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Element-based B-spline basis function spaces: construction and application in isogeometric analysis

Peng Yang, Maodong Pan, Falai Chen, Zhimin Zhang

TL;DR

The paper introduces an element-based construction of B-spline spaces, termed B-spline elements, that unifies the local, elementwise viewpoint of finite elements with the global, CAD-driven B-spline framework used in isogeometric analysis. By representing the global basis as explicit linear combinations of local element bases and employing Hermite interpolation, the method enables node- and derivative-matching without solving global systems, while maintaining optimal approximation properties even at maximal smoothness. The framework extends naturally to isogeometric analysis by decomposing geometric mappings into element contributions and enabling FEM-like assembly with nonuniform knot vectors, supported by explicit basis representations and a rigorous error theory. Numerical experiments in 2D show optimal convergence and superconvergence phenomena under uniform knots, and 3D tests demonstrate substantial computational gains with nonuniform knot configurations, validating both accuracy and efficiency. The work offers a practical, theoretically sound path to more efficient IgA implementations and provides a foundation for further adaptive refinement and higher-dimensional extensions.

Abstract

This paper develops a unified theoretical framework for constructing B-spline basis function spaces with structural equivalence to finite element spaces. The theory rigorously establishes that these bases emerge as explicit linear combinations of B-spline element bases. For any prescribed smoothness requirements, this element-wise formulation enables the Hermite interpolation at nodes, which directly utilizes function values and derivatives without solving global linear systems. By focusing on explicit interpolation properties, element-wise analysis establishes optimal approximation errors, even when the space smoothness attains its theoretical maximum for the space degree. In isogeometric analysis (IgA), the construction naturally decomposes geometric mappings into element-level representations, allowing efficient computations across elements regardless of node distribution. Notably, the same Hermite interpolation framework simultaneously handles domain parameterization and IgA solutions, allowing direct imposition of boundary conditions through function and derivative matching. Numerical tests demonstrate optimal convergence rates and superconvergence properties in 2D IgA under uniform knot configurations, and improved computational efficiency in 3D IgA with non-uniform knot distributions.

Element-based B-spline basis function spaces: construction and application in isogeometric analysis

TL;DR

The paper introduces an element-based construction of B-spline spaces, termed B-spline elements, that unifies the local, elementwise viewpoint of finite elements with the global, CAD-driven B-spline framework used in isogeometric analysis. By representing the global basis as explicit linear combinations of local element bases and employing Hermite interpolation, the method enables node- and derivative-matching without solving global systems, while maintaining optimal approximation properties even at maximal smoothness. The framework extends naturally to isogeometric analysis by decomposing geometric mappings into element contributions and enabling FEM-like assembly with nonuniform knot vectors, supported by explicit basis representations and a rigorous error theory. Numerical experiments in 2D show optimal convergence and superconvergence phenomena under uniform knots, and 3D tests demonstrate substantial computational gains with nonuniform knot configurations, validating both accuracy and efficiency. The work offers a practical, theoretically sound path to more efficient IgA implementations and provides a foundation for further adaptive refinement and higher-dimensional extensions.

Abstract

This paper develops a unified theoretical framework for constructing B-spline basis function spaces with structural equivalence to finite element spaces. The theory rigorously establishes that these bases emerge as explicit linear combinations of B-spline element bases. For any prescribed smoothness requirements, this element-wise formulation enables the Hermite interpolation at nodes, which directly utilizes function values and derivatives without solving global linear systems. By focusing on explicit interpolation properties, element-wise analysis establishes optimal approximation errors, even when the space smoothness attains its theoretical maximum for the space degree. In isogeometric analysis (IgA), the construction naturally decomposes geometric mappings into element-level representations, allowing efficient computations across elements regardless of node distribution. Notably, the same Hermite interpolation framework simultaneously handles domain parameterization and IgA solutions, allowing direct imposition of boundary conditions through function and derivative matching. Numerical tests demonstrate optimal convergence rates and superconvergence properties in 2D IgA under uniform knot configurations, and improved computational efficiency in 3D IgA with non-uniform knot distributions.
Paper Structure (14 sections, 7 theorems, 95 equations, 3 figures, 1 table)

This paper contains 14 sections, 7 theorems, 95 equations, 3 figures, 1 table.

Table of Contents

  1. Introduction
  2. Element-based B-spline basis function spaces
  3. Overview of B-spline basis function spaces
  4. Element-based construction of B-spline spaces
  5. Explicit representations of B-spline element bases
  6. Properties of element-based B-spline basis function spaces
  7. Hermite interpolation and optimal error estimates
  8. Reconstruction of basis via element-by-element representations
  9. Element-based B-spline basis function spaces with applications to isogeometric analysis
  10. Geometric mapping with element analysis
  11. Computations in isogeometric analysis via elements
  12. Numerical experiments
  13. Numerical examples in two dimension
  14. 2D Possion equation

Key Result

lemma thmcounterlemma

Let $\Xi:=\{t_i\}_{i=1}^{p+n+1}$ be a knot vector with nondecreasing sequence of knots, where $p$ is the polynomial degree of the B-spline sequence defined on $\Xi$, and $n$ is the number of B-splines. Consider a knot $t^{*}$ which satisfies $t_{m-1}\leq t^{*}< t_m$ for a certain index $m$. Through one can define the refined knot vector $\Xi^*:=\{t^*_i\}_{i=1}^{p+n+2}$. Let $\{a_i\}_{i=1}^{n}$ be

Figures (3)

  • Figure 1: Illustration of basis functions for the B-spline element $(K_j,\mathcal{S}_{p}^{\boldsymbol{k}}(\Xi_{K_j}),\mathcal{B}_{p}(\Xi_{K_j}))$.
  • Figure 2: Recurrence relation via B-spline elements
  • Figure 3: The physical domain $\Omega$ and exact solution of Example \ref{['example,2']}.

Theorems & Definitions (15)

  • lemma thmcounterlemma: Knot insertion
  • theorem 1
  • proof
  • remark thmcounterremark
  • lemma thmcounterlemma
  • lemma thmcounterlemma
  • proof
  • theorem 2
  • proof
  • definition thmcounterdefinition
  • ...and 5 more