一种适用于强非线性结构力学问题数值求解的修正小波伽辽金方法

A MODIFIED WAVELET-GALERKIN METHOD FOR COMPUTATIONS IN STRUCTURAL MECHANICS WITH STRONG NONLINEARITY

论文通过对有限区间上的任一连续函数在边界处采用基于泰勒展开的延拓处理,构造了一种与任意边界条件相协调的改进小波尺度基函数及在此基础上建立了小波逼近格式,由此可有效避免小波逼近在求解微分方程时在边界处的跳跃或抖动问题.在此基础上,结合论文后两位作者提出的广义小波高斯积分法,关于未知函数的任意非线性项的小波展开可以显式地用原未知函数的展开系数表征,据此建立了一种可适用于求解任意强非线性的梁弯曲问题的小波伽辽金方法.该方法具有解的封闭性与计算简单等特点.通过定量求解包含幂次非线性与非幂次非线性项梁的两例大挠度弯曲问题,所得结果表明论文所建立的方法具有良好的数值精度.

To avoid the undesired jump or wiggle phenomenon near the boundary points when the wavelet-based method is employed to solve a boundary-value problem,this paper presents a set of modified scaling base functions through the interval extension of an unknown continuous function defined in a finite interval on the basis of the Taylor series expansion associated with the arbitrary boundary conditions.After that,an approximate scheme of the function is proposed by the modified scaling base functions.According to the generalized-Gaussian-quadrature method in wavelet analysis,which was developed by the last two authors of this paper,the expansion constants in the approximation of arbitrary nonlinear term of the unknown function can be explicitly expressed in finite terms of the expansion ones of the approximation of the unknown function.Once the wavelet-Galerkin method on the basis of the approximation is employed to solve the nonlinear differential equation with the nonlinear term（s） of a finite beam structure with arbitrary boundary conditions,it is found that the solution has the closure property and the virtue of easy implement in calculation of solving a strong nonlinear problem.The numerical results obtained in solving two cases of large deflected beams with different nonlinear characterization,i.e.,either integer order or non-integer order nonlinear terms or both,indicate that this approach has high accuracy.