Daubechies条件小波有限元法研究

为拓展小波理论在结构工程中的应用,提高结构计算精度,提出了以Daubechies条件小波Ritz法为基础的Daubechies条件小波有限元法。该法结合广义变分原理和拉格朗日乘子法构造修正泛函,根据修正泛函的驻值条件得到全域法求解方程矩阵。根据构件的边界条件,按左右边界对求解矩阵进行相应拆分,构建条件小波单元刚度矩阵,并依据公共节点位移相等原则形成总体刚度矩阵,由此解得各单元的小波基待定系数,即可进一步求解位移场函数、内力分布函数及荷载集度函数。以工程中常见的弹性拉压杆及平面弯曲梁为例,详细阐述了该方法的构造过程。并通过典型算例将Daubechies条件小波有限元法计算值与理论解进行了对比,结果表明:在弹性拉压杆算例中,位移、应力、载荷集度的相对误差均在1.22×10-3%以内;在平面弯曲梁算例中,挠度、弯矩、载荷集度的相对误差均在8.91×10-2%以内。

Study on Daubechies conditional wavelet finite element method

In order to develop the application of wavelet theory in structure engineering,and to improve the accuracy of structural computation,Daubechies conditional wavelet Finite Element Method is proposed,which is based on Daubechies conditional wavelet Ritz method.Daubechies conditional wavelet finite element method combines generalized variational principle with the Lagrangian multiplier to build a new modified functional,according to its stationary condition,the matrix equation based on full domain is constructed.Based on the boundary condition of structural members,the coefficient matrix is split by considering the left and right boundary condition,and the conditional wavelet element stiffness matrix is constructed,then the global stiffness matrix is assembled according to the rule of equal displacement in common nodal.The undetermined coefficients of wavelet primary function of all units can be obtained,and the internal force distributed function and load intensity distributed function can be solved out too.Taking common used structural members——elastic bar and plane bending beam as examples,the construction of Daubechies conditional wavelet FEM is introduced in detail. Typical examples are taken to compare the computation results between Daubechies conditional wavelet Finite Element Method and theoretical method.In elastic bar example,the maximum relative error between Daubechies conditional wavelet Finite Element Method and theoretical method of displacement,stress,load distribution respectively is within 1.22×10-3 %,in plane bending beam example,the maximum relative error between Daubechies conditional wavelet Finite Element Method and theoretical method of deflection,bending moment,load distribution respectively is within 8.91×10-2 %.