Solid T-spline Modeling for Isogeometric Analysis

In this dissertation, a new concept of the rational T-spline is defined, which is generalized from the traditional T-spline in order to get a set of partition of unity basis functions. In our preliminary work, a novel algorithm has been developed to convert any unstructured quadrilateral or hexahedral mesh to a T-spline surface or solid T-spline, based on the rational T-spline basis functions. Our conversion algorithm consists of two stages: the topology stage and the geometry stage. In the topology stage, the input quadrilateral or hexahedral mesh is taken as the initial T-mesh and templates are applied to each type of nodes to construct a gap-free T-spline. In the geometry stage, an efficient surface fitting technique is developed to improve the surface accuracy with sharp feature preservation. The constructed T-spline surface and solid T-spline interpolate every boundary node in the input mesh, with C2-continuity everywhere except the local region around irregular nodes. Finally, a Bézier extraction technique is developed and linear independence of the constructed T-splines is studied to facilitate T-spline based isogeometric analysis.;Furthermore, another algorithm is proposed to construct solid rational T-splines for complex genus-zero geometry from boundary surface triangulations. We first build a parametric mapping between the triangulation and the boundary of the parametric domain, a unit cube. After that we adaptively subdivide the cube using an octree subdivision, project the boundary nodes onto the input triangle mesh, and at the same time relocate the interior nodes via mesh smoothing. This process continues until the surface approximation error is less than a pre-defined threshold. T-mesh is then obtained by pillowing the subdivision result one layer on the boundary and its quality is improved. Templates are implemented to handle extraordinary nodes and partial extraordinary nodes in order to get a gap-free T-mesh. The obtained solid T-spline is C2-continuous except for the local region around each extraordinary node and partial extraordinary node. The boundary surface of the solid T-spline is C2-continuous everywhere except for the local region around the eight nodes corresponding to the eight corners of the parametric cube. Finally, the constructed solid T-splines are converted to Bézier meshes, which are analysis-suitable with no negative Jacobians. Several examples are presented to show the robustness of the algorithm.;As a follow-up, a comprehensive scheme is described to construct rational solid T-splines from boundary triangulations with arbitrary topology. To extract the topology of the input geometry, we first compute a smooth harmonic scalar field defined over the mesh and saddle points are extracted to determine the topology. By dealing with the saddle points, a polycube whose topology is equivalent to the input geometry is built and it serves as the parametric domain for the solid T-spline. A polycube mapping is then used to build a one-to-one correspondence between the input triangulation and the polycube boundary. After that, we choose the deformed octree subdivision of the polycube as the initial T-mesh, and make it valid through pillowing, quality improvement and applying templates to handle extraordinary nodes and partial extraordinary nodes. The obtained T-spline is C2-continuous everywhere over the boundary surface except for the local region surrounding polycube corner nodes. The efficiency and robustness of the presented technique are demonstrated with several applications in isogeometric analysis.;To achieve a tight integration of design and analysis, conformal solid T-spline construction with the input boundary spline representation preserved is desirable. However, to the best of our knowledge, this is still an open problem. In this dissertation, we provide its first solution. The input boundary T-spline surface has genus-zero topology and only contains eight extraordinary nodes, with an isoparametric line connecting each pair. One cube is adopted as the parametric domain for the solid T-spline. Starting from the cube with all the nodes on the input surface as T-junctions, we adaptively subdivide the domain based on the octree structure until each face or edge contains at most one face T-junction or one edge T-junction. Next, we insert two boundary layers between the input T-spline surface and the boundary of the subdivision result. Finally, knot intervals are calculated from the T-mesh and the solid T-spline is constructed. The obtained T-spline is conformal to the input T-spline surface with exactly the same boundary representation and continuity. For the interior region, the continuity is C2 everywhere except for the local region surrounding irregular nodes. Several examples are presented to demonstrate the performance of the algorithm. (Abstract shortened by UMI.).