Numerische Simulationen großer Deformationen
Numerical simulations are an indispensable active field of the modern engineering and science development. Considerable effort has been devoted by researchers to develop numerical methods that are able to simulate practical applications. These are more challenging when large deformation is involved. In continuum-based models, the traditional description of kinematics is either based on Lagrangian or Eulerian approach where each has its pros and cons. Coupling the two descriptions in one approach by exploiting the best features of each is desirable. At our institute, two coupling forms are used and developed as shortly explained next.
1. Material Point Method (MPM)
The inspiration of replacing the continuum with material points able to follow the movement in a natural manner has the beauty of linearising the convection-diffusion equation. However, it is not necessary to perform the solution of the equation of motion on the material points like in meshless methods. Instead it can be accomplished on a fixed background computational mesh. The evolved version of the last method is called the Material Point Method (MPM) (Sulsky et al., 1994).
In MPM, the computational domain is discretised using two types of discretisations. The first is the Lagrangian discretisation where the continuum body is represented by material points, or particles, which are tracked during the computation. To solve the momentum equation, the computational mesh is introduced as a second discretisation that provides a convenient means of calculating discrete derivatives and carrying out integration. The solution procedure of MPM during one time step is shown in the figure (Hamad 2014).
Figure 1. Solution procedure of an MPM computational step
Applications:
As an example of applying MPM to simulate geo-systems, which consist of geotextile in combination with soil and fluid-behaviour materials, the installation procedure of geotubes and geocontainers are highlighted here. Although the analysis is performed for two-dimensional problems, the formulation is three-dimensional using four-node tetrahedral elements.
Geotube: the geotextile tube, or geotube, is a tube formed in-situ consisting of permeable but soil-tight geotextile. Geotubes are widely used for applications in coastal and hydraulic engineering where the gravity barrier type structures are required. In order to examine the applicability of combining the enhancement schemes for the liquid-behaviour materials in MPM (Stolle et al., 2014), the geotube problem is simulated numerically and compared to the closed-form solution.
Geocontainers: another application of geotextiles is for the construction of geocontainer units, which consist of a prefabricated geotextile placed in a split barge and mechanically filled with sand or slurry up to several hundred cubic meters (Hamad et al., 2013). They are subsequently dumped from the scow bed in the desired position. Geocontainer units are used for underwater structures such as breakwaters, barriers to close openings, and dams that hold contaminated sludges. The releasing, dropping and interaction of two containers are simulated in MPM as shown in the figure. As the process of dropping a sand-filled container in water is very complex, many laboratory tests have been performed as reported in literature to describe the phenomenon. One of these tests is reproduced in MPM and compared to the measured data, as depicted in the figure (Hamad 2014).
Figure 2: a. Final configuration of the geotube | b. Interaction of geocontainer | c. Dropping bags in water |
2. Finite Element Method (FEM)
Another form of exploiting the two classical reference of configurations is observed in problems dealing with fluid-solid interfaces. For such problems, two discretisations are employed instead of a single mesh. The Coupled Eulerian-Lagrangian (CEL) methods are based on coupling between Lagrangian body, which is most commonly solid material, and Eulerian for the fluid behaviour material. For such an analysis strategy, explicit coupling is obtained by applying pressure boundary condition on the Lagrangian body, whereas velocity boundary condition prescribed on the Eulerian discretisation for the fluid as illustrated in the figure. The commercial software Abaqus is used for this purpose.
Figure 3. Explicit coupling solutions of CEL methods
Applications:
The following applications are performed at the institute of Geotechnical Engineering in Stuttgart.
Landslides: the dynamic process of rock landslides induced by earthquakes is modeled, where the failure surface is predefined. Two numerical approaches are examined: the first by modelling the slope as a continuum material in which the Mohr-Coulomb failure criterion is adopted, and the second as discrete blocks where frictional contact algorithm among the blocks is introduced, as demonstrated in figure 4.
Figure 4. Rock slope failure: a. Continuum model | b. Discrete Model |
Pile installation: an important application of installing pile into granular material is simulated using CEL. The soil is modeled using the hypoplastic constitutive law, and the pile as an elastic material where the external hammer load is applied. In order to reduce the effect of reflected wave, silent boundary condition is introduced at the bottom of the model, as shown in figure 5. |
Figure 5. Pile installation in sand |
References:
Sulsky, D., Chen, Z., Schreyer, H. (1994) A particle method for history–dependent materials, Computer Methods in Applied Mechanics and Engineering 118 (1) 179–196.
Hamad, F. (2014) Formulation of a dynamic material point method and applications to soil-water-geotextile systems. PhD thesis, Institute of Geotechnical Engineering, University of Stuttgart.
Stolle, D., Maitland, K., Hamad, F. (2014) Incompressible flow strategies for ice creep, Finite Elements in Analysis and Design 89 67–76.
Hamad, F., Vermeer, P., Moormann, C. (2013): Development of a coupled FEM-MPM approach to model a 3D membrane with an application of releasing geocontainer from barge, Installation Effects in Geotechnical Engineering, 176-183.
Hamad, F., Stolle, D., Vermeer, P. (2014) Modelling of membranes in the material point method with applications, International Journal for Numerical and Analytical Methods in Geomechanics 39, 833–853.
contact: Fursan Hamad