The multiresolution capability provided by the family of Daubechies wavelets is exploited to develop a new computational approach, termed as multiresolution finite wavelet domain method for the fast and hierarchical prediction of transient dynamic problems with focus on wave propagation in one- and two-dimensional structural configurations. In the developed method, both scaling and wavelet functions are employed as basis functions for the approximation of displacements in the governing dynamic equations. Because of the orthogonal and the multiresolution properties of Daubechies wavelets, a two-scale system of mass uncoupled dynamic equations, representing the coarse and the fine spatial wave component solutions is derived. Numerical results concerning wave propagation in one- and two-dimensional elastic structures demonstrate substantial computational gains of the method in comparison with other well-established numerical methods. Moreover, additional benefits of the involved coarse and fine solutions enabling the analysis and characterization of the resultant wave response are revealed and discussed.

中文翻译：

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用于弹性固体瞬态动态波分析的多分辨率 Daubechies 有限小波域方法

Daubechies 小波家族提供的多分辨率能力被用来开发一种新的计算方法，称为多分辨率有限小波域方法，用于快速和分层预测瞬态动态问题，重点是在一维和二维结构配置中的波传播. 在开发的方法中，标度函数和小波函数都被用作控制动态方程中位移近似的基函数。由于 Daubechies 小波的正交性和多分辨率特性，推导出了一个质量非耦合动力学方程的两尺度系统，代表了空间波分量的粗解和细解。与其他行之有效的数值方法相比，关于一维和二维弹性结构中波传播的数值结果证明了该方法的大量计算增益。此外，还揭示和讨论了所涉及的粗略和精细解决方案的额外好处，这些解决方案能够分析和表征合成波响应。