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A unified stability analysis of meshless particle methods. (English) Zbl 0972.74078
Summary: We present a unified stability analysis of meshless methods with Eulerian and Lagrangian kernels. Three types of instabilities are identified in one dimension: an instability due to rank deficiency, a tensile instability, and a material instability which is also found in continua. The stability properties of particle methods with Eulerian and Lagrangian kernels are markedly different: Lagrangian kernels do not exhibit the tensile instability. In both kernels, the instability due to rank deficiency can be suppressed by stress points. In two dimensions the stabilizing effect of stress points is dependent on their locations. It is found that the best approach to stable particle discretizations is to use Lagrangian kernels with stress points. We also study the stability of the least-squares stabilization.
Reviewer: Reviewer (Berlin)

MSC:
74S30 Other numerical methods in solid mechanics (MSC2010)
65N12 Stability and convergence of numerical methods for boundary value problems involving PDEs
Software:
DYNA3D
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