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Gradient robust methods for nearly incompressible materials

Basava, Seshadri (2023)
Gradient robust methods for nearly incompressible materials.
Technische Universität Darmstadt
doi: 10.26083/tuprints-00026463
Ph.D. Thesis, Primary publication, Publisher's Version

Abstract

Mixed finite elements for incompressible Navier-Stokes equations have seen a great success in mathematical fluid dynamics [16, 50, 35, 24, 21], to name a few. However, the dependency on pressure causes numerical instability. Linke [39], proposed a cure for this by introducing the gradient-robust interpolation operator π div.

We construct necessary assumptions and conditions needed to choose the suitable finite dimensional subspace of H_div (Ω; Rd ), given a stable inf-sup finite element pair solving the linear elasticity problem. We use Raviart-Thomas (RT_1) and Brezzi-Douglas-Marini (BDM_2) elements for Q_2 × DGQ_1 and Q_2 × DGP_1 finite element pairs respectively.

For computation, we use C++ based open source finite element libraries deal.II [5] and DOpElib [25]. We develop the FEValuesInterpolated class, which is derived from FEValues class of deal.II. Our class gives the value of π_div v_h , while the latter gives v_h. In case of linear elasticity, under the influence of external thermal force, we show that the gradient-robust method gives a well represented solution with fewer elements, compared to the non-gradient robust techniques for both incompressible and nearly incompressible materials. As an extension of our work [9], we show that, for phase-field fracture models under the effect of external thermal force, a well represented solution of displacement and fracture propagation for gradient-robust methods can be obtained with fewer elements, compared to non-gradient robust techniques.

Item Type: Ph.D. Thesis
Erschienen: 2023
Creators: Basava, Seshadri
Type of entry: Primary publication
Title: Gradient robust methods for nearly incompressible materials
Language: English
Referees: Wollner, Prof. Winnifried ; Tscherpel, Prof. Tabea ; Giesselman, Prof. Jan ; Wick, Prof. Thomas
Date: 19 December 2023
Place of Publication: Darmstadt
Collation: xii, 82 Seiten
Refereed: 5 October 2023
DOI: 10.26083/tuprints-00026463
URL / URN: https://tuprints.ulb.tu-darmstadt.de/26463
Abstract:

Mixed finite elements for incompressible Navier-Stokes equations have seen a great success in mathematical fluid dynamics [16, 50, 35, 24, 21], to name a few. However, the dependency on pressure causes numerical instability. Linke [39], proposed a cure for this by introducing the gradient-robust interpolation operator π div.

We construct necessary assumptions and conditions needed to choose the suitable finite dimensional subspace of H_div (Ω; Rd ), given a stable inf-sup finite element pair solving the linear elasticity problem. We use Raviart-Thomas (RT_1) and Brezzi-Douglas-Marini (BDM_2) elements for Q_2 × DGQ_1 and Q_2 × DGP_1 finite element pairs respectively.

For computation, we use C++ based open source finite element libraries deal.II [5] and DOpElib [25]. We develop the FEValuesInterpolated class, which is derived from FEValues class of deal.II. Our class gives the value of π_div v_h , while the latter gives v_h. In case of linear elasticity, under the influence of external thermal force, we show that the gradient-robust method gives a well represented solution with fewer elements, compared to the non-gradient robust techniques for both incompressible and nearly incompressible materials. As an extension of our work [9], we show that, for phase-field fracture models under the effect of external thermal force, a well represented solution of displacement and fracture propagation for gradient-robust methods can be obtained with fewer elements, compared to non-gradient robust techniques.

Alternative Abstract:
Alternative abstract Language

Gemischte Finite-Elemente-Methoden für die inkompressiblen Navier-Stokes-Gleichungen haben in der mathematischen Fluiddynamik großen Erfolg verzeichnet [16, 50, 35, 24, 21], um nur einige zu nennen. Allerdings führt die Abhängigkeit vom Druck zu numerischer Instabilität. Linke [39], schlug hierfür eine Lösung vor, indem er den gradientenrobusten Interpolationsoperator π div einführte.

In Betrachtung eines stabilen inf-sup Finite-Elemente-Paares für das Elastizitätsproblem, untersuchen wir Annahmen und Bedingungen um einen geeigneten endlich-dimensionalen Unterraum von H_div (Ω; Rd ) zu wählen. Hierbei werden Raviart-Thomas (RT_1 ) bzw. Brezzi-Douglas-Marini (BDM_2) Elemente für die Finite-Elemente-Paare Q_2 × DGQ_1 und Q_2 × DGP_1 genutzt.

In mehreren numerischen Experimenten zeigen wir die Vorteile dieser Erweiterung. Hierzu, haben wir die C++ basierten Finite-Elemente-Pakete deal.II [5] und DOpElib [25] genutzt und die Klasse FEValuesInterpolated entwickelt, welche auf der FEValues Klasse von deal.II basiert und es ermöglicht den Wert von π div vh berechnen. Im Vergleich zu nicht-gradientenrobusten Techniken zeigen wir für den Fall der linearen Elastizität unter dem Einfluss äußerer thermischer Kräfte, dass die gradientenrobuste Methode eine gut repräsentierte Lösung mit weniger Elementen liefert. Dies können wir sowohl für inkompressibele als auch für nahezu inkompressibele Materialien beobachten. Dieses Verhalten bestätigt sich auch bei der Betrachtung von Rissausbreitungsmodellen unter Einfluss äußerer thermischer Kräfte. Hier kann im Vergleich zu nicht-gradientenrobusten Techniken eine gut repräsentierte Lösung der Verschiebung und Rissausbreitung für gradientenrobuste Methoden mit weniger Elementen erzielt werden.

German
Status: Publisher's Version
URN: urn:nbn:de:tuda-tuprints-264632
Classification DDC: 500 Science and mathematics > 510 Mathematics
Divisions: 04 Department of Mathematics
04 Department of Mathematics > Optimization
04 Department of Mathematics > Optimization > Nonlinear Optimization
Date Deposited: 19 Dec 2023 13:27
Last Modified: 20 Dec 2023 11:34
PPN:
Referees: Wollner, Prof. Winnifried ; Tscherpel, Prof. Tabea ; Giesselman, Prof. Jan ; Wick, Prof. Thomas
Refereed / Verteidigung / mdl. Prüfung: 5 October 2023
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