Adaptable Spline Element for Membrane Vibration Analysis

Abstract

SUMMARYAn adaptable curved quadrilateral element based on quadratic B-splines is developed for finite element analysis of membrane vibration problems. The approximated displacement field in each element is divided into corners, edges and internal modes. The number of knots on the edges of elements are distinct in any integer power of 2. The elements of different resolutions can be thus connected smoothly under the constraint from the 'two-scale relation' of splines bases. The displacement compatibility on the interconnected edges between adjacent elements is satisfied along the whole edge. The advantages of the proposed element also result in a more adaptable element meshing strategy for structural discretization. Some numerical examples are studied as verification and demonstration of the present work.

Construction of wavelet boundary element method for solving SIFs of two-dimensional plates

“…In addition, B-spline wavelet has been widely concerned because of its many excellent characteristics. In order to study the vibration of frame structure and membrane vibration, Chen and Wu (1995, 1996a, 1996b) combined B-spline wavelet with FEM and achieved good results. Later, the 1D and 2D nonsingular transformation matrix introduced by Xiang et al (2006) maps the wavelet expansion coefficient into the physical space, which greatly promotes the development of BSWI FEM (Xiang et al2007;Xiang et al2008).…”

Transverse Vibration and Waves in a Membrane: Frequency Domain Spectral Element Modeling and Analysis

“…However, no closed-form solutions exist for membranes with complex geometry. Thus, various solution techniques have been developed by many researchers: they include the finite difference method [7,8], the superposition method using wave-type base functions [9], the discrete singular convolution method [10,11], the Kantorovich method [12], the optimized Galerkin-Kantorovich method [13], the hybrid method composed of differential transforms and the Kantorovich method [14], the differential quadrature method [13,15], the collocation approach based on the little sinc functions [16], the Rayleigh-Ritz method [17], and the finite element method (FEM) [18][19][20][21][22][23][24].…”

Adaptive Trigonometric Wavelet Composite Beam Element Method

“…element parameters automatically by conducting error prediction of the calculation results during the analysis, and then to choose the best discrete pattern and to achieve the required calculation accuracy gradually. Later on, three types of adaptive finite element method, h, p and h-p formulations were studied systemically (Chen and Wu 1996). The accuracy of h-extension method is achieved by refining the mesh, while p-extension method implements a fixed mesh and the desired approximate result is promoted by increasing the order of polynomials in the elements.…”

“…Chen et al (1995Chen et al ( , 1996 solved the truss and membrane vibration problems by using the element constructed by the spline wavelets, and derived the lifting algorithm that takes an advantage of the "two-scale relation" of wavelets. By introducing a transformation matrix that transforms the element deflection field represented by the coefficients of wavelet expansions from wavelet space to physical space, Ma et al (2003) and Chen et al (2004) constructed the wavelet beam element based on Daubechies wavelet and B-spline wavelet, respectively.…”