Development of a Material Point Method (MPM) for Geotechnical Applications
History of MPM The Finite Element Method (FEM) has become the standard tool for the analysis of a wide range of mechanical problems. However, the classical FEM is not well suited for the treatment of large deformation problems since excessive mesh distortions require computationally timeconsuming remeshing. In order to analyse geotechnical problems that involve large deformations, an alternative meshfree approach referred to as the Material Point Method (MPM) will be made operational. 

The early beginnings of the MPM can be traced back to the work of Harlow (1964) at Los Alamos National Laboratory (USA), who studied fluid flow by material points moving through a fixed grid. Later at the University of New Mexico, Sulsky et al. (1995) extended the approach to the modelling of problems in solid mechanics. At the University of Utah, Bardenhagen et al. (2000) extended the method further to include frictional contact between deformable solid bodies. The potential of the MPM for simulating granular flow was first recognised by Więckowski (1998) in Poland and Coetzee (2004) in South Africa. At the University of Stuttgart, both a dynamic 3D MPM with explicit time integration and a quasistatic 3D MPM with implicit integration are being developed for geotechnical problems with support of Dr. Coetzee from Stellenbosch University (South Africa) and Professor Więckowski from TU Łódź (Poland). Developments at the University of Stuttgart are sponsored by Deltares and Plaxis B.V. (The Netherlands). 

Figure 1. Impact of elastic steel disc on elastoplastic aluminium target simulated with the MPM 
Current State of the MPM Development at Stuttgart University In 2007, the quasistatic MPM has been successfully applied to elementary geotechnical problems for validation of a very first version of the code. Moreover, the method has been enhanced to reach a level of accuracy that matches the Finite Element Method. In order to prepare the ground for material nonlinearity and consolidation, highorder elements with quadratic interpolation of displacements have been implemented. These elements reproduce stress and strain fields more accurately than the initially employed loworder elements that use linear interpolation. Furthermore, highorder elements are less prone to locking effects often observed when applying loworder elements to elastoplastic problems. Meanwhile, elastoplastic slope and retaining wall problems have been solved. 
Figure 2. Shear bands during punching of elastoplastic material by a rigid body simulated with MPM 
Ongoing and future development of the MPM For the dynamic MPM, a contact algorithm has been implemented for the simulation of frictional contact. The method has been tested for elementary problems with known analytical solutions as well as the elastoplastic geotechnical slope stability problem showing excellent agreement with the results of the quasistatic simulation. The obtained computational results show, that the MPM is an ideal tool for the analysis of geotechnical problems involving large deformations. The MPM is to be extended and tested for practical geotechnical problems with a special view to simulate the installation of foundation structures. For foundation piles different installation procedures are to be simulated, such as pile jacking and pile battering. Moreover, the typical offshore installation of spudcans and bucket foundations are to be studied by numerical simulations. As many problems in geomechanics involve soilstructure interaction, the implementation of interface elements is to be studied. Further efforts will go into the introduction of pore water pressures to the quasistatic code and, at a later stage, also to the dynamic MPM. 
Figure 3. Deformation of frictional slope under gravity 
Figure 4. Development of shear bands during the deformation process of a frictional slope simulated with the MPM 
References Bardenhagen S.G., Brackbill J.U., Sulsky D. 2000. The materialpoint method for granular materials. Computational Methods in Applied Mechanics and Engineering, 187, 529541. Beuth L., Benz T., Vermeer P.A., Coetzee C.J., Bonnier P., Van Den Berg P. 2007. Formulation and Validation of a QuasiStatic Material Point Method, Proceedings of the 10th International Symposium on Numerical Methods in Geomechanics (NUMOG), Rhodes, Greece, 189195. Beuth L., Benz T., Vermeer P.A. 2007. Large Deformation Analysis Using a QuasiStatic Material Point Method, Proceedings of the 17th International Conference on Computer Methods in Mechanics (CMM), ŁódźSpała, Poland, 3334. Coetzee C.J. 2004. The modelling of granular flow using the particleincell method. PhD Thesis, Department of Mechanical Engineering, University of Stellenbosch, South Africa. Coetzee C.J., Vermeer P.A., Basson A.H. 2005. The modelling of anchors using the material point method. International Journal for Numerical and Analytical Methods in Geomechanics, 29, 879895. Sulsky D., Zhou S., Schreyer H.L. 1995. Application of a particleincell method to solid mechanics. Computer Physics Communications, 87, 236252. Więckowski Z. 1998. A particleincell method in analysis of motion of a granular material in a silo. Computational Mechanics: New Trends and Applications, CIMNE, Barcelona. Więckowski Z., Youn S.K., Yeon Y.H. 1999. A particleincell solution to the silo discharging problem. Int. J. Numer. Meth. Engng., 45, 12031225. Więckowski Z. 2003. Modelling of silo discharge and filling problems by the material point method. Task Quarterly, 4, 701721. 