Numerical manifold method based on isogeometric analysis

• Published in: Science China. Technological sciences Vol. 58; no. 9; pp. 1520 - 1532
• Main Authors: Zhang, YouLiang, Liu, DengXue, Tan, Fei
• Format: Journal Article
• Language: English
• Published: Beijing Science China Press 01.09.2015
• Subjects: B样条曲线; Engineering; NURBS曲面; 几何分析; 几何形状; 拉格朗日插值多项式; 数值流形方法; 网格细化; 非均匀有理B样条; local refinement; mathematical cover; numerical manifold method (NMM); isogeometric analysis
• ISSN: 1674-7321; 1869-1900
• DOI: 10.1007/s11431-015-5900-6

In this study, numerical manifold method （NMM） coupled with non-uniform rational B-splines （NURBS） and T-splines in the context of isogeometric analysis is proposed to allow for the treatments of complex geometries and local refinement. Computational formula for a 9-node NMM based on quadratic B-splines is derived. In order to exactly represent some common free-form shapes such as circles, arcs, and ellipsoids, quadratic non-uniform rational B-splines （NURBS） are introduced into NMM. The coordinate transformation based on the basis function of NURBS is established to enable exact integration for the manifold elements containing those shapes. For the case of crack propagation problems where singular fields around crack tips exist, local refinement technique by the application of T-spline discretizations is incorporated into NMM, which facilitates a truly local refinement without extending the entire row of control points. A local refinement strategy for the 4-node mathemat- ical cover mesh based on T-splines and Lagrange interpolation polynomial is proposed. Results from numerical examples show that the 9-node NMM based on NURBS has higher accuracies. The coordinate transformation based on the NURBS basis function improves the accuracy of NMM by exact integration. The local mesh refinement using T-splines reduces the number of degrees of freedom while maintaining calculation accuracy at the same time.