Dual form of discontinuous deformation analysis

Discontinuous deformation analysis (DDA) is a numerical method for analyzing dynamic behaviors of an assemblage of distinct blocks, with the block displacements as the basic variables. The contact conditions are approximately satisfied by the open-close iteration, which needs to fix or remove repeatedly the virtual springs between blocks in contact. The results from DDA are strongly dependent upon stiffness of these virtual springs. Excessively hard or soft springs all incur numerical problems. This is believed to be the biggest obstacle to more extensive application of DDA. To avoid the introduction of virtual springs, huge efforts have been made with little progress related to low efficiency in solution. In this study, the contact forces, instead of the block displacements, are taken as the basic variables. Stemming from the equations of momentum conservation of each block, the block displacements can be expressed in terms of the contact forces acting on the block. From the contact conditions a finite-dimensional quasi-variational inequality is derived with the contact forces as the independent variables. On the basis of the projection-contraction algorithm for the standard finite-dimensional variational inequalities, an iteration algorithm, called the compatibility iteration, is designed for the quasi-variational inequality. The main processes can be highly parallelized with no need to assemble the global stiffness matrix. A number of numerical tests, including those very challenging, suggest that the proposed procedure has reached practical level in accuracy, robustness and efficiency, and the goal to abandon completely virtual springs has been reached.