An ALE method for penetration into sand utilizing optimization-based mesh motion

Abstract

The numerical simulation of penetration into sand is one of the most challenging problems in computational geomechanics. The paper presents an arbitrary Lagrangian–Eulerian (ALE) finite element method for plane and axisymmetric quasi-static penetration into sand which overcomes the problems associated with the classical approaches. An operator-split is applied which breaks up solution of the governing equations over a time step into a Lagrangian step, a mesh motion step, and a transport step. A unique feature of the ALE method is an advanced hypoplastic rate constitutive equation to realistically predict stress and density changes within the material even at large deformations. In addition, an efficient optimization-based algorithm has been implemented to smooth out the non-convexly distorted mesh regions that occur below a penetrator. Applications to shallow penetration and pile penetration are given which make use of the developments.

Introduction

Penetration into sand is one of the oldest problems in geomechanics, and it turned out to be also one of the most challenging. Rigorous modeling is very difficult because of the large local deformations in the vicinity of the penetrator, the evolution of material interfaces and free surfaces, the changing contact conditions, the large stiffness variations, and the complex nonlinear behavior of the granular material. Therefore, penetration into sand has not been extensively explored so far from a numerical viewpoint.

The finite element method (FEM) is the dominating tool in computational geomechanics because of its broad applicability and technological sophistication. Typical models are based on a Lagrangian description, which is in many respects the most attractive approach for problems where path-dependent material response and evolving interfaces are present. However, if material deformations are large severe element distortion may occur which may slow down or even terminate the calculation. Element distortion will not occur in the Eulerian methods commonly used in computational fluid dynamics (CFD) because the mesh is kept spatially fixed. However, tracking of free surfaces and material motion becomes non-trivial because the material is allowed to flow through the mesh.

The arbitrary Lagrangian–Eulerian (ALE) methods have been developed in order to overcome the difficulties arising from the purely Lagrangian and Eulerian approaches, and to combine their advantages [1], [2], [3], [4], [5], [6]. The major advantage of ALE is that the computational mesh is regarded as a independent reference domain. As a consequence, the ALE mesh can be continuously smoothed so that element quality remains acceptable during the entire calculation. Mesh connectivity is kept unchanged, hence the solution variables can be remapped onto the improved mesh by using conservative CFD advection algorithms.

Despite their popularity in fluid–structure interaction and solid mechanics, applications of ALE methods to soil mechanical problems are still rare. Examples include [7], [8], [9], [10], [11], [12], but the employed methods reveal two limitations when applied to penetration into sand. First, common algorithms for node relocation are not qualified to smooth the non-convex mesh regions that inevitably occur during penetration of blunt bodies. As a consequence, the range of problems that can be addressed by these ALE methods is not much larger than for a purely Lagrangian approach. Second, the constitutive equations involved (Mohr–Coulomb, Drucker-Prager, modified Cam-Clay, etc.) are not able to realistically reproduce the stress- and density-dependent response of sand under monotonic and cyclic loading.

The mechanical behavior of sand is very complex and has several influencing factors. Amongst others, it is generally a function of the effective stress state, the relative density, and the material history due to monotonic or cyclic loading [13]. An important characteristic that distinguishes the behavior of sand from that of common solids is dilatancy, which has been shown to depend on both the effective stress state and density state [14]. Moreover, sand shows asymptotic behavior [15] and reaches a critical state after monotonic loading paths starting from a particular stress and density state.

The best models currently available to properly reproduce this complex nonlinear behavior are phenomenological constitutive equations that rely on the continuum representation of sand. However, there are just a few constitutive equations for granular solids available which need only a single set of material constants and are then able to simulate the mechanical behavior at finite deformations and under complex loading paths over the wide range of densities and stress states present during penetration processes.

In this paper we present an ALE method which is particularly suitable for penetration into sand. Its unique features are (i) an advanced constitutive equation for sand and (ii) an efficient and robust optimization-based mesh regularization algorithm which delivers excellent results even on non-convexly distorted domains. For simplicity, we consider sand as a single-phase medium, being either dry or fully saturated and locally drained. Moreover, only plane strain and axisymmetric quasi-static problems are considered in the numerical treatment. The penetrator is assumed to be either smooth (zero friction) or perfectly rough (no sliding). The 3-node triangle element [16], [17] is used for the spatial discretization despite of its tendency to lock up. However, the impact of this undesirable feature is not much significant provided that special mesh pattern are used.

The remainder of this paper is structured as follows. Section 2 presents the governing equations associated with our ALE approach. The steps of the operator-split solution procedure are described in detail in Section 3, and Section 4 shows example applications concerning shallow penetration and pile penetration into sand. The paper closes with some concluding remarks in Section 5.