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Meshfree generalized finite difference methods in soil mechanics—part II: numerical results


In geotechnical engineering, simulations are of utmost importance. Due to large deformations, meshfree methods are more suitable than classical meshbased methods. Nevertheless, they have to be validated on the laboratory scale in order to guarantee reliable conclusions for real life applications. In this contribution, we complete the theoretical description of the two novel meshfree generalized finite difference methods Finite Pointset Method (FPM) and Soft PARticle Code (SPARC) by numerical results for the standard benchmark problems oedometric and triaxial test. We focus on the quality of the results as well as on the rate-independent character of the numerical implementation of the nonlinear barodesy model for sand.

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  1. 1.

    Although the wording is different, points in FPM and particles in SPARC, both notations stand for numerical points without mass.

  2. 2.

    Averaging is absolutely necessary in case of inhomogeneous deformation occurring in the triaxial test due to friction at the plates. Homogeneous deformation does not demand for it. However, we use the described averaging strategy in both cases.

  3. 3.

    In the sense of absolute value.

  4. 4.

    Localization of deformation means that with increasing loading the deformation of a solid body localizes in narrow zones which gradually develops to shear bands. This occurs when the stiffness approaches zero. Vanishing stiffness leads to an ill-posed initial boundary value problem inducing convergence problems in the Newton solver. For further details see Schneider-Muntau et al. (2017).


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This paper presents results of the joint research project “Ein netzfreier numerischer Zugang zu Böden in Ruhe und im Fließen (A meshfree numerical approach for soils at rest and in flow). (1) Kaiserslautern, Germany: The group consisting of Prof. Dr.-Ing. C. Vrettos, Dr.-Ing. A. Becker (Division of Soil Mechanics and Foundation Engineering, University of Kaiserslautern), Dr. J. Kuhnert, Dr. I. Michel (Fraunhofer Institute for Industrial Mathematics ITWM) is supported by the “Deutsche Forschungsgemeinschaft (DFG)”, Germany. (2) Innsbruck, Austria: The group consisting of Prof. Dr.-Ing. D. Kolymbas, S.M.I. Bathaeian, Dr.-Ing. C.-H. Chen (temporarily), Dr.-Ing. I. Polymerou (temporarily) (Division of Geotechnical and Tunnel Engineering, University of Innsbruck) is supported by the “Fonds zur Förderung der wissenschaftlichen Forschung (FWF)”, Austria.

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Correspondence to I. Michel.

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Michel, I., Bathaeian, S.M.I., Kuhnert, J. et al. Meshfree generalized finite difference methods in soil mechanics—part II: numerical results. Int J Geomath 8, 191–217 (2017). https://doi.org/10.1007/s13137-017-0096-5

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  • Generalized finite difference methods
  • Meshfree methods
  • Finite Pointset Method (FPM)
  • Soft PARticle Code (SPARC)
  • Barodesy model

Mathematics Subject Classification

  • 35D35
  • 35Q74
  • 65D05
  • 65M99