Optimization techniques are widely used in many research and engineering areas. This dissertation presents numerical and geometric optimization methods for solving geometric and solid modeling problems.
Geometric optimization methods are designed for manufacturing process planning, which optimizes the process by changing dependency relationships among geometric primitives from the original design diagram. Geometric primitives are used to represent part features, and dependencies in the dimensions between parts are represented by a topological graph. The ordering of these dependencies can have a significant effect on the tolerance zones in the part. To obtain tolerance zones from the dependencies, the conventional parametric method of tolerance analysis is de-composed into a set of geometric computations, which are combined and cascaded to obtain the tolerance zones in the geometric representations. Geometric optimization is applied to the topological graph in order to find a solution that provides not only an optimal dimensioning scheme but also an optimal plan for manufacturing the physical part. The applications of our method include tolerance analysis, dimension scheme optimization, and process planning.
Two numerical optimization methods are proposed for local and global surface parameterizations. One is the nonlinear optimization, which is used for building the local field-aware parameterization. Given a local chart of the surface, a two-phase method is proposed, which generates a folding-free parameterization while still being aware of the geodesic metric. The parameterization method is applied in a view-dependent 3D painting system, which constitutes a local, adaptive and interactive painting environment. The other is the mixed-integer quadratic optimization, which is used for generating a quad mesh from a given triangular mesh. With a given cross field, the computation of parametric coordinates is formulated to be a mixed-integer optimization problem, which parameterizes the surface with good quality by adding redundant integer variables. The mixed integer system is solved more efficiently by an improved adaptive rounding solver. To obtain the final quadrangular mesh, an isoline tracing method and a breadth-first traversal mesh generation method are proposed so that the final mesh result has face information, which is useful for further model processing
