Optimal development of doubly curved surfaces

Abstract

Surfaces of many engineering structures are commonly fabricated as doubly curved shapes to fulfill functional requirements such as hydrodynamic, aesthetic, or structural. Given a three-dimensional design surface, the first step of the fabrication process is flattening or planar development of this surface into a planar shape so that the manufacturer can not only determine the initial shape of the flat plate but also estimate the strain distribution required to form the shape. In this thesis, we develop an algorithm for optimal development of a general doubly curved surface in the sense that the strain from the surface to its planar development is minimized. A planar development corresponding to minimum stretching or shrinkage is highly desirable for the following reasons: (1) it saves material; (2) it reduces the work needed to form the planar shape to the doubly curved design surface. The development process is modeled by tensile strains isoparametric directions, or along principal curvature directions from the curved surface to its planar development. The distribution of the appropriate minimum strain field is obtained by solving a constrained nonlinear programming problem. Based on the strain distribution and the coefficients of the first fundamental form of the curved surface, another unconstrained nonlinear programming problem is solved to obtain the optimal developed planar shape. Convergence, complexity, and accuracy of the algorithm are studied. Examples show the effectiveness of this algorithm.