Computer Aided Geometric Design 79 (2020) 101847 Contents lists available at ScienceDirect Computer Aided Geometric Design www.elsevier.com/locate/cagd Isogeometric analysis for trimmed CAD surfaces using multi-sided toric surface patches Xuefeng Zhu a , Ye Ji b,c , Chungang Zhu b,c,∗ , Ping Hu a , Zheng-Dong Ma d a School of Automotive Engineering, State Key Laboratory of Structural Analysis for Industrial Equipment, Dalian University of Technology, Dalian 116024, China b School of Mathematical Sciences, Dalian University of Technology, Dalian 116024, China c Key Laboratory for Computational Mathematics and Data Intelligence of Liaoning Province, Dalian University of Technology, Dalian 116024, China d Department of Mechanical Engineering, University of Michigan, Ann Arbor, MI 48109, United States of America a r t i c l e i n f o a b s t r a c t Article history: Available online 23 April 2020 Keywords: Toric surface patches Isogeometric analysis Trimmed CAD geometries NURBS spline finite-element method Trimmed spline surfaces We propose a new isogeometric method using Toric surface patches for trimmed CAD planar surfaces. This method converts each trimmed spline element into a Toric surface patch with conforming boundary representation and converts each non-trimmed spline element into a Bézier element. Because the Toric surface patches are a multi-sided generalization of classical Bézier surface patches, all trimmed and non-trimmed elements of a trimmed CAD surface have a unified geometric representation using Toric surface patches. Toric surface patches share the advantages of isogeometric continuum elements in that they can exactly model the geometry and can be easily implemented in standard finite-element code architectures. Several numerical examples are used to demonstrate the reliability of the proposed method.  2020 Elsevier B.V. All rights reserved. 1. Introduction Isogeometric analysis (IGA) was introduced by Hughes and coworkers (Hughes et al., 2005; Cottrell et al., 2006) to bridge the gap between Computer Aided Geometric Design (CAGD) and Finite Element Analysis (FEA). The core idea of IGA is to use the same smooth and higher-order basis functions for the representation of both the geometry in CAGD and the approximation of solution fields in FEA. IGA has been successfully applied in many areas, such as structural vibrations (Kiendl et al., 2009; Goyal et al., 2013), beam and shell problems (Wang et al., 2015; Liu et al., 2016), fluid–structure interaction problems (Bazilevs et al., 2008, 2009), structural optimization (Dede et al., 2012; Dhote et al., 2013) and fracture analysis (Borden et al., 2014; Schillinger et al., 2015) et al. IGA has been based on a variety of spline basis functions, e.g., B-splines, Non-Uniform Rational B-Splines (NURBS), T-splines (da Veiga et al., 2012; Wang et al., 2011), B-plus-plus splines (B++ Splines) (Zhu et al., 2016), hierarchical B(T)-splines (Schillinger and Rank, 2011), and PHT-splines (Li et al., 2010). Among those spline methods, NURBS is the most popular mathematical tool in CAD/CAM. It is also a major geometric element in international product data-transfer standards such as the standard for the exchange of product model data (STEP) and the initial graphics exchange specification (IGES). However, due to the inflexibility of the tensor-product form of B-spline and NURBS, determining how to perform IGA on complex topological shapes remains a critical challenge for * Corresponding author at: School of Mathematical Sciences, Dalian University of Technology, Dalian 116024, China. E-mail address: cgzhu@dlut.edu.cn (C. Zhu). https://doi.org/10.1016/j.cagd.2020.101847 0167-8396/ 2020 Elsevier B.V. All rights reserved.