[Paper Review] Construction of analysis-suitable $G^1$ planar multi-patch parameterizations

Mario Kapl, Giancarlo Sangalli+1 more|arXiv (Cornell University)|Jun 10, 2017

Advanced Numerical Analysis Techniques24 references3 citations

TL;DR

This paper presents a method to construct analysis-suitable G1 (AS-G1) multi-patch parameterizations for planar domains from a given initial C0 multi-patch geometry. By solving a quadratic optimization problem with linear constraints, the approach generates AS-G1 parameterizations that match the boundary, vertices, and first derivatives at vertices while being as close as possible to the original geometry—demonstrating optimal h-convergence in numerical tests.

ABSTRACT

Isogeometric analysis allows to define shape functions of global $C^{1}$ continuity (or of higher continuity) over multi-patch geometries. The construction of such $C^{1}$-smooth isogeometric functions is a non-trivial task and requires particular multi-patch parameterizations, so-called analysis-suitable $G^{1}$ (in short, AS-$G^{1}$) parameterizations, to ensure that the resulting $C^{1}$ isogeometric spaces possess optimal approximation properties, cf. [7]. In this work, we show through examples that it is possible to construct AS-$G^{1}$ multi-patch parameterizations of planar domains, given their boundary. More precisely, given a generic multi-patch geometry, we generate an AS-$G^{1}$ multi-patch parameterization possessing the same boundary, the same vertices and the same first derivatives at the vertices, and which is as close as possible to this initial geometry. Our algorithm is based on a quadratic optimization problem with linear side constraints. Numerical tests also confirm that $C^{1}$ isogeometric spaces over AS-$G^{1}$ multi-patch parameterized domains converge optimally under mesh refinement, while for generic parameterizations the convergence order is severely reduced.

Motivation & Objective

• To develop a practical method for constructing analysis-suitable G1 (AS-G1) multi-patch parameterizations for planar domains.
• To ensure that the resulting parameterizations preserve the boundary, vertices, and first derivatives at vertices of an initial C0 multi-patch geometry.
• To generate AS-G1 parameterizations that are as close as possible to the initial geometry while satisfying AS-G1 constraints.
• To numerically validate that C1 isogeometric spaces over AS-G1 parameterizations achieve optimal h-convergence rates.
• To demonstrate the flexibility and feasibility of AS-G1 geometry construction despite sensitivity to perturbations.

Proposed method

• Formulate the construction of AS-G1 parameterizations as a quadratic optimization problem with linear equality constraints.
• Use the initial C0 multi-patch parameterization as a starting point, preserving its boundary, vertices, and first derivatives at vertices.
• Impose AS-G1 constraints derived from geometric continuity conditions at patch interfaces.
• Solve the optimization problem numerically using symbolic computation to maintain accuracy due to constraint sensitivity.
• Construct the parameterization patch-by-patch, ensuring C1 smoothness across interfaces via the AS-G1 conditions.
• Employ a basis with small local supports to improve conditioning and reduce sensitivity in future work.

Experimental results

Research questions

• RQ1 Can AS-G1 multi-patch parameterizations be constructed for arbitrary planar multi-patch domains given only their boundary?
• RQ2 To what extent do AS-G1 constraints restrict the quality and proximity of the resulting parameterization to an initial geometry?
• RQ3 Does the use of AS-G1 parameterizations guarantee optimal h-convergence in isogeometric analysis?
• RQ4 How sensitive are the AS-G1 constraints to small perturbations, and can this be mitigated through basis design?
• RQ5 Can the proposed method be extended to volumetric or surface-based multi-patch domains?

Key findings

• The proposed method successfully generates AS-G1 multi-patch parameterizations that are visually indistinguishable from initial C0 geometries but satisfy all required continuity constraints.
• Numerical experiments confirm that C1 isogeometric spaces over AS-G1 parameterizations achieve optimal convergence rates of order O(h⁴) in the L2-norm.
• In contrast, generic C0 parameterizations exhibit severely reduced convergence, with rates as low as O(h¹/²) in the L2-norm and no convergence in the L∞-norm.
• The AS-G1 constraints do not prevent high-quality parameterization construction, demonstrating significant flexibility in geometry design.
• The method is robust and effective even when the initial geometry is not AS-G1, as shown in multiple test cases including a car, puzzle piece, and deformed circle.
• Symbolic computation is essential due to the high sensitivity of AS-G1 constraints to numerical perturbations.