Application Research Of Wavelet Finite Element Method In Vibration Response Of Beam Structure

In recent years, a new numerical analysis method named wavelet finite elementmethod (WFEM) has been took shape by introducing wavelet theory into traditionalFEM. This method keeps many advantages of traditional FEM, which includes discreteapproximation and solving boundary conditions easily. Simultaneously, because of themulti-resolution property of wavelet function, WFEM can improve the accuracy ofcalculation without changing the original mesh. Both the stability of algorithm and thespeed of computing are excellent.In addition, the computational efficiency often decreases sharply with theimprovement of accuracy when traditional FEM is used to solve vibration response ofcomplex structures. Therefore, in order to overcome the difficulty, an efficientevaluation method is presented, which is based on the most common beam model inengineering practice and makes full use of the characteristics of WFEM. It can providea new idea for the research work in the field of mechanical dynamics.Considering that Daubechies wavelet is lack of the analytical mathematicalexpression, the method of calculating the low-pass filter coefficients, function value andits derivatives is given according to the basic theory of wavelet analysis. Then, thecomputational method for connection coefficient of WFEM stiffness and load matricesis derived in detail. However, the calculation results are different when the number ofcomplement equations change. In this thesis, a new equation is used to make up for thedeficiency, and both the numerical stability and the accuracy of solving process areincreased.Subsequently, a wavelet beam element used for vibration response is constructedby replacing the low order polynomial of traditional FEM as the linear combination ofscaling functions of Daubechies wavelet. Through using this element to analysis the energy of whole system and considering Hamilton principle, the dynamic model forBernoulli-Euler beam is built based on WFEM. The vibration response can be solved byNewmark-method.Finally, in order to verifying the validity of WFEM and testing the calculationaccuracy and computational efficiency of such model, free vibration and forcedvibration problem under moving load are solved through the numerical experiments inthis thesis, and the analytical results are compared with traditional FEM. According tothe research, WFEM is superior to FEM on the calculation accuracy when the totalfreedom of structure is the same. And, for identical calculation accuracy, WFEM canimprove the computational efficiency greatly. So the WFEM model provides a highlyefficient calculation method for computing complex dynamics problem of beamstructure.