A Wavelet Method For Solving Nonlinear Wave Problems And Its Application In Viscoelastic Structures

The research and application of mechanics and physics usually involve the propagation,absorption and scattering characteristics of waves in a variety of complex media.These problems can usually be described quantitatively by wave equations with different characteristics,such as space-time coupling,nonlinearity,high dimension,high-order derivative,fractional derivative,etc.For a long time,the high-precision and efficient universal solution and analysis of wave equations with these complex characteristics has been an important research topic that is of great interest to scholars in the fields of mechanics and physics,but has not been effectively solved.In order to address this tough issue,based on the high-precision basic wavelet algorithms for boundary value problems which have been developed by scholars,this dissertation has proposed an effective method which can effectively and acuractely solve wave equations containing strong nonlinear,higher-order spatial derivatives,fractional-order derivatives with time memory and weak singularity by using wavelet theory as an advanced mathematical tool,and the quantitative characterization of the solution accuracy and convergence speed of the proposed method is given.The method is applied to dynamic analysis of viscoelastic structures with different characteristics,and more efficient and accurate quantitative results are obtained than existing methods.The specific innovation results are as follows:(1)In order to solve the nonlinear integer order wave equations without time memory,a space-time fully decoupled wavelet integral collocation method with sixth order accuracy is established.As a result,these matrices generated in the spatial discretization do not need to be updated in the time integration,such that a fully decoupling between spatial and temporal discretization can be achieved.A linear multistep method based on the same wavelet approximation used in the spatial discretization is then employed to solve such a semi-discretization system.By numerically solving several widely considered benchmark problems,we demonstrate that the proposed wavelet algorithm possesses much better accuracy and a faster convergence rate than many existing numerical methods.Most interestingly,the space-associated convergence rate of the present method is always about sixth order for different nonlinearities,which is in the same order with direct approximation of a function in terms of the proposed wavelet approximation scheme.This fact implies that the accuracy of the proposed method is almost independent of the equation order and nonlinearity.(2)In order to solve various nonlinear fractional wave equations with weak singular characteristics,a high-precision wavelet explicit method is established.The original fractional wave equation is transformed into a time Volterra-type integrodifferential equation associated with a smooth time kernel and spatial derivatives of unknown function by using the technique of Laplace transform.Then,an explicit solution procedure based on the collocation method and the proposed algorithm on integral approximation is established to solve the transformed nonlinear integrodifferential equation.Eventually the nonlinear fractional wave equation can be readily and accurately solved.As examples,this method is applied to solve several fractional wave equations with various nonlinearity.Results show that the proposed method can successfully avoid the difficulty in the treatment of singularity associated with the fractional derivatives.Compared with other existing methods,this method not only has the advantages of high-order accuracy,but also does not even need to solve the nonlinear spatial system after time discretization to obtain the numerical solution,which reduces the storage and computation cost.(3)The high-precision wavelet method established in this paper is applied to the dynamic problems of viscoelastic structures with memory characteristics.A quantitative method for solving such problems is provided,and more efficient and accurate results are obtained than existing methods.The inherent laws of dynamic performance of viscoelastic damping materials are analyzed to provide theoretical basis for the research,development and performance prediction of damping materials.In addition,considering the factors of geometric nonlinearity and variable section,the large deflection of the cantilever beam composed of viscoelastic material under combined load is studied,which provides a theoretical basis for the application of viscoelastic material in modern engineering.Through the high-precision and high-efficiency solutions of various types of nonlinear wave equations,it is proved that the wavelet method established in this paper is effective for solving wave equations containing strong nonlinearity,high-order spatial derivatives,and fractional-order derivatives with time memory and weak singular characteristics.It shows that our proposed method has higher accuracy and convergence order compared with other methods,which provides an effective means for us to analyze and study the wave phenomenon and other dynamic problems.