Topology and geometry optimization of grid-shells under self-weight loading

TL;DR

This paper extends convex grid-shell topology and geometry optimization to include the structure's self-weight by embedding a catenary-equal-stress model within a second-order cone programming framework. By introducing variables for horizontal forces and end-vertical forces, and relaxing geometric coupling to a rotated conic constraint, the authors obtain a globally optimal solution for a fixed ground structure while enabling reliable reconstruction of elevations from primal/dual solutions. The approach is validated on diverse examples (barrel vault, square domains with distributed and point loads, holes, and self-intersections), demonstrating that self-weight significantly alters both topology and elevations and that the method achieves substantial speedups over conventional 3D truss formulations. The work provides a practical tool for designing efficient grid-shells under self-weight, with clear guidelines for reconstruction and interpretations when fully stressed conditions fail, and it includes accessible scripts and Grasshopper files for replication.

Abstract

This manuscript presents an approach for simultaneously optimizing the connectivity and elevation of grid-shell structures acting in pure compression (or pure tension) under the combined effects of a prescribed external loading and the design-dependent self-weight of the structure itself. The method derived herein involves solving a second-order cone optimization problem, thereby ensuring convexity and obtaining globally optimal results for a given discretization of the design domain. Several numerical examples are presented, illustrating characteristics of this class of optimal structures. It is found that, as self-weight becomes more significant, both the optimal topology and the optimal elevation profile of the structure change, highlighting the importance of optimizing both topology and geometry simultaneously from the earliest stages of design. It is shown that this approach can obtain solutions with greater accuracy and several orders of magnitude more quickly than a standard 3D layout/truss topology optimization approach.

Figures (20)

• Figure 1: Summary of methodology developed herein, compared to existing approach bolbotowski2022optimal for cases where self-weight can be neglected. Note that the initial setup process is common to both cases, with the key differences being the formulation of the optimization problem and the reconstruction of the optimal elevations. The procedures for when self-weight is negligible are recapped in Section \ref{['sec:vault']}, whilst the new procedure including self-weight is derived in Section \ref{['sec:vaultCat']}.
• Figure 2: Notation for grid-shell topology and geometry optimization without self-weight. The blue lines show elements in the ground structure defined in the horizontal plane. The highlighted element is also shown (in brown) as it would be in the optimal solution, once the end-points are projected to their optimal elevations for the final structure. The force, $\hat{q}$, acting on the nodes is aligned to the final centre-line of the element, i.e. at an angle of $\theta$ to the horizontal.
• Figure 3: Notation for the problem of truss topology optimization with continuous self-weight, a single element shown in elevation in the vertical plane containing the element. The element number subscript $i$ is omitted for clarity. The indicated forces acting on the nodes are aligned to the element centreline at the respective point.
• Figure 4: Notation for catenary elements in the vault formulation. The forces acting on the nodes are as shown. $s$, $q_A$ and $q_B$ are the corresponding optimization variables.
• Figure 5: A single catenary element AB in the catenary vault formulation, including representation of the required 'lumped mass' relaxation. Note that, as in Figure \ref{['fig:singleElement']}, $\bar{q}_A$ is shown with a positive value, whilst $\bar{q}_B$ is shown with a negative value, meanwhile $x$ and $s$ are always restricted to non-negativity.