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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].

MSC:

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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