Convergence and Exact Solutions of the Spline Finite Strip Method Using a Unitary Transformation Approach

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