# Cut Finite Element Methods for Linear Elasticity Problems

## TL;DR

This paper introduces a cut finite element method for linear elasticity problems, incorporating stabilization techniques, error analysis, and applications to various structural and material models, including fiber-reinforced composites.

## Contribution

The paper develops a novel cut finite element framework for linear elasticity with higher order elements, stabilization, and applications to complex material models.

## Key findings

- Error estimates and condition number bounds established.
- Numerical examples demonstrate method effectiveness.
- Application to fiber-reinforced materials with stabilized beam modeling.

## Abstract

We formulate a cut finite element method for linear elasticity based on higher order elements on a fixed background mesh. Key to the method is a stabilization term which provides control of the jumps in the derivatives of the finite element functions across faces in the vicinity of the boundary. We then develop the basic theoretical results including error estimates and estimates of the condition number of the mass and stiffness matrices. We apply the method to the standard displacement problem, the frequency response problem, and the eigenvalue problem. We present several numerical examples including studies of thin bending dominated structures relevant for engineering applications. Finally, we develop a cut finite element method for fibre reinforced materials where the fibres are modeled as a superposition of a truss and a Euler-Bernoulli beam. The beam model leads to a fourth order problem which we discretize using the restriction of the bulk finite element space to the fibre together with a continuous/discontinuous finite element formulation. Here the bulk material stabilizes the problem and it is not necessary to add additional stabilization terms.