Finite Elemente und nichtlineare Felder

Überblick

Entwicklung finiter Elmente für Problemstellungen, die Felder in nichtlinearen Mannigfaltigkeiten benötigen
Behebung von Problemen in bisherigen Formulierungen
Entwicklung neuer Formulierungen um Problemstellungen konsistent zu beschreiben und zu lösen

Projektbeschreibung

Häufig tauchen sowohl in der Strukturmechanik als auch in der Materialtheorie Größen auf, die in nichtlinearen Mannigfaltigkeiten definiert sind. In der Strukturmechanik sind dies beispielsweise nichtlineare Reissner-Mindlin-Schalenmodelle, die einen Direktor mit Einheitslänge haben oder nichtlineare Cosserat-Balkentheorien, für die der Balkenquerschnitt mit Hilfe einer Rotationsmatrix aus SO(3) beschrieben wird. Ein Beispiel aus der Materialtheorie ist inkompressibles Materialverhalten, für das der Deformationsgradient eine konstante Determinante von 1 hat. Ein weiteres Beispiel hierfür ist die Beschreibung von magnetischem Material, das die Einführung eines Einheitsdirektors benötigt, um die Orientierung der Magnetisierung zu beschreiben.

Herausforderungen

Für all diese Beispiele führt die Anwendung von Standardverfahren aus der Finite-Elemente-Technologie zu verschiedenen unerwünschten Symptomen, da viele dieser Verfahren nur für Vektorräume funktionieren. Ihre direkte Anwendung auf nichtlineare Mannigfaltigkeiten führt auf Probleme wie Verlust der Objektivität, artifizielle Pfadabhängigkeit und sogar Divergenz der Lösungsalgorithmen. Diese Probleme lassen sich sowohl auf eine fehlerbehafte Interpolation und Linearisierung als auch den nichtlinearen Zusammenhang zwischen den Knotengrößen und ihrer Updategröße, welche im Tangentialraum liegt, zurückführen. Die Identifikation und Behebung dieser Probleme ist unter anderem Gegenstand dieses Projekts.

Simulationsbeispiele

Iterationsverlauf für die Suche des Minimums der Magnetisierungseinheitsvektoren: Ausgehend von einer zufälligen Orientierung und dem zusätzlichen Fixieren der linken und unteren Kante.

Simulationsbeispiel Magnetisierung

Simulation von Schalenbeulen mit nichtlinearen Reissner-Mindlin-Schalenelementen mit Einheitsdirektoren: Es handelt sich jeweils um eine statische Simulation unter Zuhilfenahme eines Riemannschen Trust-Region-Verfahrens.

Die obere Kante wird nach rechts bewegt und die Schale beult aus der Ebene. Nach Experimenten von Wong YW and Pellegrino S (2006) Wrinkled membranes part I: experiments.

Simulationsbeispiel Schalenbeulen

Die obere Kante des Zylinders wird nach unten bewegt und es werden verschiedene Gleichgewichtspunkte erreicht mit unterschiedlichen Beulmustern.

Simulationsbeispiel Zylinderbeulen

Veröffentlichungen

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
- Zusammenfassung
- BibTeX
Zusammenfassung
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
- Zusammenfassung
- BibTeX
Zusammenfassung
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 }

Überblick

Projektbeschreibung

Herausforderungen

Simulationsbeispiele

Veröffentlichungen

Zusammenfassung

BibTeX

Zusammenfassung

BibTeX

Bearbeitung:

Alexander Müller

Zielgruppe

Formalia

Services

Organisation

Finite Elemente und nichtlineare Felder

Überblick

Projektbeschreibung

Herausforderungen

Simulationsbeispiele

Veröffentlichungen

Zusammenfassung

BibTeX

Zusammenfassung

BibTeX

Bearbeitung:

Alexander Müller

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Zielgruppe

Formalia

Services

Organisation