## 1 INTRODUCTION

The development of numerical methods for frictional contact problems is one of the most challenging task in the field of computational sciences. It combines several fields of research, including optimization theory to deal with arising inequality constraints, the development of appropriate discretization schemes in space and time, and tribology for the physical modeling of the surface interaction.

The numerical model to be developed can be subdivided into different tasks, using constitutive laws to describe the physical nature of the surface interaction and, on the other hand, kinematic relations to embed these laws into the generally nonlinear finite element method. We omit any small deformation considerations and focus on the large deformation frictional contact problem. For a comprehensive survey of contemporary methods, we refer to the textbooks of Laursen [1] and Wriggers [2].

Within this paper, we focus on the development of a new method for the contact kinematics. The governing constitutive laws have been extensively investigated by several authors (see among others He and Curnier [3] and Laursen and Oancea [4]). We apply the well-known Coulomb's law, using the established analogy between Coulomb friction and nonassociative plasticity (see for example [5]). However, the proposed approach is capable to embed all kinds of constitutive friction laws.

Our approach follows the lines of the work of Laursen [1, 6], referred to herein as the direct approach [7]. Therefore, in contrast to earlier works (see for example Wriggers and Simo [8] and Parisch [9]), continuum mechanical arguments are used to derive the underlying variational formulation.

In contrast to previous works dealing with large deformation frictional contact, we propose a new augmentation technique for the description of the frictional kinematics. The augmentation technique relies on the introduction of additional variables representing the local convective coordinates. Note that the terminus augmentation is widely used for different methods, including the augmented Lagrangian update algorithm. Within our newly proposed method for frictional contact, we refer to augmentation techniques as used in the context of multibody systems [10]. The additional variables are connected to the original ones by introducing additional algebraic constraints that are enforced by means of Lagrange multipliers. When compared with the traditional direct approach, the augmentation technique significantly simplifies the whole formulation.

After the discretization in space has been performed, the resulting differential-algebraic equations can be reformulated to reduce the size of the algebraic system to be eventually solved. That is, applying a size-reduction procedure within the discrete setting essentially recovers the size of the original system. It is worth noting that the size-reduction process does not affect the algorithmic conservation properties of the proposed discretization in space and time. In particular, our discretization approach inherits the conservation laws for linear and angular momentum from the underlying continuous formulation.

The development of structure-preserving time-stepping schemes for large deformation contact problems has been subject of extensive research, see Laursen and Chawla [11, 12], Armero and Petöcz [13, 14], and Hesch and Betsch [15, 16]. Similar to these works, the spatial discretization of the contact surface used in the present work is based on the node-to-surface (NTS) method. The extension to more sophisticated models using mortar-based formulations [17-22] will be dealt with in a subsequent work.

The article is organized as follows. The fundamental equations of the underlying problem in strong and weak form are outlined in Section 2. In this connection, the well-established description of the frictional kinematics is summarized, and the newly proposed augmentation technique is introduced. The spatial discretization and the NTS method together with the specific formulation of the augmentation technique is given in Section 3. In Section 4, we apply a suitable time integration scheme and verify algorithmic conservation of linear and angular momentum. Representative numerical examples are presented in Section 5. Eventually, conclusions are drawn in Section 6.