Field Scale Modelling of Explosion-Generated Crack Densities in Granitic Rocks Using Dual-Support Smoothed Particle Hydrodynamics (DS-SPH)

Abstract

A dual-support smoothed particle hydrodynamics (DS-SPH) method is developed to quantify explosion-generated crack densities within granitic rock masses in field-scale computational domains. In DS-SPH framework, coupled Eulerian total Lagrangian formulations, along with interface treatment between solid and inviscid fluid particles are fully considered. A new momentum equation formulation for interface treatment between inviscid fluids with various density ratios inside the blast borehole is also developed. The DS-SPH solutions are extended in such a way that decoupled explosions together with free surface and non-reflecting boundary conditions can be easily implemented. The three main deficiencies of conventional SPH (e.g., inconsistency, tensile instability, and hourglass mode) are removed in the stabilized DS-SPH method. In addition, GPU parallelization is adopted to accelerate the stabilized DS-SPH approach for higher efficiency. Then, the robustness of the developed DS-SPH solutions are verified by a number of theoretical and computational examples, and reproducing the full-scale blast field experiments. The developed DS-SPH solutions precisely reproduce the experimentally observed blast wave structures, and crack densities at several monitor locations. This is accomplished by addressing uncertainties in input parameters and enforcing various stabilization terms in DS-SPH formulations. Satisfactory speedup and acceptable scalability are also obtained, demonstrating that GPU-accelerated DS-SPH is a promising tool to speed up field scale particle-based simulations.

Appendices

Appendices

1.1 Appendix A

1.1.1 Algorithm for DS-SPH Implementation

The governing and constitutive equations in aforementioned sections can be solved numerically through many efficient available techniques, including the leap-frog prediction-correction and Runge–Kutta methods. As the energy equation is considered leap-frog prediction-correction scheme is an efficient technique as a second-order accurate numerical model, and is adopted herein to solve the partial differential equations.

1.1.2 Time Integration

The time step $\Delta t$ is computed by the Courant–Friedrichs–Lewy condition as follows

$$\Delta t=\min\left({\Delta t}_{f}=0.1{\min}_{f}\left(\sqrt{h/\Vert {{\varvec{a}}}_{f}\Vert }\right),{\Delta t}_{s}=0.8\frac{h}{{c}_{s}}\right)$$

(48)

where ${{\varvec{a}}}_{f}=\frac{d{{\varvec{v}}}_{f}}{dt}$ is the acceleration of fluid particle f, and ${c}_{s}=\sqrt{E/{\rho }_{0}}$ is the sound speed in solid. As identical time steps are used in the integration of fluid and solid equations, the problem of force mismatch in the fluid–solid interaction is avoided and the momentum conservation is not violated.

1.1.3 Neighbouring Particle Search

The computational performance of DS-SPH considers an efficient algorithm for searching the neighbouring particles at a specified position in the computational domain. Here, the bucket sort algorithm is used for the search of neighbourhood particles. It permits the computational time to be remarkably reduced and is efficient when particles have different smoothing lengths with different particle sizes. It is also robust for problems with large number of SPH particles and is suitable for graphics processing unit (GPU) computing (Gharehdash et al. 2020a, b). Our three-dimensional stabilized DS-SPH solver is written in C++ and implemented in OpenMP library (https://www.openmp.org/). Figure 23 shows the flowchart of the DS-SPH modelling of rock blasting in field scale for each time step.

Fig. 23

Algorithm for the DS-SPH implementation in one computational step

1.2 Appendix B

In this computational example, we simulate blast wave radiation propagation with increasing free-surface and non-reflecting boundary conditions complexity (Fig. 24). Two snapshots are produced to illustrate the ability of the DS-SPH in simulating blast wave propagation with complicated boundary conditions in computational models that include interfaces zones with large variations in SPH particle sizes. In Fig. 24 as wavefronts expand from the source, P and S waves clearly separate and show different radiation patterns. Both waves are later reflected by the free surface and travel downward. In the computational model (e.g., Fig. 24d) with two free surfaces at top and bottom, P and S waves radiations occur much more complicated. In this snapshot, the propagation and reflection of multiple blast waves give rise to a complex pattern of mechanical disturbances.

Fig. 24

Snapshots of blast wave radiation at times 3 ms and 4 ms for computational domains with a, b one (top), and c, d two (top and bottom) free surface boundary conditions. Grey colours indicate the modelling domains. Red and blue colours represent positive and negative wavefield values

1.3 Appendix C

To further demonstrate the effectiveness of the proposed formulations in DS-SPH framework, the experimental and numerical problems presented in references (Ma et al. 1998; Hu et al. 2014) is revisited. We focus on the attenuation of peak particle velocity (PPV), and peak particle acceleration (PPA) values. The first field test is an underground explosion (Ma et al. 1998). In this case, we adopted a discretization of varying particle sizes, 4,000,000 fine particles around the explosion chamber and 1,500,000 coarse particles for surrounded rock mass. The second field test is a blasting excavation of a rock slope (Hu et al. 2014). In this case, the number and size of SPH particles equaled the number and size of FEM elements (Hu et al. 2014). In Fig. 

Fig. 25

a Attenuation of PPA, and b, c PPV

25a–c, prominent features in the measured and simulated waveforms are identified for PPA, and PPV values. The calculated attenuation of velocity and acceleration agree very well with the test data due to the material viscous damping is included in the DS-SPH calculations, and the complicated site conditions are not simplified in the numerical model as an equivalent isotropic rock mass. These observations indicate the applicability of the developed DS-SPH method and the suitability of using proposed formulations and anisotropic brittle damage model to simulate underground explosions and the blasting excavation of the rock slope in rock mass medium.