^{1}

^{2}

^{1}

^{3}

^{4}

^{1}

^{2}

^{3}

^{4}

To achieve numerical simulation of large deformation evolution processes in underground engineering, the barycentric interpolation test function is established in this paper based on the manifold cover idea. A large-deformation numerical simulation method is proposed by the double discrete method with the fixed Euler background mesh and moving material points, with discontinuous damage processes implemented by continuous simulation. The material particles are also the integration points. This method is called the manifold cover Lagrangian integral point method based on barycentric interpolation. The method uses the Euler mesh as the background integral mesh and describes the deformation behavior of macroscopic objects through the motion of particles between meshes. Therefore, this method can avoid the problem of computation termination caused by the distortion of the mesh in the calculation process. In addition, this method can keep material particles moving without limits in the set region, which makes it suitable for simulating large deformation and collapse problems in geotechnical engineering. Taking a typical slope as an example, the results of a slope slip surface obtained using the manifold cover Lagrangian integral point method based on barycentric interpolation proposed in this paper were basically consistent with the theoretical analytical method. Hence, the correctness of the method was verified. The method was then applied for simulating the collapse process of the side slope, thereby confirming the feasibility of the method for computing large deformations.

The surrounding rocks of tunnels and other underground engineering structures mainly have three deformation failure mechanisms: rock burst, collapse, and large deformation [

Numerical computation methods, because of their proven effectiveness in tunnel stability analysis, have been developing rapidly and are applied widely in geotechnical engineering. The finite element method (FEM) and the finite difference method (FDM) are now the most common methods in geotechnical engineering for analyzing continuous deformations. The FEM, developed in the 1950s, is the earliest numerical method [

To resolve this problem, improved FEMs, numerical manifold methods, discrete element methods, and meshless methods have been applied widely.

In the 2000s, the FEM with Lagrangian integration points (FEMLIP) was proposed in geophysics to represent mantle convection [

Shi [

Discrete element methods (DEMs) are effective numerical computation methods, especially for solving noncontinuum problems. By the type of basic unit simulated, DEMs can be classified into two types: particle DEMs, which use discs or spheres as the basic unit, and block DEMs, which use blocky masses as the basic unit. Early DEMs include the DEM proposed by Peter Cundall and the discontinuous deformation analysis (DDA) proposed by Shi [

Research of meshless methods dates back to the 1970s. Gingold and Monaghan and Lucy [

However, numerical simulation methods for solving large-deformation problems in geotechnical engineering are still in the test-and-trial stage and are not ready for wide applications. Thus, this study proposes a numerical simulation method for solving large deformation and collapse problems in geotechnical engineering. Particularly, the method constructs barycentric interpolation trial functions based on the idea of manifold cover, realizes double discretization of Eulerian background meshes and moving material points, uses material particles as mesh integration points, and represents the macroscopic deformation behavior of objects using the motion of particles between meshes. Thus, this method is not affected by mesh distortion and allows unlimited computation in the predefined domain. The method was applied to a typical side slope. The resultof slip surface of the slope was then compared with that obtained using the Bishop method of slices. The method proposed in this study was then applied to simulate the large deformation (collapse) of the side slope, thereby confirming its validity and feasibility.

In continuum mechanics, there are two classical methods for the description of motion: Lagrangian description and Eulerian description. These two methods are also commonly used for describing the motion of finite elements.

In the reference coordinate system shown in Figure

Reference coordinate system.

In the Lagrangian description, mesh nodes (together with the material) are fixed to and move with material points. The Lagrangian description, focusing on material points and using the coordinate of material points for describing their motion, is mainly used for the stress-strain analysis of solid structures. When using the Lagrangian description for object analysis, the change in the object’s shape is completely consistent with that in the discrete meshes. This characteristic allows an accurate representation of the object’s motion and boundaries. However, when used for solving large-deformation problems, the Lagrangian description very often suffers mesh distortion-caused computation termination.

In the Eulerian description, mesh nodes and materials are independent from each other, and meshes are usually fixed [

The arbitrary Lagrangian–Eulerian (ALE) description has the advantages of both Lagrangian and Eulerian descriptions. It uses the Lagrangian description for representing the boundaries of mass and the Eulerian description for representing the interior of objects [

The FEMLIP uses fixed Eulerian background meshes and discretizes materials using particles, with the properties of materials stored in the particles. It originated from the particle-in-cell (PIC) method used in plasma research. During computation, the background meshes are fixed, and particles are used as the principal medium for transmitting the physical information of materials. Figure

Dual discretization of Eulerian meshes and material particles.

The manifold cover system is a concept originating from numerical manifold methods. A manifold cover system consists of mathematical and physical manifolds. The problem-solving domain is defined using physical manifolds, and mathematical manifolds are used to define the integration domain and construct trial functions. The accuracy of the computational solutions usually depends on the density of mathematical manifolds. The FEM is a particular form of the manifold cover method, that is, a manifold cover method with the mathematical and physical manifolds completely overlapping. The displacement functions of the cover usually use constant functions, and the weight functions usually use linear functions. The meshless methods developed in recent years treat the effect of nodes or particles as mathematical covers and construct approximation functions through mathematical approximation. Figures

The cover system of the numerical manifold method.

The cover system of the FEM.

The cover system of the meshless method.

The subdivision of an arbitrary polygon

Polygonal element.

In finite element computation, to define boundary conditions of the model and represent the rigid object displacement mode and linear displacement field accurately, the shape function,

Nonnegativity and interpolation:

where

Unit partition:

Linear completeness:

Assume a polygonal domain, _{i} then intersect at _{i}. A polygon circumscribing the circle and crossing _{i} can then be derived, as shown in Figure

Geometric structure of the barycentric interpolation function.

According to the Gauss divergence theorem, a polygon-bounded area _{1}, _{2}, ... _{n} satisfies the following equation:

When

The unit outward normal vector of a polygon _{1}, _{2}, ... _{n} can be expressed as follows:

The boundary of the polygon can be integrated as follows:

Equation (

Thus, the coordinate of point

The weight function is defined as the ratio of the side length, _{i}, _{i−1}, of the polygon _{1}, _{2}, ... _{n} to the length of _{i}:

The following equation can be derived from Figure

Substituting (

The barycentric interpolation function for the polygonal unit can then be derived from (

Assume a problem in a planar domain. Its boundary, Г, can then be expressed as follows:

Balance equation:

Velocity and stress boundaries:

The above balance equation is essentially a Navier–Stokes equation with inertia not considered:

According to the unit trial functions, the relationship between the coordinates of any point in the unit and the node in the local coordinate system can be expressed as follows:

Similarly, the relationship between the displacements (velocities) of any point in the unit and the node can be expressed as follows:

The relationship between the strain and displacement of a unit can be expressed as follows:

The constitutive relationship for representing the stress-strain behavior of objects of viscous constitution can be expressed as follows:

According to the principle of virtual work, the weak form of integration of the discretization equation can be expressed as follows:

According to (

The relationship between the velocity of any point in the unit and the nodal velocity can be expressed as the following trial function:

Thus, the following equation can be derived:

According to the arbitrariness of virtual velocity, the following equation can be derived:

The mesh node velocity was resolved and then substituted into (

Considering the temporal variation of large deformations, a viscoelastic-plastic constitutive model was adopted, which consisted of viscous, elastic, and plastic components connected in series, as shown in Figure

Viscoelastic-plastic constitutive model.

The major parameters of the constitutive model include viscous shear modulus, viscous bulk modulus, elastic shear modulus, elastic bulk modulus, cohesion, and angle of friction.

For viscoelastic-plastic models, the total strain rate can be expressed as the sum of viscous, elastic, and plastic strain rates:

Dividing the strain rate into two components, partial and spherical strain rates, the following equation can be derived:

Stress rate is affected by the rigid object rotation. The component of the stress rate that is not affected by the rigid object rotation is referred to as the Jaumann stress rate,

Designating

Thus, the following equation can be derived:

Designating

Designating

Similarly, the following equation can be derived:

According to (

Traditional FEMs usually select integration points according to predefined schemes so that the most accurate integration with respect to a minimum number of points can be realized (for example, Gauss integration). However,when mesh nodes move as the material deforms, traditional FEMs are prone to stop computation because of mesh distortion when used for computing large deformations. To resolve this problem, fixed Eulerian background meshes were used, and the particles moving inside the meshes were adopted to describe material deformation and object motion. With the moving particles as integration points, the instantaneous relationship between the material points and meshes can be established for each computational increment, thereby enabling accurate tracking of material deformation and object motion and ensuring mesh adaptability. Because the location of particles changes continuously, the integration weight needs to be updated to realize correct integration.

For _{n−1}, the following equation stands:_{n} is the weight function, and _{n} is the location of integration points.

For one-dimensional problems (linear units), the integrated function can be expressed as the following polynomial (integrated in the range of −1 to +1):

Integrating (_{n} can be obtained according to

In principle, the value of _{n} can be estimated by defining an appropriate number of constraints. However, in real operations, this requires a large amount of computation.,Simultaneously, ωn calculated maybe negative values, and reasonable convergence cannot be ensured. For real applications, the relationship between the value of ωn of a particle and the mass or volume of the material can be established only when the value is positive.

For viscous flows using bilinear units, if only steady and linear constraints are used, then the optimum convergence rate under the unlimited mesh constraint can be obtained. The steady constraint is defined as follows:

The bilinear constraint is defined as follows:

The following definitions are then given:

The value of _{n} can then be obtained through the following iteration:

The convergence condition for the iteration is defined as follows:

The right side of equation (

After the mesh node velocity is obtained, the constantly changing location of particles can be updated as follows:

The stability of the side slope presented by Dawson et al., which has been cited by numerous studies, is analyzed using the method proposed above. The slip surface of the side slope obtained using the method was then compared with the Bishop method of slices in the literature. On this basis, the large deformation (collapse) of the side slope was simulated using the method proposed in this study, thereby confirming its accuracy and feasibility.

Figure

Computational model.

Physical-mechanical parameters of the slope.

Density (kg·m^{−3}) | Bulk modulus (MPa) | Shear modulus (MPa) | Cohesion (kPa) | Angle of friction (°) | Maximum tensile strength (MPa) |
---|---|---|---|---|---|

2,000 | 100 | 30 | 0.35 | 20 | 0.0045 |

Partial strain rate yielded by the method proposed in this study.

Potential slip surfaces computed using the two methods.

Figure

Slope collapse process. (a) Step 8. (b) Step 12. (c) Step 28. (d) Step 50.

This study proposed a barycentric interpolation manifold method with Lagrangian integration points, a continuous numerical simulation method for simulating the discontinuous failure process of large deformations, by constructing barycentric interpolation trial functions based on the idea of the manifold cover, adopting double discretization of Eulerian background meshes and moving material points, and using material particles as mesh integration points. Our conclusions can be summarized as follows:

The method adopts Eulerian background integration meshes and represents the macroscopic deformation behavior of objects using the motion of particles between meshes, thereby avoiding the computation termination caused by mesh distortion that usually occurs with large deformation computation. Moreover, the method allows unlimited computation in the predefined domain and is suitable for simulating large deformations and collapses in geotechnical engineering.

The method proposed in this study was applied to a typical side slope. The slip surface yielded by the method was basically consistent with that yielded by a theoretical analysis method, thereby confirming the validity of the method proposed in this study. The method was then applied to simulate the collapse process of the side slope, confirming the feasibility of the method for computing large deformations [

The data used to support the findings of this study are available from the corresponding author upon request.

The authors declare that they have no conflicts of interest.

This work was financially supported by the National Natural Science Foundation of China (nos. 51879150 and 51809115), Qilu Construction Projects of Science and Technology in 2016 (no. 2016B20), and the Shandong Provincial Natural Science Foundation, China (nos. ZR2019QEE003 and 2018GGX104024). These financial supports are gratefully acknowledged.