Structural design is a search for the best trade-off between multiple architecture, engineering, and construction objectives, not only mechanical efficiency or construction rationality. Producing hybrid designs from single-objective optimal designs to explore multi-objective trade-offs is common in the design of structural forms, constrained to one parametric design space. However, producing topological hybrids offers a more complex challenge, as a combinatorial problem that is not encoded as a finite set of real numbers but as an unbonded series of grammar rules. This paper presents a strategy for the generation of hybrid designs of quad-mesh pattern topologies for surface structures. Based on a quad-mesh grammar, an algebra is introduced to measure the distance between designs, find their similar features, and enumerate designs with different degrees of topological similarity. Structural design applications are shown to highlight the use of topologically hybrid designs as a surrogate for obtaining multi-objective trade-offs.

The rigid-block equilibrium (RBE) method uses a penalty formulation to measure structural infeasibility or to guide the design of stable discrete-element assemblies from unstable geometry. However, RBE is a purely force-based formulation, and it incorrectly describes stability when complex interface geometries are involved. To overcome this issue, this paper introduces the coupled rigid-block analysis (CRA) method, a more robust approach building upon RBE’s strengths. The CRA method combines equilibrium and kinematics in a penalty formulation in a nonlinear programming problem. An extensive benchmark campaign is used to show how CRA enables accurate modelling of complex three-dimensional discrete-element assemblies formed by rigid blocks. In addition, an interactive stability-aware design process to guide user design towards structurally-sound assemblies is proposed. Finally, the potential of our method for real-world problems are demonstrated by designing complex and scaffolding-free physical models.

The patterns of many structural systems must fulfil a property of two-colourability to partition their elements into two groups. Such examples include top versus bottom layers of continuous beams in elastic gridshells, corrugated versus non-corrugated directions in corrugated shells or warp versus weft threads in woven structures. Complying with such constraints does not depend on the geometry but on the topology of the structure, and, more specifically, on its singularities. This paper presents a search strategy to obtain patterns that fulfil this topological requirement, which represent only a fraction of the general design space. Based on an algebra for the exploration of the topology of quad meshes, including a grammar and a distance, a topology-finding algorithm is proposed to find the closest two-colour quad-mesh patterns from an input quad-mesh pattern. This approach is expressed as the projection to the two-colourable subspace of the design space. The distance underlying the definition of the projection measures the similarity between designs as the minimum number of topological grammar rules to apply to modify one design into another. A design application illustrates how two-colour topology finding can complement workflows for the exploration of structural patterns with singularities informed by the system’s topological requirements.

This paper presents a method that employs parametric surfaces as surface geometry representations at any stage of a computational process to compute self-supporting surfaces. This approach can be differentiated from existing relevant methods because such methods represent surfaces by a triangulated mesh surface or a network consisting of lines. The proposed method is based on the theory of Airy stress functions. Although some existing methods are also based on this theory, they apply its discrete version to discrete geometries. The proposed method simultaneously applies the theory to parametric surfaces directly and the discrete theory to the edges of parametric patches. The discontinuous boundary between continuous patches naturally corresponds to ribs seen in traditional vault masonry buildings. We use nonuniform rational B-spline surfaces in this study; however, the basic idea can be applied to other parametric surfaces. A variety of self-supporting surfaces obtained by the proposed computational scheme is presented.

Self-supporting structures are prominent in historical and contemporary architecture due to advantageous structural properties and efficient use of material. Computer graphics research has recently contributed new design tools that allow creating and interactively exploring self-supporting freeform designs. However, the physical construction of such freeform structures remains challenging, even on small scales. Current construction processes require extensive formwork during assembly, which quickly leads to prohibitively high construction costs for realizations on a building scale. This greatly limits the practical impact of the existing freeform design tools. We propose to replace the commonly used dense formwork with a sparse set of temporary chains. Our method enables gradual construction of the masonry model in stable sections and drastically reduces the material requirements and construction costs. We analyze the input using a variational method to find stable sections, and devise a computationally tractable divide-and-conquer strategy for the combinatorial problem of finding an optimal construction sequence. We validate our method on 3D printed models, demonstrate an application to the restoration of historical models, and create designs of recreational, collaborative self-supporting puzzles.

We present a complete design pipeline that allows non-expert users to design and analyze masonry structures without any structural knowledge. We optimize the force layouts both geometrically and topologically, finding a self-supported structure that is as close as possible to a given target surface. The generated structures are tessellated into hexagonal blocks with a pattern that prevents sliding failure. The models can be used in physically plausible virtual environments or 3D printed and assembled without reinforcements.