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在现有的Daubechies小波Ritz法中,为方便边界条件的引入,借助于位移转换矩阵将Daubechies小波待定系数转换为节点位移。但该方法会降低计算精度,并且计算结果是多个离散的单点位移,不利于进一步解得弯矩、剪力、荷载集度。为寻求更为高效精确的弹性地基梁计算方法,对现有的Daubechies小波Ritz法进行改进,以避免位移转换矩阵的出现,从而提高了计算精度。结合广义变分原理,采用Lagrange乘子法,将边界条件作为附加条件引入自然变分条件下的泛函表达式,构造新的修正泛函。以该修正泛函的驻值条件建立求解矩阵方程组,进而解得未知场函数。此法称为Daubechies条件小波Ritz法。该法计算结果直接是小波基函数待定系数,单元内部任意点的位移均可通过小波基函数得到,也可进一步解得弯矩、剪力、荷载集度,因此比原有方法更为有效。最后,采用受均布荷载的两端铰支弹性地基梁算例,将Daubechies条件小波Ritz法计算结果与基于弹性地基梁理论的解析解进行比较,挠度值(保留小数点后6位小数)与解析解完全一致,弯矩值的相对误差为0.03%,说明Daubechies条件小波Ritz法具有较高计算精度。
In the existing Daubechies wavelet Ritz method, to facilitate the introduction of boundary conditions, the Daubechies wavelet undetermined coefficients are converted to node displacements by means of displacement transformation matrices. However, this method will reduce the calculation accuracy, and the calculation result is a number of discrete single-point displacements, which is not conducive to further solution of bending moment, shear force, load concentration. In order to find a more efficient and accurate method for calculating elastic foundation girder, the existing Daubechies wavelet Ritz method is improved to avoid the emergence of displacement transformation matrix and improve the calculation accuracy. Combined with generalized variational principle, the Lagrange multiplier method is adopted to introduce the boundary conditions as additive conditions into the functional expressions under the condition of natural variational conditions to construct a new modified functional. The equations for solving matrix equations are established based on the condition of the modified functional terms, and the unknown field functions are solved. This method is called Daubechies conditional wavelet Ritz method. The result of this method is directly the undetermined coefficient of the wavelet basis function. The displacement of any point inside the element can be obtained by the wavelet basis function, and the bending moment, shear force and load concentration can be further solved, so it is more effective than the original method. Finally, by using the example of elastic beam with hinged elastic ends under uniform load, the calculated results of Daubechies condition wavelet Ritz method are compared with the analytic solution based on the elastic foundation beam theory. The deflection values (retaining 6 decimal places after decimal point) The solution is completely consistent. The relative error of bending moment is 0.03%, which shows that the Daubechies condition wavelet Ritz method has higher accuracy.