A semi-smooth Newton and Primal–Dual Active Set method for Non-Smooth Contact Dynamics

Abstract

Multi-rigid-body dynamic contact systems, in other words Non Smooth Contact Dynamics (NSCD) problems, generate some inherent difficulties to multivocal laws, which results in non-linearities and non-smoothness associated to frictional contact models. Recently, Primal–Dual Active Set strategies (PDAS) have emerged as a promising method for solving contact problems. These methods are based on the following principle: the frictional contact conditions are restated as non-linear complementary functions for which the solution is provided by the iterative semi-smooth Newton method. Based on these prerequisites, this contribution aims to provide a generalization of the NSCD-PDAS for dynamic frictional contact problems. Several numerical experiments are reported for algorithm validation purposes and also to assess the efficiency and performances of PDAS methods with respect to the Newton/Augmented Lagrangian and the Bi-Potential methods.

Introduction

Predicting snow avalanches, railway ballast track deformations or designing effective process for C02 capture requires taking into account local interactions modeling to assess macroscopic properties. This research issue is particularly topical because of its wide range of applications and the computational resources constantly increasing. As a matter of fact, multi-rigid-body dynamic contact field, as granular materials, illustrates some inherent difficulties to multivocal laws, which results in non-linearities and non-smoothness associated to frictional contact models. Numerical simulations of discrete materials, and specifically granular ones, was first initiated within the framework of the Discrete Element Method (DEM) [1], [2], [3]. DEM relies on appropriate regularization techniques for non-linear contact laws while using an explicit time scheme to ease their numerical treatments. Currently, this approach is still widely used in the engineering process design (see for example the MFIX open source software [4]). Some DEM weaknesses, reported in [2], include the small time steps inherent in the explicit time scheme or the damping introduced for stability matters.

Moreau established the theoretical concepts to properly handle the non-smooth mechanics featured by non-differentiable laws in its common definition. Thus, he introduced the second-order sweeping process as the framework of mathematical analysis and numerical schemes designed for granular materials. The cornerstone is the formulation for the Signorini problem in terms of velocities and impulses, leading to the Moreau–Jean time-stepping approach [5]. This numerical scheme is implicit and the energy conservation property holds unlike the DEM framework. The local conditions of contact between rigid bodies are ensured by means of a Non-Linear Gauss–Seidel (NLGS) developed by M. Jean and J. J. Moreau [5], [6], [7], [8]. Also, The numerical aspects and certain algorithmic developments for the Non-Smooth Contact Dynamics (NSCD) have been proposed in [9]. Some alternatives such as conjugate gradient solvers have been proposed [10], with a specific analysis for parallel computing [11], [12]. As pointed out by Dubois et al. [2], this non-linear formulation is computationally expensive compared to DEM, especially for compressed granular materials, but larger time steps can be considered. Another point to consider in NSCD is the way to ensure the local contact condition. Usual approaches are based on the bi-potential or the augmented Lagrangian theory [13], [14], [15]. Efficiency of such methods strongly depends on the penalty coefficients [14], [16], [17]. Some improvements have been proposed to overcome this issue [18], [19], but they are more time consuming.

Numerical simulations carried out for hyper-elastic bodies with frictional contact have led to specific approaches [16], [20], [21], [22], [23], [24], [25], [26], [27], [28], [29]. Recently, Primal–Dual Active Set strategies (PDAS) have emerged as a promising method for solving contact problems (see [30], [31], [32], [33]). These methods are based on the following principle: the frictional contact conditions are restated as non-linear complementary functions for which the solution is provided by the iterative semi-smooth Newton method [30], [31]. In practice, contact with Coulomb friction conditions can be formulated in terms of a fixed point problem related to a quasi-optimization problem.

Therefore, the whole elastic body’s problem, including the frictional contacts, consists of successive solutions of many elastic problems with simpler boundary conditions: Dirichlet, Neumann or Robin boundary conditions depending on the pseudo-transient status of the contact nodes [34], [35]. This method is also known as the stabilized or Nitsche’s methods [23], [36], [37].

According to our knowledge, few references on Active Set methods for solving multi-rigid-body dynamic contact problems are available. Based on a pseudo-rigid body assumption, Koziara and Bicanic [38] modified a semi-smooth Newton technique to effectively deal with the frictional contact problem. Within a framework of optimization, Sharaf [39] proposes an Active Set method for solving positive definite and positive semi-definite Linear Complementary Problems (LCP). The effectiveness of its approach has been tested on the toy problem of static vertical rigid spheres assuming frictionless contacts. The resulting discrete model for the Newton–Euler dynamics equations with Signorini’s conditions is formulated as a LCP. For multi-rigid-body dynamic contact problems, Barboteu and Dumont [40] have developed a PDAS type method to address the local treatment of the contact conditions using the Non-Smooth Contact Dynamics (NSCD) formalism [5], [6], [8], [41], [42]. Comparisons with the well-established methods bi-potential and augmented Lagrangian outline the efficiency of the NSCD-PDAS methods for the considered validation tests [40], dealing with frictionless model of multi-rigid-bodies contacts. The noted efficiency in [40] must be confirmed for problems involving friction. This contribution aims to provide a generalization of the NSCD-PDAS for dynamic frictional contact problems and to assess its efficiency with respect to the bi-potential and augmented Lagrangian theories.

The article is organized as follows. First, the mathematical model for the frictional contact conditions as formulated in contact dynamics is stated in Section 2. The complementary functions for both contact and friction are formulated in Section 3. In addition, their generalized derivatives are derived to provide the core of the Semi-Smooth Newton method. Section 4 presents the NSCD framework, especially the equations of motion, the local–global mapping and the NLGS iterative solver. Based on these prerequisites, Section 3 introduces two versions of the PDAS to tackle the frictional contact for solid rigid bodies. Several numerical experiments are reported in Section 6 for algorithm validation purposes and also to assess the efficiency and performances of PDAS methods with respect to the Newton/Augmented Lagrangian and the Bi-Potential methods. Finally, some perspectives and future works are presented in Section 7.