Abstract
This paper presents a numerical computation procedure to solve the minimal surface problem based on discrete differential form. Since membrane structures possess structural stiffness by introducing prestress, an initial form is needed to be stable after introducing appropriate tension stress. Therefore, a form-finding for membrane structures is required to decide the initial form. The form-finding is classified into methods based on mechanical description and geometric description. The former methods are the analysis of an equilibrium form with uniform tension stress distribution in a membrane. The latter methods are variational problems to find a form of minimal area with a boundary, making use of the property that the uniform stress surface is equal to the minimal surface. However, It is well known that the numerical computation procedure is unstable in convergence process when three-dimensional coordinate values are set as unkown variables in the minimal surface problem. It is necessary to know the reason of unstability in a computational procedure.
In this paper, we formalize the form-finding analysis based on discrete differential form which provides the simple formulation for cable-reinforced membrane structures in Chapter 3 by Ref. 9). As a result, it reveals that the coefficient matrix of linearized equations for convergence corresponds with the geometrical stiffness matrix as shown in Ref. 10). Numerical results of discrete minimal surface with fixed boundary and with free boundary constraint by cables in Fig. 6-8. These results agree very well with the previous research of minimal surface problem. Next, we compare with the present method and the initial stress method focused on a geometrical stiffness matrix. This result shows that the present method corresponds with the initial stress method without material property and the nonlinear term of strain-displacement matrix in calculating unbalance force affects also the result as shown in Fig. 9.
Finally, we confirm that obtained solutions possess the uniform stress distribution on discrete surfaces by using the geometrically nonlinear analysis by finite element method. In the case of the model with fixed boundary, exact uniform stress distribution is obtained as shown in Table2. In the case of the model with free boundary constraint with cables, the slight error can be found around the central part of cables. However, these results approximate the minimal surface very well, it is verified that present method is useful for the form-finding of membrane structures and cable-reinforced membrane structures. In future, we develop the form-finding analysis with good convergence procedure.
