^{1}

^{2}

^{1}

^{3}

^{2}

^{1}

^{2}

^{3}

Based on the general Biot theory of saturated porous media, a modified time-discontinuous Galerkin finite element method (MDGFEM) is presented to simulate the structural dynamics and wave propagation problems of gas-saturated coal subjected to impact loading. Numerical results of one dimension and two dimensions show that the present MDGFEM possesses better abilities and provides much more accurate solutions than the traditional Newmark method and previous DGFEM for the impact problem. It can effectively capture the discontinuities of the wave and filter out the effects of spurious numerical oscillation induced by high-frequency impulsive load. The results can provide a technological basis for the research of the prevention of coal and gas dynamic disasters under deep mining. And the method could be useful for the further numerical research of coal-rock-gas coupling problems and coal-gas-heat coupling problems subjected to impact loading.

With the increase of coal mining depth, coal and gas outburst disasters are becoming more and more severe [

In the past, efforts have been made to find a porous media model, such as Biot’s theory, to discover the reflection and transmission of waves in a porous medium [

Developing a robust and efficient numerical method for the coupled problems subjected to the high-frequency impact loading is challenging, and the simulation of the problems is another motivation of our research. This is particularly the case for the problems of wave propagation in gas-saturated coal where discontinuities exist both in the time domain and the space domain. Despite the popularity, the traditional time integration method (such as Newmark method) [

Li et al. presented a novel time-discontinuous Galerkin finite element method (DGFEM) for solving dynamics and wave propagation in nonlinear solids and saturated porous media and heat wave propagation problem subjected to impact loading [

For the high-speed motion and high pressure of the gas-saturated coal problem under high-frequency loading, the acceleration of porous fluid should be considered [

This section summarizes the basic governing equations of gas-saturated coal porous media subjected to impact loading. As the porous fluid is compressible, the equations of motion for the coal skeleton can be expressed as [

For the pore fluid, the equation of motion can be written as

Insertion of

Equation (

Then, the mass conservation equation of the fluid flow can be written as

Using the definition of equation (

It should be noted that the appropriate boundary conditions associated with the governing equations (

On the other hand, if surface loadings are applied to the corresponding surfaces

In the present paper, surface impulse loading is applied with the form as

The standard Galerkin discretized equation (

The main features of the discontinuous Galerkin integration method in time domain have been described in our previous articles [

The global displacement vectors of solid and fluid at arbitrary time

The global velocity vector of solid and fluid at arbitrary time

The weak forms of the semidiscretized equation (

Substituting equations (

It should be noted that the global nodal displacement vector of solid and fluid is continuous at any time level and the global nodal velocity vector is still discontinuous at any time level [

When simulating problems of structure dynamics and wave propagation under impulse load, the modified time-continuous Galerkin finite element method shows better ability to filter out the effects of spurious numerical oscillations. Having demonstrated the applicability and advantage of the MDGFEM as indicated in our articles [

As we know that the selections of a stiffness proportional and mass proportional damping coefficient are effective for high-frequency oscillations and low-frequency oscillations, respectively, we modify the present DGFEM by using an artificial stiffness proportional Rayleigh-type damping scheme. The equations of the damping scheme can be written as follows:

Using the stiffness proportional artificial damping matrix equation (

Equations (

In this section, three numerical examples of gas-saturated coal based on Biot’s model subjected to impulse load are investigated. Various results in 1-D and 2-D are presented to demonstrate that the MDGFEM formulations are capable of producing reliable results in the problem [

Values of relevant parameters for calculations.

Parameters | Cortical bone |
---|---|

Hydraulic permeability | |

Biot parameter | |

Porosity | |

Young’s modulus | |

Bulk modulus of the fluid | |

Bulk modulus of the solid | |

Densities of the fluid | |

Densities of the solid | |

Poisson’s ratio |

We first consider a one-dimensional stress wave propagation problem in saturated gas-saturated coal. The length of the column is 200 m, and the area of the cross section is 1 m^{2}. The column, as shown in Figure

Model of example 1.

The column is meshed in elements of 0.4 m along the column axis. Figures

Stress profiles of different methods at

Gas pressure profiles of different methods at

Secondly, we consider a column of gas-saturated coal rock with the same geometry, boundary condition, load, and finite element mesh as the first example. The Drucker–Prager criterion and the linear strain hardening rule

Elastic-plastic stress profiles of different methods at

Pressure profiles of different methods at

As the third example, we consider elastic-plastic wave propagation problem of gas-saturated coal rock in two dimensions. A square domain, which is 10 m deep, 10 m wide, and of infinite length in the horizontal direction, is subjected to an impulse inclined compression load at top left corner (0–1.4 m) as depicted in Figure

Modeling and mesh discretization for example 3.

In the present example, the material property data are defined as follows:

Figures

Comparison of the stress distributions between the Newmark method and the DGFEM. (a) Newmark method 0.001 s, (b) MDGFEM 0.001 s, (c) Newmark method 0.003 s, (d) MDGFEM 0.003 s, (e) Newmark method 0.005 s, and (f) MDGFEM 0.005 s.

Comparison of the pressure distributions between the Newmark method and the MDGFEM. (a) Newmark method 0.001 s, (b) MDGFEM 0.001 s, (c) Newmark method 0.003 s, (d) MDGFEM 0.003 s, (e) Newmark method 0.005 s, and (f) MDGFEM 0.005 s.

The traditional Galerkin finite element method such as the Newmark method fails to capture the discontinuities or sharp gradients of the stress wave in solving the impact problem. An additional artificial damping discontinuous Galerkin numerical algorithm was incorporated into the final finite element discretised form to reduce the numerical oscillations in the wave-front stage for the impact problem of gas-saturated coal. Based on the MDGFEM, series of numerical examples of dynamic problem of gas-saturated coal under impact loading show that the modified discontinuous Galerkin finite element method can effectively filter out the spurious numerical oscillations in the wave front of the elastic-plastic stress wave and the gas pressure wave. The study of the method in this paper demonstrates that it may also serve as a viable method for the coal-rock-gas coupling problem and coal-gas-heat coupling problems subjected to impact loading.

The data used to support the findings of this study are included within the article.

The authors declare that they have no conflicts of interest.