Discontinuous deformation analysis 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 . 6 dimensional generalized force vector in Eq . matrix formed by all the local frames at block Eq . matrix of the th contact local frame at block Eq . cohesion at the th contact Eq . incremental displacement vector of block Eq . time interval of a time step Eq . Young s modulus in all the examples stiffness matrix of block Eq . tolerance of contact force vector in the compatibility iteration direction normal strain of block Eq . direction normal strain of block Eq . flexibility matrix Eq . 6 dimensional flexibility vector Eq . vector valued gap function in Eq . acceleration of gravity in all the examples a part of normal gap of the contact pair formed by an edge of and a vertex of in Eq . a part of tangential slide of the contact pair formed by an edge of and a vertex of in Eq . normal gap of the th contact Eq . tangential slide of the th contact Eq . shear strain component of block Eq . equivalent stiffness matrix of block Eq . allowable maximum displacement within a time step in all the example density of block Eq . mass matrix of block Eq . friction coefficient at the th contact Eq . the number of contact pairs within a time step in Section 4 unit normal vector of local frame of a contact pair and and dimensional vectors. 


Summary: The dual form of discontinuous deformation analysis abbreviated as DDA d with the contact forces as the basic variables is established. DDA d does not need the virtual springs the stiffness of which has strong influence on the computational results. An iterative algorithm for the finite dimensional quasi variational inequality in DDA d called the compatibility iteration is designed. DDA d is more accurate and robust than the classic discontinuous deformation analysis and comparable in efficiency with DDA.