Kirchhoff-Love shell representation and analysis using triangle configuration B-splines

This paper presents the application of triangle configuration B-splines (TCB-splines) for representing and analyzing the Kirchhoff-Love shell in the context of isogeometric analysis (IGA). The Kirchhoff-Love shell formulation requires global $C^1$-continuous basis functions. The nonuniform rational B-spline (NURBS)-based IGA has been extensively used for developing Kirchhoff-Love shell elements. However, shells with complex geometries inevitably need multiple patches and trimming techniques, where stitching patches with high continuity is a challenge. On the other hand, due to their unstructured nature, TCB-splines can accommodate general polygonal domains, have local refinement, and are flexible to model complex geometries with $C^1$ continuity, which naturally fit into the Kirchhoff-Love shell formulation with complex geometries. Therefore, we propose to use TCB-splines as basis functions for geometric representation and solution approximation. We apply our method to both linear and nonlinear benchmark shell problems, where the accuracy and robustness are validated. The applicability of the proposed approach to shell analysis is further exemplified by performing geometrically nonlinear Kirchhoff-Love shell simulations of a pipe junction and a front bumper represented by a single patch of TCB-splines

Espaces de splines réproduisant les polynômes à partir de pavages de zonotopes

Revised version March 2021Given a point configuration A, we uncover a connection between polynomial-reproducing spline spaces over subsets of conv(A) and fine zonotopal tilings of the zonotope Z(V) associated to the corresponding vector configuration. This link directly generalizes a known result on Delaunay configurations and naturally encompasses, due to its combinatorial character, the case of repeated and affinely dependent points in A. We prove the existence of a general iterative construction process for such spaces. Finally, we turn our attention to regular fine zonotopal tilings, specializing our previous results and exploiting the adjacency graph of the tiling to propose a set of practical algorithms for the construction and evaluation of the associated spline functions.Étant donné une configuration de points A, on explore une connexion entre les espaces de splines reproduisant les polynômes sur certains sous-ensembles de conv(A) et les pavages fins du zonotope Z(V) associé à la configuration de vecteurs correspondante. Ce lien généralise directement un résultat connu sur les configurations de Delaunay et inclut naturellement, grâce à son charactère combinatoire, le cas de points en répétés et affinement dépendants en A. On prouve l'existence d'un processus de construction itératif général pour ces espaces. Enfin, on tourne notre attention vers les pavages de zonotopes fins et réguliers, en spécialisant nos résultats précédentes et en exploitant le graphe d'adjacence du pavage afin de proposer un ensemble d'algorithmes utiles en pratique pour la construction et l'évaluation des fonctions splines associées

Computations of Delaunay and Higher Order Triangulations, with Applications to Splines

Digital data that consist of discrete points are frequently captured and processed by scientific and engineering applications. Due to the rapid advance of new data gathering technologies, data set sizes are increasing, and the data distributions are becoming more irregular. These trends call for new computational tools that are both efficient enough to handle large data sets and flexible enough to accommodate irregularity. A mathematical foundation that is well-suited for developing such tools is triangulation, which can be defined for discrete point sets with little assumption about their distribution. The potential benefits from using triangulation are not fully exploited. The challenges fundamentally stem from the complexity of the triangulation structure, which generally takes more space to represent than the input points. This complexity makes developing a triangulation program a delicate task, particularly when it is important that the program runs fast and robustly over large data. This thesis addresses these challenges in two parts. The first part concentrates on techniques designed for efficiently and robustly computing Delaunay triangulations of three kinds of practical data: the terrain data from LIDAR sensors commonly found in GIS, the atom coordinate data used for biological applications, and the time varying volume data generated from from scientific simulations. The second part addresses the problem of defining spline spaces over triangulations in two dimensions. It does so by generalizing Delaunay configurations, defined as follows. For a given point set P in two dimensions, a Delaunay configuration is a pair of subsets (T, I) from P, where T, called the boundary set, is a triplet and I, called the interior set, is the set of points that fall in the circumcircle through T. The size of the interior set is the degree of the configuration. As recently discovered by Neamtu (2004), for a chosen point set, the set of all degree k Delaunay configurations can be associated with a set of degree k plus 1 splines that form the basis of a spline space. In particular, for the trivial case of k equals 0, the spline space coincides with the PL interpolation functions over the Delaunay triangulation. Neamtu’s definition of the spline space relies only on a few structural properties of the Delaunay configurations. This raises the question whether there exist other sets of configurations with identical structural properties. If there are, then these sets of configurations—let us call them generalized configurations from hereon—can be substituted for Delaunay configurations in Neamtu’s definition of spline space thereby yielding a family of splines over the same point set

Unstructured Multi-Patch DG-IGA Formulation for Wave Propagation

International audienceWave propagation problems in geophysics and in engineering often require different tools. In geophysics, one has to contend with heterogeneous and often discontinuous physical properties determined by subsoil structures such as strata and salt domes, often represented via unstructured meshes. Recent works highlighted the advantages of discontinuous Galerkin (DG) schemes, able to achieve high-order approximations while relying on block-diagonal matrices, well-suited for parallelization. Engineering simulations, on the other hand, often involve homogeneous materials with complex, but known, geometries. Isogeometric analysis (IGA) [1], which replaces polynomial bases by B-spline (or NURBS) bases coming from CAD models, has been shown to have higher efficiency per degree of freedom, better convergence in high energy modes and an improved CFL condition for wave propagation.REFERENCES[1] T. J. R. Hughes, J. A. Cottrell, and Y. Bazilevs, Isogeometric analysis: CAD, finite elements,NURBS, exact geometry and mesh refinement, CMAME., 2005.[2] J. Chan and J.A. Evans, Multi-patch discontinuous Galerkin isogeometric analysis for wave propagation:Explicit time-stepping and efficient mass matrix inversion, CMAME, 2018.[3] Y. Liu and J. Snoeyink, Quadratic and cubic B-splines by generalizing higher-order Voronoi diagrams,proceedings of the XXIII annual symposium on Computational geometry, ACM, 2007