Meshfree generalized finite difference methods in soil mechanics. I: Theory. (English) Zbl 1277.74076

Summary: In soil mechanics, laboratory tests are typically used to classify soils or to test new material laws such as the barodesy model. The results of these tests provide the theoretical basis for subsequent simulations and analysis in geotechnical engineering (e.g., cuts, embankments, foundations). Simulation tools which are reliable as well as economical concerning the computing time are indispensable for applications. In this contribution we introduce two novel meshfree generalized finite difference methods – Finite Pointset Method and Soft PARticle Code – to simulate the standard benchmark problems “oedometric test” and “triaxial test”. One of the most important ingredients of both meshfree approaches is the weighted moving least squares method used to approximate the required spatial partial derivatives of arbitrary order on a finite pointset.


74S20 Finite difference methods applied to problems in solid mechanics
74L10 Soil and rock mechanics
35D35 Strong solutions to PDEs
35Q74 PDEs in connection with mechanics of deformable solids
65D05 Numerical interpolation
65M99 Numerical methods for partial differential equations, initial value and time-dependent initial-boundary value problems
Full Text: DOI


[1] Bardenhagen, S.G., Brackbill, J.U., Sulsky, D.: The material-point method for granular materials. Comput. Methods Appl. Mech. Eng. 187, 529–541 (2000) · Zbl 0971.76070
[2] Beuth, L., Więckowski, L., Vermeer, P.A.: Solution of quasi-static large-strain problems by the material point method. Int. J. Numer. Anal. Meth. Geomech. 35, 1451–1465 (2011)
[3] Blanc, T.: Numerical Simulation of Debris Flows with the 2D-SPH Depth Integrated Model. Institute for Mountain Risk Engineering, University of Natural Resources and Applied Life Sciences, Vienna, Master Thesis (2008)
[4] Blanc, T., Pastor, M.: A stabilized smoothed particle hydrodynamics, Taylor-Galerkin algorithm for soil dynamics problems. Int. J. Numer. Anal. Meth. Geomech. 37, 1–30 (2013)
[5] 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. Meth. Geomech. (2011). doi: 10.1002/nag.1084
[6] Coetzee, C.J., Vermeer, P.A., Basson, A.H.: The modelling of anchors using the material point method. Int. J. Numer. Anal. Meth. Geomech. 29, 879–895 (2005) · Zbl 1104.74040
[7] 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)
[8] Desrues, J., Zweschper, B., Vermeer, P.A.: Database for Tests on Hostun RF Sand. Institute of Geotechnical Engineering, University of Stuttgart, Technical Report (2000)
[9] Fries, T.P., Matthies, H.G.: Classification and Overview of Meshfree Methods. Institute of Scientific Computing, Technical University Braunschweig, Technical Report (2003) · Zbl 1354.76131
[10] Hietel, D., Junk, M., Kuhnert, J., Tiwari, S.: Meshless Methods for Conservation Laws. In: Warnecke G (ed) Analysis and Numerics for Conservation Laws, pp. 339–362. Springer, Berlin (2005) · Zbl 1088.76036
[11] 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. Meth. Geomech. 61(7), 1–6 (2011)
[12] Iliev, O., Tiwari, S.: A generalized (meshfree) finite difference discretization for elliptic interface problems. In: Dimov, I., Lirkov, I., Margenov, S., Zlatev, Z. (eds.) Numerical Methods and Applications. Lecture Notes in computer Sciences, pp. 480–489. Springer, Berlin (2002)
[13] Jassim, I., Stolle, D., Vermeer, P.: Two-phase dynamic analysis by material point method. Int. J. Numer. Anal. Meth. Geomech. (2012). doi: 10.1002/nag.2146
[14] Khoshghalb, A., Khalili, N.: A stable meshfree method for fully coupled flow-deformation analysis of saturated porous media. Comput. Geotech. 37, 789–795 (2010)
[15] Khoshghalb, A., Khalili, N.: A meshfree method for fully coupled analysis of flow and deformation in unsaturated porous media. Int. J. Numer. Anal. Meth. Geomech. (2012). doi: 10.1002/nag.1120
[16] Kolymbas, D.: Barodesy: a new hypoplastic approach. Int. J. Numer. Anal. Meth. Geomech. 36, 1220–1240 (2011)
[17] Kolymbas, D.: Barodesy: a new constitutive frame for soils. Géotech. Lett. 2, 17–23 (2012)
[18] Kuhnert, J., Schäfer, M., Gerstenberger, R.: Meshfree Numerical Scheme for Time Dependent Problems in Continuum Mechanics. Preprint, Fraunhofer ITWM (2012)
[19] Lancaster, P., Salkauskas, K.: Surfaces generated by moving least squares methods. Math. Comp. 37(155), 141–158 (1981) · Zbl 0469.41005
[20] 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)
[21] 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. Meth. Geomech. 33, 143–172 (2008) · Zbl 1272.74464
[22] Tiwari, S., Antonov, S., Hietel, D., Kuhnert, J., Olawsky, F., 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) · Zbl 1111.76042
[23] Tiwari, S., Kuhnert, J.: A meshfree method for incompressible fluid flows with incorporated surface tension. Revue Européenne des Éléments 11(7–8), 965–987 (2002) · Zbl 1120.76355
[24] 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) · Zbl 1090.76566
[25] 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, USA (2004)
[26] 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.) Meshfree Methods for Partial Differential Equations II, pp. 191–206. Springer, Berlin (2005) · Zbl 1063.76078
[27] Tiwari, S., Kuhnert, J.: Modeling of two-phase flows with surface tension by finite pointset method (FPM). J. Comp. Appl. Math. 203(2), 376–386 (2007) · Zbl 1113.76074
[28] Vermeer, P.A., Beuth. L., Benz, T.: A quasi-static method for large deformation problems in geomechanics. In: Proceedings of 12th IACMAG, pp. 55–63 (2008)
[29] Zhu, H.H., Miao, Y.B., Cai, Y.C.: Meshless natural neighbour method and its application in elasto-plastic problems. In: Lin, G.R., Tan, V.B.C., Han, X. (eds.) Computational Methods, pp. 1465–1475. Springer, Netherlands (2006)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.