Design and structural optimization of topological interlocking assemblies

Abstract

We study assemblies of convex rigid blocks regularly arranged to approximate a given freeform surface. Our designs rely solely on the geometric arrangement of blocks to form a stable assembly, neither requiring explicit connectors or complex joints, nor relying on friction between blocks. The convexity of the blocks simplifies fabrication, as they can be easily cut from different materials such as stone, wood, or foam. However, designing stable assemblies is challenging, since adjacent pairs of blocks are restricted in their relative motion only in the direction orthogonal to a single common planar interface surface. We show that despite this weak interaction, structurally stable, and in some cases, globally interlocking assemblies can be found for a variety of freeform designs. Our optimization algorithm is based on a theoretical link between static equilibrium conditions and a geometric, global interlocking property of the assembly---that an assembly is globally interlocking if and only if the equilibrium conditions are satisfied for arbitrary external forces and torques. Inspired by this connection, we define a measure of stability that spans from single-load equilibrium to global interlocking, motivated by tilt analysis experiments used in structural engineering. We use this measure to optimize the geometry of blocks to achieve a static equilibrium for a maximal cone of directions, as opposed to considering only self-load scenarios with a single gravity direction. In the limit, this optimization can achieve globally interlocking structures. We show how different geometric patterns give rise to a variety of design options and validate our results with physical prototypes.

State of the Art on Computational Design of Assemblies with Rigid Parts

“…Designing assemblies in equilibrium: (a) masonry building [WOD09], (b) equilibrium puzzle [FMB15], and (c) free‐form architectural assembly [WSIP19]. …”

“…An assembly is interlocking if it can be in equilibrium under arbitrary external forces when the key is held by other means [WSIP19]. According to the static‐kinematic duality, the interlocking property of a given assembly can be tested by using kinematic methods described in Section 2.3.…”

“…In practice, equilibrium under gravity might be insufficient since an assembly could be exposed to different forces (e.g., live loads). This motivates the stability measure based on the tilt analysis described in Section 2.3, where an assembly can be in equilibrium for a cone of gravity directions, as well as the work of optimizing free‐form architectural assemblies to maximize the stability measure [WSIP19]; see Figure 16(c). On the other hand, global interlocking might impose too strict constraints on the assembly's geometry, as real assemblies usually do not have to experience arbitrary external forces.…”

Free-form structures from topologically interlocking masonries

“…The Upper bound theorem states that the given load factor is superior to the actual load factor. The case of interlocked masonries does not consider an external load, unlike equilibrium-based methods [32], which provide stable solutions for only one load case. The kinematic method is dual to an equilibrium-based approach, which looks for stress fields satisfying equilibrium and resistance constraints.…”

Integrated computational framework for the design and fabrication of bending-active structures made from flat sheet material

“…The complexity originates from the different constraints that shape aesthetics, structural design, and engineering aspects of a desired structure pose. Thus, the paneling of free-form shapes [6,7], the assembly of self-supporting and interlocking structures [8,9,10] or the structural design of shells and grid shells [11,12,13,14,15,16,17] have become non-trivial prob-lems. A new generation of computational design workflows takes advantage of geometry processing methods to support the exploration of the space of solutions that conform to mechanics, fabrication, or budget limitations.…”