Most existing Finite Element Method and the Material Point Method (FEM-MPM) coupling is designed for explicit solvers. By contrast, implicit schemes offer the advantage of substantially larger time steps while maintaining enhanced stability, particularly beneficial for tackling stiff nonlinear problems. Despite this, the development of implicit FEM-MPM coupling has not been extensively explored, leaving a notable gap in the context of contact and elastoplastic deformation challenges. Thus, this paper proposes a novel unified FEM-MPM coupling approach within implicit time integration under the framework of multivariable variational principle and convex cone programming, termed CP-FEMP. The CP-FEMP is the first successful attempt to impose the contact constraints via Lagrange multiplier and barrier method under convex cone programming, which can tackle not only the tie constraints but also the frictional contact between MPM and FEM domains with ensuring convergence and feasibility regardless of the time step size or the mesh resolutions. The contact locking issue in tie contact is circumvented using a well-defined interpolation space. The governing equations, associated frictional contact model, and associated elastoplastic constitutive law are formulated into a global convex optimisation problem, which is efficiently solved using primal-dual interior-point method. Through a succession of standard contact and elastoplastic benchmarks, the CP-FEMP demonstrates its proficiency in the precise transference of contact forces across MPM and FEM domains while showcasing commendable energy conservation attributes. Finally, the CP-FEMP is applied to a slope-retaining wall interaction problem. All results demonstrate CP-FEMP provides a comprehensive solution for FEM-MPM coupling, allowing for large incremental step under nonlinear contact and elastoplastic large deformation and guaranteeing strict, hard non-penetration conditions without convergence issues.

中文翻译：

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使用凸锥规划解决弹塑性问题的新型隐式 FEM-MPM 耦合框架


大多数现有的有限元法和质点法 (FEM-MPM) 耦合都是为显式求解器设计的。相比之下，隐式方案具有时间步长明显更大的优点，同时保持增强的稳定性，特别有利于解决刚性非线性问题。尽管如此，隐式 FEM-MPM 耦合的发展尚未得到广泛探索，在接触和弹塑性变形挑战方面留下了显着的差距。因此，本文在多变量变分原理和凸锥规划框架下提出了一种新颖的隐式时间积分内的统一FEM-MPM耦合方法，称为CP-FEMP。 CP-FEMP是第一个在凸锥规划下通过拉格朗日乘子和势垒法施加接触约束的成功尝试，它不仅可以解决束缚约束，还可以解决MPM和FEM域之间的摩擦接触，并确保收敛性和可行性，无论时间步长或网格分辨率。使用明确定义的插值空间可以避免连接接触中的接触锁定问题。将控制方程、相关的摩擦接触模型和相关的弹塑性本构定律表述为全局凸优化问题，并使用原对偶内点法有效求解。通过一系列标准接触和弹塑性基准测试，CP-FEMP 展示了其在 MPM 和 FEM 域之间精确传递接触力的能力，同时展示了值得称赞的节能属性。最后，将 CP-FEMP 应用于斜坡-挡土墙相互作用问题。 所有结果表明，CP-FEMP 为 FEM-MPM 耦合提供了全面的解决方案，允许在非线性接触和弹塑性大变形下进行大增量步长，并保证严格、硬的非穿透条件，而不会出现收敛问题。