[Paper Review] Isogeometric analysis with $C^1$ functions on unstructured quadrilateral meshes

Mario Kapl, Giancarlo Sangalli+1 more|arXiv (Cornell University)|Dec 21, 2018

Advanced Numerical Analysis Techniques31 references4 citations

TL;DR

This paper presents a method for constructing $C^1$-smooth isogeometric spline spaces over unstructured quadrilateral meshes using analysis-suitable $G^1$ multi-patch parametrizations, enabling optimal Galerkin discretization of fourth-order PDEs. It extends the approach to non-analysis-suitable $G^1$ geometries by enriching splines near extraordinary vertices, achieving optimal convergence rates in numerical tests.

ABSTRACT

In the context of isogeometric analysis, globally $C^1$ isogeometric spaces over unstructured quadrilateral meshes allow the direct solution of fourth order partial differential equations on complex geometries via their Galerkin discretization. The design of such smooth spaces has been intensively studied in the last five years, in particular for the case of planar domains, and is still task of current research. In this paper, we first give a short survey of the developed methods and especially focus on the approach [26]. There, the construction of a specific $C^1$ isogeometric spline space for the class of so-called analysis-suitable $G^1$ multi-patch parametrizations is presented. This particular class of parameterizations comprises exactly those multi-patch geometries, which ensure the design of $C^1$ spaces with optimal approximation properties, and allows the representation of complex planar multi-patch domains. We present known results in a coherent framework, and also extend the construction to parametrizations that are not analysis-suitable $G^1$ by allowing higher-degree splines in the neighborhood of the extraordinary vertices and edges. Finally, we present numerical tests that illustrate the behavior of the proposed method on representative examples.

Motivation & Objective

• To develop globally $C^1$-smooth isogeometric spaces over unstructured planar multi-patch domains for direct Galerkin solution of fourth-order PDEs.
• To address the challenge of constructing $C^1$-smooth splines across patch interfaces in complex, unstructured quadrilateral meshes with extraordinary vertices.
• To extend existing $C^1$ construction methods beyond analysis-suitable $G^1$ parametrizations by allowing higher-degree splines near extraordinary vertices.
• To validate the method numerically with optimal convergence rates for biharmonic and related PDEs on representative multi-patch domains.

Proposed method

• Uses the equivalence between $C^1$-smooth isogeometric functions and $G^1$-smooth graph surfaces to enforce global $C^1$ continuity across patch interfaces.
• Applies the framework of analysis-suitable $G^1$ multi-patch parametrizations, which ensure optimal approximation properties and allow complex planar domain representation.
• Extends the construction to non-analysis-suitable $G^1$ geometries by locally increasing spline degree in the neighborhood of extraordinary vertices and edges.
• Employs a refined space construction that reduces degree in high-degree regions to minimize the number of high-degree elements, though resulting in non-nested spaces.
• Implements standard Galerkin discretization for the biharmonic equation with strong enforcement of Dirichlet and Neumann boundary conditions via $L^2$ projection.
• Uses numerical refinement studies with $h = 1/4, 1/8, 1/16, 1/32$ to verify convergence rates for polynomial degrees $p = 3$ and $p = 4$.

Experimental results

Research questions

• RQ1 How can $C^1$-smooth isogeometric spaces be constructed over unstructured quadrilateral meshes with extraordinary vertices while preserving optimal approximation properties?
• RQ2 What modifications are required to extend $C^1$ construction beyond analysis-suitable $G^1$ parametrizations to general multi-patch geometries?
• RQ3 What is the convergence behavior of the resulting $C^1$ isogeometric spaces when solving fourth-order PDEs like the biharmonic equation?
• RQ4 How does local degree enrichment near extraordinary vertices affect the conditioning and approximation quality of the discrete space?
• RQ5 Can the method maintain optimal convergence rates despite non-nested spaces arising from degree reduction?

Key findings

• The proposed method achieves optimal convergence rates of $\mathcal{O}(h^{p+1})$ in the $L^2$-norm, $\mathcal{O}(h^p)$ in the $H^1$-norm, and $\mathcal{O}(h^{p-1})$ in the $H^2$-norm for both $p=3$ and $p=4$ on the tested AS-$G^1$ multi-patch geometries.
• Numerical results confirm that the $C^1$ isogeometric spaces constructed via the extended method maintain optimal approximation properties even on non-analysis-suitable $G^1$ parametrizations.
• The use of higher-degree splines near extraordinary vertices enables $C^1$ continuity across interfaces in complex geometries where standard methods fail.
• The space construction leads to a non-nested family of discrete spaces due to degree reduction, but convergence is still optimal across refinement levels.
• The method successfully enables Galerkin discretization of the biharmonic equation with exact solutions recovered up to machine precision for the chosen test cases.