Automated Discretization of Digital Curves through Local or Global Constrained Optimization
McLaurin, D. (2012). Automated Discretization of Digital Curves through Local or Global Constrained Optimization. Physics-Based Modeling in Design & Development (NDIA 3170). Denver, CO.
Computational design and analysis has become a fundamental part of research for development and manufacturing. In general, the accuracy of a computational analysis depends heavily on the fidelity of the computational representation of a real-world object or phenomenon. However, the task of creating high fidelity models of actual geometries can be time-consuming. If geometry repair (gluing, trimming, defeaturing, etc…) is not considered, the amount of user input required to generate a volume grid can be concentrated on the lowest levels of the grid generation hierarchy—i.e. manually setting point spacing values at the end points of curves. A reduction of the overall time required for creating a computational model can be accomplished by accelerating the process of mesh generation. Shorter turn-around times for generating solutions directly addresses the conference objective: “…to reduce acquisition time, cost and risk and improve system performance” by addressing the call for methods “…for generating weapon system geometries and meshes for design and analysis.”
The development of the proposed methods is justified by the need for an automated way of generating or refining edge grids. This process can only be automated if some way of judging “how well” an edge grid represents a curve is present. To this end, a detailed discussion of the classification and calculation of the various types of discretization error associated with discrete representations of digital curves is presented.
Edge grid generation via arc-length-deficit minimization is presented in two ways: a global optimization approach where the number of points is included implicitly in the definition of the optimization problem; a local optimization approach which uses a “divide-and-conquer” strategy to refine the edge grid. Arc-length-deficit is defined here as the combined length of the segments that define the discretization subtracted from the arc length of the digital curve. The optimization function using global method seeks to place n number of points on the interior of a curve such that the arc length is maximized. For the local method the optimization function seeks to locally refine the edge grid based on maximizing the local change in arc length for a refinement step. The relative advantages and disadvantages of each method are discussed with respect to each other and relevant literature.
Error bounds for this method are rigorously developed and proven. Further discussion of element quality, robustness, and a framework for implementing the information associated with an optimal edge grid into an existing grid generator is also presented. Results using both of the proposed methods for generating optimal edge grids will be presented—as well as the corresponding surface grids. Discussion of where the presented methods are most appropriate will also be given. Convergence performance will be compared between the two methods with respect to a desired tolerance. Estimates for time saved by using the presented methods will be discussed for the example geometries.