Block Models For Discontinuous Deformation Analysis

The discontinuous deformation analysis(DDA)is a recently widely used numerical method for both rock engineering and engineering geology.This method provides a series of com-prehensive computational tools for analyzing the contact interactions of rock blocks,includ-ing contact detections,contact type judgments,contact calculations and block deformations.Thus,this method has been increasingly researched in both science and engineering soci-eties.In this dissertation,the DDA is considered as the computing framework comprising the contact algorithms and block deformation models.Therefore,either the improvement of its contact algorithms or the promotion of the block models is the way to facilitate the accuracy and efficiency of this method.Although DDA contact algorithms have been exten-sively improved in recent years,the development of block models is so slow that the contact simulations using this method have been restricted.For this reason,five novel block models are proposed in this dissertation,where DDA contact algorithms are inherited.Besides,high performance solution schemes are also developed using the conjugate gradient method.The study of this dissertation is consist of the following six aspects:1.The Galerkin weighted residual method based three-dimensional block model with higher-order polynomial basis functions is presented,and is applied to analyze the deflec-tions of clamped thin,mid-thick and thick plates.This model is distinguished from the existing higher-order models in that:the global equilibrium equations of the new model are assembled through the variational formulations derived by the Galerkin weighted residual method;the essential boundary conditions are imported via the integral-form penalty method instead of the springs implemented at fixed points,and in this way the difficulties due to fixed point springs are avoided.Numerical experiments indicate that the proposed block model applies to the transversely loaded clamped square plates with various thickness.By compar-ing the results with analytical theories and ADINA solutions,the new model and computing formulations are demonstrated to be effective and accurate.2.The bonding block model as a novel fictitious discontinuous mesh based block model is presented.In the proposed model,blocks are discretized into the assemblies of triangular or quadrilateral elements.The overlapped edges of the neighboring elements within the same block are separated from each other and glued together by bonding springs,while the element edges along the boundaries of different blocks are connected through contact springs.The open-close iteration is inherited from the DDA to control the implementation of contact springs.Compared with existing fictitious discontinuous mesh based block models,the new model is more efficient owing to the chop for considering the contact detentions and open-close iterations of the sub-blocks within the same block.Besides,the proposed model introduces bonding springs to preserve nodal displacement continuity,and thus more accurate block stress'es can be figured out.Numerical experiments indicate that the presented model and computing scheme are effective and accurate.3.A new bonding block model is presented based on the augmented Lagrangian method.The original model is expressed into an optimization problem,in which gluing spring strain energies are used as the penalty item and Lagrangian multipliers are introduced to represent gluing spring forces.In this way,the total potential energy as the objective function becomes an augmented Lagrangian functional.The algorithm integrated with augmented Lagrangian iterations is designed to obtain the optimal solution during every time step.A numerical example is conducted to verify the accuracy and effectiveness of the proposed model and iterative algorithm.4.A novel block model is presented for contact simulations by coupling the DDA and discontinuous Galerkin finite element methods.In this model,the interior penalty Galerkin method is implemented on the blocks whose stresses have to be refined,while the contacts among the elements edges along block boundaries are dealt with using the DDA.The simul-taneous equilibrium equations of this coupling model are assembled in a mixed strategy:the entries are derived from both discontinuous Galerkin variational formulations and the mini-mization of contact spring strain energies.The contact algorithms of the DDA are inherited and generalized for the contacts of the elements within different blocks.Besides,two new criteria for contact detections are developed to further filter contacts and determine contact types.Three representative numerical examples are conducted,and the comparative inves-tigations demonstrate the accuracy and effectiveness of the proposed model and computing formulations.5.A new post-processing procedure is presented to overcome the drawback of the block models using fictitious discontinuous meshes,which can not obtain accurate nodal stresses.In the proposed procedure,the computing formulations for stress recovery are firstly derived out based on the moving least-square interpolation method,and then the recovered stress at any point can be straightforwardly obtained by simply substituting DDA results into these formulas.In contrast with the existing DDA post-processing procedures,the new one is capable of giving not only nodal stresses but also the accurate stresses at any point of a block.Numerical experiments are conducted to demonstrate the effectiveness and accuracy of the proposed procedure.6.Three preconditioners for the conjugate gradient method are proposed to solve the linear algebraic equations arising from the block models proposed in this dissertation,including the block Jacobi preconditioning matrix,the block symmetric Gauss-Seidel preconditioning matrix and the block symmetric super over-relaxation preconditioning matrix.The algo-rithms for computing the inverse matrices of these preconditioners are also designed.In the numerical experiments,the performance of the presented preconditioned conjugate gradientmethod are carefully compared,and the conclusion is valuable for engineering practices.