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.

中文翻译：

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基于Runge-Kutta方案的岩土应用有限元方法算法与自动误差控制

摘要 本文介绍了一种新的显式算法来求解将单元节点位移与通过非线性全局刚度矩阵施加的外力联系起来的有限元方程。所提出的方法使用具有自动误差控制的 Runge-Kutta 方案求解方程。该方法允许任何 Runge-Kutta 方案，论文证明了 Runge-Kutta 方案的算法效率，精度为二阶到五阶。本文讨论了理论背景、实现步骤并验证了所提出的算法。数值试验表明所提出的方法具有鲁棒性和稳定性。与迭代隐式方法（例如 Newton-Raphson 方法）相比，新算法克服了偶尔发散的问题。此外，考虑到计算时间，五阶精确方案证明与迭代方法具有竞争力。对于具有强非线性的情况，在典型算法可能会出现分歧的情况下，所提出的算法似乎可以成为标准求解程序的有力替代方案。