# Stabilization techniques for immersogeometric analysis of plate and shell problems in explicit dynamics

## TL;DR

This paper develops stabilization techniques for immersogeometric analysis of plates and shells in explicit dynamics, enabling stable simulations with lumped mass matrices despite challenges posed by high-order PDEs and cut elements.

## Contribution

The authors extend previous methods to achieve stable immersogeometric analysis of plates and shells with lumped mass matrices, using polynomial extensions to improve accuracy.

## Key findings

- Stable immersogeometric analysis achieved for plates and shells
- Lumped mass matrices can be used without spurious oscillations
- Accuracy comparable to boundary-fitted discretizations

## Abstract

Finite element plate and shell formulations are ubiquitous in structural analysis for modeling all kinds of slender structures, both for static and dynamic analyses. The latter are particularly challenging as the high order nature of the underlying partial differential equations and the slenderness of the structures all impose a stringent constraint on the critical time step in explicit dynamics. Unfortunately, badly cut elements in immersed finite element discretizations further aggravate the issue. While lumping the mass matrix often increases the critical time step, it might also trigger spurious oscillations in the approximate solution thereby compromising the numerical solution. In this article, we extend our previous work in \cite{voet2025stabilization} to allow stable immersogeometric analysis of plate and shell problems with lumped mass matrices. This technique is based on polynomial extensions and restores a level of accuracy comparable to boundary-fitted discretizations.