Fast parametric analysis of trimmed multi-patch isogeometric Kirchhoff-Love shells using a local reduced basis method

TL;DR

The paper tackles the challenge of real-time, parametric analysis of trimmed, multi-patch isogeometric Kirchhoff-Love shells, where geometry-dependent operators are non-affine. It introduces a local reduced basis framework that combines snapshot extension, clustering, POD, and DEIM to construct affine local surrogates, enabling efficient offline/online decomposition. The authors demonstrate the method on several benchmarks, including trimmed multi-patch geometries and parametric shape optimization, using both projected super-penalty and Nitsche coupling, with substantial online speedups while preserving accuracy. The work highlights the practical potential of ROMs for rapid design and optimization in complex shell geometries, and outlines paths for future extensions to more general shell theories and error certification.

Abstract

This contribution presents a model order reduction framework for real-time efficient solution of trimmed, multi-patch isogeometric Kirchhoff-Love shells. In several scenarios, such as design and shape optimization, multiple simulations need to be performed for a given set of physical or geometrical parameters. This step can be computationally expensive in particular for real world, practical applications. We are interested in geometrical parameters and take advantage of the flexibility of splines in representing complex geometries. In this case, the operators are geometry-dependent and generally depend on the parameters in a non-affine way. Moreover, the solutions obtained from trimmed domains may vary highly with respect to different values of the parameters. Therefore, we employ a local reduced basis method based on clustering techniques and the Discrete Empirical Interpolation Method to construct affine approximations and efficient reduced order models. In addition, we discuss the application of the reduction strategy to parametric shape optimization. Finally, we demonstrate the performance of the proposed framework to parameterized Kirchhoff-Love shells through benchmark tests on trimmed, multi-patch meshes including a complex geometry. The proposed approach is accurate and achieves a significant reduction of the online computational cost in comparison to the standard reduced basis method.

Figures (23)

• Figure 1: Exemplary trimmed domain. The final rectangular domain $\Omega({\bm{\mu}})$ is a result of trimming away the green regions $\Omega_1({\bm{\mu}})$ and $\Omega_2({\bm{\mu}})$ from the non-trimmed domain $\Omega_0$.
• Figure 2: Example 7.1: Geometry and parameterization of the Scordelis-Lo roof with holes.
• Figure 3: Example 7.1: Singular values decay and relative error in $H^2$ norm vs. maximum number of reduced basis functions $N$ over all the clusters, for different numbers of clusters.
• Figure 4: Example 7.1: K-means variance over number of clusters $N_c$.
• Figure 5: Example 7.1: Vertical displacement solutions computed with the FOM (top) and local ROM (bottom) with $N_c = 8$ clusters for three parameter values $\mu= \lbrace0.0, 0.05, 0.1\rbrace$.

Theorems & Definitions (4)

• Remark 1
• Remark 2
• Remark 3
• Remark 4