A Novel Unified λ-subdivision Scheme with Optimal G2 Bézier Extraction and Optimal Convergence for Isogeometric Analysis Using Unstructured Meshes

Description

This project investigates tuned subdivision schemes and develops a novel unifiedl- subdivision scheme with a continuous family of subdivisions for quadrilateral meshes of arbitrary topology. Subdivision stencils of the unified subdivision scheme will be represented directly or indirectly as B-spline functions of the subdominant eigenvaluel of the respective subdivision matrix that can be arbitrarily selected within a wide range to meet desired properties of refined control meshes and resulting limit surfaces. The main subdivision stencil coefficients will be constructed as B-spline functions through discrete optimization that minimizes the eigenbasis functions corresponding to the subsubdominant eigenvalues of the unifiedl-subdivision scheme towardsC2continuity. The resulting subdivision scheme will produce globalC2 continuity in regular regions. Ifl for a particular valence N is chosen within a priority interval, the unifiedl- subdivision scheme produces limit surfaces withC1continuity and with optimal bounded curvature at extraordinary positions. By selecting an appropriatelvalue, the resulting subdivision scheme can also produce desired optimal convergence in extraordinary regions consistent with that in regular regions of the subdivision scheme for isogeometric analysis (IGA) while maintainingC1continuity with optimized curvature property at extraordinary positions. Several other existing tuned subdivision schemes for quadrilateral meshes also fall in the family of the proposed unified subdivision scheme as special cases.  In addition, optimalG2Bézier extraction operations in extraordinary regions will also be developed, which is particularly useful for solving IGA problems having a need of higher order continuity conditions in extraordinary regions, such as that for solving higher order partial differential equations (PDEs). The proposed unifiedl-subdivision scheme and its optimalG2Bézier extraction operations will also be integrated with unstructured T-splines for producing refined meshes in extraordinary regions with desired properties for isogeometric analysis. A comprehensive study will also be conducted through the project for a systematic evaluation and validation of the above properties of the proposed unifiedl-subdivision scheme, optimalG2Bézier extraction and their integration with unstructured T-splines for IGA applications. While the proposed unifiedl-subdivision scheme can be useful for many applications in computer aided design (CAD) and computer graphics (CG) in general, the emphasis of this project will be for applications in isogeometric analysis. The competitive advantages of the proposed unifiedl-subdivision and other relevant algorithms for isogeometric analysis will be three-folds. It provides unified and universal refinement operations for unstructured T-splines commonly used for IGA applications. Thel-subdivision scheme can conveniently address the challenging problem of sub-optimal convergence of both subdivision schemes and unstructured T-splines in extraordinary regions for IGA. The developed optimal  G2Bézier extraction operations for both thel-subdivision and its integration with unstructured T-splines will further benefit isogeometric analysis in providing higher order solutions needed for solving some engineering problems. The proposed project can thus further benefit both the CAD/CG and IGA communities with added flexibility and capability for modeling complex engineering models, quality IGA solutions of complex engineering system, and in addressing some of the well-known issues in using both subdivision schemes and unstructured T-splines for IGA.