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Improving the DDA Solution Time by Modifying the Contact Enforcement Technique

Improving the DDA Solution Time by Modifying the Contact Enforcement Technique M.S. Khan2, A. Riahi1, & J.H. Curran1, 3 1. Rocscience Inc., Toronto 2. Schlumberger Canada Ltd. 3. Professor Emeritus, Civil Engineering Department & Lassonde Institute, University of Toronto ABSTRACT This paper will focus on the efficiency of Discontinuous Deformation Analysis (DDA) applied to jointed rock problems. DDA uses an implicit time-integration scheme to solve the governing equations of motion through time. It therefore requires forming global system of equations. In recent studies, the authors have shown that although theoretically there is no restriction on DDA time-step size, DDA-Shi is in fact slower than methods that utilize an explicit time integration scheme. The observed slowness in the DDA solution of real-scale geotechnical problems is caused by the combination of the highly nonlinear nature of contact problems and the size of the global system of equations in implicit time integration techniques. This paper will focus on improving the computational efficiency of DDA-Shi by adopting a concept similar to the soft contact approach. It will be shown that by removing the no-penetration constraint of the contact penalty enforcement in DDA-Shi, the solution time decreases significantly. The conclusions are verified in examples that examine the stability of slopes in jointed rock masses. RÉSUMÉ Cet article mettra l'accent sur l'efficacité de l'analyse de la déformation discontinue (ADD) appliqués aux problèmes des roches démontables. L’ADD utilise un système d'intégration temporelle implicite pour résoudre les équations qui gouvernent le mouvement à travers le temps. Il faut donc former un système global des équations. Dans des études récentes, il a été démontré que même si théoriquement il n'y a aucune restriction sur la taille du temps-étape de l’ADD, l’AAD-Shi est effectivement plus lente que les méthodes qui utilisent un système d'intégration temporelle explicit. La lenteur observée dans la solution des problèmes géotechniques d’échelle réelle dans l’ADD est causée par la combinaison de la nature des problèmes de contact, qui sont extrêmement non-linéaire, et de la taille du système global des équations dans les techniques d'intégration temporelle implicite. Cet article se concentrera sur l'amélioration de l'efficacité de calcul de l’ADD-Shi en prennant un concept similaire à l'approche du contact souple. Il sera démontré que, en supprimant la contrainte de non-pénétration de l'application des peines du contact, le temps de la solution diminue de manière considérable. Les conclusions sont vérifiées dans les exemples qui examinent la stabilité des pentes dans les masses de roches démontables. 1 INTRODUCTION In recent years, much attention has been focused on efficient numerical techniques for analyzing jointed rock problems. The presence of joints in rock masses results in discontinuous behavior, and gives rise to failure mechanisms that vary with scale. There are two approaches to the numerical analysis of jointed rock problems: continuum-based methods and discrete techniques. This paper will focus on the latter category of computational methods. In discrete element techniques, jointed rock masses are modelled as assemblies of discrete blocks that may be rigid or deformable. In geotechnical problems, the Discontinuous Deformation Analysis (DDA) and the Distinct Element Method (DEM) are often identified as the two most widely-used discrete element techniques (Jing, 2007). DDA uses an implicit time-integration scheme to solve the governing equations of motion, namely conservation of linear and angular momentum, through time. It therefore requires forming a global system of equations. DEM, on the other hand, applies explicit time integration and thus can solve the equations of motion locally. The computational efficiency of DDA and DEM is the main determinant in choosing between them as a solution tool for engineering scale geotechnical problems. Computational efficiency is defined as the computational speed in terms of CPU time required for the analysis. It has been argued in literature that DDA is unconditionally stable due to its use of implicit time integration, and is faster than conditionally stable time integration techniques because it can accommodate considerably larger time steps. Theoretically, it is therefore expected that DDA will solve jointed rock problems more efficiently than DEM. In a recent work, we investigated the expected benefits of DDA (Khan, 2010; Khan et al., 2010). We performed a systematic study on the factors that affect the solution time, including the time step size, type of matrix solver, contact search algorithm, and contact resolution technique. The comparison was performed using two representative software tools, namely the original DDA developed by Shi (DDA-Shi), and UDEC, developed by Itasca (2004). It was concluded that although theoretically there is no restriction on DDA time- step size, DDA-Shi is actually slower than methods that utilize an explicit time integration scheme. The observed slowness in the DDA solution of real- scale geotechnical problems is caused by the combination of the highly nonlinear nature of contact problems and the size of the global system of equations in implicit time integration techniques. In DDA-Shi, the no- penetration constraint is satisfied in an iterative manner referred to as the open-close iterations. Due to its use of implicit time integration, the global system of equations needs to be rebuilt at each iteration of the DDA solution, which demands considerable time. This paper will focus on improving the computational efficiency of DDA-Shi by adopting a concept similar to the soft contact approach used in UDEC. It will be shown that by removing the no-penetration constraint of the contact penalty enforcement, the solution time decreases significantly. These conclusions will be verified in examples that examine the stability of slopes in jointed rock masses. 2 INTRODUCTION TO DEM AND DDA The equations governing the behavior of a system of discrete blocks are the conservation of mass, conservation of linear and angular momentum, and the material constitutive equations. The motion and deformation of each individual block in a discrete system follows from the conservation of linear and angular momentum equations. The conservation of linear momentum equation is expressed by 2 , 2 0 j ij i j u b t         , (1) where u is the displacement, σ , the Cauchy stress tensor, b , the body force, and , the density of material. The original DEM assumes that the blocks are rigid. The displacement field over each rigid block can then be represented by the rigid body displacement of a reference point on the block and the rotation of the discrete body about this point. To determine block rotations, the equation of angular momentum needs to be explicitly solved. DDA-Shi uses a first-order interpolation function to approximate the deformation field of a block from block vertex displacements. The displacement field over a block is then represented by displacement of a reference point on the block, rotation of the block about the axes passing through the reference point and a constant strain field. However, similar to rigid block DEM, since the motion of a volume is expressed by motion of a point, equilibrium of angular momentum needs to be explicitly satisfied to determine the rotation of the block. In this work we have assumed that the DDA blocks are rigid. Details on how to implement this assumption are discussed in [Koo, 1998; Khan, 2010]. By enforcing this assumption the number of degrees of freedom for a given block assembly will become identical in rigid block DDA and DEM. This eliminates the contributions that differences in the size of the system of equations can have on the solution time. DEM uses an explicit central difference time- marching scheme to solve the governing equations of motion through time. In the central difference scheme, the equilibrium of the system at time t is considered to calculate the displacement at time t t . The solution for the nodal point displacements at time t t  is obtained using the central difference approximation for the accelerations, ut i . In general, the mass matrix on the left-hand side can be represented as a diagonal matrix. Equation (1) therefore can be rearranged as ( ) u F t t out of balance i i i m    , (2) where i represents the index of the degree of freedom. For the particular case of rigid block DEM, i m becomes the mass of each block. The uncoupling of the equations of motion, which is one of the major advantages of explicit integration schemes, eliminates the need for assembly of global mass or stiffness matrices and inversion of the global matrices. In rigid block DEM, at each time step, the kinematic variables, i.e., accelerations, velocities and displacements, are first calculated using a central difference scheme, and the dynamic quantities (contact forces or stresses, as well as internal stresses of the elements) are then obtained by invoking the constitutive relations for the contacts and the block materials.   ( / 2) ( / 2) / t t t i i i t t t u u F m g t        , and (3) ( / 2) ( / 2) ( / ) i i t t t t uploads/Voyage/ improving-the-dda-solution-time-by-modifying-the-contact-enforcement-technique-pdf.pdf