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

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Abstract

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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Notes

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

  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. In the sense of absolute value.

  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).

References

  • Abe, K., Nakamura, S., Nakamura, H., Shiomi, K.: Numerical study on dynamic behavior of slope models including weak layers from deformation to failure using material point method. Soils Found 57(2), 155–175 (2017)

    Article  Google Scholar 

  • Bandara, S., Soga, K.: Coupling of soil deformation and pore fluid flow using material point method. Comput. Geotech. 63, 199–214 (2015)

    Article  Google Scholar 

  • Bardenhagen, S.G., Brackbill, J.U., Sulsky, D.: The material-point method for granular materials. Comput. Methods Appl. Mech. Eng. 187, 529–541 (2000)

    Article  MATH  Google Scholar 

  • Bathe, K.-J.: Finite Element Procedures, 2nd edn. Bathe K-J, Watertown (2014). ISBN 978-0-9790049-5-7

    MATH  Google Scholar 

  • Beuth, L., Więckowski, L., Vermeer, P.A.: Solution of quasi-static large-strain problems by the material point method. Int. J. Numer. Anal. Methods Geomech. 35, 1451–1465 (2011)

    Google Scholar 

  • Beuth, L., Ceccato, F., Rohe, A.: Modelling of cone penetration testing with the matrial point method. Bemessen mit numerischen Methoden, Workshop 24./25.09.2013, TU Hamburg-Harburg, pp. 8–25 (2013)

  • Bhandari, T., Hamad, F., Moormann, C., Sharma, K.G., Westrich, B.: Numerical modelling of seismic slope failure using MPM. Comput. Geotech. 75, 126–134 (2016)

    Article  Google Scholar 

  • Blanc T (2008) Numerical simulation of debris flows with the 2D-SPH depth integrated model. Master Thesis, Institute for Mountain Risk Engineering, University of Natural Resources and Applied Life Sciences, Vienna

  • Blanc, T., Pastor, M.: A stabilized smoothed particle hydrodynamics, Taylor–Galerkin algorithm for soil dynamics problems. Int. J. Numer. Anal. Methods Geomech. 37, 1–30 (2013)

    Article  Google Scholar 

  • Bui, H.H., Fukagawa, R.: An improved SPH method for saturated soils and its application to investigate the mechanisms of embankment failure: case of hydrostatic pore-water pressure. Int. J. Numer. Anal. Methods Geomech. (2011). doi:10.1002/nag.1084

    Google Scholar 

  • Bui, H.H., Kodikara, J.A., Pathegama, R., Bouazza, A., Haque, A.: Large deformation and post-failure simulations of segmental retaining walls using mesh-free method (SPH). CoRR (2015). arXiv:1501.04000

  • Carbonell, J.M., Oñate, E., Suárez, B.: Modelling of tunnelling processes and rock cutting tool wear with the particle finite element method. Comput. Mech. 52(3), 607–629 (2013)

    Article  MATH  Google Scholar 

  • Chen, C.-H.: Development of Soft Particle Code (SPARC). Ph.D. Thesis, University of Innsbruck, Logos Verlag Berlin GmbH (2014)

  • Chen, J.-S., Hillman, M., Chi, S.-W.: Meshfree methods: progress made after 20 years. J. Eng. Mech. 143(4), 04017001 (2017)

    Article  Google Scholar 

  • Coetzee, C.J., Vermeer, P.A., Basson, A.H.: The modelling of anchors using the material point method. Int. J. Numer. Anal. Methods Geomech. 29, 879–895 (2005)

    Article  MATH  Google Scholar 

  • Cuéllar, P., Baeßler, M., Rücker, W.: Ratcheting convective cells of sand grains around offshore piles under cyclic lateral loads. Granul. Matter 11, 379–390 (2009)

    Article  Google Scholar 

  • Dang, H.K., Meguid, M.A.: An efficient finite-discrete element method for quasi-static nonlinear soil–structure interaction problems. Int. J. Numer. Anal. Methods Geomech. 37(2), 130–149 (2013)

    Article  Google Scholar 

  • Desrues, J., Zweschper, B., Vermeer, P.A.: Database for Tests on Hostun RF Sand. Technical Report, Institute of Geotechnical Engineering, University of Stuttgart (2000)

  • Drumm, C., Tiwari, S., Kuhnert, J., Bart, H.J.: Finite pointset method for simulation of the liquid–liquid flow field in an extractor. Comput. Chem. Eng. 32(12), 2946–2957 (2008)

    Article  Google Scholar 

  • Dufour, F., Mühlhaus, H.-B., Moresi, L.: A particle-in-cell formulation for large deformation in Cosserat continua. In: Mühlhaus, H.-B., Dyskin, A., Pasternak, E. (eds.) Bifurcation and Localization in Soils and Rocks, pp. 133–138. Balkema, Leiden (2001)

    Google Scholar 

  • Gabrieli, F., Cola, S., Calvetti, F.: Use of an up-scaled DEM model for analysing the behavior of a shallow foundation on a model slope. Geomech. Geoeng. 4(2), 109–122 (2009)

    Article  Google Scholar 

  • Holmes, D.W., Williams, J.R., Tilke, P., Leonardi, C.R.: Characterizing flow in oil reservoir rock using SPH: absolute permeability. Int. J. Numer. Anal. Methods Geomech. 61(7), 1–6 (2011)

    Google Scholar 

  • Holtz, R.D., Kovacs, W.D.: An Introduction to Geotechnical Engineering. Prentice Hall, Englewood Cliffs (1981)

    Google Scholar 

  • Hu, M., Liu, M.B., Xie, M.W., Liu, G.R.: Three-dimensional run-out analysis and prediction of flow-like landslides using smoothed particle hydrodynamics. Environ. Earth Sci. 73(4), 1629–1640 (2015)

    Article  Google Scholar 

  • Jassim, I., Stolle, D., Vermeer, P.: Two-phase dynamic analysis by material point method. Int. J. Numer. Anal. Methods Geomech. (2012). doi:10.1002/nag.2146

    Google Scholar 

  • Jefferies, A., Kuhnert, J., Aschenbrenner, L., Giffhorn, U.: Finite pointset method for the simulation of a vehicle travelling through a body of water. In: Griebel, M., Schweitzer, M.A. (eds.) Meshfree Methods for Partial Differential Equations VII, pp. 205–221. Springer, Berlin (2015)

    Google Scholar 

  • Jiang, M., Yin, Z.-Y.: Analysis of stress redistribution in soil and earth pressure on tunnel lining using discrete element method. Tunn. Undergr. Sp. Tech. 32, 251–259 (2012)

    Article  Google Scholar 

  • Kardani, O., Nazem, M., Kardani, M., Sloan, S.: On the application of the maximum entropy meshfree method for elastoplastic geotechnical analysis. Comput. Geotech. 84, 68–77 (2017)

    Article  Google Scholar 

  • Khoshghalb, A., Khalili, N.: A Meshfree method for fully coupled analysis of flow and deformation in unsaturated porous media. Int. J. Numer. Anal. Methods Geomech. (2012). doi:10.1002/nag.1120

    Google Scholar 

  • Khoshghalb, A., Khalili, N.: A meshfree method for fully coupled analysis of flow and deformation in unsaturated porous media. Int. J. Numer. Anal. Methods Geomech. 37(7), 716–743 (2013)

    Article  Google Scholar 

  • Khoshghalb, A., Khalili, N.: An alternative approach for quasi-static large deformation analysis of saturated porous media using meshfree method. Int. J. Numer. Anal. Methods Geomech. 39, 913–936 (2015)

    Article  Google Scholar 

  • Kolymbas, D.: Barodesy: a new hypoplastic approach. Int. J. Numer. Anal. Methods Geomech. 36, 1220–1240 (2011)

    Article  Google Scholar 

  • Kolymbas, D.: Barodesy: a new constitutive frame for soils. Géotech. Lett. 2, 17–23 (2012)

    Article  Google Scholar 

  • Kolymbas, D.: Introduction to barodesy. Géotechnique 65(1), 52–65 (2015). doi:10.1680/geot.14.P.151

    Article  Google Scholar 

  • Komoróczi, A., Abe, S., Urai, J.L.: Meshless numerical modeling of brittle-viscous deformation: first results on boundinage and hydrofracturing using a coupling of discrete element method (DEM) and smoothed particle hydrodynamics (SPH). Comput. Geosci. 17(2), 373–390 (2013)

    Article  MathSciNet  Google Scholar 

  • Kuhnert, J.: Finite Pointset Method (FPM): meshfree flow solver with applications to elastoplastic material laws. In: Ońate, E., Owen, D.R.J. (eds) Proceedings of International Conference on Particle Based Methods: Fundamentals and Applications, Particles 2009, CIMNE, pp. 423–426 (2009)

  • Kuhnert, J.: Meshfree numerical scheme for time dependent problems in fluid and continuum mechanics. In: Sundar, S. (ed.) Advances in PDE Modeling and Computation, pp. 119–136. Ane Books, New Delhi (2014)

    Google Scholar 

  • Kuhnert, J., Ostermann, I.: Finite Pointset Method (FPM) and an application in soil mechanics. In: Pardo-Igúzquiza, E., Guardiola-Albert, C., Heredia, J., Moreno-Merino, L., Durán, J., Vargas-Guzmán, J. (eds.) Mathematics of Planet Earth. Lecture Notes in Earth System Sciences, pp. 815–818. Springer, Berlin (2014)

  • Lim, K.-W., Andrade, J.E.: Granular element method for three-dimensional discrete element method calculations. Int. J. Numer. Anal. Methods Geomech. 38(2), 167–188 (2014)

    Article  Google Scholar 

  • Medicus, G., Fellin, W., Kolymbas, D.: Barodesy for clay. Géotech. Lett. 2, 173–180 (2012)

    Article  Google Scholar 

  • Medicus, G., Kolymbas, D., Fellin, W.: Proportional stress and strain paths in barodesy. Int. J. Numer. Anal. Methods Geomech. 40(4), 509–522 (2016)

    Article  Google Scholar 

  • Medicus, G., Fellin, W.: An improved version of barodesy for clay. Acta Geotech. 12(2), 365–376 (2017)

    Article  Google Scholar 

  • Michel, I., Kuhnert, J.: Meshfree numerical simulation in soil mechanics with the Finite Pointset Method (FPM). In: Schaeben, H., Tolosana Delgado, R., van den Boogaart, K.G., van den Boogaart, R. (eds) Proceedings of IAMG 2015 Freiberg, pp. 652–658 (G0102) (2015). ISBN 978-3-00-050337-5 (DVD)

  • Michel, I., Kuhnert, J., Kolymbas, D.: Meshfree simulation of avalanches with the Finite Pointset Method (FPM). Geophysical Research Abstracts, vol. 19, EGU2017-13203, EGU General Assembly 2017, Vienna (2017)

  • Murakami, A., Setsuyasu, T., Arimoto, S.: Mesh-free method for soil–water coupled problem within finite strain and its numerical validity. Soils Found. 45(2), 145–154 (2005)

    Google Scholar 

  • Obermayr, M., Dressler, K., Vrettos, C., Eberhard, P.: A bonded-particle model for cemented sand. Comput. Geotech. 49, 229–313 (2013)

    Article  Google Scholar 

  • Obermayr, M., Vrettos, C.: Anwendung der Diskrete Elemente Methode zur Vorhersage von Kräften bei der Bodenbearbeitung. Geotechnik 36(4), 231–242 (2013)

    Article  Google Scholar 

  • Oñate, E., Idelsohn, S.R., Celigueta, M.A., Rossi, R., Marti, J., Carbonell, J.M., Ryzhakov, P., Suárez, B.: Advances in the particle finite element method (PFEM) for solving coupled problems in engineering. In: Oñate, E., Owen, R. (eds.) Particle-Based Methods, pp. 1–49. Springer, Dordrecht (2011)

    Chapter  Google Scholar 

  • Oñate, E., Labra, C., Zárate, F., Rojek, J.: Modelling and simulation of the effect of blast loading on structures using an adaptive blending of discrete and finite element methods. In: Escuder-Bueno, I., Altarejas-García, L., Castillo-Rodríguez, J.T., Matheu, E. (eds.) Risk Analysis, Dam Safety, Dam Security and Critical Infrastructure Management, pp. 365–371. Taylor & Francis Group, London (2011)

    Chapter  Google Scholar 

  • Ostermann, I., Kuhnert, J., Kolymbas, D., Chen, C.-H., Polymerou, I., Šmilauer, V., Vrettos, C., Chen, D.: Meshfree generalized finite difference methods in soil mechanics—part I: theory. Int. J. Geomath. 4, 167–184 (2013)

    Article  MATH  MathSciNet  Google Scholar 

  • Pastor, M., Haddad, B., Sorbino, G., Cuomo, S., Drempetic, V.: A depth-integrated, coupled SPH model for flow-like landslides and related phenomena. Int. J. Numer. Anal. Methods Geomech. 33, 143–172 (2008)

    Article  MATH  Google Scholar 

  • Peng, C., Wu, W., Yu, H., Wang, C.: A SPH approach for large deformation analysis with hypoplastic constitutive model. Acta Geotech. 10(6), 703–717 (2015)

    Article  Google Scholar 

  • Polymerou, I.: Untersuchung großer Verformungen in der Vertushka. PhD Thesis, University of Innsbruck, Logos Verlag Berlin GmbH (2017)

  • Schenkengel, K.-U., Vrettos, C.: Modelling of liquefaction-induced lateral spreading using the Lattice Boltzmann method. In: Proceedings 5th International conference on Earthquake Geotechnical Engineering, Paper No MOLSC-922621417 (2011)

  • Schneider-Muntau, B., Chen, C.-H., Bathaeian, S.M.I.: Simulation of shear bands with Soft PARticle Code (SPARC) and FE. Int. J. Geomath. 8, 135–151 (2017)

    Article  MathSciNet  Google Scholar 

  • Sloan, S.W., Nazem, M., Zakrzewski, N., Cassidy, M.: On application of the maximum entropy meshless method for large deformation analysis of geotechnical problems. Appl. Mech. Mater. 846, 331–335 (2016)

    Article  Google Scholar 

  • Soga, K., Alonso, E., Yerro, A., Kumar, K., Bandara, S.: Trends in large-deformation analysis of landslide mass movements with particular emphasis on the material point method. Géotechnique 66(3), 1–26 (2015)

    Google Scholar 

  • Suchde, P., Kuhnert, J.: Point cloud movement for fully Lagrangian meshfree methods. (2017) arXiv:1704.00618

  • Suchde, P., Kuhnert, J., Tiwari, S.: On meshfree GFDM solvers for the incompressible Navier–Stokes equations. (2017) arXiv:1701.03427

  • Suchde, P., Kuhnert, J., Schröder, S., Klar, A.: A flux conserving meshfree method for conservation laws. Int. J. Numer. Methods Eng. (2017). doi:10.1002/nme.5511

    Google Scholar 

  • Tiwari, S., Antonov, S., Hietel, D., Kuhnert, J., Wegener, R.: A meshfree method for simulations of interactions between fluids and flexible structures. In: Griebel, M., Schweitzer, M.A. (eds.) Meshfree Methods for Partial Differential Equations III, pp. 249–264. Springer, Berlin (2007)

    Chapter  Google Scholar 

  • Tiwari, S., Kuhnert, J.: A meshfree method for incompressible fluid flows with incorporated surface tension. Revue Eur. Élém. 11(7–8), 965–987 (2002)

    MATH  Google Scholar 

  • Tiwari, S., Kuhnert, J.: Finite Pointset Method based on the projection method for simulations of the incompressible Navier–Stokes equations. In: Griebel, M., Schweitzer, M.A. (eds.) Lecture Notes in Computational Science and Engineering, vol. 26, pp. 373–387. Springer, Berlin (2002)

    Google Scholar 

  • Tiwari, S., Kuhnert, J.: Grid free method for solving poisson equation. In: Rao, G.S. (ed.) Wavelet Analysis and Applications, pp. 151–166. New Age International Publishers, New Delhi (2004)

    Google Scholar 

  • Tiwari, S., Kuhnert, J.: A numerical scheme for solving incompressible and low mach number flows by Finite Pointset Method. In: Griebel, M., Schweitzer, M.A. (eds.) Lecture Notes in Computational Science and Engineering, vol. 43, pp. 191–206. Springer, Berlin (2005)

    Google Scholar 

  • Tiwari, S., Kuhnert, J.: Modeling of two-phase flows with surface tension by Finite Pointset Method (FPM). J. Comput. Appl. Math. 203(2), 376–386 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  • Tramecon, A., Kuhnert, J.: Simulation of advanced folded airbags with VPS-PAMCRASH/FPM: development and validation of turbulent flow numerical simulation techniques applied to curtain bag deployments. SAE Technical Paper 2013-01-1158 (2013) doi:10.4271/2013-01-1158

  • Tootoonchi, A., Liu, G.R., Khalili, N.: A cell-based smoothed point interpolation method for flow-deformation analysis of saturated porous media. Comput. Geotech. 75, 159–173 (2016)

    Article  Google Scholar 

  • Uhlmann, E., Gerstenberger, R., Kuhnert, J.: Cutting simulation with the meshfree Finite Pointset Method. Procedia CIRP 8, 391–396 (2013)

    Article  Google Scholar 

  • Vermeer, P.A., Beuth, L., Benz, T.: A Quasi-static method for large deformation problems in geomechanics. In: Proceedings 12th IACMAG, pp. 55–63 (2008)

  • Wu, C.T., Chen, J.S., Chi, L., Huck, F.: Lagrangian meshfree formulation for analysis of geotechnical materials. J. Eng. Mech. 127(5), 440–449 (2001)

    Article  Google Scholar 

  • Zhu, H.H., Miao, Y.B., Cai, Y.C.: Meshless natural neighbour method and its application in elasto-plastic problems. In: Liu, G., Tan, V., Han, X. (eds.) Computational Methods, pp. 1465–1475. Springer, Berlin (2006)

    Chapter  Google Scholar 

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Acknowledgements

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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Authors and Affiliations

  1. Fraunhofer Institute for Industrial Mathematics ITWM, Fraunhofer-Platz 1, 67663, Kaiserslautern, Germany

    I. Michel & J. Kuhnert

  2. Division of Geotechnical and Tunnel Engineering, University of Innsbruck, Technikerstr. 13, 6020, Innsbruck, Austria

    S. M. I. Bathaeian, D. Kolymbas, C.-H. Chen & I. Polymerou

  3. Division of Soil Mechanics and Foundation Engineering, University of Kaiserslautern, P.O. Box 3049, 67653, Kaiserslautern, Germany

    C. Vrettos & A. Becker

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  1. 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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