Complex Surface Optimization Expand Research

This paper deals with the optimal development of curved surfaces in 3-dimesion Euclidean space. The major works of this paper include the following aspects:1. A review of the researches on curved surfaces development is presented. The development methods are classified based on the strategies adopted.2. The bending status of curved surfaces is analyzed with differential geometrical theory and the concept curve of a surface is defined. As a result, the essential features of a developable surface are presented. A new method that can be used to analyze the developability of a curved surfaces' mesh is proposed based on Gaussian theory of mapping a curved surface onto a unit global surface.3. A new method is proposed to homo-topologically map a triangular finite element mesh into a planar shape based on the assumption that the surface deformation is continuous and homogeneous. With this method, developable surfaces and non-developable surfaces can be developed in a unique way. The developable surfaces can be developed submitted to certain error, and the non-developable surfaces developed homo-topologically.4. The deformation distributed in the developed surface is analyzed, and a Lagrange deformation displacement is proposed to describe the deformation of an element. Then, an ideal elastic stress-strain relation model is debated to buildup the equivalent load from the development deformation. The potential energy fonctionelle is built with the structural displacements introduced. The optimization equations are built by making the variation of the potential energy fonctionelle be zero.5. The convergence and complexity of all the algorithms proposed in the paper are debated. Examples are given to show the effectiveness of the algorithms.