A $C^1$-continuous Trace-Finite-Cell-Method for linear thin shell analysis on implicitly defined surfaces

TL;DR

This paper introduces a novel $C^1$-continuous Trace-Finite-Cell-Method for thin shell analysis on implicitly defined surfaces, combining TraceFEM and Finite-Cell-Method concepts, with verification through convergence and benchmark tests.

Contribution

It presents a new $C^1$-continuous shell analysis method using tensor product cubic splines on implicit surfaces, unifying TraceFEM and Finite-Cell-Method approaches.

Findings

Method achieves optimal convergence rates.

Verification confirms accuracy on complex geometries.

Benchmark tests demonstrate robustness and efficiency.

Abstract

A Trace-Finite-Cell-Method for the numerical analysis of thin shells is presented combining concepts of the TraceFEM and the Finite-Cell-Method. As an underlying shell model we use the Koiter model, which we re-derive in strong form based on first principles of continuum mechanics by recasting well-known relations formulated in local coordinates to a formulation independent of a parametrization. The field approximation is constructed by restricting shape functions defined on a structured background grid on the shell surface. As shape functions we use on a background grid the tensor product of cubic splines. This yields $C^1$-continuous approximation spaces, which are required by the governing equations of fourth order. The parametrization-free formulation allows a natural implementation of the proposed method and manufactured solutions on arbitrary geometries for code verification. Thus, the implementation is verified by a convergence analysis where the error is computed with an exact manufactured solution. Furthermore, benchmark tests are investigated.