Stability in Lagrangian and semi-Lagrangian reproducing kernel discretizations using nodal integration in nonlinear solid mechanics. en. (English) Zbl 1323.74104

Leitão, V. M. A. (ed.) et al., Advances in meshfree techniques. Invited contributions based on the presentation at the ECCOMAS thematic conference on meshless methods, Lisbon, Portugal, July 11–14, 2005. Dordrecht: Springer (ISBN 978-1-4020-6094-6/hbk; 978-90-481-7533-8/pbk; 978-1-4020-6095-3/ebook). Computational Methods in Applied Sciences (Springer) 5, 55-76 (2007).
Summary: Stability analyses of Lagrangian and Semi-Lagrangian Reproducing Kernel (RK) approximations for nonlinear solid mechanics are performed. It is shown that the semi-Lagrangian RK discretization yields a convective term resulting from the non-conservative coverage of material points under the kernel support. The von Neumann stability analysis shows that the discrete equations of both Lagrangian and semi-Lagrangian discretizations are stable when they are integrated using stabilized conforming nodal integration. On the other hand, integrating the semi-Lagrangian discretization with a direct nodal integration yields an unstable discrete system which resembles the tensile instability in SPH. Under the framework of semi-Lagrangian discretization, it is shown that the inclusion of convective term yields a more stable discrete system compared to the semi-Lagrangian discretization without convective term as was the case in SPH.
For the entire collection see [Zbl 1125.74003].


74S30 Other numerical methods in solid mechanics (MSC2010)
65M12 Stability and convergence of numerical methods for initial value and initial-boundary value problems involving PDEs
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[1] Belytschko T. and Xiao S.P. Stability analysis of particle methods with corrected derivatives. \(Computers and Mathematics with Applications\), 43:329-350, 2002. · Zbl 1073.76619
[2] Belytschko T., Guo Y., Liu W.K. and Xiao S.P. A unified stability analysis of meshless particle methods. \(International Journal for Numerical Methods in Engineering\), 48:1359-1400, 2000. · Zbl 0972.74078
[3] Bonet J. and Kulasegaram S. Remarks on tension instability of Eulerian and Lagrangian corrected smooth particle hydrodynamics (CSPH) methods. \(International Journal for Numerical Methods in Engineering\), 52:1203-1220, 2001. · Zbl 1112.74562
[4] Chen J.S., Pan C. and Wu C.T. Large deformation analysis of rubber based on a reproducing kernel particle method. \(Computational Mechanics\), 19:211-217, 1997. · Zbl 0888.73073
[5] Chen J.S., Pan C., Roque C. and Wang H.P. A Lagrangian reproducing kernel particle method for metal forming analysis. \(Computational Mechanics\), 22:289-307, 1998. · Zbl 0928.74115
[6] Chen J.S., Pan C., Wu C.T. and Liu W.K. Reproducing kernel particle methods for large deformation analysis of non-linear structures. \(Computer Methods in Applied Mechanics and Engineering\), 139:195-227, 1996. · Zbl 0918.73330
[7] Chen J.S., Roque C.M.O.L., Pan C. and Button S.T. Analysis of metal forming process based on meshless method. \(Journal of Materials Processing Technology\), 80/81:642-646, 1998.
[8] Chen J.S., Wu C.T. and Belytschko T. Regularization of material instability by meshfree approximations with intrinsic length scales. \(International Journal for Numerical Methods in Engineering\), 47:1301-1322, 2000. · Zbl 0987.74079
[9] Chen J.S., Wu C.T., Yoon S. and You Y. A stabilized conforming nodal integration for Galerkin mesh-free methods. \(International Journal for Numerical Methods in Engineering\), 50:435-446, 2001. · Zbl 1011.74081
[10] Chen J.S., Yoon S. and Wu C.T. Non-linear version of stabilized conforming nodal integration for Galerkin mesh-free methods. \(International Journal for Numerical Methods in Engineering\), 53:2587-2615, 2002. · Zbl 1098.74732
[11] Chen J.S., Yoon S., Wang H.P. and Liu W.K. An improved reproducing kernel particle method for nearly incompressible hyperelastic solids. \(Computer Methods in Applied Mechanics and Engineering\), 181:117-145, 2000. · Zbl 0973.74088
[12] Gingold R.A. and Monaghan J.J. Smoothed particle hydrodynamics: Theory and application to non-spherical stars. \(Monthly Notices of the Royal Astronomical Society\), 181:375-389, 1977. · Zbl 0421.76032
[13] Lancaster P. and Salkauskas K. Surfaces generated by moving least-squares methods. \(Mathematics of Computation\), 37:141-158, 1981. · Zbl 0469.41005
[14] Liu W.K., Jun S. and Zhang Y.F. Reproducing kernel particle methods. \(International Journal for Numerical Methods in Fluids\), 20:1081-1106, 1995. · Zbl 0881.76072
[15] Lucy L.B. A numerical approach to the testing of the fission hypothesis. \(The Astronomical Journal\), 82:1013-1024, 1977.
[16] Libersky L.D. and Petschek A.G. Smooth particle hydrodynamics with strength of materials. In: \(Advances in the Free Lagrange Method\), Lecture Notes in Physics, Vol. 395, Springer-Verlag, 1990. · Zbl 0791.76066
[17] Monaghan J.J. SPH without a tensile instability. \(Journal of Computational Physics\), 159:290-311, 2000. · Zbl 0980.76065
[18] Rabczuk T., Belytschko T. and Xiao S.P. Stable particle methods based on Lagrangian kernels. \(Computer Methods in Applied Mechanics and Engineering\), 193:1035-1063, 2004. · Zbl 1060.74672
[19] Randles P.W. and Libersky L.D. Smoothed particle hydrodynamics: Some recent improvements and applications. \(Computer Methods in Applied Mechanics and Engineering\), 139:375-408, 1996. · Zbl 0896.73075
[20] Swegle J.W., Hicks D.L. and Attaway S.A. Smoothed particle hydrodynamics stability analysis. \(Journal of Computational Physics\), 116:123-134, 1995. · Zbl 0818.76071
[21] Dyka C.T. and Ingel R.P. An approach for tension instability in smoothed particle hydrodynamics. \(Computers & Structures\), 57:573-580, 1995. · Zbl 0900.73945
[22] Guenther C., Hicks D.L. and Swegle J.W. Conservative smoothing versus artificial viscosity. Sandia Report SAND94-1853, Sandia National Lab., 1994.
[23] Swegle J.W., Attaway S.W., Heinstein M.W., Mello F.J. and Hicks D.L. An analysis of smoothed particle hydrodynamics. Sandia Report SAND94-2513, Sandia National Lab., 1994.
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