A modified wavelet approximation for multi-resolution AWCM in simulating nonlinear vibration of MDOF systems
A modified wavelet approximation for multi-resolution AWCM in simulating nonlinear vibration of MDOF systems
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DOI：
10.1016/j.cma.2007.11.017
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发表时间：
2008-03
影响因子：
7.2
通讯作者：
Youhe Zhou;Jùn Zhou
中科院分区：
文献类型：
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作者：
Youhe Zhou;Jùn Zhou
小波方法是近年来发展起来的一种求解含时常微分方程初值问题的方法。在应用现有小波方法之前，通常需要将非线性振动方程转化为状态方程，这会增加存储和计算量。本文提出了线性/非线性多自由度系统振动解的一种改进的小波近似形式，并在此基础上建立了多自由度系统的多分辨率自适应小波配置方法（AWCM）。修正的小波近似的小波系数显式地包括边界的一阶时间导数的解决方案，除了解决方案的定义上的时域的内部时间位置上的值。这就避免了将线性/非线性振动多自由度系统的动力学方程转化为状态方程。因此，采用改进的小波近似，小波配置法所需的计算量和存储量最多可减少到基于状态方程的小波配置法的12;并且计算效率可以比基于状态方程的小波配置方法快至少两倍，因为需要计算的解分量的数量仅为相应状态方程的12个。另一方面，通过使用本文提出的改进的小波近似，初始条件的实现是直接的。通过将其应用于线性和非线性10自由度振动系统来测试修改的小波近似的有效性。
Wavelet methods have been presented recently to solve the initial value problems of time-dependent ODEs. Before applying the current wavelet methods, a nonlinear vibration equation usually needs to be transformed to state equations, which may raise the storage and computation cost. In this paper, a modified form of wavelet approximation for the vibration solutions of linear/nonlinear MDOF systems is presented, based on which a multi-resolution AWCM (adaptive wavelet collocation method) is established for MDOF systems. The modified wavelet approximation’s wavelet coefficients explicitly include the boundary first-order time derivatives of the solutions, besides values of solutions on inner time locations of the time domain on which the solutions are defined. This can avoid to transform the dynamical equations of a linear/nonlinear vibrational MDOF system to state equations. Therefore, by using the modified wavelet approximation, the computation and storage cost required for wavelet collocation method may be reduced to at most 12 of that needed by those based on state equations; and the computation efficiency may be at least twice faster than wavelet collocation methods based on state equations, because the number of solution components that are needed to be computed is only 12 of the corresponding state equations. On the other hand, the implementation of initial conditions is straightforward by using the modified wavelet approximation proposed in this paper. The effectiveness of the modified wavelet approximation is tested by applying it to a linear and a nonlinear 10-DOF vibrational system.