Showing papers on "Discontinuous Deformation Analysis published in 2015"

A two-dimensional coupled hydromechanical discontinuum model for simulating rock hydraulic fracturing

Abstract: Summary Within the framework of our discontinuous deformation analysis for rock failure algorithm, this paper presents a two-dimensional coupled hydromechanical discontinuum model for simulating the rock hydraulic fracturing process. In the proposed approach, based on the generated joint network, the calculation of fluid mechanics is performed first to obtain the seepage pressure near the tips of existing cracks, and then the fluid pressure is treated as linearly distributed loads on corresponding block boundaries. The contribution of the hydraulic pressure to the initiation/propagation of the cracks is considered by adding the components of these blocks into the force matrix of the global equilibrium equation. Finally, failure criteria are applied at the crack tips to determine the occurrence of cracking events. Several verification examples are simulated, and the results show that this newly proposed numerical model can simulate the hydraulic fracturing process correctly and effectively. Although the numerical and experimental verifications focus on one unique preexisting crack, because of the capability of discontinuous deformation analysis in simulating block-like structures, the proposed approach is capable of modeling rock hydraulic fracturing processes. Copyright © 2014 John Wiley & Sons, Ltd.

Discontinuous deformation and displacement analysis: From continuous to discontinuous

Abstract: A discontinuous deformation and displacement (DDD) analysis method is proposed for modelling the rock failure process. This method combines the rock failure process analysis (RFPA) method (based on finite element method) and discontinuous deformation analysis (DDA) method. RFPA is used to simulate crack initiation, propagation and coalescence processes of rock during the small deformation state. The DDA method is used to simulate the movement of blocks created by the multiple cracks modelled by the RFPA. The newly developed DDD method is particularly suitable for modelling both crack propagation and block movement during the rock failure process because of the natural and convenient coupling of continuous and discontinuous deformation analyses. The proposed method has been used to simulate crack initiation, propagation and coalescence within a slope as well as the block movement during the landslide process. Numerical modelling results indicate that the proposed DDD method can automatically simulate crack propagation and block movement during the rock failure process without degrading accuracy.

An improved three-dimensional spherical DDA model for simulating rock failure

Abstract: In this paper, a discontinuous numerical model, namely SDDARF3D (three-dimensional spherical discontinuous deformation analysis for rock failure), is proposed for simulating the whole process of rock failure. Firstly, within the framework of the classical discontinuous deformation analysis (DDA) method, the formulation of three-dimensional spherical DDA (3D SDDA) is deduced; secondly, a bonding and cracking algorithm is constructed and the SDDARF3D model is proposed; thirdly, corresponding VC++ calculation code is developed and some verification examples are calculated. The simulated results can intuitively reproduce the failure phenomena of rock mass, indicating that the proposed SDDARF3D numerical model is correct and effective.

Boundary setting method for the seismic dynamic response analysis of engineering rock mass structures using the discontinuous deformation analysis method

Abstract: Summary Large deformations and discontinuous problems can be calculated using the discontinuous deformation analysis (DDA) method by solving time steps, and this method is suitable for simulating the seismic dynamic response of engineering rock mass structures. However, the boundary setting must be carefully analyzed. In this paper, four boundary settings for the DDA method are investigated. First, the contributions to the DDA equations for nonreflecting boundaries (including the viscous boundary and the viscoelastic boundary) are deduced based on the Newmark method. Second, a free-field boundary is introduced in the DDA method with boundary grid generation and coupling calculation algorithms to accurately simulate external source wave motion, such as earthquakes. Third, seismic input boundary treatments are intensively examined, and the force input method is introduced based on nonreflecting boundaries. Finally, the static-dynamic unified boundary is implemented to ensure consistent boundary transformation. The boundary setting method in the DDA method is discussed, and the suggested treatments are used to analyze the seismic dynamic response of underground caverns. Copyright © 2015 John Wiley & Sons, Ltd.

An Angle-Based Method Dealing with Vertex–Vertex Contact in the Two-Dimensional Discontinuous Deformation Analysis (DDA)

Abstract: Due to the discontinuity of the direction of a normal vector at the corner, vertex-vertex contact is indeterminate in numerical modelling. Two classes are indeterminate in two-dimensional discontinuous deformation analysis (DDA), designated as "genuine indeterminacy" and "pseudo indeterminacy". While examining these indeterminacies, it is necessary to pre-process quasi-vertex-vertex contact and determine contact or entrance edges. On one hand, improper pre-processing not only changes the equilibrium equations of the system, but also increases the burden on open-close iteration. On the other hand, there are two kinds of quasi-contact edges, i.e. the same and different block quasi-contact edges. Certain previous distance-based methods, which are used to seek the correct entrance edge, lead to some forced contacts, including vertex-vertex or vertex-edge. New pre-processing and angle-based methods for determining entrance edges are developed in this study. Additionally, the new concept of strengthening the movement trend of blocks is presented in DDA. The new pretreatment method combining the angle-based method and a strategy for strengthening the movement trend of blocks are intended to reduce forced contacts, maintain the movement trend of blocks, and speed up open-close iterations to a certain extent. Furthermore, the new method only depends on the orientation of quasi-contact edges and the movement trend of convex corners. The effectiveness of these improvements is verified by translational motion, rotation trend, and time-step size effect tests.

Comparisons between centrifuge and numerical modeling results for slope toppling failure

Abstract: This paper presents series studies on the toppling mechanism by centrifuge tests and numerical simulations. Two different discrete element methods, i.e., the continuum-based discrete element method (CDEM) and the discontinuous deformation analysis (DDA), are adopted. The modeling results show that both the methods can accurately capture the failure modes of the centrifuge tests, including three distinct zones and two failure surfaces. Comparisons are made between the physical test and numerical simulation results. The critical inclination angle of the tilting table where the slope models are fixed on can be moderately predicted by the two methods, with different degrees of precision. The error between the test results and the simulated results is within 1% for the slope models without rock-bridges by both CDEM and DDA. However, it is amplified for the staggered-joint models that simulate the rock-bridges. With DDA, the average error is about 5%, and the maximum error is up to 17%. While with CDEM, the errors for the aligned-joint models are ranged from 1% to 6%, and it is from 10% to 29% for the staggered- joint models. The two numerical methods show the capability in simulating toppling failure of blocky rock mass with and without rock-bridges. The model with rock-bridges which provides a certain bending resistance is more stable than the one without any rock-bridge. In addition, the two failure surfaces were observed, which is different from the common understanding that only one failure surface appears.

Analysis of Continuous and Discontinuous Cases of a Contact Problem Using Analytical Method and FEM

Abstract: In this paper, continuous and discontinuous cases of a contact problem for two elastic layers supported by a Winkler foundation are analyzed using both analytical method and finite element method. In the analyses, it is assumed that all surfaces are frictionless, and only compressive normal tractions can be transmitted through the contact areas. Moreover, body forces are taken into consideration only for layers. Firstly, the problem is solved analytically using theory of elasticity and integral transform techniques. Then, the finite element analysis of the problem is carried out using ANSYS software program. Initial separation distances between layers for continuous contact case and the size of the separation areas for discontinuous contact case are obtained for various dimensionless quantities using both solutions. In addition, the normalized contact pressure distributions are calculated for both cases. The analytical results are verified by comparison with finite element results. Finally, conclusions are presented.

Augmented Numerical Manifold Method with implementation of flat-top partition of unity

Abstract: This paper formulates the flat-top partition of unity (PU)-based numerical manifold method (NMM). Through modifying the finite element PU into the flat-top PU, the linear dependence problem involved in finite element PU-based high-order polynomial approximations has been successfully alleviated. Moreover, the procedure to construct the flat-top PU is substantially simplified by taking full advantage of the unique features of the NMM, compared to that in other flat-top PU-based methods. The proposed method could also be treated as an improved version of the discontinuous deformation analysis (DDA) by completely avoiding the additional strategies previously used to enforce the displacement compatibility between adjacent elements. The formulations are presented in details for 1-D, 2-D and 3-D cases. The discrete equations of the flat-top PU-based NMM are derived based on the minimum potential energy principle. The simplex integration technique and its variation are employed to evaluate the integration of the matrices of the discrete equations over arbitrarily shaped manifold elements. Seven representative examples have validated the proposed method in the aspects of approximation accuracy and efficiency.

Condensed form of complementarity formulation for discontinuous deformation analysis

Abstract: While the classical discontinuous deformation analysis (DDA) is applied to the analysis of a given block system, one must preset stiffness parameters for artificial springs to be fixed during the open-close iteration. To a great degree, success or failure in applying DDA to a practical problem is dependent on the spring stiffness parameters, which is believed to be the biggest obstacle to more extensive applications of DDA. In order to evade the introduction of the artificial springs, this study reformulates DDA as a mixed linear complementarity problem (MLCP) in the primal form. Then, from the fact that the block displacement vector of each block can be expressed in terms of the contact forces acting on the block, the condensed form of MLCP is derived, which is more efficient than the primal form. Some typical examples including those designed by the DDA inventor are reanalyzed, proving that the procedure is feasible.

Equal Order Discontinuous Finite Volume Element Methods for the Stokes Problem

Abstract: The aim of this paper is to develop and analyze a family of stabilized discontinuous finite volume element methods for the Stokes equations in two and three spatial dimensions. The proposed scheme is constructed using a baseline finite element approximation of velocity and pressure by discontinuous piecewise linear elements, where an interior penalty stabilization is applied. A priori error estimates are derived for the velocity and pressure in the energy norm, and convergence rates are predicted for velocity in the $$L^2$$L2-norm under the assumption that the source term is locally in $$ H^1$$H1. Several numerical experiments in two and three spatial dimensions are presented to validate our theoretical findings.

Performance of DDA time integration

Abstract: Discontinuous deformation analysis (DDA) is a numerical method for analyzing the deformation of block system. It employs unified dynamic formulation for both static and dynamic analysis, in which the so-called kinetic damping is adopted for absorbing dynamic energy. The DDA dynamic equations are integrated directly by the constant acceleration algorithm of Newmark family integrators. In order to have an insight into the DDA time integration scheme, the performance of Newmark time integration scheme for dynamic equations with kinetic damping is systematically investigated, formulae of stability, bifurcation, spectral radius, critical kinetic damping and algorithmic damping are presented. Combining with numerical examples, recognition and suggestions of Newmark integration scheme application in the DDA static and dynamic analysis are proposed.

Reducing spurious oscillations in discontinuous wave propagation simulation using high-order finite elements

Abstract: In this paper, a modified explicit time integration scheme is proposed for simulating the propagation of discontinuous waves. The spatial domain is discretized by the finite element method. To obtain accurate results, the standard finite element method requires a very fine mesh, which is why the computational effort can be very time consuming. The use of high-order finite element methods-such as the spectral element method based on Lagrange polynomials through Gauss-Lobatto-Legendre points or the iso-geometric analysis using non-uniform rational B-splines-can reduce the enormous computational costs significantly, compared to the standard finite element method. However, explicit time integration schemes such as the central difference method cannot eliminate the spurious oscillation near the front wave. The procedure proposed here is basically similar to the explicit method of Noh and Bathe, but the semi-discrete equation of motion is modified by introducing a damping parameter to suit the high-order FEM analysis of the discontinuous wave propagation. The performance due to this modification is tested for one-dimensional wave propagation problems using both lumped and consistent mass matrices. Then, the idea of a combination of the consistent and row sum lumped mass matrices is evaluated. The proposed method is studied also in two-dimensions by considering the Lamb problem of wave propagation. The results are promising enough to provide a better simulation of the discontinuous wave propagation problem.

Strength Reduction Method for Stability Analysis of Local Discontinuous Rock Mass with Iterative Method of Partitioned Finite Element and Interface Boundary Element

Abstract: SRM (strength reduction method) with iterative method of PFE (partitioned fnite element) and IBE (interface boundary element) is proposed to solve the safety factor of local discontinuous rock mass. Slope system is divided into several continuous bodies and local discontinuous interface boundaries. Each block is treated as a partition of the system and contacted by discontinuous joints. The displacements of blocks are chosen as basic variables and the rigid displacements in the centroid of blocks are chosen as motion variables. The contact forces on interface boundaries and the rigid displacements to the centroid of each body are chosen as mixed variables and solved iteratively using the interface boundary equations. Flexibility matrix is formed through PFE according to the contact states of nodal pairs and spring flexibility is used to reflect the influence of weak structural plane so that nonlinear iteration is only limited to the possible contact region. With cohesion and friction coefficient reduced gradually, the states of all nodal pairs at the open or slip state for the first time are regarded as failure criterion, which can decrease the effect of subjectivity in determining safety factor. Examples are used to verify the validity of the proposed method.

Some modifications to the process of discontinuous deformation analysis

Abstract: This paper presents a modified method of discontinuous deformation analysis (DDA). In the presented method, open-close iteration may not be needed, small penetration is permitted among blocks, and springs are added between contacting block pairs only when a penetration takes place. The three contact patterns (i.e. sliding, locking and opening) in original DDA method are not involved, and the recognition of these contact patterns and treatment of transformation among patterns are not required either, significantly saving the computing time. In a convex to concave contact, there are two candidate entrance edges which may cause uncertainty. In this case, we propose the angle bisector criterion to determine the entrance edge. The spring stiffness is much larger than Young's modulus in the original DDA, however we find that the correct results can still be obtained when it is much smaller than Young's modulus. Finally, the penetrations by using penalty method and augmented Lagrangian method are compared. Penetration of the latter is 1/4 of the former. The range of spring stiffness for the latter is wider than the former, being 0.01–1 of the former. Both methods can lead to correct contact forces.

Dual reciprocity boundary element based block model for discontinuous deformation analysis

Abstract: This paper presents a further development of the dual reciprocity boundary element method (DRBEM) with stepwise updating to pave the way for the introduction of boundary element mesh into the discontinuous deformation analysis (DDA). The advantage of the proposed method lies in its adoption of static fundamental solutions and reduction in the size of the governing equations by transforming the inertial term domain integrals to boundary integrals in the dynamic large displacement analysis. The unconditionally stable Newmark- β time integration method involving numerical damping to enhance the numerical stability is implemented for the dynamic analysis. In order to be coupled with the DDA to improve the deformability of the DDA block domains, a stepwise updating algorithm of the system variables is introduced. The stress updating in the analysis involved in the calculation of a domain integral and internal cells are used for the integration of the initial stress term. Several examples are used to verify the geometry-updated DRBEM model and satisfactory results have been obtained.

Discontinuous Deformation Analysis Enriched by the Bonding Block Model

Abstract: The discontinuous deformation analysis (DDA) has been extensively applied in geotechnical engineering owing to its salient merits in the modeling of discontinuities. However, this method assumes a constant stress field within every block and hence cannot provide reliable estimation for block deformations and stresses. This paper proposes a novel scheme to improve the accuracy of the DDA. In our method, advanced subdivision is introduced to represent a block as an assembly of triangular or quadrilateral elements, in which overlapped element edges are separated from each other and are glued together by bonding springs. The accuracy and the effectiveness of the proposed method are illustrated by three numerical experiments for both continuous and discontinuous problems.

Development of the 3D DDA method with rate-and state-friction laws

Abstract: The elastic medium in lithosphere should be considered as discontinuities since the complex tectonic background has produced many active faults as separation.The Mohr-Coulomb joint failure criterion with a constant friction coefficient,adopted in the original 3Ddiscontinuous deformation analysis(DDA)method,cannot meet the requirement of highly accurate calculations for motion and deformation of tectonic block systems.The rate-and state-friction laws,which are capable of reproducing virtually the entire range of observed fault behaviors,are combined into the 3D DDA method.Firstly,the formula of computing coefficient of friction on the interface with rate-and state friction laws is derived.In order to calculate the value of state variableθ,slip velocity Vand friction coefficientμin every time step,a first-order differential equation about Vis deduced.The increment of Vis calculated by the second-order Taylor series expansion in our scheme.Secondly,the evolution law is determined by the Runge-Kutta scheme with adaptive step-size control.The friction submatrix,which consists of discrete forms of V andθ,is rewritten and then combined into the 3D DDA method.Finally,on the basis of reasonable geometry and mechanical properties of the numerical model,slide-hold-slide tests and velocity stepping tests are designed to examine the accuracy of the modified method.Suits of numerical slide-hold-slide tests are performed using hold time from 1to 10000 seconds and then we take the numerical test of the 10-second hold time as an example to make a brief illustration.A rigid block moves along the base in uniform linear motion at the first five seconds,because of the equality between friction and point loading.At the fifth second,the point loading is set to zero,and the friction strength drops immediately.After ten seconds,the block is reset as the previous loading.Strength increases,reaches a peak value and returns to its previous steady-state value subsequently.The numerical results are consistent with the laboratory data measured by Beeler et al.(1994)with a 0.999-goodness of fits.Even though both data sets are mixed together,there is also a 0.997-goodness of fits.In addition,the friction healing rateβcan be evaluated by the friction parameter b when the slip rate approximates to 0.The slope of these numerical data comes close to the b value of assumption with a 5% relative error.Besides,velocity stepping tests are also modeled using the improved 3D DDA.A slider keeps a steady sliding under a loading rate of 10μm/s with a distance of 25μm,and then decreases to 1μm/s during a characteristic distance.The results show a strong velocity dependence of friction which is consistent with laboratory data.Comparison between numerical results and laboratory data shows that the 3DDDA method in combination with rata-and state-dependent friction laws is capable of simulating velocity dependence of sliding friction and time dependence of static friction,which resolves a basic problem when using 3DDDA in geodynamics research.The improved 3D DDA method still has the limitation such as unbalanced embedment due to the uncertainty of stiffness on the discontinuous interface,which leads to inexact results of contact force determination.In the future,this modification can be used in quantitative simulation of regional crustal deformation in combination with observations,such as GPS data.