Development of a Composite Implicit Time Integration Scheme for Three-Dimensional Discontinuous Deformation Analysis

Abstract

Discontinuous deformation analysis (DDA) is a discontinuum-based and implicit method for investigating the deformational behavior of block systems. The constant acceleration integration (CAI) scheme characterized by unconditional stability is employed in the traditional DDA. In this study, the problems of the CAI scheme regarding the time step in DDA are pointed out. A too large or too small time step size adopted in the CAI scheme will have adverse effects on the DDA computation. To overcome the weaknesses, an alternative composite implicit time integration (CITI) scheme, which is a combination of the trapezoidal rule and the three-point backward Euler method, is implemented in the three-dimensional (3D) DDA method. Verification examples and slope numerical simulations are presented to illustrate the accuracy and effectiveness of the proposed methodology. The results showed that the CITI scheme can overcome the numerical error caused by the large time step size, and the algorithm damping is closely related to the time step size and the selected splitting parameter.

1. Introduction

Discontinuous deformation analysis (DDA) [1] is an implicit discrete numerical method, which can analyze the discontinuous deformation behavior and mechanical response of rock block systems under general loading and boundary conditions simultaneously. DDA parallels, but is superior to both the finite element method and discrete element method with its own unique characteristics [1], i.e., complete kinematic theory and its numerical realization, perfect first-order displacement approximation, strict postulate of equilibrium, correct energy consumption, and high computing efficiency. Thus, DDA has drawn more and more attention in the past few decades. As for two-dimensional (2D) DDA, since the formulations and program code were provided early on, it has been extensively developed in terms of rock mechanics study [2,3,4], rock engineering application [5,6,7,8], the modification of the DDA method itself [9,10,11,12,13], and even coupling with other methods [14,15,16]. As for three-dimensional (3D) DDA, its basic formulations were derived in 2001 by Shi [17]. Many contact models [18,19,20,21,22] were then proposed to contribute to the promotion of 3D DDA, but it is still in the fundamental research stage due to a limited amount of works on 3D DDA having been presented [23]. There is an urgent need for 3D DDA because of the highly directional nature of the jointed rock mass behavior, making 2D DDA inapplicable to many practical problems [24].

It can be seen from previous studies that the time step size directly affects the accuracy and efficiency of the DDA computation. As a nonlinear kinematic method, DDA considers the inertia force making the global equilibrium equations contain an artificial parameter, i.e., time step size Δ, which is used by both statics and dynamics. For large deformations, each step of DDA starts with the deformed block shape and position resulting from the previous step, and the equilibrium equations are re-written and re-solved for the updated block geometry [25]. DDA adopts an implicit and direct time integration method, i.e., the constant acceleration integration (CAI) scheme, which is from the generalized Newmark family time integration and characterized by unconditional stability within each time step. Regarding static problems, the CAI scheme can use a large time size, and the computational results approach the static state quickly and remain stable [26]. Concerning acceleration-variation dynamic problems, the computational accuracy is greatly affected by the time step size. If the time step size is small, the acceleration within one time step tends to be a constant and the CAI scheme is available; however, if the time step size is large, the acceleration within one time step may be variable and the assumption of constant acceleration may be incorrect, and this will result in excessive integration error. Based on the above disadvantages of the CAI scheme in the acceleration-variation dynamic computation, a more-applicable time integration method as a substitute for the CAI scheme in the DDA solution is desirable.

To solve the dynamic response problems, many researchers have been striving to propose more-effective step-by-step implicit integration methods to obtain the transient analyses of a long time duration by solving differential equations of motion. Implicit time integration schemes, such as the classical Newmark method [27], the Wilson-θ method [28], the HHT-α method [29], the ρ-method [30], and the generalized-α method [31], have been originated or extended and successfully applied in structural dynamics. These schemes employ numerical dissipation and are unconditionally stable in linear dynamic analysis; however, unstable and unreliable results may be encountered when they are utilized in nonlinear dynamic analysis, and appropriate integration constants are required to solve a specific problem [32].

Bathe and Baig [33] proposed a composite implicit time integration (CITI) scheme, which is the coupling of the trapezoidal rule and the three-point backward Euler method and is also a self-starting and single-step (but having two sub-steps) method [34]. The CITI scheme is attractive because it only operates on the usual global vectors, only uses the usual symmetric matrices, shows good stability characteristics, and has second-order accuracy [33]. By utilizing the scheme, the dynamic equilibrium equations of structural systems including linear and nonlinear are solved [35,36], and some applications have been made, e.g., wave propagation [37], shell structure [32,38], and stiff pendulum and propfan [39]. A large deformation block system of DDA is a high nonlinear system, and there is nonlinearity within blocks and between blocks; the CITI scheme is applicable to this large deformation problem [33]. Since 2D DDA cannot consider the lateral movement of rock blocks and it is difficult to estimate the runout area and the effect of the 3D shape of both rock blocks and the slope on the trajectory and movement [40], this study developed a 3D DDA computer program and made a daring attempt to implement this CITI scheme in the 3D DDA integration solution.

The structure of the rest of the paper is as follows. In Section 2, the fundamentals of 3D DDA are described and the problems of the CAI scheme related to the time step in DDA are pointed out. In Section 3, the basic idea of the CITI scheme is introduced and the strategy of the CAI scheme replacing the CITI scheme of 3D DDA is proposed. In Section 4, the accuracy and correctness of the developed 3D DDA are verified by numerical examples including the five basic forms of block movement. In Section 5, the developed technique is successfully applied to slope failure models, and in Section 6, conclusions are provided.

2. Fundamentals of 3D DDA and Time Integration Scheme

Three-dimensional DDA calculates 3D deformable block systems, and the final displacement variables are actually obtained by iteratively solving the simultaneous equilibrium equations, which are derived by minimizing the total potential energy of single blocks and contacts between two blocks, until no tension and no penetration are found in all contacts, namely “open-close” iteration. When the blocks are in contact, Mohr–Coulomb’s criterion is applied to the contact interfaces. In this method, large displacements are the accumulation of incremental displacements and deformations at each time step. Within each time step, the incremental displacements of all points are small, so the complete first-order approximation of the displacements needs to be employed [17].

2.1. Block Deformation and Displacement

In the 3D DDA method, a block system is formed by n blocks through contacts between blocks and displacement constraints on individual blocks, and the deformation variables of block i can be represented by twelve unknowns:

$$[D_i]={[u_c,v_c,w_c,r_x,r_y,r_z,{\epsilon }_x,{\epsilon }_y,{\epsilon }_z,{\gamma }_{yz},{\gamma }_{zx},{\gamma }_{xy}]}_i^T,\hspace{1em}(i=1,2,\cdots ,n)$$

(1)

where u_c, v_c, and w_c are the rigid body translations at the centroid point (x_c, y_c, z_c) within the block, r_x, r_y, and r_z are the rotation angles of the block with the rotation center (x_c, y_c, z_c), and ε_x, ε_y, ε_z, γ_xy, γ_yz, and γ_zx are the normal and shear strains of the block. By using the complete first-order polynomial as the displacement function for a block, the distribution of the strains and stresses is constant throughout. The complete first-order approximation of block displacements at any point (x, y, z) of block i can be written as

$${\left[u, v, w\right]}_i^T=\left[T_i\right(x, y, z\left)\right][D_i], (i=1, 2, \cdots , n)$$

(2)

where

$$[T_i(x,y,z)]={\left(\begin{array}{cccccccccccc}1& 0& 0& 0& z-z_c& -(y-y_c)& x-x_c& 0& 0& 0& (z-z_c)/2& (y-y_c)/2\\ 0& 1& 0& -(z-z_c)& 0& x-x_c& 0& y-y_c& 0& (z-z_c)/2& 0& (x-x_c)/2\\ 0& 0& 1& y-y_c& -(x-x_c)& 0& 0& 0& z-z_c& (y-y_c)/2& (x-x_c)/2& 0\end{array}\right)}_i$$

(3)

2.2. Simultaneous Equilibrium Equations with the CAI Scheme

The simultaneous equilibrium equations of the block system composed of n blocks are derived by minimizing the total potential energy with respect to the variables [D], and the following equations can be deduced:

$$[M][\ddot{D}]+[C][\dot{D}]+[K][D]=[F]$$

(4)

where [M], [C], [K], and [F] are the mass matrix, the damping matrix, the stiffness matrix, and the loading vector, respectively; $[\ddot{D}]$, $[\dot{D}]$, and $[D]$ are the acceleration vector, the velocity vector, and the displacement vector, respectively. No physical damping is considered in the 3D DDA computation, i.e., [C] = 0.

Assume the acceleration vector, the velocity vector, and the displacement vector at the beginning of the time step are $[{\ddot{D}}_0]$, $[{\dot{D}}_0]$, and $[D_0]$, respectively, which result from the previous time step, and the time step size is Δ. Based on the Newmark direct integration method, $\dot{[D}]$ and $[D]$ can be represented by

$$[\dot{D}]=[{\dot{D}}_0]+{\displaystyle {\int }_0^{\Delta }[\ddot{D}]d\tau }\approx [{\dot{D}}_0]+(1-\gamma )\Delta [{\ddot{D}}_0]+\gamma \Delta [\ddot{D}]$$

(5)

$$[D]=[D_0]+\Delta [{\dot{D}}_0]+{\displaystyle {\int }_0^{\Delta }(\Delta -\tau )}[\ddot{D}]d\tau \approx [D_0]+\Delta [{\dot{D}}_0]+(\frac{1}{2}-\beta ){\Delta }^2[{\ddot{D}}_0]+\beta {\Delta }^2[\ddot{D}]$$

(6)

where τ is the time increment and β and γ are the integration parameters of the Newmark method. The DDA time integration scheme is the CAI algorithm from the classical Newmark family time integration, that is β = 0.5 and γ = 1.0. Then, substituting (5) and (6) into (4), the simultaneous equilibrium equations can be derived:

$$([K]+\frac{2}{{\Delta }^2}[M])[D]=[F]+\frac{2[M]}{\Delta }[{\dot{D}}_0]+\frac{2[M]}{{\Delta }^2}[D_0]$$

(7)

Collecting terms on both sides, (7) can be conveniently written as [41]:

$$[\widehat{K}][D]=[\widehat{F}]$$

(8)

where $\left[\widehat{K}\right]$ and $\left[\widehat{F}\right]$ are the equivalent global stiffness and load matrices, respectively. (8) can be expressed in matrix form:

$$\left(\begin{array}{cccc}[K_{11}]& [K_{12}]& \cdots & [K_{1n}]\\ [K_{21}]& [K_{22}]& \cdots & [K_{2n}]\\ ⋮& ⋮& \ddots & ⋮\\ [K_{n1}]& [K_{n2}]& \cdots & [K_{nn}]\end{array}\right)\left(\begin{array}{c}[D_1]\\ [D_2]\\ ⋮\\ [D_n]\end{array}\right)=\left(\begin{array}{c}[F_1]\\ [F_2]\\ ⋮\\ [F_n]\end{array}\right)$$

(9)

where [K_ij] (i, j = 1, 2, …, n) are 12 × 12 sub-matrices, [K_ii] depends on the material properties of block i, and [K_ij] (i ≠ j) is defined by the contacts between blocks i and j; [D_i] and [F_i] (i = 1, 2, …, n) are 12 × 1 sub-matrices; [D_i] represents the deformation variables of block i; [F_i] is the loading distributed to the twelve deformation variables. For more details on the interpretation and implementation of 3D DDA, refer to [17].

2.3. Problems of the CAI Scheme concerning Time Step in the Traditional DDA

DDA considers both statics and dynamics by a time step marching scheme and an implicit algorithm formulation [42]. All the geometric and physical parameters have to be transferred from the end of the previous time step to the beginning of the next time step, e.g., stresses and strains in each element, the geometry of the joint boundaries and elements, and all closed contacts [43]. The only difference is that the static analysis assumes the velocity as zero at the beginning of each time step, while the dynamics analysis inherits the velocity of the previous time step. The DDA time integration scheme is the CAI scheme from the classical Newmark family time integration, which is a direct integration method with open–close iterations and is unconditionally stable within each time step [44]. Thus, a larger time step size results in fewer time steps required for analysis, indicating that less computing time is required for DDA [45]. However, problems of the CAI scheme may arise with respect to the time step size, the value of which is artificial and uncertain:

(1)

If the time step size is larger than a certain value, the assumption that the acceleration is a constant within a time step may be incorrect. As shown in Figure 1, the acceleration $[\ddot{D}]$ at the end of time step t₀ + Δ is equal to or approximately equal to the acceleration $[{\ddot{D}}_0]$ at the beginning of time step t₀, which leads to linear velocity and quadratic displacement. There can be no doubt that this is errorless or the numerical error is small. However, when a larger time step size is adopted, the acceleration may be variable, and the results of the DDA computation will be inaccurate. The numerical error will grow as the calculation step size increases. Meanwhile, the effect of the inertia force on the block movement may be weakened or even ignored within a larger time step.

(2)

If the time step size is too small, the square terms Δ² ((6)) on the right side of the equal sign are ignored, and the inertia force has a great influence on the diagonal entries [K_ii] of the global stiffness matrix ((9)). These also bring to DDA computational inaccuracy and even a high computational cost.

It can be said that the time step size has an important effect on the accuracy and efficiency of the DDA computation, but it cannot be well selected. We need a larger time step size to improve the computational efficiency, but the larger time step size must be sufficiently small to capture the nonlinear nature of the DDA computation and allow the first-order displacement function, which is an inherent feature of DDA, to be appropriate. It is necessary to diminish the numerical error brought by the problem that the constant acceleration is used within a larger time step with variable acceleration; consequently, the alternative CITI scheme is implemented in 3D DDA, trying to reduce the constraint of the time step size on the DDA computation.

3. CITI Scheme for 3D DDA

The CITI scheme is a direct and implicit integration method to solve nonlinear structural dynamics and shows good stability characteristics because of the numerical dissipation and its second-order accuracy [33]. The CITI scheme divides each time step into two sub-steps. It was assumed that the time step size was chosen as Δ, and t₀ and t₀ + Δ are the initial and end moment of the time step. The first sub-step is equal to δΔ (0 < δ < 1), and the trapezoidal rule was applied; the second sub-step is equal to (1 − δ)Δ, and the three-point backward Euler method was applied.

3.1. Trapezoidal Rule

Assume that the solutions, i.e., the acceleration vector $[{\ddot{D}}_0]$, the velocity vector $[{\dot{D}}_0]$, and the displacement vector $[D_0]$, are known at time t₀ and the solutions at time t₀ + Δ are expected to be calculated. As shown in Figure 2, the middle time between t₀ and t₀ + Δ is t₀ + δΔ. Using the trapezoidal rule over the first sub-step, time interval δΔ, the assumptions on the acceleration vector, the velocity vector, and the displacement vector at time t₀ + δΔ are as follows:

$$[{\dot{D}}_{0+\delta \Delta }]=[{\dot{D}}_0]+\frac{1}{2}([{\ddot{D}}_0]+[{\ddot{D}}_{0+\delta \Delta }])\delta \Delta$$

(10)

$$[D_{0+\delta \Delta }]=[D_0]+\frac{1}{2}([{\dot{D}}_0]+[{\dot{D}}_{0+\delta \Delta }])\delta \Delta$$

(11)

Substituting (10) into (11) and rearranging (11),

$$[D_{0+\delta \Delta }]=[D_0]+[{\dot{D}}_0]\delta \Delta +\frac{1}{4}([{\ddot{D}}_0]+[{\ddot{D}}_{0+\delta \Delta }]){\delta }^2{\Delta }^2$$

(12)

Thus, the acceleration vector and velocity vector can be written as

$$[{\ddot{D}}_{0+\delta \Delta }]=([D_{0+\delta \Delta }]-[D_0]-[{\dot{D}}_0]\delta \Delta )\frac{4}{{\delta }^2{\Delta }^2}-[{\ddot{D}}_0]$$

(13)

$$[{\dot{D}}_{0+\delta \Delta }]=([D_{0+\delta \Delta }]-[D_0])\frac{2}{\delta \Delta }-[{\dot{D}}_0]$$

(14)

The movement Equation (4) at time t₀ + δΔ is

$$[M][{\ddot{D}}_{0+\delta \Delta }]+[K][D_{0+\delta \Delta }]=[F_{0+\delta \Delta }]$$

(15)

Once the displacement vector has been obtained, the acceleration vector and velocity vector are determined by (13) and (14), respectively. The velocity vector at time t₀ + δΔ is inherited by the second sub-step, and the global stiffness matrix and loading vector vary with the movement of the blocks.

3.2. Three-Point Backward Euler Method

On the basis of the three-point backward Euler method [46], the velocity vector at time t₀ + Δ can be represented by the displacement vectors at time t₀, t₀ + δΔ, and t₀ + Δ, and the acceleration vector at time t₀ + Δ can be represented by the velocity vectors at time t₀, t₀ + δΔ, and t₀ + Δ. Then, in the second sub-step,

$$[\dot{D}]=c_1[D_0]+c_2[D_{0+\delta \Delta }]+c_3[D]$$

(16)

$$[\ddot{D}]=c_1[{\dot{D}}_0]+c_2[{\dot{D}}_{0+\delta \Delta }]+c_3[\dot{D}]$$

(17)

where

$$c_1=\frac{1-\delta }{\delta \Delta }, c_2=\frac{-1}{(1-\delta )\delta \Delta }, c_3=\frac{2-\delta }{(1-\delta )\delta \Delta }$$

Substituting (16) into (17) and rearranging (17),

$$[\ddot{D}]=c_1([{\dot{D}}_0]+c_3[D_0])+c_2([{\dot{D}}_{0+\delta \Delta }]+c_3[D_{0+\delta \Delta }])+c_3^2[D]$$

(18)

Then, combining (18) and (4) ([C] = 0), the displacement vector can be obtained, and the acceleration vector and the velocity vector are determined. The velocity vector at time t₀ + Δ is inherited by the next time step, and the global stiffness matrix and loading vector vary with the movement of the blocks.

3.3. Implementation in 3D DDA

The CITI scheme was introduced into 3D DDA to replace the original CAI scheme, and the flowchart outlining the procedures of the CITI scheme is shown in Figure 3. The algorithm is easy to implement in a computer program, and the convergence accuracy analysis was performed by Zhang et al. [34].

4. Verifications of the Developed 3D DDA Method

The block motion encountered in practical engineering can be treated as five basic motion forms (i.e., freefalling, sliding, oblique projectile, rolling, and bouncing) or combinations of several of them. The basic forms of block motion can serve as numerical examples to verify the correctness and accuracy of the newly developed 3D DDA. The developed 3D DDA is more advantageous in calculating the acceleration-variation motion, e.g., the rolling and bouncing of a single block in this section, compared with the original DDA with the CAI scheme. The gravity acceleration g = 10.0 m/s², and the normal spring stiffness and shear spring stiffness for the following 3D DDA examples are k_n = 10⁶ kN/m and k_s = 10⁵ kN/m, respectively. The splitting parameter δ = 0.5 recommended by Bathe and Baig [33] is used in this CITI scheme for 3D DDA in Section 4.1 and Section 4.2.

4.1. Freefalling, Sliding, and Oblique Projectile

The first example is a single cube with an edge length of 1.0 m on an inclined surface with a slope angle of α = 20°, and the 3D DDA model is shown in Figure 4. The cube slides from rest along the slope surface with a friction angle of φ. The sliding distance d = 11.0 m, and the falling height h = 5.0 m. The analytical solution for the sliding displacement d₁(t) at time t of the cube can be derived based on Newton’s second law as

$$d_1(t)=\left\{\begin{array}{ll}(\sin \alpha -\cos \alpha \tan \phi )g{t}^2/2, & \phi \le \alpha \\ 0& \phi >\alpha \end{array}\right.$$

(19)

and the motion transforms sliding into oblique projectile motion when the cube falls from the slope surface with the velocities in the x, y, and z directions being v_0x, v_0y, and v_0z, respectively. The displacement d₂(t) of oblique projectile motion is

$$d_2(t)=\sqrt{{(v_{0x}t)}^2+{(v_{0z}t+g{t}^2/2)}^2}$$

(20)

Three friction angle values, φ = 0°, 10°, and 21°, were studied under three different time step sizes, e.g., Δ = 0.0001 s, 0.0005 s, and 0.001 s. The comparisons of the displacements of sliding and oblique projectile motion between the 3D DDA results and the analytical solutions ((19) and (20)) are shown in Figure 5. It can be seen that the 3D DDA results were stable under the three different time step sizes and showed a highly satisfactory agreement with the time-dependent displacement functions predicted by the analytical solutions. The correctness of the freefalling simulation was also verified by oblique projectile motion indirectly, since the cube is only affected by gravity in the process of oblique projectile motion.

4.2. Rolling

Rolling, a motion process with variable acceleration, was verified by comparing the solutions of the “critical rolling initiation angle (CRIA)” defined by Chen et al. [40]:

$$\theta =\frac{{360}^{\circ }}{2n}$$

(21)

As shown in Figure 6, assume a rolling block i on a fixed block j with a slope angle of α, and the friction angle φ is larger than α to ensure block i is at rest while no rolling occurs. The block system is only affected by gravity. The CRIA θ is half of the corresponding central angle for one edge of an equilateral polygon with n edges. For block i in Figure 6, n = 8 and α = 22.5°.

Only when the slope angle α is equal to the CRIA θ does the rolling motion start. Six cases, e.g., n = 4, 6, 8, 10, 12, 18, were computed by 3D DDA and (21). Figure 7 displays the comparison between the 3D DDA results and the analytical solutions from (21) and shows that the analytical solutions agreed well with the results computed by 3D DDA.

To illustrate the superiority of the CITI scheme relative to the CAI scheme in the rolling simulation of 3D DDA, four time step sizes, Δ = 0.0001 s, 0.001 s, 0.01 s, and 0.1 s, were studied. Figure 8 shows the rolling angular velocity–time response of block i around the y axis under different time integration schemes and different time step sizes. As can be seen from Figure 8a, the changing law curves of the rolling angular velocities are disordered, and when Δ = 0.01 s and 0.1 s, the angular velocities are variable, but not constant within one stage, which should have been smooth and steady. Figure 8b shows good performance in the changing law of the rolling angular velocities. The results of the CITI scheme were more stable and precise than the CAI scheme in terms of large time step size cases.

4.3. Bouncing

The 3D DDA bouncing model is shown in Figure 9. A single block undergoes freefalling from a height of 5.0 m from the ground and collides with the ground repeatedly. First, the splitting parameter was assumed to be δ = 0.5. The bouncing displacement of the freefalling block was recorded up to 10.0 s, and the time-dependent displacement curves calculated by the CAI scheme and CITI scheme are plotted in Figure 10. The bouncing process of the freefalling block with the two schemes is summarized and discussed as follows:

(1)

When a small time step size (e.g., Δ = 0.0001 s) was adopted, the calculation results of the two schemes were similar, and the corresponding bouncing displacements were rarely dissipated with the collision with the ground, whose change trends were basically consistent. When a large time step size (e.g., Δ = 0.001 s) was adopted, the bouncing displacement was completely dissipated to rest at about 8.9 s with the CAI scheme, whereas with the CITI scheme, the bouncing displacement was slowly dissipated and there was still large bouncing displacement until 10.0 s.

(2)

When a larger time step size (e.g., Δ = 0.01 s) was adopted, with the CAI scheme, the amplitudes of bouncing displacement were close to the ones when a small time step (e.g., Δ = 0.0001 s) was adopted before the third collision with the ground. However, upon the third collision with the ground, the amplitude of bouncing displacement sharply decayed, and then, the bouncing displacement was dissipated to be resting at 6.2 s. With the CITI scheme, the amplitudes of bouncing displacement decayed gradually with the collision between the block and the ground.

(3)

For the CAI scheme, the change of the decay trend of bouncing displacement was significant with the increase of the time step size (e.g., from 0.0001 s to 0.01 s), and even the decay was abrupt when a larger time step size (e.g., Δ = 0.01 s) was adopted. For the CITI scheme, the bouncing displacement gradually attenuated, and this trend can be maintained even if the time step size is large, which conforms to the phenomenon of elastic cases encountered in practice and is also correct. Thus, the CITI scheme is more suitable for bouncing calculation and was less constrained by the time step.

(4)

From the perspective of damping, on the one hand, the algorithm damping of the CAI scheme was larger than the CITI scheme under the same large time step, so that the energy dissipation and bouncing displacement decay of the CAI scheme were faster than the CITI scheme. On the other hand, the larger the time step used in both the CAI scheme and the CITI scheme, the greater the algorithm damping used in DDA, the faster the energy dissipation and the bouncing displacement attenuation were. One of the possible reasons is that the effect of the inertia force on block movement is weakened within a larger time step.

To explore the influence of the sub-step sizes (i.e., δΔ and (1 − δ)Δ) on the energy dissipation of the bouncing block, four splitting parameters, δ = 0.4, 0.5, 0.6, and 0.7, were studied under time step size Δ = 0.001 s, and the corresponding bouncing displacement–time curves are shown in Figure 11. The larger the splitting parameter was, the faster the bouncing displacement decayed, and the more significant the dissipated energy. For a small splitting parameter (e.g., δ = 0.4), the bouncing displacement decay was subtle; for a large splitting parameter (e.g., δ = 0.6 or δ = 0.7), the bouncing displacement decay was obvious; for δ = 0.5, the bouncing displacement decay was gradual. The recommended splitting parameter δ = 0.5 by Bathe and Baig [33] is efficient for the 3D DDA calculation. It can be seen that the splitting parameter had an influence on algorithm damping, i.e., the larger the splitting parameter, the greater the algorithm damping was.

5. Numerical Examples

5.1. Collision between Rolling Blocks and Protective Wall

Three static rolling blocks roll along an inclined plane with a slope angle of α = 40° and collide with a protective wall composed of 14 blocks. The details of the 3D DDA model are shown in Figure 12. The rolling blocks are the same approximate spheres; the number of both their longitudinal and latitudinal segments is six, the approximate radii are r_b = 0.5 m, and their unit mass was assumed to be 2.5 kg/m³. The 3D DDA calculation parameters were as follows: time step size Δ = 0.00001 s; cohesion c = 0; friction angle φ = 30°; the normal and shear spring stiffnesses are 10⁵ kN/m and 10⁴ kN/m, respectively.

The final states of the collisions between the three blocks and the protective wall under different integration schemes are shown Figure 13, and

Table 1. Ultimate stability time of collisions under different integration schemes.

Integration Schemes	CAI Scheme	CITI Scheme
		δ = 0.5	δ = 0.6	δ = 0.7
Time	6.55 s	6.70 s	6.45 s	6.15 s

lists the corresponding time to ultimate stability. The final states obtained by the CAI scheme (Figure 13a) were more destructive than those by the CITI scheme (Figure 13b), which is conservative. The stabilizing time of the CITI scheme (δ = 0.5) was slower than that of the CAI scheme, and thus, the damping of the CAI scheme was larger in this case. For the CITI scheme, with the increase of splitting parameter δ, the stabilizing time of the rolling blocks tended to be faster and faster, and thus, the damping increased as the splitting parameter δ increased. Remarkably, the calculation efficiency of the CITI scheme became lower than that of the CAI scheme in the case of many rolling blocks being involved. The probable reason is that the time step of the CITI scheme is divided into two steps. The CITI scheme was also generalized to n sub-steps by Bathe and Baig [33]; the trapezoidal rule can be applied n − 1 times, and the solution can be obtained at the end of the time step by an (n + 1)-point backward difference scheme. Although the n-sub-step CITI scheme is theoretically available in DDA, the computational efficiency may be significantly decreased.

5.2. Wedge Failure

Wedge failure, caused by numerous intersecting discontinuities, is an important component of slope failure and often occurs in both civil and mining engineering. A 3D DDA ideal model of an open pit with a high and steep slope undergoing wedge failure is shown in Figure 14a. Half of the wedge failure model and its details are shown in Figure 14b; the pit depth h = 50.0 m; the side length of square pit bottom e = 20.0 m; the slope angles of Slopes 1 and 2 were 10° and 70°, respectively. There were four wedge blocks with equal spacing at each junction of Slopes 1 and 2, which were subjected to wedge failure due to geological discontinuities. The open pit and its wedge failure were geometrically symmetric. Four junctions had 16 blocks in total, which slid from their parent rocks and tended to rest on the pit bottom after sliding, rolling, freefalling, bouncing, colliding, oblique projectile motion, or combinations of them. The 3D DDA calculation parameters were as follows: time step size Δ = 0.0001 s; cohesion c = 0; friction angle φ = 10°; the normal and shear spring stiffnesses were 10⁶ kN/m and 10⁵ kN/m, respectively.

Figure 15, Figure 16, Figure 17 and Figure 18 show the movement states of the processive moment (t = 10.0 s, 15.0 s, and 20.0 s) of wedge failure under different integration schemes. The visualization and animation of the falling wedges were carried out by a display software developed by the authors [47].

Table 2. Ultimate stability time of wedge failure under different integration schemes.

Integration Schemes	CAI Scheme	CITI Scheme
		δ = 0.5	δ = 0.6	δ = 0.7
Time	29.0 s	30.5 s	28.5 s	23.5 s

gives the corresponding ultimate stability time. Similar conclusions to Section 5.1 can be obtained. The time calculated by the CITI scheme (δ = 0.5) when wedge blocks tended to be stable was later than the CAI scheme, and thus, the damping of the CAI scheme was larger in this case. For the CITI scheme, the larger the splitting parameter, the easier the wedge blocks became stable, because the numerical dissipations grew proportionally to the splitting parameter δ values. The movement process and ultimate stability time of wedge blocks calculated by the CAI scheme were similar to those calculated by the CITI scheme (δ = 0.6). Note also that the calculation efficiency of the CITI scheme was lower than that of the CAI scheme when many wedge blocks were considered.

6. Conclusions

In this paper, the CITI scheme was successfully introduced into 3D DDA to replace the original CAI scheme. The effectiveness of the developed 3D DDA was verified by basic forms of block motion. Furthermore, two numerical examples of the failure processes of rock slopes were simulated. From this study, the following conclusions can be drawn:

(1)

The basic idea of the CITI scheme is easy to understand, and the algorithm can be easily implemented into the 3D DDA program.

(2)

The CITI scheme can effectively compute the large displacement and deformation of block systems and can solve the nonlinear dynamic failure problems of rock slopes. The computational results of the CITI scheme may be more conservative than those of the CAI scheme.

(3)

For a large time step size, the CITI scheme can capture the nonlinear nature of the DDA computation; its results were still stable, and its computational accuracy was higher than the CAI scheme. The CITI scheme expands the value range of the time step size and reduces the numerical error brought by a large time step size. The CITI scheme is more applicable to variable acceleration motion than the CAI scheme.

(4)

The CITI scheme is more time-consuming than the CAI scheme in terms of computational efficiency, because there are two sub-steps involved within each time step. Especially, when the number of blocks was large, the computational efficiency reduced obviously.

(5)

No matter what integration schemes 3D DDA adopts, the larger the time step, the greater the damping, the faster the energy dissipation, and the shorter the time becoming stable are. For the CITI scheme, the damping is related to the splitting parameter δ, i.e., the greater the splitting parameter, the greater the damping is.