Exact conversion from Bézier tetrahedra to Bézier hexahedra

International audienceModeling and computing of trivariate parametric volumes is an important research topic in the field of three-dimensional isogeo-metric analysis. In this paper, we propose two kinds of exact conversion approaches from Bézier tetrahedra to Bézier hexahedra with the same degree by reparametrization technique. In the first method, a Bézier tetrahedron is converted into a degenerate Bézier hexahedron, and in the second approach, a non-degenerate Bézier tetrahedron is converted into four non-degenerate Bézier hexahedra. For the proposed methods, explicit formulas are given to compute the control points of the resulting tensor-product Bézier hexahedra. Furthermore, in the second method, we prove that tetrahedral spline solids with C k-continuity can be converted into a set of tensor-product Bézier volumes with G k-continuity. The proposed methods can be used for the volumetric data exchange problems between different trivariate spline representations in CAD/CAE. Several experimental results are presented to show the effectiveness of the proposed methods

Constructing IGA-suitable planar parameterization from complex CAD boundary by domain partition and global/local optimization

In this paper, we propose a general framework for constructing IGA-suitable planar B-spline parameterizations from given complex CAD boundaries consisting of a set of B-spline curves. Instead of forming the computational domain by a simple boundary, planar domains with high genus and more complex boundary curves are considered. Firstly, some pre-processing operations including B\'ezier extraction and subdivision are performed on each boundary curve in order to generate a high-quality planar parameterization; then a robust planar domain partition framework is proposed to construct high-quality patch-meshing results with few singularities from the discrete boundary formed by connecting the end points of the resulting boundary segments. After the topology information generation of quadrilateral decomposition, the optimal placement of interior B\'ezier curves corresponding to the interior edges of the quadrangulation is constructed by a global optimization method to achieve a patch-partition with high quality. Finally, after the imposition of C1=G1-continuity constraints on the interface of neighboring B\'ezier patches with respect to each quad in the quadrangulation, the high-quality B\'ezier patch parameterization is obtained by a C1-constrained local optimization method to achieve uniform and orthogonal iso-parametric structures while keeping the continuity conditions between patches. The efficiency and robustness of the proposed method are demonstrated by several examples which are compared to results obtained by the skeleton-based parameterization approach

Adaptive isogeometric analysis with hierarchical box splines

Isogeometric analysis is a recently developed framework based on finite element analysis, where the simple building blocks in geometry and solution space are replaced by more complex and geometrically-oriented compounds. Box splines are an established tool to model complex geometry, and form an intermediate approach between classical tensor-product B-splines and splines over triangulations. Local refinement can be achieved by considering hierarchically nested sequences of box spline spaces. Since box splines do not offer special elements to impose boundary conditions for the numerical solution of partial differential equations (PDEs), we discuss a weak treatment of such boundary conditions. Along the domain boundary, an appropriate domain strip is introduced to enforce the boundary conditions in a weak sense. The thickness of the strip is adaptively defined in order to avoid unnecessary computations. Numerical examples show the optimal convergence rate of box splines and their hierarchical variants for the solution of PDEs

Extended Powell–Sabin finite element scheme for linear elastic fracture mechanics

Data availability: No data was used for the research described in the article.Powell–Sabin B-splines, which are based on triangles, are employed in the framework of the extended finite element method (XFEM) for fracture analysis. This avoids the necessity of remeshing in discrete fracture models and increases the solution accuracy around the crack tip. Powell–Sabin B-splines are -continuous throughout the whole domain. The stresses around crack tips are captured more accurately than when using elements with a standard Lagrangian interpolation. Although Powell–Sabin B-splines do not hold the Kronecker-delta property, the Heaviside function and the tip enrichment function are confined to the cracked elements only, similar to the traditional XFEM but different from the extended isogeometric method. In addition, Powell–Sabin B-splines still hold -continuous throughout cracked elements. There is no need to lower the continuity at element boundaries, to confine basis function support in cracked elements. Shifting is used to ensure compatibility with the surrounding discretization. The sub-triangle technique is employed for the numerical integration over crack elements. The versatility and accuracy of the approach to simulate crack problems are assessed in case studies, featuring mode-I and mixed-mode crack problems.Engineering and Physical Sciences Research Council (EPSRC: EP/T033940/1)

Optimizing domain parameterization in isogeometric analysis based on Powell-Sabin splines

We address the problem of constructing a high-quality parameterization of a given planar physical domain, defined by means of a finite set of boundary curves. We look for a geometry map represented in terms of Powell-Sabin B-splines. Powell-Sabin splines are C1 quadratic splines defined on a triangulation, and thus the parameter domain can be any polygon. The geometry map is generated by the following three-step procedure. First, the shape of the parameter domain and a corresponding triangulation are determined, in such a way that its number of corners matches the number of corners of the physical domain. Second, the boundary control points related to the Powell-Sabin B-spline representation are chosen so that they parameterize the boundary curve of the physical domain. Third, the remaining inner control points are obtained by solving a nimble optimization problem based on the Winslow functional. The proposed domain parameterization procedure is illustrated numerically in the context of isogeometric Galerkin discretizations based on Powell-Sabin splines. It turns out that the flexibility rising from the generality of the parameter domain has a beneficial effect on the quality of the parameterization and also on the accuracy of the computed approximate solution