In the FWF project "Isogeometric multi-patch shells and multigrid solvers" we develop and analyze isogeometric discretizations and multigrid solvers for plates and thin shell structures. On the one hand, we develop easily refinable splines over unstructured meshes. We focus on splines that reproduce tensor-product B-splines in regular regions of the mesh and that possess similar regularity and approximation properties near extraordinary vertices or near patch interfaces. Constructions of interest are analysis-suitable G1 multi-patch parameterizations, cf. (Collin, Sangalli, Takacs. CAGD, 2016), and almost-C1 splines, cf. (Takacs, Toshniwal. CMAME, 2023). Moreover, we analyze multigrid solvers for such isogeometric discretizations, with a special focus on robustness properties.

We investigate the regularity and approximation properties of multi-patch discretizations, unstructured spline spaces, spline manifolds and subdivision based constructions in IGA. The common feature of those constructions is that complex geometries, surface domains and volumes, are represented by joining multiple spline patches or polynomial elements. A key challenge lies in ensuring smoothness across patch interfaces and near extraordinary vertices and edges while maintaining a high approximation order. We analyze how different continuity conditions between patches affect the regularity and convergence behavior of the discretization. The goal is to analyze the geometric flexibility as well as the regularity and approximation properties of the different constructions. We intend to develop discretizations that balance geometric flexibility with good approximation properties, enabling accurate and efficient numerical solutions in non-tensor-product IGA frameworks.

We explore non-standard adaptivity strategies for spline-based discretizations, focusing on r-adaptivity (adaptivity by reparameterization of the domain), isogeometric h-adaptivity based on unstructured T-splines and on hierarchical B-splines, as well as adaptive refinement guided by neural networks. We compare our isogeometric adaptivity methods with traditional h-, p- and hp-adaptive finite element methods, which adjust element size and/or polynomial degree. The goal is to improve solution accuracy and efficiency by dynamically adjusting the geometry and resolution of the computational mesh. We investigate the use of neural networks to identify regions requiring refinement or mesh movement, enabling problem-specific adaptation strategies that require little prior knowledge.

We contribute to the G+Smo (Geometry + Simulation Modules, pronounced "gismo") open-source C++ library, which brings together mathematical tools for geometric design and numerical simulation. It is one of the leading libraries for isogeometric analysis, making use of a unified framework for the design and analysis pipeline. G+Smo is an object-oriented, cross-platform, template C++ library and follows the generic programming principle, with a focus on both efficiency and ease of use. The library is partitioned into smaller entities, called modules. The library is licensed under the Mozilla Public License v2.0. It aims at providing access to high quality, open-source software to the isogeometric numerical simulation community and beyond.