Accurately modelling the interactions between continuous and discontinuous materials is essential for advancing engineering solutions across a wide range of fields. Owing to fundamental differences in the governing equations of discontinuous and continuous numerical models, distinct solution schemes have been developed, presenting challenges for their coupling. This paper proposes a unified scheme for modelling continuous and discontinuous materials, integrating them monolithically by adopting a variational framework for an implicit Discrete Element Method (DEM) and the Finite Element Method (FEM). The seamlessly coupled FEM-DEM model is formulated as a convex optimisation problem, which can be transformed into a standard second-order cone program (SOCP) and solved efficiently using modern optimisation algorithms. The proposed approach has been validated through a series of numerical examples. Additionally, a numerical model test on granular flow against an elastic barrier has been conducted, demonstrating the scheme's capability in modelling the impact dynamics of granular flows and the resulting deformation and stress distribution in structures, which has significant implications for engineering design involving granular material handling.

中文翻译：

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用于对连续和不连续媒体及其相互作用进行建模的统一计算框架


准确模拟连续和非连续材料之间的相互作用对于推进广泛领域的工程解决方案至关重要。由于非连续和连续数值模型的控制方程存在根本差异，人们开发了不同的求解方案，这为它们的耦合带来了挑战。本文提出了一种对连续和不连续材料进行建模的统一方案，通过采用隐式离散元法 （DEM） 和有限元法 （FEM） 的变分框架将它们整体集成。无缝耦合的 FEM-DEM 模型被表述为凸优化问题，可以将其转换为标准的二阶锥程序 （SOCP），并使用现代优化算法进行有效求解。所提出的方法已经通过一系列数值示例进行了验证。此外，还对弹性屏障下的颗粒流进行了数值模型测试，证明了该方案能够模拟颗粒流的冲击动力学以及由此产生的结构变形和应力分布，这对涉及颗粒物料处理的工程设计具有重要意义。