A Lagrangian/semi-Lagrangian coupling approach for accelerated meshfree modelling of extreme deformation problems

Abstract

In the reproducing kernel particle method (RKPM), the approximation is achieved through construction of shape functions in the physical domain and the interaction of neighbouring nodes. When modelling large deformation problems, Lagrangian and semi-Lagrangian formulations have been proposed, where the RK functions are evaluated in the reference initial coordinates and in the current configuration, respectively. The Lagrangian RKPM breaks down when the mapping between current and reference configuration is not one-to-one and the deformation gradient is not invertible, such as in modelling high velocity impact and penetration processes. On the other hand, the semi-Lagrangian RKPM formulation with RK functions evaluated in the current configuration eliminates the requirement on the deformation mapping but results in a high computational cost. In order to retain the advantages of both formulations, a blending-based spatial coupling scheme is proposed to allow transition from the Lagrangian to the semi-Lagrangian RKPM meshfree formulation. Spatial consistency and temporal stability of the proposed coupling approach are analysed. The analysis results show that the critical time step of the coupled formulation is larger than the worst case between the pure Lagrangian and pure semi-Lagrangian RK cases. Moreover, an algorithm to update three-dimensional semi-Lagrangian kernel supports and smoothing cells is introduced to enhance the solution accuracy of the semi-Lagrangian portion of the proposed coupling scheme. The proposed update algorithm avoids the numerical issues of either insufficient support coverage or loss of locality associated with the conventional pure semi-Lagrangian formulation under large deformations. The performance of the proposed Lagrangian/semi-Lagrangian coupling approach is verified through a suite of benchmark problems, including extreme deformation problems.