Cut Finite Element Methods for Linear Elasticity Problems
This work provides a robust numerical framework for solving linear elasticity problems on complex geometries without conforming meshes, which is important for engineering applications involving thin structures and fiber-reinforced materials.
The paper develops a cut finite element method for linear elasticity using higher-order elements on a fixed background mesh, with a stabilization term to control derivative jumps near boundaries. The method is applied to displacement, frequency response, and eigenvalue problems, and extended to fiber-reinforced materials, demonstrating accuracy and stability through numerical examples.
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.