# Finite elements and non-linear fields

Research project

Development and application of finite elements for problems with non-linear fields.

## Overview

• Development of finite elements for applications where fields are defined on non-linear manifolds
• Identification and removal of issues in existing formulations
• Development of new formulations to consistently formulate and solve such problems

## Project Description

Many models in structural mechanics and material theory contain quantities which live on non-linear manifolds. In structural mechanics a classic example is the geometrically non-linear Reissner-Mindlin shell model. This model contains a field of unit directors to describe the shell body from the midsurface. Another example is a geometrically non-linear Cosserat beam model. Here, the orientation of the rigid cross-section is described by a field of rotation matrices living in SO(3). Material theory also contains several examples for such non-linear spaces, e.g. incompressible material behavior, where the deformation gradient has a constant determinant of 1. Another examples is the description of magnetic material, which also contains a unit vector to describe the orientation of magnetization.

### Challenges

For all these examples, application of standard procedures of finite element technology yields various undesired symptoms, since most of them are developed for vector spaces only. Direct application of these procedures to non-linear manifolds leads to non-objective formulations, artificial path dependence and even non-convergence of the solution algorithms. These problems can be traced back to a faulty interpolation, a faulty linearization or a wrong interpretation of the relationship between the discrete quantity itself and its corresponding update -- which necessarily lives in the tangent space of the manifold. The Identification and removal of these drawbacks is also a main ingredient of this project.

### Numerical Examples

Iteration process to minimize the exchange energy of magnetization: Starting from a random initialization and a fixed left and bottom edge.

Simulation of shell buckling using a geometrically non-linear Reissner-Mindlin shell formulation with unit directors: These simulations are static simulations using a Riemannian trust-region minimization algorithm.

The upper and lower edges are clamped. The upper edge is homogeneously moved to the right. During this process, the shell buckles out of plane. Following the experiments of Wong YW and Pellegrino S (2006) Wrinkled membranes part I: experiments.

The upper edge of the zylinder witnesses a vertical displacement. During the process several equilibria are reached and mode jumping occurs.

### Publications

1. Müller, A., Bischoff, M., & Keip, M.-A. (2023). Thin cylindrical magnetic nanodots revisited: Variational formulation, accurate solution and phase diagram. Journal of Magnetism and Magnetic Materials, 586(171095), Article 171095. https://doi.org/10.1016/j.jmmm.2023.171095
2. Müller, A., & Bischoff, M. (2022). A Consistent Finite Element Formulation of the Geometrically Non-linear Reissner-Mindlin Shell Model. Archives of Computational Methods in Engineering. https://doi.org/10.1007/s11831-021-09702-7

### Researcher: M. Sc.

Scientific Staff