@phdthesis{Hossain,
  author    = {Hossain, Md Naim},
  title     = {Isogeometric analysis based on Geometry Independent Field approximaTion (GIFT) and Polynomial Splines over Hierarchical T-meshes},
  doi       = {10.25643/bauhaus-universitaet.4037},
  url       = {http://nbn-resolving.de/urn:nbn:de:gbv:wim2-20191129-40376},
  school      = {Bauhaus-Universit{\"a}t Weimar},
  pages     = {157},
  abstract  = {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.},
  subject      = {Finite-Elemente-Methode},
  language  = {en}
}
@phdthesis{Kessler2018,
  author    = {Keßler, Andrea},
  title     = {Matrix-free voxel-based finite element method for materials with heterogeneous microstructures},
  doi       = {10.25643/bauhaus-universitaet.3844},
  url       = {http://nbn-resolving.de/urn:nbn:de:gbv:wim2-20190116-38448},
  school      = {Bauhaus-Universit{\"a}t Weimar},
  pages     = {113},
  year      = {2018},
  abstract  = {Modern image detection techniques such as micro computer tomography (μCT), magnetic resonance imaging (MRI) and scanning electron microscopy (SEM) provide us with high resolution images of the microstructure of materials in a non-invasive and convenient way. They form the basis for the geometrical models of high-resolution analysis, so called image-based analysis. However especially in 3D, discretizations of these models reach easily the size of 100 Mill. degrees of freedoms and require extensive hardware resources in terms of main memory and computing power to solve the numerical model. Consequently, the focus of this work is to combine and adapt numerical solution methods to reduce the memory demand first and then the computation time and therewith enable an execution of the image-based analysis on modern computer desktops. Hence, the numerical model is a straightforward grid discretization of the voxel-based (pixels with a third dimension) geometry which omits the boundary detection algorithms and allows reduced storage of the finite element data structure and a matrix-free solution algorithm. This in turn reduce the effort of almost all applied grid-based solution techniques and results in memory efficient and numerically stable algorithms for the microstructural models. Two variants of the matrix-free algorithm are presented. The efficient iterative solution method of conjugate gradients is used with matrix-free applicable preconditioners such as the Jacobi and the especially suited multigrid method. The jagged material boundaries of the voxel-based mesh are smoothed through embedded boundary elements which contain different material information at the integration point and are integrated sub-cell wise though without additional boundary detection. The efficiency of the matrix-free methods can be retained.},
  subject      = {Dissertation},
  language  = {en}
}