تجزیه و تحلیل حساسیت برای مسائل تماس اصطکاکی در فرمول بندی لاگرانژی افزوده

Direct differentiation method of sensitivity analysis is developed for frictional contact problems. As a result of the augmented Lagrangian treatment of contact constraints, the direct problem is solved simultaneously for the displacements and Lagrange multipliers using the Newton method. The main purpose of the paper is to show that this formulation of the augmented Lagrangian method is particularly suitable for sensitivity analysis because the direct differentiation method leads to a non-iterative exact sensitivity problem to be solved at each time increment. The approach is applied to a general class of three-dimensional frictional contact problems, and numerical examples are provided involving large deformations, multibody contact interactions, and contact smoothing techniques.

Sensitivity analysis provides quantitative information on the variation of the response of a system, e.g. a mechanical system, related to a variation of parameters, e.g. material or shape parameters, on which the system depends. Response sensitivity is thus essentially a derivative of the response with respect to these parameters, however, the dependence is implicit—through the governing equations of the problem. Typically, the need for response sensitivities arises in optimization and inverse problems whenever gradient-based minimization algorithms are used, however, numerous other applications appear in the engineering practice, e.g. imperfection sensitivity analysis, error analysis, and others. Efficient analytical methods of sensitivity analysis are already available for a variety of problems, see the monographs of Kleiber et al. [16], Choi and Kim [6], Kowalczyk [22]. In particular, this concerns the path-dependent problems of elasto-plasticity, cf. Tsay and Arora [42], Kleiber [15], Michaleris et al. [28]. Frictional contact problems, being the main concern of this work, also belong to the class of path-dependent problems. Continuum formulations for the most general case of three-dimensional multibody, large deformation frictional contact problems have been developed by Laursen and Simo [25], Klarbring [14], Pietrzak and Curnier [31] and [32]. In these formulations, an essential role is played by the contact constraints enforced on the potential contact surfaces, and several approaches have been developed for the treatment of these constraints and for the discretization of the contact interaction terms, see the recent monographs of Laursen [24] and Wriggers [43]. The penalty method is the simplest and apparently the most widely used method of enforcing contact constraints. It leads to a purely displacement formulation and straightforward implementation, however the constraints cannot be enforced exactly. Furthermore, ill-conditioning appears when the penalty parameter is increased in order to improve satisfaction of the constraints. Recently, the augmented Lagrangian method has become a popular alternative, free of the main drawbacks of the penalty method. There are two solution schemes commonly used in the context of the augmented Lagrangian method. The so-called Uzawa method, which seems to be more popular, combines the augmented Lagrangian regularization with a first-order update of Lagrange multipliers, cf. Simo and Laursen [36]. Alternatively, a Newton-like solution scheme can be applied to solve the saddle-point problem simultaneously for the displacements and Lagrange multipliers, cf. Alart and Curnier [2], Pietrzak and Curnier [31] and [32]. The latter approach is adopted in this work, however, the penalty and Uzawa methods are also discussed from the point of view of sensitivity analysis. The most widely used contact discretization scheme is the node-to-segment approach. In order to avoid convergence problems caused by discontinuity of the normal vector, which is characteristic for the simple node-to-segment implementations, contact smoothing techniques are often introduced, e.g. Pietrzak [31], Puso and Laursen [33], Krstulović-Opara et al. [23]. This approach is used in the present work. An alternative approach, employing the mortar method and leading to a segment-to-segment strategy, has recently been developed, cf. Puso and Laursen [34]. Several developments of sensitivity analysis for frictional contact problems have been reported in the literature. If the penalty approach is adopted, then the general structure of the frictional contact problem is similar to that of the elasto-plasticity, so that the respective formalism of sensitivity analysis by the direct differentiation method can be directly applied, cf. Kim et al. [12] and [13], Stupkiewicz et al. [40]. In the case of the augmented Lagrangian method with the Uzawa-type algorithm, the exact sensitivity analysis requires iterations corresponding to the iterative update scheme for Lagrange multipliers. In order to avoid this, Srikanth and Zabaras [37] have introduced an approximate non-iterative sensitivity problem in which oversized penalties are used. Sensitivity analysis methods specialized for frictionless contact problems have been developed, e.g., by Bendsoe et al. [4], Haslinger et al. [10], Fancello et al. [8], Tardieu and Constantinescu [41], Hilding et al. [11]. On the other hand, approaches applicable to optimization of metal forming processes with a rigid-plastic material behaviour have been proposed, e.g., by Kleiber and Sosnowski [17], Antunez and Kleiber [3], and Fourment et al. [9]. In this work, the direct differentiation method (DDM) of sensitivity analysis is applied to frictional contact problems in the augmented Lagrangian formulation. In particular, we show that the augmented Lagrangian treatment of contact constraints, proposed by Alart and Curnier [2] and Pietrzak and Curnier [31] and [32], is particularly suitable for sensitivity analysis because it leads to a non-iterative exact sensitivity problem at each time step. In order to illustrate that, sensitivity analysis is developed for a general class of three-dimensional contact problems including friction, multibody contact interaction and contact smoothing. Preliminary results concerning sensitivity analysis for the augmented Lagrangian formulation and for contact smoothing techniques have been published in conference papers [39] and [26], respectively. The paper is organized as follows. A framework for the sensitivity analysis for path-dependent problems is presented in Section 2. The direct problem is introduced in a general discretized form, and then the direct differentiation method is used to derive the corresponding sensitivity problem. In Section 3 we show that this general framework applies also to the contact problems. Contact kinematics and contact constraints are specified in Section 3.1. Detailed discussion of the penalty and augmented Lagrangian methods for a simplified case of elastic frictionless problems is provided in Section 3.2. Finally, the general frictional contact problem is formulated in Section 3.3. Remarks concerning the finite element implementation are provided in Section 3.4, and illustrative numerical examples are given in Section 4.

The direct differentiation method of sensitivity analysis has been applied to the frictional contact problems in the augmented Lagrangian formulation. It has been shown that the augmented Lagrangian treatment of contact constraints, as proposed in [2], [31] and [32], is particularly suitable for sensitivity analysis. This is because the tangent matrix, used for the solution of the direct problem simultaneously for the displacements and Lagrange multipliers, is the exact tangent operator of the problem, including the sensitivity problem. As a result, a non-iterative exact sensitivity formulation is obtained and the cost of computation of a response sensitivity with respect to one parameter constitutes only a small fraction of the solution cost of the direct problem. This beneficial feature of the usual sensitivity formulations based on the direct differentiation method is thus preserved. The present implementation of sensitivity analysis relies on the use of the AceGen symbolic code generation system. Accordingly, the finite element implementation of the direct differentiation method has been efficiently performed for an advanced three-dimensional smooth contact formulation. The formulation is applicable to a general class of problems including finite-deformation, finite-slip frictional contact of deformable bodies. The applicability of the specific contact smoothing technique, employing the B-spline patches [31], used in the reported numerical examples is restricted to structured meshes only. Nevertheless, the general approach can be directly applied to other smooth node-to-segment formulations, e.g. [23] and [33]. Applicability and accuracy of the approach have been demonstrated by two numerical examples featuring large deformations in 3D. Since frictional contact problems are path-dependent, the sensitivity analysis is carried out incrementally, as is the primal analysis. In both examples, the loading is essentially monotonic (though the contact, slip and stick zones develop in a complex manner), however, the approach is fully applicable to general loading paths, see, e.g., the cyclic loading example in [40].