Isogeometric analysis is aimed to mitigate the gap between Computer-Aided Design (CAD) and analysis by using a unified geometric representation. Thanks to the exact geometry representation and high smoothness of adopted basis functions, isogeometric analysis demonstrated excellent mathematical properties and successfully addressed a variety of problems. In particular, it allows to solve higher order Partial Differential Equations (PDEs) directly omitting the usage of mixed approaches. Unfortunately, complex CAD geometries are often constituted by multiple Non-Uniform Rational B-Splines (NURBS) patches and cannot be directly applied for finite element analysis.parIn this work, we presents a dual mortaring framework to couple adjacent patches for higher order PDEs. The development of this formulation is initiated over the simplest 4th order problem-biharmonic problem. In order to speed up the construction and preserve the sparsity of the coupled problem, we derive a dual mortar compatible C1 constraint and utilize the Bezier dual basis to discretize the Lagrange multipler spaces. We prove that this approach leads to a well-posed discrete problem and specify requirements to achieve optimal convergence. After identifying the cause of sub-optimality of Bezier dual basis, we develop an enrichment procedure to endow Bezier dual basis with adequate polynomial reproduction ability. The enrichment process is quadrature-free and independent of the mesh size. Hence, there is no need to take care of the conditioning. In addition, the built-in vertex modification yields compatible basis functions for multi-patch coupling.To extend the dual mortar approach to couple Kirchhoff-Love shell, we develop a dual mortar compatible constraint for Kirchhoff-Love shell based on the Rodrigues' rotation formula. This constraint provides a unified formulation for both smooth couplings and kinks. The enriched Bezier dual basis preserves the sparsity of the coupled Kirchhoff-Love shell formulation and yields accurate results for several benchmark problems.Like the dual mortaring formulation, locking problem can also be derived from the mixed formulation. Hence, we explore the potential of Bezier dual basis in alleviating transverse shear locking in Timoshenko beams and volumetric locking in nearly compressible linear elasticity. Interpreting the well-known B projection in two different ways we develop two formulations for locking problems in beams and nearly incompressible elastic solids. One formulation leads to a sparse symmetric symmetric system and the other leads to a sparse non-symmetric system.
| Identifer | oai:union.ndltd.org:BGMYU2/oai:scholarsarchive.byu.edu:etd-8731 |
| Date | 01 August 2019 |
| Creators | Miao, Di |
| Publisher | BYU ScholarsArchive |
| Source Sets | Brigham Young University |
| Detected Language | English |
| Type | text |
| Format | application/pdf |
| Source | Theses and Dissertations |
| Rights | http://lib.byu.edu/about/copyright/ |
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