A simple strategy for defining polynomial spline spaces over hierarchical T-meshes

Abstract

h i g h l i g h t s• A strategy for defining cubic tensor product spline functions is proposed. • Simple rules for inferring local knot vectors to define blending functions for a given T-mesh.• Examples of application of the strategy for adaptive refinement in isogeometric analysis and CAD. a b s t r a c tWe present a new strategy for constructing spline spaces over hierarchical T-meshes with quad-and octree subdivision schemes. The proposed technique includes some simple rules for inferring local knot vectors to define C 2 -continuous cubic tensor product spline blending functions. Our conjecture is that these rules allow to obtain, for a given T-mesh, a set of linearly independent spline functions with the property that spaces spanned by nested T-meshes are also nested, and therefore, the functions can reproduce cubic polynomials. In order to span spaces with these properties applying the proposed rules, the T-mesh should fulfill the only requirement of being a 0-balanced mesh. The straightforward implementation of the proposed strategy can make it an attractive tool for its use in geometric design and isogeometric analysis. In this paper we give a detailed description of our technique and illustrate some examples of its application in isogeometric analysis performing adaptive refinement for 2D and 3D problems.

Isogeometric solution of Helmholtz equation with Dirichlet boundary condition: numerical experiences

“…The function g(x, y) is the restriction of u(x, y) to the boundary of Ω. This problem is solved in [Bro16] on the unit square [0, 1] 2 . Here we solve it on several irregular regions.…”

Spline parameterization method for 2D and 3D geometries based on T-mesh optimization

“…Due to their simplicity, tree-structured meshes are an attractive tool for performing adaptive refinement in IGA and geometric modelling. For spline representation of the object we use polynomial spline spaces constructed via the technique described in [17]. This strategy allows to define easily a cubic spline space with nice properties over a given strongly balanced quadtree/octree T-mesh.…”

“…We have to obtain a global one-to-one spline transformation S : Ω = [0, 1] 2 → Ω that maps the parametric domain into the physical one. For this purpose, we use polynomial spline spaces constructed via the strategy proposed in our previous work [17,37]. This strategy allows to define easily cubic spline spaces with nice properties over a given strongly balanced quadtree/octree T-mesh.…”

“…Then, spline representation of the object is calculated by imposing interpolation conditions using the data provided by one-to-one correspondence between the mesh of the parametric domain and the mesh of the physical object. The interpolation is performed using polynomial spline spaces constructed via a technique proposed in our previous work [17]. This simple strategy allows to define easily a cubic spline space with nice properties over a given strongly balanced quadtree/octree T-mesh.…”

A Versatile Strategy for the Implementation of Adaptive Splines

“…Several other spaces and alternative bases exist, e.g., [36,13,8,27,6]. On the one hand, the mentioned spaces contain piecewise polynomials over box-shaped subdomains and allow for smooth functions.…”