On the Implementation of augmented Lagrangian method in the two‐dimensional discontinuous deformation Analysis

Abstract

The discontinuous deformation analysis (DDA) is a discontinuum-based method, which employs a penalty method to represent the contact between blocks. The penalty method is easy to be implemented in the program, but the contact constraint is only approximately satisfied. Penetrations between contacting blocks are unavoidable even if the penalty value is very large. To improve the contact precision in the DDA, an augmented Lagrangian method is introduced, which can make use of advantages of both the Lagrangian multiplier method and the penalty method. This paper provides a detailed implementation of the augmented Lagrangian method in the DDA program and compares it with the standard DDA on the computational efficiency and contact precision.Compared with the Lagrangian multiplier method [3,4], penalty methods have an important advantage that no additional unknowns are introduced into the final system of equations and the global stiffness matrix is positive definite, which makes the implementation of penalty methods easy and straightforward [5][6][7]. Hence, penalty methods have been widely used in computer codes for the treatment of constraint conditions [8][9][10].In penalty methods, the contact force is assumed to be proportional to the penetration distance. Therefore, a penetration between blocks is inevitable if contact force exists. A drawback of this method is that the accuracy of the solution depends strongly on the choice of penalty values. In static problems, the penalty parameter should be, in principle, an arbitrarily large number to enforce the constraint condition. However, for a computer with a finite number of digits, it should not be so large that the governing equations become ill-conditioned. On the other hand, too small a penalty parameter may result in an unacceptable penetration of one body into the other, and the overall response is distorted.The augmented Lagrangian method (ALM) is an iterative method to obtain the exact solutions for contact forces with important advantages over the more traditional Lagrangian multiplier and penalty methods. It was initially proposed by Hestenes [11] and Powell [12] for solving non-linear programming problems with equality constraints. It was extended to treat convex differentiable optimization problems with inequality constraints (such as the frictionless contact problem) by Rockafellar [13]. Later, it was widely used in the contact problems of numerical methods [14][15][16][17][18][19][20][21][22][23][24][25].The essential concept behind the ALM is to use an augmented Lagrangian multiplier that can be iteratively calculated by adding the penalty force onto the Lagrangian multiplier obtained in last iteration. In the ALM, the Lagrangian multiplier is not unknown any more as that in the Lagrangian multiplier method. Hence, the simplicity of the penalty method is retained, and its disadvantages are minimized. The precision of solutions depends on residual forces, that is, the penalty forces produced during the current iteration. The residual force should be as ...

Investigation of highly efficient algorithms for solving linear equations in the discontinuous deformation analysis method

“…The standard DDA programme consists of three main loops: time‐step iteration, open‐close iteration and linear equation solver iteration . The linear equation solver iteration is the smallest loop, and Section 2 of this paper proves that the efficiency of the DDA method is primarily controlled by this aspect.…”

An implicitly coupled hydro-geomechanical model for hydraulic fracture simulation with the discontinuous deformation analysis

“…(7), is detailed in [20], while incorporation of both the normal and shear terms is discussed in [21]. More information regarding the original form of the contact constraints and how they are incorporated into the global equations of motion may be found in [22,23].…”

A new algorithm to identify contact types between arbitrarily shaped polyhedral blocks for three-dimensional discontinuous deformation analysis

“…As a discontinuum-based method, discontinuous deformation analysis (DDA) [3,4] was originally developed to investigate the mechanical behaviour of jointed or blocky systems, such as rock masses. DDA is similar to finite element method (FEM) [5], but it additionally imposes the no-penetration contact constraints at block boundaries using the penalty method [4], the Lagrange multiplier method [6,7], the augmented Lagrangian method [8][9][10][11] or the complementarity method [12][13][14][15][16]. The sliding behaviour along the joints is controlled by the Coulomb friction law [4].…”