Convergence and exact solutions of spline finite strip method using unitary transformation approach

Abstract

The spline finite strip method (FSM) is one of the most popular numerical methods for analyzing prismatic structures. Efficacy and convergence of the method have been demonstrated in previous studies by comparing only numerical results with analytical results of some benchmark problems. To date, no exact solutions of the method or its explicit forms of error terms have been derived to show its convergence analytically. As such, in this paper, the mathematical exact solutions of spline finite strips in the plate analysis are derived using a unitary transformation approach (abbreviated as the U-transformation method herein). These exact solutions are presented for the first time in open literature. Unlike the conventional spline FSM which involves assembly of the global matrix equation and its numerical solution, the U-transformation method decouples the global matrix equation into the one involving only two unknowns, thus rendering the exact solutions of the spline finite strip to be derived explicitly. By taking Taylor's series expansion of the exact solution, error terms and convergence rates are also derived explicitly and compared directly with other numerical methods. In this regard, the spline FSM converges at the same rate as a non-conforming finite element, yet involving a smaller number of unknowns compared to the latter. The convergence rate is also found superior to the conventional finite difference method.

On the Validity of Flat-Plate Finite Strip Approximation of Circular Cylindrical Shells

“…In essence, the Utransformation provides a mathematical tool for decoupling a cyclic symmetric structural system and making it possible for obtaining the solution in an explicit form. The method has been successfully applied to study the convergence of finite strips for rectangular plates [13,14]. In this paper, the method is further extended to study the vibration of cylindrical shells using the thin flat-plate finite strips; the latter was originally developed by Cheung [15] and remains as one of the most effective elements for analyzing prismatic structures [16].…”