A comparison of smooth basis constructions for isogeometric analysis

In order to perform isogeometric analysis with increased smoothness on complex domains, trimming, variational coupling or unstructured spline methods can be used. The latter two classes of methods require a multi-patch segmentation of the domain, and provide continuous bases along patch interfaces. In the context of shell modelling,...

Algebraic methods to study the dimension of supersmooth spline spaces

Multivariate piecewise polynomial functions (or splines) on polyhedral complexes have been extensively studied over the past decades and find applications in diverse areas of applied mathematics including numerical analysis, approximation theory, and computer aided geometric design. In this paper we address various challenges arising in the...

Almost-C¹ splines: Biquadratic splines on unstructured quadrilateral meshes and their application to fourth order problems

Isogeometric Analysis generalizes classical finite element analysis and intends to integrate it with the field of Computer-Aided Design. A central problem in achieving this objective is the reconstruction of analysis-suitable models from Computer-Aided Design models, which is in general a non-trivial and time-consuming task. In this article,...

Quadratic splines on quad-tri meshes: Construction and an application to simulations on watertight reconstructions of trimmed surfaces

Given an unstructured mesh consisting of quadrilaterals and triangles (we allow both planar and non-planar meshes of arbitrary topology), we present the construction of quadratic splines of mixed smoothness — C¹ smooth away from the unstructured regions of T and C⁰ smooth otherwise. The splines have several useful B...

An optimally convergent smooth blended B-spline construction for semi-structured quadrilateral and hexahedral meshes

Easy to construct and optimally convergent generalisations of B-splines to unstructured meshes are essential for the application of isogeometric analysis to domains with non-trivial topologies. Nonetheless, especially for hexahedral meshes, the construction of smooth and optimally convergent isogeometric analysis basis functions is still an...

Isogeometric discrete differential forms: Non-uniform degrees, Bezier extraction, polar splines and flows on surfaces: Non-uniform degrees, Bézier extraction, polar splines and flows on surfaces

Spaces of discrete differential forms can be applied to numerically solve the partial differential equations that govern phenomena such as electromagnetics and fluid mechanics. Robustness of the resulting numerical methods is complemented by pointwise satisfaction of conservation laws (e.g., mass conservation) in the discrete setting. Here we...

Polynomial spline spaces of non-uniform bi-degree on T-meshes: combinatorial bounds on the dimension

Polynomial splines are ubiquitous in the fields of computer-aided geometric design and computational analysis. Splines on T-meshes, especially, have the potential to be incredibly versatile since local mesh adaptivity enables efficient modeling and approximation of local features. Meaningful use of such splines for modeling and approximation...

Counting the dimension of splines of mixed smoothness: A general recipe, and its application to planar meshes of arbitrary topologies

In this paper, we study the dimension of bivariate polynomial splines of mixed smoothness on polygonal meshes. Here, “mixed smoothness” refers to the choice of different orders of smoothness across different edges of the mesh. To study the dimension of spaces of such splines, we use tools from homological algebra. These tools were first applied...

The divergence-conforming immersed boundary method: Application to vesicle and capsule dynamics

We extend the recently introduced divergence-conforming immersed boundary (DCIB) method [1] to fluid-structure interaction (FSI) problems involving closed co-dimension one solids. We focus on capsules and vesicles, whose discretization is particularly challenging due to the higher-order derivatives that appear in their formulations. In two...

A General Class of C¹ Smooth Rational Splines: Application to Construction of Exact Ellipses and Ellipsoids

In this paper, we describe a general class of C¹ smooth rational splines that enables, in particular, exact descriptions of ellipses and ellipsoids — some of the most important primitives for CAD and CAE. The univariate rational splines are assembled by transforming multiple sets of NURBS basis functions via so-called design...

Dimension of polynomial splines of mixed smoothness on T-meshes

In this paper we study the dimension of splines of mixed smoothness on axis-aligned T-meshes. This is the setting when different orders of smoothness are required across the edges of the mesh. Given a spline space whose dimension is independent of its T-mesh's geometric embedding, we present constructive and sufficient conditions that ensure...