Related papers: Basis Construction for Spline Spaces over Arbitrar…
This paper presents the first method for constructing bases for polynomial spline spaces over an arbitrary T-meshes (PT-splines for short). We construct spline basis functions for an arbitrary T-mesh by first converting the T-mesh into a…
We present computational methods for constructing orthogonal/orthonormal polynomials over arbitrary polygonal domains in $\mathbb{R}^2$ using bivariate spline functions. Leveraging a mature MATLAB implementation which generates spline…
Multi-degree splines are piecewise polynomial functions having sections of different degrees. For these splines, we discuss the construction of a B-spline basis by means of integral recurrence relations, extending the class of multi-degree…
This paper introduces the concept of dimensional stability for spline spaces over T-meshes, providing the first mathematical definition and a preliminary classification framework. We define dimensional stability as an invariant within the…
Multi-degree splines are piecewise polynomial functions having sections of different degrees. They offer significant advantages over the classical uniform-degree framework, as they allow for modeling complex geometries with fewer degrees of…
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…
We consider a class of non-polynomial spline spaces over T-meshes, that is, of spaces locally spanned both by polynomial and by suitably-chosen non-polynomial functions, which we will refer to as generalized splines over T-meshes. For such…
We introduce here Cartesian splines or, for short, C-splines. C- splines are piecewise polynomials which are defined on adjacent Cartesian coordinate systems and are Cr continuous throughout. The Cr continuity is enforced by constraining…
Multi-degree splines are smooth piecewise-polynomial functions where the pieces can have different degrees. We describe a simple algorithmic construction of a set of basis functions for the space of multi-degree splines, with similar…
With the renewed and growing interest in geometric continuity in mind, this article gives a general definition of geometrically continuous polygonal surfaces and geometrically continuous spline functions on them. Polynomial splines defined…
We analyze the space of geometrically continuous piecewise polynomial functions or splines for quadrangular and triangular patches with arbitrary topology and general rational transition maps. To define these spaces of G 1 spline functions,…
This paper presents a general framework for calculating the dimension of spline spaces over arbitrary rectilinear partitions using the smoothing cofactor method. The approach extends existing dimension theory for polynomial splines over…
Spline functions have long been used in numerically solving differential equations. Recently it revives as isogeometric analysis, which uses NURBS for both parametrization and element functions. In this paper, we introduce some multivariate…
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…
In this paper, we construct a bijective mapping between a biquadratic spline space over the hierarchical T-mesh and the piecewise constant space over the corresponding crossing-vertex-relationship graph (CVR graph). We propose a novel…
We study the module $C^r(\mathcal{P})$ of piecewise polynomial functions of smoothness $r$ on a pure $n$-dimensional polytopal complex $\mathcal{P}\subset\mathbb{R}^n$, via an analysis of certain subcomplexes $\mathcal{P}_W$ obtained from…
In this paper, we address the problem of constructing $C^2$ cubic spline functions on a given arbitrary triangulation $\mathcal{T}$. To this end, we endow every triangle of $\mathcal{T}$ with a Wang-Shi macro-structure. The $C^2$ cubic…
A method is proposed for constructing a spline curve of the Bezier type, which is continuous along with its first derivative by a piecewise polynomial function. Conditions for its existence and uniqueness are given. The constructed curve…
The construction of $C^m$ conforming finite elements on simplicial meshes has recently advanced through the groundbreaking work of Hu, Lin, and Wu (Found. Comput. Math. 24, 2024). Their framework characterizes smoothness via moments of…
Spline functions have long been used in numerical solution of differential equations. Recently it revives as isogeometric analysis, which offers integration of finite element analysis and NURBS based CAD into a single unified process.…