Critical time-step size analysis and mass scaling by ghost-penalty for immersogeometric explicit dynamics

• Published in: Computer methods in applied mechanics and engineering Vol. 412; p. 116074
• Main Authors: Stoter, Stein K.F., Divi, Sai C., van Brummelen, E. Harald, Larson, Mats G., de Prenter, Frits, Verhoosel, Clemens V.
• Format: Journal Article
• Language: English
• Published: Elsevier B.V 01.07.2023
• Subjects: Critical time step; Explicit dynamics; Finite cell method; Ghost penalty; Immersogeometric analysis; Mass scaling; Critical time step; Ghost penalty; Finite cell method; Explicit dynamics; Immersogeometric analysis; Mass scaling
• ISSN: 0045-7825; 1879-2138
• DOI: 10.1016/j.cma.2023.116074

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.