Discontinuous deformation analysis based on complementary theory

Abstract

The contact between blocks is treated by the open-close iteration in the conventional discontinuous deformation analysis (DDA), which needs to introduce spurious springs between two blocks in contactand to assume the normal stiffness and the tangential stiffness (the penalty factors). Unreasonable values of stiffness would result in numerical problems. To avoid the penalty factors and the open-close iteration, we reformulate the DDA as a mixed complementary problem (MiCP) and then choose the path Newton method (PNM) to solve the problem. Some examples including those originally designed by Shi are reanalyzed, which proves feasibility of the proposed procedure. discontinuous deformation analysis (DDA), contact problems, open-close iteration, complementary theory, non-smooth analysisIn numerical modeling geotechnical problems discontinuities and nonlinearities would cause a lot of difficulties and effective treatment of these difficulties has become a major focus of current research. Beginning with the conventional finite element procedures with interface elements, great progress has been made. The discrete models of highly discontinuous blocks, such as the discontinuous deformation analysis (DDA) proposed by Shi [1] , mark an epoch in modeling the discontinuities. The DDA is recognized as an efficient method based on the rigorous mathematical and mechanical theories, and has been solving diverse problems in geotechnical engineering (see refs. [2, 3] and many others).Since the object analyzed by the DDA is usually a system of blocks, the treatment of contact between blocks is a major task in the DDA. In the conventional DDA contacts between blocks are divided into three types: a convex angle to an edge, a convex angle to a concave angle and a convex angle to a convex angle. The three types of contact are finally reduced to contact of a pair of "angle to edge", referred to as a contact-pair in this study. If 3D problems are analyzed, the contact behaviors would become far more complicated. There have been many studies on the treatment of 3D contact, e.g., Chen and Zheng's penetration edge algorithm (PE) [4] , Wang and Lin's incision body algorithm (IB) [5] , Jiang and Yeung's "point to plane" algorithm [6] and "edge to edge" algorithm [7] , Wu and Juang's "angle to plane" algorithm [8] , Liu's fast common plane (FCP) algorithm [9] , Ali's main plane algorithm (MP) [10] . All these works are contributions to the development of robust programs of 3D DDA.In principle, a contact problem can be reduced to an optimization problem (or a variational problem) with inequality constraints. Now the penalty function method and the Lagrange multiplier method or its variants are generally utilized to solve the contact problems. Each method has its own merits and demerits. In the penalty function method, adopted in the original DDA and its improved version [11] , small penetration between blocks is allowed, and even necessary. Although the penetration is inappreciable, it is not easy to select reasonably the penalty factors (the st...

An efficient remedy for the false volume expansion of DDA when simulating large rotation

“…(18) and (20), the total strain vector e iþ1 and the total stress vector r iþ1 in the global system are obtained. Fig.…”

“…To simulate progressive failure, Jiang et al introduced a viscous damping component to absorb the kinetic energy of discrete blocks [18]. To alleviate the sensibility of the penalty parameters, in addition to the Lagrange multiplier method [19], the complementarity method [20] and the variational inequality theory [21] have been successfully applied to reconfigure DDA without requiring penalty parameters. Recently, DDA has been reformulated as a linear complementarity problem [22], which further enhanced the convergence and solution efficiency.…”

Complementarity problem arising from static growth of multiple cracks and MLS-based numerical manifold method

“…Usually, system G (u, v) = 0 represents the equilibrium condition; while the complementarity relationship in system (2) represents the constraints [32,33].…”

Condensed form of complementarity formulation for discontinuous deformation analysis

“…However, we have to admit that the procedure in ref. [23] is suitable to the general MiCP, which does not exploit the structure of CDDA-c. As a result, the efficiency has not reached the maximum.…”

“…To avoid the usage of artificial springs, Zheng and Jiang [23] reformulated DDA as a mixed complementarity problem where the contact forces are independent variables and the contact conditions are expressed by the complementarity equations. In the formulation, however, the complementarity equations representing the tangential contact condition are nonlinear, which compromises the convergence and accuracy of this formulation.…”