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Combination of an adaptive remeshing technique with a coupled FEM–DEM approach for analysis of crack propagation problems

Abstract

This paper presents an enhanced coupled approach between the finite element method (FEM) and the discrete element method (DEM) in which an adaptive remeshing technique has been implemented. The remeshing technique is based on the computation of the Hessian of a selected nodal variable, i.e. the mesh is refined where the curvature of the variable field is greater. Once the Hessian is known, a metric tensor is defined node-wise that serves as input data for the remesher (MmgTools) that creates a new mesh. After remeshing, the mapping of the internal variables and the nodal values is performed and a regeneration of the discrete elements on the crack faces of the new mesh is carried out. Several examples of fracturing problems using the enhanced FEM–DEM formulation are presented. Accurate results in comparison with analytical and experimental solutions are obtained.

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Acknowledgements

This work has been supported by the Spanish Government program FPU: FPU16/02697. The authors gratefully acknowledge the received support.

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Appendix

Appendix

Kratos multiphysics

The FEM–DEM formulation presented has been implemented in the Kratos multiphysics framework [58] that has been specially designed for helping the development of multi-disciplinary finite element programs. We can summarize the following features:

  • Kernel The kernel and application approach is used to reduce the possible conflicts arising between developers of different fields.

  • Object oriented The modular design, hierarchy and abstraction of these approaches fit to the generality, flexibility and re-usability required for the current and future challenges in numerical methods. The main code is developed in C++ and the Python language is used for scripting

  • Open source The BSD (Berkeley Software Distribution) license allows to use and distribute the existing code without any restriction, but with the possibility to develop new parts of the code on an open or close basis depending on the developers. Additionally, Kratos can be freely used.

Mmg library

What is Mmg and how does it work?

Mmg is an open source software for anisotropic automatic remeshing for unstructured meshes based on Delaunay triangulations. It is licensed under a LGPL license and it has been integrated in Kratos,kratos via the mmg_process.h in the MeshingApplication. It provides three applications and four libraries (Figs. 27, 28):

  • The mmg2d application and the libmmg2d library: adaptation and optimization of a two-dimensional triangulation and generation of a triangulation from a set of points or from given boundary edges.

  • The mmgs application and the libmmgs library: adaptation and optimization of a surface triangulation and isovalue discretization.

  • The mmg3d application and the libmmg3d library: adaptation and optimization of a tetrahedral mesh and implicit domain meshing.

  • The libmmg library gathering the libmmg2d, libmmgs and libmmg3d libraries.

Fig. 27
figure 27

Kratos Multiphysics logo. https://github.com/KratosMultiphysics/Kratos

Fig. 28
figure 28

Image from Mmg web

Mmg logo

The Mmg remeshing process modifies the mesh [59, 60] iteratively until it is in agreement with the prescribed sizes on the idealized (Fig. 29) contour (and directions in case of anisotropic mesh). The software reads the mesh and the metric; then, the mesh is modified using local mesh modifications of which an intersection procedure based on anisotropic Delaunay kernel.

Fig. 29
figure 29

Image from [60]

Mmg idealized geometry

We can resume the remeshing algorithm in the following steps:

  1. 1.

    Mmg tries to have a good approximation of the surface (with respect to the Hausdorff parameter).

  2. 2.

    It remeshes according to a geometric criterion. Mmg scans the surface tetrahedra and splits the tetrahedra using predefined patterns if the Hausdorff distance [61] between the surface triangle of the tetrahedra and its curve representation does not respect the Hausdorff parameter.

  3. 3.

    The library scans again the surface tetrahedra and collapses all the edges at a Hausdorff distance smaller than a threshold defined in terms of the Hausdorff parameter.

  4. 4.

    Next, it intersects the provided metric and a surface metric computed at each point from the Hausdorff parameter and the curvature tensor at the point.

  5. 5.

    Then, Mmg smooths the metric to respect the gradation parameter. The metrics are iteratively propagated until the respect of the gradation everywhere.

  6. 6.

    Next, it remeshes the surface tetrahedra in order to respect the new metric.

  7. 7.

    Finally, it remeshes both the volume and surface to have edges between 0.6 and 1.3 (in the metric). The long edges are cut and short ones are deleted (collapsed).

Integration between Mmg and Kratos

In order to understand the integration between Kratos and Mmg is important to understand the data structure of Kratos. In Fig. 30 an example of the data structure of the Model can be analysed. The Model stores the whole model to be analysed and manages the different ModePart used on the simulation. The ModelPart holds all data related to an arbitrary part of model. It stores all existing components and data like Nodes, Properties, Elements, Conditions and solution data related to a part of the Model.

Fig. 30
figure 30

Model data structure

Fig. 31
figure 31

Concept of colours

The entities stored on the ModelPart are:

  • Node It is a point with additional facilities. Stores the nodal data, historical nodal data, and list of DoF.

  • Condition encapsulates data and operations necessary for calculating the local contributions of Condition to the global system of equations.

  • Element encapsulates the elemental formulation in one object and provides an interface for calculating the local matrices and vectors necessary for assembling the global system of equations. It holds its geometry that meanwhile is its array of Nodes.

  • Properties encapsulate data shared by different Elements or Conditions. It can store any type of data.

In our implementations, we used a process to set the BC (both Neumann and Dirichlet). In order to preserve that information after remeshing we need to create an identification system, so we are able to create an unique ID that will allow us to reconstruct the submodelpart structure after remeshing, this methodologies are commonly called colour identification. Figure 31 shows the concept of this idea.

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Cornejo, A., Mataix, V., Zárate, F. et al. Combination of an adaptive remeshing technique with a coupled FEM–DEM approach for analysis of crack propagation problems. Comp. Part. Mech. 7, 735–752 (2020). https://doi.org/10.1007/s40571-019-00306-4

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Keywords

  • Remeshing technique
  • Coupled formulation
  • Fracture mechanics
  • FEM–DEM
  • Finite element method
  • Discrete element method