Date: Tuesday, July 15, 2025, Time: 11:00 am, S2 416-1

Fatima Hasanova: Multigrid methods for the biharmonic equation on multi-patch domains

In this talk, we present an ongoing investigation into multigrid solvers for biharmonic equations discretized using isogeometric analysis (IGA). Our primary interest lies in handling $C^1$-smooth multi-patch domains, which are relevant for fourth-order partial differential equations (PDEs) arising in structural modeling of thin plates and shell structures, discretized with multi-patch spline parameterizations. Motivated by the works [1] and [3], we explore analysis-suitable $G^1$ multi-patch parametrizations that facilitate $C^1$-smooth discretizations. Additionally, we outline a multigrid framework inspired by [4], focusing on efficient two-level refinement relations and the structure of smoothing matrices to optimize computational efficiency. We also discuss prospects for extending these techniques to arbitrary multi-patch surfaces, following the ideas in [2].
This research is part of the ongoing project ’Isogeometric multi-patch shells and multigrid solvers’.

[1] A. Collin, G. Sangalli, T. Takacs. Analysis-suitable $G^1$ multi-patch parametrizations for $C^1$ isogeometric spaces, Computer Aided Geometric Design, 47 (2016) 93–113.
[2] A. Farahat et al. Isogeometric analysis with $C^1$-smooth functions over multi-patch surfaces, Computer Methods in Applied Mechanics and Engineering, 403, Part A (2023) 115706.
[3] M. Kapl, G. Sangalli, T. Takacs. Dimension and basis construction for analysis-suitable $G^1$ two-patch parameterizations, Computer Aided Geometric Design, 5253 (2017) 7589.
[4] J. Sogn, S. Takacs. Robust multigrid solvers for the biharmonic problem in isogeometric analysis, Computers & Mathematics with Applications, 77, Issue 1 (2019) 105124.

Date: Tuesday, July 15, 2025, Time: 11:45 pm, S2 416-1

Syeda Hijab Zahra: Construction of bi-cubic unstructured splines with exact and approximate $C^1$-smoothness for shell simulations

Ensuring the continuity and smoothness of surfaces remains a significant challenge in geometric modeling, particularly when dealing with complex configurations such as extraordinary vertices [1]. In this talk, we examine different techniques for constructing smooth surfaces, with a focus on approaches like approximate $C^1$ constructions, and the almost-$C^1$ formulation [2]. Almost-$C^1$ splines are a type of bi-quadratic spline designed for fully unstructured quadrilateral meshes, without limitations on the number or placement of extraordinary vertices. Another technique is based on bi-cubic Coons patches with Hermite interpolation curves [3]. The construction utilizes cubic Hermite curves and bi-linear Coons interpolation, ensuring $G^1$-continuity at the nodes and $C^0$-continuity across the edges. Each of these approaches presents specific challenges, especially in ensuring continuity and effectively handling unstructured meshes for spline generation in surface modeling. We propose a bi-cubic surface construction that ensures exact $C^1$ continuity in regular regions while preserving $G^1$ continuity at extraordinary vertices. This construction is designed to work with unstructured quadrilateral meshes and can be applied to fourth-order problems, such as Kirchhoff–Love shells. This talk is based on ongoing research conducted within the project ”Isogeometric Multi-Patch Shells and Multigrid Solvers”.