Extension of spline wavelets element method to membrane vibration analysis

The B-spline wavelets element technique developed by Chen and Wu (1995a) is extended to the membrane vibration analysis. The tensor product of the finite splines and spline wavelets expansions in different resolutions is applied in the development of a curved quadrilateral element. Unlike the process of direct wavelets adding in the previous work, the elemental displacement field represented by the coefficients of wavelets expansions is transformed into edges and internal modes via elemental geometric conditions and two-scale relations. The multiple stages two-scale sequence of quadratic B-spline function is provided to accelerate the sequential transformations between different resolution levels of wavelets. The hierarchical property of wavelets basis approximation is also reserved in this extension. For membrane vibration problems where variations lack regularity at certain lower vibration modes, the present element can still effectively provide accurate results through a multi-level solving procedure. Some numerical examples are studied to demonstrate the proposed element.