A Haar Wavelet Method And Its Application In Mechanics

Wavelet analysis is a mathematical tool that has been developed rapidly in recent decades.It has been applied to numerical solutions of differential equations.Most of the problems in structural analysis and engineering mechanics are expressed in the form of differential equations.Those kinds of equations have difficulties of high dimension,high order and nonlinearity,so effective numerical methods are extremely needed.An integral collocation method based on Coiflet wavelet that has been proposed previously by our group shows high accuracy in numerical solutions.However,the Coiflet wavelet with support [0,17] has no analytical expression and its value or integral can only be obtained at specific points through a series of complex calculations,that increases the computational complexity and limits the use of this method.As a simplest wavelet,Haar wavelet has explicit expression as well as some properties like normal orthogonality and compact support.In this thesis,the integral collocation method is constructed based on Haar wavelet,aiming at equations that demand lower accuracy.Firstly,the feasibility of solving differential equations by wavelet integral is analyzed based on function expansion theorem of Haar wavelet.Then,a method is given,by which partial derivatives of functions and equations can be expressed using Haar wavelet and its integral.How to process boundary conditions is also discussed.Finally,the procedure of discretizing equations through collocation method and solving algebraic equations after discretization is given,so does the method of reconstruction.In order to test the performance of this method,static boundary value problems like one-dimensional Bratu equation and bending of square plate are selected as examples.The Bratu equation contains different parameters on behalf of different nonlinearity.The bending problem of square plate includes linear and nonlinear equations corresponding to small deflection and large deflection with different types of loads.By solving those equations of different kinds and analyzing the error,we find that the Haar wavelet integral collocation method has stable second-order convergence accuracy,which is not related to the order of the equation and the strength of nonlinearity.For initial boundary value problems of dynamics,we select the channel flow and cavity flow as examples and solve the original variable Navier-Stocks equation of twodimensional viscous incompressible flow by the combination of Haar wavelet integral collocation method and artificial compression method.The time is treated as a variable equivalent to space components,and boundary conditions is embedded in initial conditions.The results show that the flow can be simulated well with only number of nodes,which proves the feasibility of the method in solving complex nonlinear equations of dynamic problems.