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.
Highlights
The Finite Element Method approximates the solution of the continuous mechanical balance equation by calculating the unknown displacement u at a set of discrete points only, typically at the element nodes, as a response to the external load fext
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
This paper introduced a new method to solve the load-displacement finite element equations based on Runge-Kutta scheme
Summary
The Finite Element Method approximates the solution of the continuous mechanical balance equation by calculating the unknown displacement u at a set of discrete points only, typically at the element nodes, as a response to the external load fext This requires solving a large set of equations, with coefficients given in the global stiffness matrix K where: Ku fext = 0. If the load increment is large compared to the nonlinearity of the problem, the initial iteration result may fall outside of the convergence radius and the Newton-Raphson scheme may fail to reach the solution. The study included, in addition to the iterative procedures, the explicit first-order forward Euler scheme (tangent stiffness method) They concluded that there is no generic statement on the recommended method to be used and it is necessary to equip the numerical code with different solution methods so that the most suitable one can be used depending on the solved problem. This opens new possibilities for novel numerical algorithms, more stable and robust than existing ones
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