Finite element method algorithm for geotechnical applications based on Runge-Kutta scheme with automatic error control
Ayman Abed, Wojciech T. Sołowski
Abstract
This paper introduces a novel explicit algorithm to solve the finite element equation linking the nodal displacements of the elements with the external forces applied via means of non-linear global stiffness matrix. The proposed method solves the equation using Runge-Kutta scheme with automatic error control. The method allows any Runge-Kutta scheme, with the paper demonstrating the algorithm efficiency for Runge-Kutta schemes of second to fifth order of accuracy. The paper discusses the theoretical background, the implementation steps and validates the proposed algorithm. The numerical tests show that the proposed method is robust and stable. In comparison to the iterative implicit methods (e.g. Newton-Raphson method), the new algorithm overcomes the problem of occasional divergence. Furthermore, considering the computation time, the fifth-order accurate scheme proves to be competitive with the iterative method. It seems that the proposed algorithm could be a powerful alternative to the standard solution procedures for the cases with strong nonlinearity, where the typical algorithms may diverge.