Large Deformation Analysis in Geotechnics Using the Node-Based Smoothed Rpim and Second Order Cone Programming

Abstract

This paper proposes a novel method based on second-order cone programming (SOCP) and node-based smoothed radial basis point interpolation (NSRPIM) to address the limitations of traditional Newton-type methods in solving elastoplastic boundary value problems, as well as the numerical instability caused by mesh distortion. The displacement field is first discretized using the Radial Point Interpolation Method (RPIM), and the smoothed strain is computed within a node-based smoothing domain, with the assumption that stress remains constant within the smoothing domain. Based on this framework, the generalized Hellinger-Reissner variational principle is discretized, and by incorporating the flow rule, the elastoplastic boundary value problem is transformed into a second-order cone programming model and solved using a primal-dual interior-point algorithm. Furthermore, the SOCP-based node smoothing radial basis point interpolation method is applied to the particle method for solving small deformations at each step, enabling large deformation simulations and analyses in geotechnical engineering. Numerical analyses of four examples classified as two categories demonstrate that the proposed method offers high accuracy, computational efficiency, is unaffected by mesh distortion, and is well-suited for large deformation problems. Additionally, the advantage of NSRPIM lies in that all physical fields are represented by nodes in large deformation problems, eliminating the need for variable mapping. This increases computational precision and reduces computational cost. The analysis of the mechanical characteristics of layered sensitive clay landslides using the new method reveals that the strength ratio between different soil layers is one of the key factors influencing the failure mode.

Keywords: Node-Based Smoothed Radial Basis Point Interpolation (NSRPIM), Elastoplastic Boundary Value Problems, Second-Order Cone Programming (SOCP), Large Deformation Simulation, Layered Sensitive Clay Landslides