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 | ||||
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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. |
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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 |
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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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