Scattered Data Approximation by LR B-Spline Surfaces: A Study on Refinement Strategies for Efficient Approximation

Locally refined B-spline (LRB) surfaces provide a representation that is well suited to scattered data approximation. When a data set has local details in some areas and is largely smooth elsewhere, LR B-splines allow the spatial distribution of degrees of freedom to follow the variations of the data set. An LRB surface approximating a data set is refined in areas where the accuracy does not meet a required tolerance. In this paper we address, in a systematic study, different LRB refinement strategies and polynomial degrees for surface approximation. We study their influence on the resulting data volume and accuracy when applied to geospatial data sets with different structural behaviour. The relative performance of the refinement strategies is reasonably coherent for the different data sets and this paper concludes with some recommendations. An overall evaluation indicates that bi-quadratic LRB are preferable for the use cases tested, and that the strategies we denote as “full span" have the overall best performance.publishedVersio

Adaptive refinement for unstructured T-splines with linear complexity

We present an adaptive refinement algorithm for T-splines on unstructured 2D meshes. While for structured 2D meshes, one can refine elements alternatingly in horizontal and vertical direction, such an approach cannot be generalized directly to unstructured meshes, where no two unique global mesh directions can be assigned. To resolve this issue, we introduce the concept of direction indices, i.e., integers associated to each edge, which are inspired by theory on higher-dimensional structured T-splines. Together with refinement levels of edges, these indices essentially drive the refinement scheme. We combine these ideas with an edge subdivision routine that allows for I-nodes, yielding a very flexible refinement scheme that nicely distributes the T-nodes, preserving global linear independence, analysis-suitability (local linear independence) except in the vicinity of extraordinary nodes, sparsity of the system matrix, and shape regularity of the mesh elements. Further, we show that the refinement procedure has linear complexity in the sense of guaranteed upper bounds on a) the distance between marked and additionally refined elements, and on b) the ratio of the numbers of generated and marked mesh elements. © 2022 The Author(s

Weakening the tight coupling between geometry and simulation in isogeometric analysis: from sub- and super- geometric analysis to Geometry Independent Field approximaTion (GIFT)

This paper presents an approach to generalize the concept of isogeometric analysis (IGA) by allowing different spaces for parameterization of the computational domain and for approximation of the solution field. The method inherits the main advantage of isogeometric analysis, i.e. preserves the original, exact CAD geometry (for example, given by NURBS), but allows pairing it with an approximation space which is more suitable/flexible for analysis, for example, T-splines, LR-splines, (truncated) hierarchical B-splines, and PHT-splines. This generalization offers the advantage of adaptive local refinement without the need to re-parameterize the domain, and therefore without weakening the link with the CAD model. We demonstrate the use of the method with different choices of the geometry and field splines, and show that, despite the failure of the standard patch test, the optimum convergence rate is achieved for non-nested spaces

Mesh Refinement Strategies for the Adaptive Isogeometric Method

This thesis introduces four mesh refinement algorithms refine_hb, refine_thb, refine_ts2D and refine_tsnD for the Adaptive Isogeometric Method using multivariate Hierarchical B-splines, multivariate Truncated Hierarchical B-splines, bivariate T-splines, and multivariate T-splines, respectively. We address, for refine hb and refine_thb, boundedness of the overlap of basis functions, boundedness of mesh overlays and linear complexity in the sense of a uniform upper bound on the ratio of generated and marked elements. The existence of an upper bound on the overlap of basis functions implies that the system matrix of the linear equation system to solve is sparse, i.e., it has a uniformly bounded number of non-zero entries in each row and column. The upper bound on the number of elements in the coarsest common refinement (the overlay) of two meshes, as well as linear complexity are crucial ingredients for a later proof of rate-optimality of the method. For refine_ts2D and refine_tsnD, the overlap of basis functions is bounded a priori and did not need further investigation. We investigate the boundedness of mesh overlays, linear independence of the T-splines, nestedness of the T-spline spaces, and linear complexity as above. Nestedness of the spline spaces is crucial in the sense that it implies the so-called Galerkin orthogonality, which characterizes the approximate solution as a best-approximation of the exact solution with respect to a norm that depends on the problem. Altogether, this work paves the way for a proof of rate-optimality for the Adaptive Isogeometric Method with HB-splines, THB-splines, or T-splines in any space dimension. In order to justify the proposed methods and theoretical results in this thesis, numerical experiments underline their practical relevance, showing that they are not outperformed by currently prevalent refinement strategies. As an outlook to future work, we outline an approach for the handling of zero knot intervals and multiple lines in the interior of the domain, which are used in CAD applications for controlling the continuity of the spline functions, and we also sketch basic ideas for the local refinement of two-dimensional meshes that do not have tensor-product structure

Mathematical foundations of adaptive isogeometric analysis

This paper reviews the state of the art and discusses recent developments in the field of adaptive isogeometric analysis, with special focus on the mathematical theory. This includes an overview of available spline technologies for the local resolution of possible singularities as well as the state-of-the-art formulation of convergence and quasi-optimality of adaptive algorithms for both the finite element method (FEM) and the boundary element method (BEM) in the frame of isogeometric analysis (IGA)

An immersed methodology for fluid-structure interaction using NURBS and T-splines: theory, algorithms, validation, and application to blood flow at small scales

[Abstract] Mesh-based immersed approaches shine in a variety of fluid-structure interaction (FSI) applications such as, e.g., simulations where the solid undergoes large displacements or rotations, particulate flow problems, and scenarios where the topology of the region occupied by the fluid varies in time. In this thesis, a new mesh-based immersed approach is proposed which is based on the use of di erent types of splines as basis functions. This approach is put forth for modeling and simulating di erent types of biological cells in blood flow at small scales. The specific contributions of this thesis are outlined as follows. Firstly, a hybrid variational-collocation immersed technique using nonuniform rational B-splines (NURBS) is presented. Newtonian viscous incompressible fluids and nonlinear hyperelastic incompressible solids are considered. Our formulation boils down to three coupled equations which are the linear momentum balance equation, the mass conservation equation, and the kinematic equation that relates the Lagrangian displacement with the Eulerian velocity. The latter is discretized in strong form using isogeometric collocation and the other two equations are discretized using the variational multiscale (VMS) paradigm. As usual in immersed FSI approaches, we define a background mesh on the whole computational domain and a Lagrangian mesh tailored to the region occupied by each solid. Besides of using NURBS for creating these meshes, the data transfer between the background mesh and the Lagrangian meshes is carried out using NURBS functions in such a way that no interpolation or projection is needed, thus avoiding the errors associated with these procedures. Regarding the time discretization, the generalized- method is used which leads to a fully-implicit and second-order accurate method. The methodology is validated in two- and three-dimensional settings comparing the terminal velocity of free-falling bulky solids obtained in our simulations with its theoretical value. Secondly, we extend our algorithms in order to use analysis-suitable T-splines (ASTS) as basis functions instead of NURBS. This required to develop isogeometric collocation methods for ASTS which was an open problem. The data transfer between meshes changes significantly from NURBS to ASTS due to the fact that their geometrical mappings are local to patches and elements, respectively. ASTS possess two main advantages with respect to NURBS: (1) ASTS support local h-refinement and (2) ASTS are unstructured. The ASTSbased method is validated solving again the aforementioned benchmark problems and showing the potential of ASTS to decrease the amount of elements needed, thus enhancing the e ciency of the method. Thirdly, capsules, modeled as solid-shell NURBS elements, are proposed as numerical proxies for representing red blood cells (RBCs). The dynamics of capsules are able to reproduce the main motions and shapes observed in experiments with RBCs in both shear and parabolic flows. Hemorheological properties as the Fåhræus and Fåhræus-Lindqvist e ects are captured in our simulations. In order to obtain the aforementioned results, it is essential to adequately satisfy the incompressibility constraint close to the fluid-solid interface, which is an arduous task in immersed approaches for fluid-structure interaction. Finally, compound capsules are presented as numerical proxies for cells with nucleus such as, e.g., white blood cells (WBCs) and circulating tumor cells (CTCs). The dynamics of hyperelastic compound capsules in shear flow are studied in both two- and three-dimensional settings. Moreover, we simulate how CTCs manage to pass through channel narrowings, which is an interesting characteristic of CTCs since it is used in experiments to sort CTCs from blood samples