Isogeometric analysis based on Geometry Independent Field approximaTion (GIFT) and Polynomial Splines over Hierarchical T-meshes

  • This thesis addresses an adaptive higher-order method based on a Geometry Independent Field approximatTion(GIFT) of polynomial/rationals plines over hierarchical T-meshes(PHT/RHT-splines). In isogeometric analysis, basis functions used for constructing geometric models in computer-aided design(CAD) are also employed to discretize the partial differential equations(PDEs) for numerical analysis.This thesis addresses an adaptive higher-order method based on a Geometry Independent Field approximatTion(GIFT) of polynomial/rationals plines over hierarchical T-meshes(PHT/RHT-splines). In isogeometric analysis, basis functions used for constructing geometric models in computer-aided design(CAD) are also employed to discretize the partial differential equations(PDEs) for numerical analysis. Non-uniform rational B-Splines(NURBS) are the most commonly used basis functions in CAD. However, they may not be ideal for numerical analysis where local refinement is required. The alternative method GIFT deploys different splines for geometry and numerical analysis. NURBS are utilized for the geometry representation, while for the field solution, PHT/RHT-splines are used. PHT-splines not only inherit the useful properties of B-splines and NURBS, but also possess the capabilities of local refinement and hierarchical structure. The smooth basis function properties of PHT-splines make them suitable for analysis purposes. While most problems considered in isogeometric analysis can be solved efficiently when the solution is smooth, many non-trivial problems have rough solutions. For example, this can be caused by the presence of re-entrant corners in the domain. For such problems, a tensor-product basis (as in the case of NURBS) is less suitable for resolving the singularities that appear since refinement propagates throughout the computational domain. Hierarchical bases and local refinement (as in the case of PHT-splines) allow for a more efficient way to resolve these singularities by adding more degrees of freedom where they are necessary. In order to drive the adaptive refinement, an efficient recovery-based error estimator is proposed in this thesis. The estimator produces a recovery solution which is a more accurate approximation than the computed numerical solution. Several two- and three-dimensional numerical investigations with PHT-splines of higher order and continuity prove that the proposed method is capable of obtaining results with higher accuracy, better convergence, fewer degrees of freedom and less computational cost than NURBS for smooth solution problems. The adaptive GIFT method utilizing PHT-splines with the recovery-based error estimator is used for solutions with discontinuities or singularities where adaptive local refinement in particular domains of interest achieves higher accuracy with fewer degrees of freedom. This method also proves that it can handle complicated multi-patch domains for two- and three-dimensional problems outperforming uniform refinement in terms of degrees of freedom and computational cost.show moreshow less

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Metadaten
Document Type:Doctoral Thesis
Author:Dr. Md Naim Hossain
DOI (Cite-Link):https://doi.org/10.25643/bauhaus-universitaet.4037Cite-Link
URN (Cite-Link):https://nbn-resolving.org/urn:nbn:de:gbv:wim2-20191129-40376Cite-Link
Title Additional (German):Die isogeometrische Analysis basierend auf der geometrieunabhängigen Feldnäherung (GIFT)und polynomialen Splines über hierarchischen T-Netzen
Referee:Prof. Dr. rer. nat. Tom LahmerORCiDGND, Prof. Dr. Stéphane Pierre Alain Bordas
Advisor:Prof. Dr.-Ing. Timon RabczukORCiDGND
Language:English
Date of Publication (online):2019/11/29
Date of first Publication:2019/11/29
Date of final exam:2019/10/21
Release Date:2019/11/29
Publishing Institution:Bauhaus-Universität Weimar
Granting Institution:Bauhaus-Universität Weimar, Fakultät Bauingenieurwesen
Institutes:Fakultät Bauingenieurwesen / Institut für Strukturmechanik
Pagenumber:157
Tag:Geometry Independent Field Approximation; Isogeometrc Analysis; Polynomial Splines over Hierarchical T-meshes; Recovery Based Error Estimator
GND Keyword:Finite-Elemente-Methode
Dewey Decimal Classification:600 Technik, Medizin, angewandte Wissenschaften
BKL-Classification:50 Technik allgemein / 50.03 Methoden und Techniken der Ingenieurwissenschaften
50 Technik allgemein / 50.31 Technische Mechanik
Licence (German):License Logo Creative Commons 4.0 - Namensnennung-Nicht kommerziell (CC BY-NC 4.0)