A comparison of smooth basis constructions for isogeometric analysis

TL;DR

This paper systematically compares four unstructured spline constructions for multi-patch isogeometric analysis—AS-$G^1$, Approximate-$C^1$, D-Patch, and Almost-$C^1$—across biharmonic and Kirchhoff-Love shell problems, including stress analyses on curved shells and large industrial geometries. It provides both qualitative assessments of degree, regularity, continuity, and nesting, and quantitative benchmarks (biharmonic, shells, spectra, and modal analyses) to reveal each method’s strengths and limitations. The key findings are that Approximate-$C^1$ and AS-$G^1$ can achieve optimal convergence and accurate stress fields on AS geometries, while D-Patch and Almost-$C^1$ offer easier construction on complex geometries but exhibit issues near extraordinary vertices or degree restrictions; no single method dominates across all scenarios. The work highlights practical guidance and future directions, including geometry pre-processing for AS-$G^1$ and Approximate-$C^1$, preconditioning for D-Patch, and extending Almost-$C^1$ to higher degrees, with implications for robust, smooth multi-patch IGA in engineering applications.

Abstract

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 modeling, variational methods are widely used, whereas the application of unstructured spline methods on shell problems is rather scarce. In this paper, we therefore provide a qualitative and a quantitative comparison of a selection of unstructured spline constructions, in particular the D-Patch, Almost-$C^1$, Analysis-Suitable $G^1$ and the Approximate $C^1$ constructions. Using this comparison, we aim to provide insight into the selection of methods for practical problems, as well as directions for future research. In the qualitative comparison, the properties of each method are evaluated and compared. In the quantitative comparison, a selection of numerical examples is used to highlight different advantages and disadvantages of each method. In the latter, comparison with weak coupling methods such as Nitsche's method or penalty methods is made as well. In brief, it is concluded that the Approximate $C^1$ and Analysis-Suitable $G^1$ converge optimally in the analysis of a bi-harmonic problem, without the need of special refinement procedures. Furthermore, these methods provide accurate stress fields. On the other hand, the Almost-$C^1$ and D-Patch provide relatively easy construction on complex geometries. The Almost-$C^1$ method does not have limitations on the valence of boundary vertices, unlike the D-Patch, but is only applicable to biquadratic local bases. Following from these conclusions, future research directions are proposed, for example towards making the Approximate $C^1$ and Analysis-Suitable $G^1$ applicable to more complex geometries.

Figures (18)

• Figure 1: General workflow for solving a physics problem and optimizing a geometry or topology coming from CAD and CAE processes. Starting from CAD and CAE, the IGA Setup is performed. In this block, a computational basis is extracted from the geometry, to be used for simulation. Then, the Simulation block involves assembly of the operators of the physics problem on the computational basis coming from the IGA Setup. In case of shape or topology optimization problems, the simulation results are evaluated and the shape/topology is modified. From this changed shape/topology, a new computational basis can be obtained and the process can be repeated. The IGA Setup block is marked to be elaborated further on in \ref{['fig:IGA_Setup']}.
• Figure 2: Inside the IGA Setup block from \ref{['fig:workflow']}, three methods are distinguished. Firstly, trimmed domain approaches use trimming curves or surfaces to identify parts of a tensor-product domain as the actual domain. However, since elements can be trimmed poorly, specialized quadrature rules and solver preconditioners are typically needed. Alternatives to trimming are weak coupling or unstructured spline methods. For both classes of methods, a geometry with a given topology needs to be decomposed into multiple sub-domains (i.e. patches) via quadrilateral meshing. Given a quadrilateral mesh, weak methods assemble extra penalty terms into the equation to be solved, or add extra equations to be solved to satisfy continuity constraints. Lastly, unstructured spline constructions can be used to couple multiple domains by constructing a continuous basis. These methods, however can only be used on manifold geometries and conforming meshes. When these requirements are satisfied, unstructured spline pre-processing is required before the unstructured spline construction can take place. The pre-processing is highlighted and will be elaborated on more in \ref{['fig:USconstraints']} in \ref{['sec:qualitative']}.
• Figure 3: Given an initial geometry $\Omega$ (a), trimming (b) uses the curves of the boundary of the original geometry to define the interior domain $\Omega_{\text{int}}$ and the exterior domain $\Omega_{\text{ext}}$. An alternative approach for modelling the domain is to use domain segmentation (c). Here, the domain is decomposed into several patches $\Omega_i$ which together define the full domain $\Omega$.
• Figure 4: Procedure to find a multi-patch segmentation from a given mesh. The original mesh in (a) has 46 vertices, 81 edges and 45 faces and the final multi-patch (c) has 20 patches.
• Figure 5: Schematic representation of the continuity across element boundaries and patch interfaces for the (\ref{['fig:continuity_ASG1']}) AS-$G^1$ construction, (\ref{['fig:continuity_approxC1']}) Approx. $C^1$ constructions. Thin lines indicate element boundaries and thick lines indicate patch interfaces. Solid lines represent $C^{p-1}$ continuity, dashed lines represent $C^{p-2}$ continuity, thick dashed lines represent $C^1$ interfaces and loosely dashed lines represent approximate $C^1$ interfaces. A double lined circle represent a $C^2$ continuous vertex, a filled circle represent a singular vertex and a white filled circled with a single line represents a $C^1$ continuous vertex. The gray shaded area for the Approx. $C^1$ represents local reduced continuity.