Structural form finding using the Stress Density Method: Well-posedness and convergence of numerical solutions

Abstract

In this study, we investigate computational form finding of funicular arches, membrane shells, and hybrid compositions of these two types using the Stress Density Method, formulated as coupled systems of linear ordinary, partial or mixed differential equations. These problems demand numerical solution techniques and discretization of the problem. We propose an extension of the Natural Force Density Method into a unified discretization scheme to transform all types of problems into a discrete formalism of the classical Force Density Method. A crucial question of numerical solutions is convergence upon mesh refinement. Three common reasons for the lack of convergence, and the emergence of irrelevant solutions are reviewed, and possible workarounds are proposed. Well-posedness of the continuous problem depends on the prescribed 2nd Piola–Kirchhoff stress function, affecting the canonical form of the differential equations, and the appropriate types of boundary conditions. In the case of hyperbolic stress states, a generalization of the Force Density Method is proposed in order to handle boundary conditions ensuring well-posedness. We uncover formal analogy with initial value problems of evolution equations with a time-like variable, corresponding to an unusual design strategy in which the positions of some free edges of a shell are prescribed by the designer, whereas the positions of some supports are not. The importance of mesh selection for the stability of some numerical computation schemes is also highlighted. Additionally, models for membranes with singular Cauchy stress fields due to point forces are studied, and various solution methods are presented to obtain relevant structural shapes in this case.