Discontinuous Deformation Analysis Method Based On Complementary Theory

The Discontinuous Deformation Analysis(DDA) method is suitable for solving geotechnical problems with many discontinuous interfaces inside the concerned domain,such as the stability analysis of fractured rock mass.The history of various discontinuous deformation analysis methods including DDA is stated.The penalty function method and the Lagrange multiplier method or its variants are generally utilized to solve contact problems in DDA.Each method has the merits and demerits and uses a process called the open-close iteration to realize the satisfaction of the contact conditions.To avoid the penalty factors and the open-close iteration,we reformulate the conventional DDA and build CDDA and VIDDA by introducing the complementary theory and the variational inequalty theory respectively.Then,CDDA and VIDDA are transfered to non-smooth equations by the C-function and the natural projection map.Various algorithms are ulitized to solve the equations and some practical examples originally designed by Shi are reanalyzed,which prove the recreated DDA methods are feasible.Finally,the CDDA and the VIDDA are applied in some common fields.The detailed contents are as follows:In chapter 2,the mathematical theory of penalty function method and Lagrange multiplier method is stated,then we analysis the merits and demerits of each method and point out that the open-close iteration is uncertainly convergent.In chapter 3,firstly we introduce some important concepts about complementary problems related to this dissertation.Secondly,we add the contact forces to basic unknowns,from the variational formulation of momentum conservation instead of the minimizing the potential function,we derive a system of equations for momentum conservation.Then we utilize C-functions to transform the inequalities reflecting contact conditions on all contact-pairs into the equivalent equations called the contact equations.Finally we combine the momentum conservation equations with the contact equations and thus obtain a system of nonlinear equations some of which are continuous but non-smooth.We call the proposed equations as the Complementary Discontinuous Deformation Analysis(CDDA).In chapter 4 we choose the hot C-functions and non-smooth equations algorithms to slove CDDA.Firstly we test the simplest C-function,the min function,which is smooth over the plane except on a line is ulitized,and use the Path Newton Method to slove equations.Then we designate the C-function as the FB function that is smooth everywhere except at the original point,and improve the FB Line Search Algorithm for solveing the system of nonlinear equations.Some examples are analyzed,which prove the procedure is feasible.In chapter 5 the variational inequalities theory is applied to this study.The DDA is reformulated as a box-constrained variational inequality problem.Based on the fact that the solution of variational inequality must be a fixed point of the natural projection map,we transfer this problem to the solution of non-smooth equations.We call the equations as the VIDDA.Then,the Path Newton Method(PNM) is used to solve the equations.Two examples are selected validate the procedure.Finally the precsion,efficiency and robustness of the three algorithms we proposed are compared.Finally in chapter 6,the methods proposed in this dissertation are used to solve some geotechnical problems,such as slop stability,the stability of gravity dam against sliding and modeling pre-tension bolt.Conclusions and prospects are put in Chapter 7.