# A hierarchical approach to the a posteriori error estimation of isogeometric Kirchhoff plates and Kirchhoff-Love shells

## TL;DR

This paper introduces a hierarchical a posteriori error estimator for isogeometric Kirchhoff plates and shells, enabling adaptive refinement with optimal convergence rates and accurate error approximation in complex structural simulations.

## Contribution

It presents a novel hierarchical spline-based error estimator for fourth-order PDEs, specifically tailored for Kirchhoff plates and shells, improving adaptive simulation accuracy.

## Key findings

- Achieves optimal convergence rates in energy norm
- Provides accurate true error approximation
- Effective on problems with smooth and singular solutions

## Abstract

This work focuses on the development of a posteriori error estimates for fourth-order, elliptic, partial differential equations. In particular, we propose a novel algorithm to steer an adaptive simulation in the context of Kirchhoff plates and Kirchhoff-Love shells by exploiting the local refinement capabilities of hierarchical B-splines. The method is based on the solution of an auxiliary residual-like variational problem, formulated by means of a space of localized spline functions. This space is characterized by $C^1$ continuous B-splines with compact support on each active element of the hierarchical mesh. We demonstrate the applicability of the proposed estimator to Kirchhoff plates and Kirchhoff-Love shells by studying several benchmark problems which exhibit both smooth and singular solutions. In all cases, we obtain optimal asymptotic rates of convergence for the error measured in the energy norm and an excellent approximation of the true error.