Finite elements and 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.

Numerical example magnetization

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.

Numerical example shell buckling

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

Numerical cylinder buckling

Publications

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
- Abstract
- BibTeX
Abstract
We investigate the variational formulation and corresponding minimizing energies for the detection of energetically favorable magnetization states of thin cylindrical magnetic nanodots. Opposed to frequently used heuristic procedures found in the literature, we revisit the underlying governing equations and construct a rigorous variational approach that takes both exchange and demagnetization energy into account. Based on a combination of Ritz’s method and a Fourier series expansion of the solution field, we are able to pinpoint the precision of solutions, which are given by vortex modes or single-domain states, down to an arbitrary degree of precision. Furthermore, our model allows to derive an expression for the demagnetization energy in closed form for the in-plane single-domain state, which we compare to results from the literature. A key outcome of the present investigation is an accurate phase diagram, within the problem class of constant magnetization through the thickness and rotational symmetry, which we obtain by comparing the vortex mode’s energy minimizers with those of the single-domain states. This phase diagram is validated with data of two- and three-dimensional models from literature. By means of the phase diagram, we particularly find the critical radius at which the vortex mode becomes unfavorable with machine precision. All relevant data and codes related to the present contribution are available at Müller (2023).
BibTeX
@article{muller2023cylindrical, abstract = {We investigate the variational formulation and corresponding minimizing energies for the detection of energetically favorable magnetization states of thin cylindrical magnetic nanodots. Opposed to frequently used heuristic procedures found in the literature, we revisit the underlying governing equations and construct a rigorous variational approach that takes both exchange and demagnetization energy into account. Based on a combination of Ritz’s method and a Fourier series expansion of the solution field, we are able to pinpoint the precision of solutions, which are given by vortex modes or single-domain states, down to an arbitrary degree of precision. Furthermore, our model allows to derive an expression for the demagnetization energy in closed form for the in-plane single-domain state, which we compare to results from the literature. A key outcome of the present investigation is an accurate phase diagram, within the problem class of constant magnetization through the thickness and rotational symmetry, which we obtain by comparing the vortex mode’s energy minimizers with those of the single-domain states. This phase diagram is validated with data of two- and three-dimensional models from literature. By means of the phase diagram, we particularly find the critical radius at which the vortex mode becomes unfavorable with machine precision. All relevant data and codes related to the present contribution are available at Müller (2023).}, author = {Müller, Alexander and Bischoff, Manfred and Keip, Marc-André}, doi = {10.1016/j.jmmm.2023.171095}, journal = {Journal of Magnetism and Magnetic Materials}, number = 171095, preprinturl = {https://doi.org/10.48550/arXiv.2302.02793}, title = {Thin cylindrical magnetic nanodots revisited: Variational formulation, accurate solution and phase diagram}, url = {https://doi.org/10.18419/darus-3103}, volume = 586, year = 2023 }
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
- Abstract
- BibTeX
Abstract
We present an objective, singularity-free, path independent, numerically robust and efficient geometrically non-linear Reissner-Mindlin shell finite element formulation. The formulation is especially suitable for higher order ansatz spaces. The formulation utilizes geometric finite elements presented by Sander 74 and Grohs 34 for the interpolation on non-linear manifolds. The proposed method is objective and free from artificial singularities and spurious path dependence. Due to the fact that the director field lives on the unit sphere, a special linearization procedure is required to obtain the stiffness matrix. Here, we use the simple constructions of as reported by Absil et al. 2, 3, which yields an easy way to obtain the correct tangent operator of the potential energy. Additionally, we compare three different interpolation schemes for the shell director that can be found in the literature, where one of them is applied for the first time for the Reissner-Mindlin shell model. Furthermore, we compare the exponential map to the radial return normalization as procedure to update the nodal directors and conclude the superiority of the latter, in terms of fewer load steps. We also investigate the construction of a consistent tangent base update scheme. Path independence, efficiency and objectivity of the formulation are verified via a set of numerical examples.
BibTeX
@article{muller2022consistent, abstract = {We present an objective, singularity-free, path independent, numerically robust and efficient geometrically non-linear Reissner-Mindlin shell finite element formulation. The formulation is especially suitable for higher order ansatz spaces. The formulation utilizes geometric finite elements presented by Sander [74] and Grohs [34] for the interpolation on non-linear manifolds. The proposed method is objective and free from artificial singularities and spurious path dependence. Due to the fact that the director field lives on the unit sphere, a special linearization procedure is required to obtain the stiffness matrix. Here, we use the simple constructions of as reported by Absil et al. [2, 3], which yields an easy way to obtain the correct tangent operator of the potential energy. Additionally, we compare three different interpolation schemes for the shell director that can be found in the literature, where one of them is applied for the first time for the Reissner-Mindlin shell model. Furthermore, we compare the exponential map to the radial return normalization as procedure to update the nodal directors and conclude the superiority of the latter, in terms of fewer load steps. We also investigate the construction of a consistent tangent base update scheme. Path independence, efficiency and objectivity of the formulation are verified via a set of numerical examples.}, author = {Müller, Alexander and Bischoff, Manfred}, doi = {10.1007/s11831-021-09702-7}, journal = {Archives of Computational Methods in Engineering }, title = {A Consistent Finite Element Formulation of the Geometrically Non-linear Reissner-Mindlin Shell Model}, year = 2022 }

Overview

Project Description

Challenges

Numerical Examples

Publications

Abstract

BibTeX

Abstract

BibTeX

Researcher:

Alexander Müller

Audience

Formalities

Services

Organization

Finite elements and non-linear fields

Overview

Project Description

Challenges

Numerical Examples

Publications

Abstract

BibTeX

Abstract

BibTeX

Researcher:

Alexander Müller

Here you can reach us

Audience

Formalities

Services

Organization