Mid-Surfaces of Profile-based Freeforms for Mold Design

Mid-surfaces of complex thin objects are commonly used in CAD applications for the analysis of casting and injection molding. However, geometrical representation in CAD typically takes the form of a solid representation rather than a mid-surface; therefore, a process for extracting the mid-surface is essential. Contemporary methods for extracting mid-surfaces are based on numerical computations using offsetting techniques or Voronoi diagram processes where the data is discrete and piecewise linear. These algorithms usually have high computational complexity, and their accuracy is not guaranteed. Furthermore, the geometry and topology of the object are not always preserved. To overcome these problems, this paper proposes a new approach for extracting a mid-surface from a freeform thin object. The proposed method reduces the mid-surface problem into a parametrization problem that is based on a matching technique in which a nonlinear optimization function is defined and solved according to mid-surface criteria. Then, the resulting mid-surface is dictated by a reparametrization process. The algorithm is implemented for freeform ruled, swept, and rotational surfaces, that are commonly used in engineering products. Reducing the problem to the profile curves of these surfaces alleviates the computational complexity of the 3D case and restricts it to a 2D case. Error is controlled globally through an iterative refinement process that utilizes continuous symbolic computations on the parametric representation. The feasibility of the proposed method is demonstrated through several examples.