Isogeometric analysis is an innovative numerical paradigm with the potential to bridge the gap between Computer-Aided Design and Computer-Aided Engineering. However, constructing analysis-suitable parameterizations from a given boundary representation remains a critical challenge in the isogeometric design-through-analysis pipeline, particularly for computational domains with complex geometries, such as high-genus cases. To tackle this issue, we propose a multi-patch parameterization method for computational domains grounded in the singular structure of cross-fields. Initially, the vector field functions over the computational domain are solved using the boundary element method. The cross-field is then obtained through the one-to-one mapping between the vector field and the cross-field. Subsequently, we acquire the position information and topological connection relations of singularities and streamlines by analyzing the singular structure of the cross-field. Moreover, we introduce a simple and effective method for computing streamlines. We propose a novel segmentation strategy to divide the computational domain into several quadrilateral NURBS sub-patches. Once the multi-patch structure is established, we develop two methods to construct analysis-suitable multi-patch parameterizations. The first method is a direct generalization of the barrier function-based approach, while the second method yields smoother parameterizations by incorporating the interface control points of sub-patches into the optimization model. Numerical experiments demonstrate the effectiveness and robustness of the proposed method.@en

The Geometry-Independent Field approximaTion (GIFT) technique, an extension of isogeometric analysis (IGA), allows for separate spaces to parameterize the computational domain and approximate solution field. Based on the GIFT approach, this paper proposes a novel IGA methodology that incorporates toric surface patches for multi-sided geometry representation, while utilizing B-spline or truncated hierarchical B-spline (THB-spline) basis for analysis. By creating an appropriate bijection between the parametric domains of distinct bases for modeling and approximation, our method ensures smoothness within the computational domain and combines the compact support of B-splines or the local refinement potential of THB-splines, resulting in more efficient and precise solutions. To enhance the quality of parameterization and consequently boost the accuracy of downstream analysis, we suggest optimizing the composite toric parameterization. Numerical examples validate the effectiveness and superiority of our suggested approach.

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