The numerical modelling of natural disasters such as landslides presents several challenges for conventional mesh-based methods such as the finite element method (FEM) due to the presence of numerically challenging phenomena such as severe material deformation and fragmentation. In contrast, meshfree methods such as the reproducing kernel particle method (RKPM) possess unique features conducive to modelling extreme events such as the absence of a structured mesh and the ease of adaptive refinement, among others. While the semi-Lagrangian reproducing kernel (SL-RK) shape functions of RKPM defined in the current configuration have proven to be effective in extreme event modelling, the computational cost for the re-evaluation of the shape functions at every time step is costly. In this work, a deformation-dependent coupling of the Lagrangian reproducing kernel (L-RK) and SL-RK approximations is proposed for the solution of a hydro-mechanical formulation for effective simulations of landslides. The ramp function is constructed based on an equivalent plastic strain as a deformation-dependent transition from L-RK shape functions to SL-RK ones as the deformation progresses. The particular focus of the paper will be on modelling seepage-induced landslides with a mixed $$u$$ u –$$p$$ p formulation to couple the solid and fluid phases. Examples are presented to examine the effectiveness of this coupled Lagrangian/semi-Lagrangian reproducing kernel (L–SL RK) formulation and to highlight its performance in landslide modelling.

The numerical modelling of natural disasters such as landslides presents several challenges for conventional mesh-based methods such as the finite element method (FEM) due to the presence of numerically challenging phenomena such as severe material deformation and fragmentation. In contrast, meshfree methods such as the reproducing kernel particle method (RKPM) possess unique features conducive to modelling extreme events such as the absence of a structured mesh and the ease of adaptive refinement, among others. While semi-Lagrangian RKPM (SL RK) where the kernel functions are defined in the current configuration has proven to be effective in extreme event modelling, the computational cost for the re-evaluation of the shape functions every time step is a drawback. A selective employment of SL RK approximation can be a remedy, but the region where the SL RK approximation is needed cannot be pre-determined in most cases. In this work, a space-time evolving coupling of the Lagrangian and SL RK approximations is proposed for the solution of a hydro-mechanical formulation for efficient and accurate simulations of landslides where regions of severe deformation cannot be predetermined. This formulation utilizes a Lagrangian formulation for regions of the model where deformation is not severe while those regions with severe deformation are described by a semiLagrangian formulation where the shape functions are updated at each time step. The coupling is achieved using equivalent plastic strain as a space-time transition from Lagrangian shape functions to semi-Lagrangian ones as deformation progresses. The particular focus of the paper will be on modelling seepage-induced landslide with a mixed 𝑢-𝑝 formulation to effectively couple the solid and fluid phases. Examples which demonstrate the effectiveness of this coupled L-SL RKPM, include a wave propagation in poroelastic media, a poroplastic soil cylinder impact problem, and a seepage-induced levee failure leading to landslide.