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. 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

Suitably graded THB-spline refinement and coarsening: Towards an adaptive isogeometric analysis of additive manufacturing processes

In the present work we introduce a complete set of algorithms to efficiently perform adaptive refinement and coarsening by exploiting truncated hierarchical B-splines (THB-splines) defined on suitably graded isogeometric meshes, that are called admissible mesh configurations. We apply the proposed algorithms to two-dimensional linear heat transfer problems with localized moving heat source, as simplified models for additive manufacturing applications. We first verify the accuracy of the admissible adaptive scheme with respect to an overkilled solution, for then comparing our results with similar schemes which consider different refinement and coarsening algorithms, with or without taking into account grading parameters. This study shows that the THB-spline admissible solution delivers an optimal discretization for what concerns not only the accuracy of the approximation, but also the (reduced) number of degrees of freedom per time step. In the last example we investigate the capability of the algorithms to approximate the thermal history of the problem for a more complicated source path. The comparison with uniform and non-admissible hierarchical meshes demonstrates that also in this case our adaptive scheme returns the desired accuracy while strongly improving the computational efficiency.Comment: 20 pages, 12 figure

Scan-based immersed isogeometric analysis

Scan-based simulations contain innate topologically complex three-dimensional geometries, represented by large data sets in formats which are not directly suitable for analysis. Consequently, performing high-fidelity scan-based simulations at practical computational costs is still very challenging. The main objective of this dissertation is to develop an efficient and robust scan-based simulation strategy by acquiring a profound understanding of three prominent challenges in scan-based IGA, viz.: i) balancing the accuracy and computational effort associated with numerical integration; ii) the preservation of topology in the spline-based segmentation procedure; and iii) the control of accuracy using error estimation and adaptivity techniques. In three-dimensional immersed isogeometric simulations, the computational effort associated with integration can be the critical component. A myriad of integration strategies has been proposed over the past years to ameliorate the difficulties associated with integration, but a general optimal integration framework that suits a broad class of engineering problems is not yet available. In this dissertation we provide a thorough investigation of the accuracy and computational effort of the octree integration technique. We quantify the contribution of the integration error using the theoretical basis provided by Strang’s first lemma. Based on this study we propose an error-estimate-based adaptive integration procedure for immersed IGA. To exploit the advantageous properties of IGA in a scan-based setting, it is important to extract a smooth geometry. This can be established by convoluting the voxel data using B-splines, but this can induce problematic topological changes when features with a size similar to that of the voxels are encountered. This dissertation presents a topology-preserving segmentation procedure using truncated hierarchical (TH)B-splines. A moving-window-based topological anomaly detection algorithm is proposed to identify regions in which (TH)B-spline refinements must be performed. The criterion to identify topological anomalies is based on the Euler characteristic, giving it the capability to distinguish between topological and shape changes. A Fourier analysis is presented to explain the effectiveness of the developed procedure. An additional computational challenge in the context of immersed IGA is the construction of optimal approximations using locally refined splines. For scan-based volumetric domains, hierarchical splines are particularly suitable, as they optimally leverage the advantages offered by the availability of a geometrically simple background mesh. Although truncated hierarchical B-splines have been successfully applied in the context of IGA, their application in the immersed setting is largely unexplored. In this dissertation we propose a computational strategy for the application of error estimation-based mesh adaptivity for stabilized immersed IGA. The conducted analyses and developed computational techniques for scan-based immersed IGA are interrelated, and together constitute a significant improvement in the efficiency and robustness of the analysis paradigm. In combination with other state-of-the-art developments regarding immersed FEM/IGA (\emph{e.g.}, iterative solution techniques, parallel computing), the research in this thesis opens the doors to scan-based simulations with more sophisticated physical behavior, geometries of increased complexity, and larger scan-data sizes.Scan-based simulations contain innate topologically complex three-dimensional geometries, represented by large data sets in formats which are not directly suitable for analysis. Consequently, performing high-fidelity scan-based simulations at practical computational costs is still very challenging. The main objective of this dissertation is to develop an efficient and robust scan-based simulation strategy by acquiring a profound understanding of three prominent challenges in scan-based IGA, viz.: i) balancing the accuracy and computational effort associated with numerical integration; ii) the preservation of topology in the spline-based segmentation procedure; and iii) the control of accuracy using error estimation and adaptivity techniques. In three-dimensional immersed isogeometric simulations, the computational effort associated with integration can be the critical component. A myriad of integration strategies has been proposed over the past years to ameliorate the difficulties associated with integration, but a general optimal integration framework that suits a broad class of engineering problems is not yet available. In this dissertation we provide a thorough investigation of the accuracy and computational effort of the octree integration technique. We quantify the contribution of the integration error using the theoretical basis provided by Strang’s first lemma. Based on this study we propose an error-estimate-based adaptive integration procedure for immersed IGA. To exploit the advantageous properties of IGA in a scan-based setting, it is important to extract a smooth geometry. This can be established by convoluting the voxel data using B-splines, but this can induce problematic topological changes when features with a size similar to that of the voxels are encountered. This dissertation presents a topology-preserving segmentation procedure using truncated hierarchical (TH)B-splines. A moving-window-based topological anomaly detection algorithm is proposed to identify regions in which (TH)B-spline refinements must be performed. The criterion to identify topological anomalies is based on the Euler characteristic, giving it the capability to distinguish between topological and shape changes. A Fourier analysis is presented to explain the effectiveness of the developed procedure. An additional computational challenge in the context of immersed IGA is the construction of optimal approximations using locally refined splines. For scan-based volumetric domains, hierarchical splines are particularly suitable, as they optimally leverage the advantages offered by the availability of a geometrically simple background mesh. Although truncated hierarchical B-splines have been successfully applied in the context of IGA, their application in the immersed setting is largely unexplored. In this dissertation we propose a computational strategy for the application of error estimation-based mesh adaptivity for stabilized immersed IGA. The conducted analyses and developed computational techniques for scan-based immersed IGA are interrelated, and together constitute a significant improvement in the efficiency and robustness of the analysis paradigm. In combination with other state-of-the-art developments regarding immersed FEM/IGA (\emph{e.g.}, iterative solution techniques, parallel computing), the research in this thesis opens the doors to scan-based simulations with more sophisticated physical behavior, geometries of increased complexity, and larger scan-data sizes

Adaptive analysis-aware defeaturing

Removing geometrical details from a complex domain is a classical operation in computer aided design for simulation and manufacturing. This procedure simplifies the meshing process, and it enables faster simulations with less memory requirements. But depending on the partial differential equation that one wants to solve, removing some important geometrical features may greatly impact the solution accuracy. For instance, in solid mechanics simulations, such features can be holes or fillets near stress concentration regions. Unfortunately, the effect of geometrical simplification on the accuracy of the problem solution is often neglected, or its evaluation is based on engineering expertise only due to the lack of reliable tools. It is therefore important to have a better understanding of the effect of geometrical model simplification, also called defeaturing, to improve our control on the simulation accuracy along the design and analysis phase. In this work, we consider the Poisson equation as a model problem, we focus on isogeometric discretizations, and we build an adaptive strategy that is twofold. Firstly, it performs standard mesh refinement in a (potentially trimmed multipatch) defeatured geometry described via truncated hierarchical B-splines. Secondly, it is also able to perform geometrical refinement, that is, to choose at each iteration step which geometrical feature is important to obtain an accurate solution. To drive this adaptive strategy, we introduce an a posteriori estimator of the energy error between the exact solution defined in the exact fully-featured geometry, and the numerical approximation of the solution defined in the defeatured geometry. The reliability of the estimator is proven for very general geometric configurations, and numerical experiments are performed to validate the presented theory and to illustrate the capabilities of the proposed adaptive strategy.Comment: 49 page