Finite element method algorithm for geotechnical applications based on Runge-Kutta scheme with automatic error control

• Published in: Computers and geotechnics Vol. 128; p. 103841
• Main Authors: Abed, Ayman A., Sołowski, Wojciech T.
• Format: Journal Article
• Language: English
• Published: New York Elsevier Ltd 01.12.2020; Elsevier BV
• Subjects: Algorithms; Aquatic reptiles; Automatic control; Automatic error control; Computation; Divergence; Explicit methods; FEM; Finite element method; Implicit methods; Iterative methods; Mathematical analysis; Matrix methods; Newton-Raphson method; Nonlinear systems; Nonlinearity; Robustness (mathematics); Runge-Kutta method; Runge-Kutta scheme; Stiffness matrix; Explicit methods; Runge-Kutta scheme; Automatic error control; FEM
• ISSN: 0266-352X; 1873-7633; 1873-7633
• DOI: 10.1016/j.compgeo.2020.103841

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.