Critical time-step size analysis and mass scaling by ghost-penalty for immersogeometric explicit dynamics. (English) Zbl 1539.74520
Summary: In this article, we study the effect of small-cut elements on the critical time-step size in an immersogeometric explicit dynamics context. We analyze different formulations for second-order (membrane) and fourth-order (shell-type) equations, and derive scaling relations between the critical time-step size and the cut-element size for various types of cuts. In particular, we focus on different approaches for the weak imposition of Dirichlet conditions: by penalty enforcement and with Nitsche’s method. The conventional stability requirement for Nitsche’s method necessitates either a cut-size dependent penalty parameter, or an additional ghost-penalty stabilization term. Our findings show that both techniques suffer from cut-size dependent critical time-step sizes, but the addition of a ghost-penalty term to the mass matrix serves to mitigate this issue. We confirm that this form of ‘mass-scaling’ does not adversely affect error and convergence characteristics for a transient membrane example, and has the potential to increase the critical time-step size by orders of magnitude. Finally, for a prototypical simulation of a Kirchhoff-Love shell, our stabilized Nitsche formulation reduces the solution error by well over an order of magnitude compared to a penalty formulation at equal time-step size.
MSC:
| 74S22 | Isogeometric methods applied to problems in solid mechanics |
| 74S05 | Finite element methods applied to problems in solid mechanics |
| 65D07 | Numerical computation using splines |
| 65M60 | Finite element, Rayleigh-Ritz and Galerkin methods for initial value and initial-boundary value problems involving PDEs |
Keywords:
immersogeometric analysis; explicit dynamics; critical time step; finite cell method; ghost penalty; mass scalingReferences:
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