Dual form of discontinuous deformation analysis

• Published in: Computer methods in applied mechanics and engineering 2016-06, Vol.305, p.196-216
• Main Authors: Zheng, Hong, Zhang, Peng, Du, Xiuli
• Format: Article
• Language: English
• Subjects: Algorithms; Contact; Contact problems; Deformation; Discontinuous deformation analysis; Finite-dimensional variational inequalities; Inequalities; Iterative methods; Mathematical analysis; Mathematical models; Open–close iteration; Projection–contraction algorithm; Springs
• ISSN: 0045-7825; 1879-2138
• DOI: 10.1016/j.cma.2016.03.008

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. •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.