In the present high-order 3-D DDA method, block contact constraints are enforced using the penalty method. This approach is quite simple, but may lead to inaccuracies that may be large for small values of the penalty number. The penalty method also creates block contact overlap, which violates the physical constraints of the problem. These limitations are overcome by using the augmented Lagrangian method that is used for normal contacts in this research. In this paper, contact constraints are enforced in high-order 3-D DDA using the augmented Lagrangian method and the formulations are presented. Moreover, a code has been programmed by Visual C++ and an illustrative example is used to validate the new formulations and the code. Using the augmented Lagrangian method to enforce contact restraints retains the simplicity of the Penalty method and reduces the disadvantages of it.
The discontinuous deformation analysis (DDA), which is an energy-based method, is an alternative to the distinct element method (DEM) for discontinuity-based problems [1]. This method can be used to solve problems involving discontinuous media. Original DDA formulation utilizes first order displacement functions to describe the block movement and deformation. Therefore, stress or strain is assumed constant through the block and the capability of block deformation is limited. This may yield unreasonable results when the block deformation is large and geometry of the block is irregular. There are some published papers on deformable blocks in 3-D DDA. Beyabanaki et al. [2-4] implemented Trilinear and Serendipity hexahedron FEM Meshes into 3-D DDA. Beyabanaki et al. [5-7] presented 3-D DDA with second-and third-order displacement functions. Beyabanaki et al. [8] presented 3-D DDA with nth-order displacement functions. Recently, contact theory of nth-order 3-D DDA is presented by Beyabanaki et al. [9]. The penalty method was originally used by the abovementioned 3-D DDA researchers to enforce contact constraints at the block interface. The accuracy of the contact solution depends highly on the choice of the penalty number and the optimal number cannot be explicitly found beforehand. Obviously, the penalty number should be very large to achieve zero interpenetration distance. However, a very high penalty number leads to progressive ill-conditioning of the resulting system and thus one cannot hope to achieve high-accuracy solutions with this approach. A well-known method to overcome these problems for equality constrained problems is the augmented Lagrangian method [10]. The augmented Lagrangian method has been advocated by Lin et al. [11] in two dimensional discontinuous deformation analysis. In this research, the same method has been implemented in high-order three-dimensional discontinuous deformation analysis and an illustrative example is presented for demonstrating this new approach.
In the original 3-D DDA, the block displacements function is equivalent to the complete first-order displacement approximation; constant strains and constant stresses are assumed within each block. When displacement functions are taken as nth-order functions: The high-order function is necessary in most engineering analyses since it can represent stress concentrations within one block. In two dimensions, the contact types between blocks include corner-to-corner, corner-to-edge and edge-toedge;
