Variably-curved, meshed thin shells: Geometry, mechanics, and fabrication

Engineered designs of complex sculptured surfaces are often illustrated with cross-section curves called loft lines. These lines, and their orthogonal trajectories, form curvilinear meshes that enhance the perception of shape. In the past, such lines for a design were obtained by using thin wooden slats or splines that were passed through points on a large floor. Now, complex sculptured surfaces are displayed with computer-aided-design (CAD) parametric meshes. These are computationally efficient Bezier and NURBS curves and surfaces whose lines are arranged to convey a shape by the manner the lines lay on a surface - how the lines twist and curve, and how they intersect. Surface lines, however, can also be arranged for other purposes, such as providing optimal cutting paths for machining.; A new manufacturing process is developed that makes use of a special and uniquely arranged set of network lines on a surface we call a "geodesic net". The lines are geodesic lines that are orthogonally arrayed from two orthogonal geodesic axes. They are used as paths for spline-like material strips to follow, and connect together where the lines cross. These strips form thin-meshed surfaces, which we call "geodesic thin shells". They can be used for a variety of purposes such as design prototypes, or when covered with material, as solid thin shells. These can serve as automobile and aircraft body panels, as marine hulls and receiving dishes, and perhaps as curvilinear civil structures. They can also be used as manufacturing molds themselves, or as curvilinear "preforms" for composite fabrication. Differential geometry, structural mechanics, and manufacturing automation are investigated to develop the process.; Differential geometry explains how these geodesic net lines behave on curved surfaces, and how the geodesic net system cannot be parameterized since its Lie derivative is non-zero. Accordingly, the Riemannian metric cannot be established, so a numerical mapping system that induces the Euclidean metric is developed to extract the metric data. Structural mechanics is then investigated to design the material strips so they follow the geodesic net lines and so the developed strain energy from twisting and bending, which equally apportions among the strips, bends the mesh into its correct shape. Lastly, manufacturing automation is investigated for ways to fabricate single, one-of-a-kind, thin-shell structures directly from a computer-aided-design (CAD) surface.