strong nonlinear problems because they do not have the closed property.As a result,according to specific problems,they need to be supplemented with special processing techniques.In response to these shortcomings,in recent years,our research group has proposed the closed wavelet method for nonlinear problems,which can solve the weak and strong nonlinear problems in a unified way.However,the existing wavelet solution can only be applied to the very regular problem domain,such as the one-dimensional domain,the two-dimensional rectangular domain and the three-dimensional cubic domain,which severely limits the applicable scope of the method.At the same time,this wavelet algorithm only allows the use of a globally uniform node arrangement,which is often unacceptable in solving problems with local large gradient.In fact,the efficient processing of complex regions and local large gradients is one of the hot and difficult issues in the field of computational mechanics.Typically,such as the finite element method,when the problematic study area is highly complex or the solution has a steep gradient locally,generating a set of available high quality meshes can be a very tedious and time consuming task,and can be often difficult to automate.Generation requires professional and technical personnel to intervene throughout the process.Although the meshless method can avoid the problem of mesh generation,it will bring new problems,such as low computational efficiency,which is using artificial parameters that have a significant impact on computational accuracy but lack quantitative determination theory.In order to address these above problems,this PhD thesis establishes an interpolation format that can approximate a continuous function defined on any two-dimensional regions by the second generation wavelet,and the wavelet interpolation format allows arbitrary local node encryption.On the basis of this,combined with the existing wavelet algorithm of nonlinear problem established by our research group,a wavelet multi-resolution with closed property method which can be applied to any problem domain and allows arbitrary refinement of local node distribution is proposed.Based on this,a corresponding adaptive format is further established.At the same time,the accuracy,computational efficiency,convergence and stability of this method are systematically studied by theoretical analysis and numerical experiments.Some typical mechanical and physical problems are quantitatively studied by using this adaptive wavelet algorithm.Firstly,based on the second generation wavelet,this paper presents an approximation format of square-integrable functions according to the multi-resolution analysis.The wavelet format not only has interpolation but also can be locally enhanced.By using the basic properties of the scale function,the precise solution process of the derivative,integral and joint coefficients of the shape function is introduced in detail,which provides the numerical basis for the function approximation and differential equations' wavelet solution.Then,we give a rigorous theoretical analysis on the errors of the function approximation,which are verified by some numerical examples.Secondly,combining the approximation format with the Galerkin method,based on the basic wavelet algorithms which have been developed by the present research group,a set of locally enhanced wavelet algorithm for strong nonlinear elliptic differential equations is proposed.We use this unified format to study the convection-diffusion equation,the Bratu equation and a singularity problem in L-region.When the y order shape function is used to solve the problem,if the solution of the problem is smooth enough,the current wavelet algorithm with closed property will have a y order convergence speed,whether it is a linear or nonlinear problem.When y is equal to 6,the numerical results will be significantly better than many existing methods.If the solution is singular,the convergence speed of the approximate solution will be limited by the smoothness of the solution.Subsequently,the large gradient region is locally encrypted and sprinkled.The numerical results show that local refinement can greatly reduce the number of sprinkling points under the premise of ensuring the accuracy of calculation.In order to estimate the computational complexity of the final algebraic equations,we compare the sparsity of the coefficient matrix under different y.The smaller y,the better sparsity of the matrix.Then the universal solution format for the linear elastic mechanics problem is given.The effectiveness of the local enhanced wavelet algorithm is verified by several classical examples such as the plate with a hole,L-shaped plate and plate with a crack,and the strain energy we get will be the upper limit of the actual strain energy.Further,based on the wavelet coefficient reflect the smoothness of the function at this point,we construct the wavelet optimal approximation format of the function.For a local singularity function in L-region,when ?=6,only 371 points are needed to obtain the same result as 28033 uniform sprinkling points,and the effect is remarkable.Then based on the optimal approximation format,an adaptive method is proposed for the elliptic problem,and the wavelet-adaptive format is used to solve several problems with local large gradient or singularity.By comparing the numerical results with the uniform spitting point,it is found that for the local singularity problem,we will need 39042 points to achieve the accuracy of 3.98E-03,and in the adaptive format,only 621 points are needed.Then,we applied the adaptive algorithm to solid mechanics also obtained satisfactory results.Finally,we use the local enhanced wavelet format for spatial discretization for the nonlinear initial boundary value problems.And then we use an implicit-explicit hybrid format or Runge-Kutta method to solve the semi-discrete equations.The unified solution formats of Burgers equations,KDV equations,nonlinear Schr?dinger equations and nonlinear Klein-Gordon equations are given respectively.The algorithm is proved to be conservative.The numerical examples can be used to find that the wavelet algorithm can deal with arbitrary nonlinear problems simply and efficiently.The spatial convergence speed can be kept at about 6th order,and the calculation accuracy is higher than many existing numerical methods.It is shown that the wavelet algorithm with closed property of this paper is not sensitive to nonlinear terms,and it can accurately capture the shock wave,the motion of the solitary wave,and simulate the whole process of multiple solitary wave interactions,even the blasting solution.