Simultaneous analysis of curved Kirchhoff beams and Kirchhoff--Love shells embedded in bulk domains

TL;DR

This paper introduces a novel numerical method for analyzing multiple Kirchhoff--Love beams and shells embedded in a bulk domain, enabling higher-order accuracy and efficient computation.

Contribution

It develops a Bulk Trace FEM that uses standard Lagrange elements to model complex curved structures within a bulk domain, simplifying implementation.

Findings

The method achieves higher-order convergence in numerical tests.

Numerical results confirm the accuracy and robustness of the proposed approach.

Benchmark cases demonstrate the method's potential for complex structural analysis.

Abstract

A set of curved beams and shells is geometrically implied by level sets of a scalar function over some bulk domain. The mechanical model for each structure is based on the Kirchhoff--Love theory, that is, small displacements without shear deformations are considered. These models for individual geometries are extended to bulk models, simultaneously modeling the whole set of beams/shells on all level sets. A major focus is on the numerical analysis of such models. A mixed-hybrid and higher-order accurate Bulk Trace FEM is proposed that enables the use of standard $C^0$-continuous Lagrange elements with dimensionality of the bulk domain. That is, the higher-order continuity requirements of displacement-based formulations in context of the Kirchhoff--Love theory are successfully alleviated. Several numerical tests confirm the accuracy and higher-order convergence of the proposed methodology, also qualifying as benchmark test cases in future studies.