Wavelet Methods For Highly Nonlinear Problems Of Mechanics In Complex Geometry

Accompanied by the expansion of human knowledge,the significance of nonlinear science is rapidly rising.Because nonlinear models cannot satisfy the superposition principle,it is impossible to analyze a problem by decomposition and assemblage.Thus obtaining an exact solution with general analytic methods is infeasible.Numerical methods become the only avilable option in most cases when solving nonlinear problems and occupy a crucial position.Besides the difficulty from nonlinearity,the actual problems in science and engineering also require more algorithms available in arbitrary geometries and boundaries.But conventional methods cannot properly deal with nonlinear problems defined in complex geometries.Noticing the wavelet method given by our research group in previous work has potential ability to solve nonlinear equations,we develop and spread its theory and application to complex geometries in this dissertation to propose a wavelet numerical method of high accuracy for nonlinear problems.To establish a general framework for the computation of boundary value problems and initial value problems overall,a family of wavelet multistep method is revealed.Furthermore,several modified methods are shown to improve the accuracy for nonlinear problems based on the work of predecessors.The Coiflet wavelet has good properties for numerical computation.As the fundamental work in the construction of filter coefficients,some new Coiflet wavelets belonging to 3N+2 family are built by means of the adjustment in vanishing moment.The supporrt interval is elongated by 2 in exchange for higher smoothness than classical3N family to improve the convenience order of wavelet base by 1.In theory,a formula suitable for Coiflet wavelet to extract the multi-fold integral outside the support interval is given.A direct polynomial type of analytical expression can be used to fast calculate the integral value at any point without the usage of filter coefficients.Therefore,the round-off error is reduced and the efficiency raises.It is also the groundwork for the wavelet numerical quadrature formula in later parts.For the amelioration of wavelet approximation,to reduce the additional error brought by boundary extension,a new formula merging Lagrange interpolation of high-order is devised.As a result,the mismatch between the high accuracy of wavelet scheme and the original difference formula of low-order is overcame.The error of approximation to tanh?x?with 15 nodes achieves 10^-8 beyond some other formulas.Its expansion in two-dimensional region does not exhibit any obvious growth of errors.Because of the severe requirement for accurate approximation of nonlinear problems,a wavelet collocation method of high accuracy is proposed with the help of Richardson extrapolation technique.The introduction of half steps and the adjustment of coefficients neutralize the lower-order errors leading to a promoted convergence order.This scheme modifies the approximation formula but keeps the wavelet base unchanged.Hence all advantages of wavelet base that the interpolation and smoothness properties can be inherited so that the original formula is replaced seamlessly.Low-order and high-order terms in equations are decoupled to separate the error and approximation and easy to impose the boundary conditions.The last is a wavelet quadrature scheme.Anticipating that the following works may be executed in common intervals,the number of available nodes varies depending on the boundary shape.The unidirectional extension of Newton's formula remove the interval limit and allow the utilization of any number of inner nodes inside any interval for approximation,which is the core calculation in complex regions in this dissertation.We compare the effect from 4,5 and6 nodes schemes and discover that the last almost fully repress the fluctuations near the boundary.In the domain with complex geometry,classical algorithms often encounter the difficulties for lack of accuracy,which is adverse for nonlinear problems.On the other side,many algorithms of high accuracy are weak in treatment for complicated boundaries and employment of boundary conditions.To take account of both sides,as the main work in this dissertation,a wavelet quadrature method which can be implemented in the domain with arbitrary shapes is proposed.This scheme is versatile enough for many cases and free from the additional disposition of boundary conditions.Because the domain is embedded into the Cartesian grid,there is no need to match the curve boundary.So that the mesh generation consuming most of calculation time is avoided and simple mesh generators are allowed.The smoothness property of wavelet base provides a fast convergence allowing it to be implemented on course grids and still outputs acceptable solutions.The interpolation property of wavelet base grant it the capability of manipulation of nonlinear operators.As a collocation method of strong formula,it can solve the equation directly without the modification into a weak formula and support some nonlinear problems in which the variation principle fails.The high stability and proper extension prevent this method from the ill-posed coefficient matrix and oscillatory boundary.Various boundary conditions are exactly satisfied rather than some inexact transform in other methods.Afterwards,the sparse matrix for total discretization will be exported,averting the low efficiency in traditional schemes that adapt the elements and reshape the discretization matrix according to the imposed boundaries.Moreover,to analyze the dynamic evaluation problems,a family of wavelet linear multistep method is proposed to solve the initial value equations,which can be executed in explicit or implicit form.A strongly stable implicit formula is designed after the adjustment of selective vanishing moment.The derivation comes from Coiflet wavelet approximation rather than conventional theories of the multistep method.However,its consistency,convergence and stability pass all examinations necessitated by the classical theory.Figures of the absolute stable region and order star demonstrate such characteristics once again.This method may also be modified as an explicit predictor-corrector scheme if it merge another wavelet forecast formula.In case that the join of Richardson extrapolation,it will be extra accelerated.We will combine these schemes and the wavelet method in complex geometry into a unified framework for initial-boudary problems in following works.Finally,some typical numerical examples are revealed to show the advantages of wavelet methods above.Because p-Laplacian equation contains a lot of difficulties in numerical computation and has strong practicability.During the derivation of new algorithms,its solution is studied as a representation of nonlinear equation.The basic idea of the previous wavelet Galerkin method and the current wavelet quadrature method are applied.Wavelet methods clearly exhibit the feature of high efficiency.One of examples shows an accuracy level of 10^-7 from wavelet method,better than the finite element method.Another example reveals that wavelet method only uses 70 percent of nodes to give a solution similar to finite element method.Compared with two finite volume methods,wavelet method holds a faster convergence.In the case of the wavelet quadrature method,its solution is identical to the shooting method and finite difference method.But wavelet quadrature method only uses step size 1/32 to achieve an accuracy equal to the solution of finite difference method given by step size 1/800.It indicates that the numerical scheme based on wavelet is very accurate.Using the wavelet Richardson collocation method,some typical nonlinear equations and a incompressible steady laminar flow problem on a stretching surface are computed.Numerical results manifest that its behaviors of accuracy promotion fully correspond to the theoretical anticipation,arriving an convergence of order 5.One of examples displays that the accuracy of numerical solution under 16 nodes from new formula gets close to the solution under 32 nodes in old formula.Another example plots a smoother solution given by new formula than old one.In regions of different shapes,nonlinear Poisson's equation,several bar torsion and thin plate bending problems are computed for example.Wavelet quadrature method is very accurate and insensitive to the boundary shapes.A comparison between finite element method and wavelet quadrature method shows that the latter has the faster convergence and the higher accuracy.It achieves the commensurate quality of solution with less than 1000 nodes when finite element method consumes more than 6000 nodes,showing a good efficiency in numerical computation.One of examples offers a case in which the convergence of finite element method is bad but wavelet quadrature method still provide a reliable sonution.Some examples of nonlinear initial value problems exhibit the convergence and accuracy of wavelet multistep method.Under certain conditions,wavelet multistep method gives better performance than other algorithms with the identical order.