Refinement of Unstructured Meshes with Optimized Properties for Isogeometric Analysis

適用於等幾何分析的非結構化網格細化及參數優化

Student thesis: Doctoral Thesis

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Award date15 Jun 2020

Abstract

Isogeometric analysis (IGA) was first introduced as an alternative approach for Finite Element Analysis (FEA). The main idea of IGA is that the basis functions used for defining the geometry will also act as the basis in the process of analysis, which enables the use of the exact geometry for engineering analysis. Most of the prevailing efforts are mainly focused on analysis-suitable parameterization, smooth basis functions with relevant properties for IGA, and applications of IGA in engineering fields, such as computational mechanics, fluid flow, thermal analysis and various coupled problems. However, the lack of smooth basis functions over unstructured meshes with superior approximation property is still a great challenge. This research mainly focuses on the refinement of unstructured meshes with optimized properties in extraordinary regions for isogeometric analysis. The improvements are mainly three folds, namely improved refinement in extraordinary regions with optimal bounded curvature or desired distribution of control points, G2 spline patches with optimal refinement in extraordinary regions, and improved convergence in extraordinary regions for isogeometric analysis.

We first present a novel method to construct subdivision stencils and smooth basis functions with optimal bounded curvature near extraordinary vertices. The refinement rules are optimized to obtain optimal bounded curvature at extraordinary positions. The subdivision stencils for newly inserted and updated vertices near extraordinary vertices are firstly constructed to ensure G1 continuity with bounded curvature. They are further optimized with respect to G2 continuity conditions. The method is demonstrated by replacing subdivision stencils near extraordinary vertices for Catmull-Clark subdivision. Compared with the original Catmull-Clark subdivision and other previous tuning subdivision schemes known with small curvature variation near extraordinary positions, the proposed method produces better or comparable curvature behavior around extraordinary vertices with comparatively simple subdivision stencils. The limit surface has optimal bounded curvature / local quadratic precision (OBC/LQP) at extraordinary positions with uniformly distributed reflection lines. We also constructed other tuned refinement schemes with desired distribution of control vertices in extraordinary regions while maintaining G1 continuity with bounded curvature. By requiring that the subdominant eigenvalue λ equals or approaches to ideal value 0.5, subdivision stencils can be obtained through optimization with the least polar artifact (LPA) in extraordinary regions. Further decreasing the subdominant eigenvalue λ = 0.39 yields optimal subdivision stencils which could result in optimal convergence rate (OCR) in L2 norm useful for isogeometric analysis.

To achieve higher order smoothness at extraordinary positions, we also construct a G2 spline representation scheme for extraordinary regions. The scheme is constructed with G2 continuity and is further optimized to achieve the least geometry consistency error in local refinement and smooth shape representation, which is compatible with bounded curvature subdivision with the least polar artifact with λ = 0.5 for all valences, namely the G2-LPA-GC scheme. The resulting scheme and the set of smooth basis functions are globally G2 continuous with optimal refinement in extraordinary regions. All developed tuned refinement/subdivision schemes and G2 spline representations can be applied to both subdivision of unstructured quadrilateral meshes and unstructured T-splines in general.

Finally, we investigate the approximation property of the resulting tuned subdivision schemes and G2 spline representations for isogeometric analysis. The results show that, while both of them demonstrate sub-optimal convergence rate, the LQP scheme can consistently reduce the L2 norm error comparing to classical Catmull-Clark subdivision. The OCR scheme produces optimal convergence rate in terms of the L2 norm error that is consistent with that of cubic spline patches in regular regions. The LPA scheme, on the other hand, produces improved convergence rate for isogeometric analysis compared with the LQP and Catmull-Clark subdivision. Combined with the LPA scheme, the G2-LPA-GC scheme can also produce improved convergence rate for isogeometric analysis, which is consistent with that of the LPA scheme.

    Research areas

  • Isogeometric Analysis, Unstructured Meshes, Subdivision Schemes, G2-splines, Optimal Bounded Curvature / Local Quadratic Precision (OBC/LQP), Least Polar Artifact (LPA), Optimal Convergence Rate (OCR)