A tool for form finding using a constrained forced density method

Funicular analysis is widely adopted to cope with the design of arcuated structures. Following this approach, spatial structures such as three-dimensional trusses and shells can be modelled as statically indeterminate networks of vertices and edges of given topology, with prescribed nodal loads and restraints. The equilibrium of funicular networks can be conveniently handled through the force density method, i.e. writing the problem in terms of the ratio of force to length in each branch of the network. As investigated in the literature for the case of vertical loads, independent sets of branches can be detected for networks with fixed plan geometry. However, enforcing the nodes to lie within a prescribed range of heights (the design domain) is not a trivial task from a numerical point of view. To this goal, a multi-constrained minimization problem has been formulated to enforce bounds for the vertical coordinates of the vertices of the network. Both the independent force densities and the coordinates of the restrained nodes are used as unknowns, whereas suitable norms of the horizontal thrusts can be adopted as objective function. Constraints on the sign / magnitude of the force density in each branch of the network may be accounted for, as well. Due to its peculiar form, this problem can be efficiently solved through techniques of sequential convex programming that were originally conceived to handle large scale multi-constrained formulations of size optimization for elastic structures. Self-weight, i.e. a design-depend load case, can be straightforwardly included in the optimization, taking full advantage of the direct analytical method to compute sensitivities. Figure 1 shows a preliminary assessment of the proposed procedure for form-finding, considering a bay whose four corners and the central point are supported in the vertical direction, whereas symmetry conditions are enforced along each external side of the grid. Crosses and circles stand for nodes whose heights match the prescribed upper and lower bounds, respectively. Networks that are fully feasible with respect to the local enforcements on the height of the vertices are retrieved in a limited number of iterations, with no need to initialize the procedure with a feasible starting guess. The algorithm applies to general networks with any type of geometry, loads and restraints.