| Mesh is widely used to represent 3D models in computer graphics and industrial commu-nity.For physical simulation,the element quality and the global structure affect the efficiency and accuracy of algorithms directly.For animation production,the shape and alignment of ele-ments are very important.Compared with triangular and tetrahedral meshes,high-quality struc-tured meshes,i.e.semi-regular quad-meshes and hex-meshes,have significant advantages in 3d model representation and physical simulation,which have higher accuracy with less elements.Remeshing technique tries to transform low-quality ill-suited meshes into high-quality suitable ones,which has widespread applications and far-reaching influence in the industrial community.Remeshing technique focuses on the quality controllability of mesh elements and the robust-ness of algorithms.The key difficulty is how to control the global topology structure efficiently and robustly in order to satisfy the requirements for element quality control.The typical compu-tational geometry-based methods use local topology modification and local mesh element adjust-ment,which lack of global topology control and cannot lead to pure quad-and hex-meshes at most time.Such methods lack of controllability.Differential geometry-based methods have a certain global topology control,but when facing complicated boundary/feature alignment,especially for 3D volume models,they usually cannot get inversion-free and non-degenerated parametrizations.To sum up,these methods have many drawbacks in controllability and robustness.Aiming at effects from geometry boundary,topology and metric field of input,this thesis uses differential geometry as the core tool,based on Morse theory,Morse-Smale complex,co-variant derivative,parallel transport,cohomology,Riemannian metric and connection,proposes a series of methods with solid theoretical foundation to control the global topology structure of meshing and improve controllability and robustness.The main contributions are listed as below.· Propose a method to generate periodic 4D vector fields using arbitrary 2D frame fields as input,which combines parametrization and Morse-Smale complex(MSC),and also pro-pose a method to extract quad-meshes from the field robustly.This strategy has advantages of both the efficiency of parametrization and the theoretical guarantee of MSC.Using 2D frame fields as input,i.e.having high controllability,this method gets a 4D vector field which has automatical inserted singularities and satisfies holonomy condition.By such a field,a pure quad mesh with no degenerated element is extracted efficiently,which is strictly aligned with input feature/boundary lines and fits well with the input frame field.This method is more efficient than typical MSC methods.· Propose a method to generate inner singularity-free 3D cross-frame fields,and based on such fields,propose a method to generate closed-form induced polycubes,i.e.general polycubes,which can be used to extract high-quality all-hex meshes.Polycube-based methods are the most robust all-hex meshing techniques,resulting inner singularity-free hex-meshes,so the difficulty about controlling the global singularity structure for valid pure hex-meshes is avoided.The general polycube is corresponding to the full space of the inner singularity-free hex mesh,thus fully broadens the advantage of poly cube-based meth-ods.Based on cohomology theory and considering the degrees of freedom about global topology of input,it extends the exact forms about the typical polycube to closed forms about the general polycube,i.e.locally integrable everywhere.This method also provides better boundary/feature alignment because of using frame fields as guidance.· In order to improve the controllability about 3D remeshing,based on 3D Riemannian metic and 3D connection,through analysing the local integrability,provide a method to generate arbitrary 3D frame fields,which are very useful for 3D remeshing.An arbitrary 3D frame field can be decomposed into a Riemannian metric field and a cross-frame field.3D con-nection is a bridge between the metric field and the cross-frame field,and the smoothness of cross field is got by using covariant derivative under the metric field.This method has two steps:metric field optimization and cross-frame field optimization.At last,combine them to get final frame fields.By using inner-product and parallel this two different princi-ples,user-specified and boundary alignment constraints can be formulated with three kinds of elementary constraints and applied into the first and second step respectively.The theoretical analysis is also useful for any other field about parametrization. |
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