Temporary bar structures made of reusable standardized components are widely used in construction, events, and exhibitions. They are economical, easy to assemble, and can be disassembled and reused in various structural arrangements for various purposes. However, existing reusable temporary structures are either limited to modular yet repetitive designs or require bespoke components, which restricts their reuse potential. Instead of designing bespoke kit of parts for limited reuse, this paper investigates how to design and build diverse freeform structures from one homogeneous kit of parts. We propose a computational framework to generate multi-tangent bar structures, a widely used jointing system, which allows bars to be joined at any point along their length with standard connectors. We present a mathematical formulation and a numerical scheme to optimize the bar spatial positions and contact assignment simultaneously, while ensuring that the constraints of tangency, collision, joint connectivity, and bar length are satisfied. Together with simulated case studies, we present two physical prototypes that reuse the same kit of parts using an augmented reality-guided assembly workflow. • Computational design of diverse freeform structures from uniform kit of parts. • Modeling of tangency, collision, connectivity, and bar-length constraints. • Simultaneous optimization of geometry and connectors for uniform bars and couplers. • Augmented reality-assisted rapid prefabrication and on-site assembly process. • Two pavilion-scale prototypes demonstrate reusability of one parts kit.

We present a method to approximate free-form surfaces using assemblies of rotational patches for architectural rationalization. Rotational surface patches inherently allow for the simultaneous repetition of multiple building elements along the arc direction. By assembling multiple patches, we can create diverse free-form-like geometries to satisfy broad design intents, while preserving local symmetry to enable cost-effective element fabrication. The main challenge lies in the strict constraint of maintaining local rotational symmetry, while ensuring the final tessellated form is seamless, smooth, and closely resembles the target surface. To address this, we propose a patch layout creation approach that segments the input surface into patches, resembling untrimmed rotational patches within a prescribed error threshold. Additionally, we develop a B-spline-based optimization framework to refine the fitted rotational patches for smooth connections and faithful surface approximation. To facilitate practical architectural applications, we provide a post-processing tool that converts the discrete patch assembly into a seamless, smooth quad mesh composed of locally repeated elements. We demonstrate that our approach is applicable to a variety of free-form surfaces, including those that mimic iconic architectural designs, and can address various practical requirements for a wide range of application scenarios.

We present a reinforcement learning framework for constructing assemblies composed of rigid parts, which are commonly seen in many historical masonry buildings and bridges. Traditional construction methods for such structures often depend on dense scaffolding to stabilize their intermediate assembly steps, making the process both labor-intensive and time-consuming. This work utilizes multiple robots to collaboratively assemble structures, offering temporary support by holding parts in place without additional scaffolding. Precomputing the robotic assembly process to ensure structural stability involves a time-consuming offline process due to the combinatorial nature of its search space. However, the precomputed assembly plans may get disrupted during real-world execution due to unforeseen changes, such as setup modifications or delays in part delivery. Recomputing these plans using traditional offline methods results in significant project delays. Therefore, we propose a reinforcement learning-based approach in which a neural network is trained to efficiently generate alternative assembly plans for a given structure online, enabling adaptation to external changes. To enable effective and efficient training, we introduce three key innovations: a GPU-based stability simulator for parallelizing simulations, a novel curriculum-based training scheme to address sparse rewards during training, and a new graph neural network architecture for efficiently encoding assembly geometry. We validate our approach by training reinforcement learning agents on various assemblies and evaluating their performance on unseen assembly tasks. Furthermore, we demonstrate the effectiveness of our framework in planning multi-robot assembly processes, effectively handling disruptions in both simulation and physical environments.

This paper proposes a method to model masonry shell structures where the shell elements fall into a set of discrete equivalence classes. Such shell structure can reduce the fabrication cost and simplify the physical construction due to reuse of a few template shell elements. Given a freeform surface, our goal is to generate a small set of template shell elements that can be reused to produce a seamless and buildable structure that closely resembles the surface. The major technical challenge in this process is balancing the desire for high reusability of template elements with the need for a seamless and buildable final structure. To address the challenge, we define three error metrics to measure the seamlessness and buildability of shell structures made from discrete equivalence classes and develop a hierarchical cluster-and-optimize approach to generate a small set of template elements that produce a structure closely approximating the surface with low error metrics. We demonstrate the feasibility of our approach on various freeform surfaces and geometric patterns, and validate buildability of our results with four physical prototypes. Code and data of this paper are at https://github.com/Linsanity81/TileableShell.

We present a computational framework for planning the assembly sequence of bespoke frame structures. Frame structures are one of the most commonly used structural systems in modern architecture, providing resistance to gravitational and external loads. Building frame structures requires traversing through several partially built states. If the assembly sequence is planned poorly, these partial assemblies can exhibit substantial deformation due to self-weight, slowing down or jeopardizing the assembly process. Finding a good assembly sequence that minimizes intermediate deformations is an interesting yet challenging combinatorial problem that is usually solved by heuristic search algorithms. In this paper, we propose a new optimization-based approach that models sequence planning using a series of topology optimization problems. Our key insight is that enforcing temporal coherent constraints in the topology optimization can lead to sub-structures with small deformations while staying consistent with each other to form an assembly sequence. We benchmark our algorithm on a large data set and show improvements in both performance and computational time over greedy search algorithms. In addition, we demonstrate that our algorithm can be extended to handle assembly with static or dynamic supports. We further validate our approach by generating a series of results in multiple scales, including a real-world prototype with a mixed reality assistant using our computed sequence and a simulated example demonstrating a multi-robot assembly application.

Interlocking puzzles are intriguing geometric games where the puzzle pieces are held together based on their geometric arrangement, preventing the puzzle from falling apart. High-level-of-difficulty , or simply high-level , interlocking puzzles are a subclass of interlocking puzzles that require multiple moves to take out the first subassembly from the puzzle. Solving a high-level interlocking puzzle is a challenging task since one has to explore many different configurations of the puzzle pieces until reaching a configuration where the first subassembly can be taken out. Designing a high-level interlocking puzzle with a user-specified level of difficulty is even harder since the puzzle pieces have to be interlocking in all the configurations before the first subassembly is taken out. In this paper, we present a computational approach to design high-level interlocking puzzles. The core idea is to represent all possible configurations of an interlocking puzzle as well as transitions among these configurations using a rooted, undirected graph called a disassembly graph and leverage this graph to find a disassembly plan that requires a minimal number of moves to take out the first subassembly from the puzzle. At the design stage, our algorithm iteratively constructs the geometry of each puzzle piece to expand the disassembly graph incrementally, aiming to achieve a user-specified level of difficulty. We show that our approach allows efficient generation of high-level interlocking puzzles of various shape complexities, including new solutions not attainable by state-of-the-art approaches.

Abstract An assembly refers to a collection of parts joined together to achieve a specific form and/or functionality. Designing assemblies is a non‐trivial task as a slight local modification on a part's geometry or its joining method could have a global impact on the structural and/or functional performance of the whole assembly. Assemblies can be classified asstructuresthat transmit force to carry loads andmechanismsthat transfer motion and force to perform mechanical work. In this state‐of‐the‐art report, we focus on computational design of structures with rigid parts, which generally can be formulated as a geometric modeling and optimization problem. We broadly classify existing computational design approaches, mainly from the computer graphics community, according to high‐level design objectives, including fabricability, structural stability, reconfigurability, and tileability. Computational analysis of various aspects of assemblies is an integral component in these design approaches. We review different classes of computational analysis and design methods, discuss their strengths and limitations, make connections among them, and propose possible directions for future research.

We present a computational framework for modeling and optimizing complex assemblies using cone joints. Cone joints are integral joints that generalize traditional single-direction joints such as mortise and tenon joints to support a general cone of directions for assembly. This additional motion flexibility not just reduces the risk of deadlocking for complex joint arrangements, but also simplifies the assembly process, in particular for automatic assembly by robots. On the other hand, compared to planar contacts, cone joints restrict relative part movement for improved structural stability. Cone joints can be realized in the form of curved contacts between associated parts, which have demonstrated good mechanical properties such as reduced stress concentration. To find the best trade-off between assemblability and stability, we propose an optimization approach that first determines the optimal motion cone for each part contact and subsequently derives a geometric realization of each joint to match this motion cone. We demonstrate that our approach can optimize cone joints for assemblies with a variety of geometric forms, and highlight several application examples.

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.

Interlocking assemblies have a long history in the design of puzzles, furniture, architecture, and other complex geometric structures. The key defining property of interlocking assemblies is that all component parts are immobilized by their geometric arrangement, preventing the assembly from falling apart. Computer graphics research has recently contributed design tools that allow creating new interlocking assemblies. However, these tools focus on specific kinds of assemblies and explore only a limited space of interlocking configurations, which restricts their applicability for design. In this paper, we propose a new general framework for designing interlocking assemblies. The core idea is to represent part relationships with a family of base Directional Blocking Graphs and leverage efficient graph analysis tools to compute an interlocking arrangement of parts. This avoids the exponential complexity of brute-force search. Our algorithm iteratively constructs the geometry of assembly components, taking advantage of all existing blocking relations for constructing successive parts. As a result, our approach supports a wider range of assembly forms compared to previous methods and provides significantly more design flexibility. We show that our framework facilitates efficient design of complex interlocking assemblies, including new solutions that cannot be achieved by state of the art approaches.

This paper presents CofiFab, a coarse-to-fine 3D fabrication solution, combining 3D printing and 2D laser cutting for cost-effective fabrication of large objects at lower cost and higher speed. Our key approach is to first build coarse internal base structures within the given 3D object using laser cutting, and then attach thin 3D-printed parts, as an external shell, onto the base to recover the fine surface details. CofiFab achieves this with three novel algorithmic components. First, we formulate an optimization model to compute fabricatable polyhedrons of maximized volume, as the geometry of the internal base. Second, we devise a new interlocking scheme to tightly connect the laser-cut parts into a strong internal base, by iteratively building a network of nonorthogonal joints and interlocking parts around polyhedral corners. Lastly, we optimize the partitioning of the external object shell into 3D-printable parts, while saving support material and avoiding overhangs. Besides cost saving, these components also consider aesthetics, stability and balancing. Hence, CofiFab can efficiently produce large objects by assembly. To evaluate CofiFab, we fabricate objects of varying shapes and sizes, and show that CofiFab can significantly outperform previous methods.