一维算子自定义小波有限元的构造方法研究

The construction method of one-dimensional operator custom-design wavelet finite elements

针对工程结构中的共性问题,即算子,提出了工程结构的算子自定义小波有限元构造方法及自适应解耦计算方法。在建立了多分辨有限元空间的基础上,根据工程结构问题中的算子,即小波函数与尺度函数之间的内积关系式,提出了基于稳定完备法的算子自定义小波单元构造方法;提出自适应算子自定义小波有限元法,该方法的优点是在保持初始尺度分析结果的基础上,向局域添加所有大于局部误差阈值的算子自定义小波基,实现工程结构问题的高效分析。数值算例表明:算子自定义小波有限元法所推导的多尺度刚阵解耦比率为89.65%,自适应算子自定义小波有限元法的计算量最大降低比率为49.23%,适合工程结构问题的高效多尺度计算。

An important problem of multiscale wavelet algorithm is that the coupling terms of the scaling functions and wavelets increase rapidly as the scale increases,which results in low convergence rate for multiscale wavelet solution.According to the common feature of the operator in engineering structures,a constructive method of operator custom-design wavelet finite elements and its adaptive decoupling algorithm are proposed for engineering structural problems.For the operators defined by the inner product of scaling functions and wavelets,a new kind of one-dimensional operator custom-design wavelets is constructed in multiresolution finite element space based on stable completion.An adaptive operator custom-design wavelet finite element method is presented for engineering structural problems.The advantage of the proposed method is that engineering problems can be solved efficiently by adding operator custom-design wavelets into the local domain while keeping analysis results on the initial scale.Numerical example demonstrates that the decoupling ratio of multiscale stiffness matrix derived by operator custom-design wavelet finite element method is 89.65% and the maximum decreasing ratio of computation complexity for adaptive operator custom-design wavelet finite element method is 49.23%.It is proved that the proposed method is suitable for efficient computation of engineering structural problems.