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Multi-layer spatial structures usually take considerable external loads with a small material usage at all scales. Polyhedral graphic statics (PGS) provides a method to design multi-layer funicular polyhedral structures, and the structural forms are usually materialized as space frames. Our previous research shows that the intrinsic planarity of the polyhedral geometries can be harnessed for efficient fabrication and construction processes using flat-sheet materials. Sheet-based structures are advantageous over conventional space frame systems because sheets can provide more load paths and constrain the kinematic degrees of freedom of the nodes. Therefore, they are more capable of taking a wider variety of load cases compared to space frames. Moreover, sheet materials can be fabricated into complex shapes using CNC milling, laser cutting, water jet cutting, and CNC bending techniques. However, not all sheets are necessary as long as the load paths are preserved and the system does not have kinematic degrees of freedom. To find an efficient set of faces that satisfies the requirements, this paper first incorporates and adapts the matrix analysis method to calculate the kinematic degrees of freedom for sheet-based structures. Then, an iterative algorithm is devised to help find a reduced set of faces with zero kinematic degrees of freedom. To attest to the advantages of this method over bar-node construction, a comparative study is carried out using finite element analysis. The results show that, with the same material usage, the sheet-based system has improved performance than the framework system under a range of loading scenarios. 

Nature has always been the master of design skills to which humans only aspire to, but new approaches bring that aspiration closer to our reach than ever before. Through 4.5 billion years of iterations, nature has shown us its extraordinary craftsmanship, breeding a variety of species whose body structures have gradually evolved to adapt to natural phenomena and make full use of their unique characteristics. The dragonfly wing, among body structure is an extreme example of efficient use of materials and minimal weight while remaining strong enough to withstand the tremendous forces of flight. It has long been the object of scientific research examining its structural advantages to applying their principles to fabricated designs.1 We can imitate its form and create duplicates, but thoroughly understanding the dragonfly wing’s mechanism, behavior and design logic is no trivial task. 

This paper introduces an interactive form-finding technique to design and explore continuous Shellular Funicular Structures in the context of Polyhedral Graphic Statics (PGS). Shellular funicular forms are two-manifold shell-based geometries dividing the space into two interwoven sub-spaces, each of which can be represented by a 3D graph named labyrinth [1]. Both form and force diagrams include labyrinths, and the form finding is achieved by an iterative subdivision of the force diagram across its labyrinths [2]. But this iterative process is computationally very expensive, preventing interactive exploration of various forms for an initial force diagram. The methodology starts with identifying three sets of labyrinth graphs for the initial force diagram and immediately visualizing their form diagrams as smooth and continuous surfaces. Followed by exploring and finalizing the desired form, the force diagram will be subdivided across the desired labyrinth graph to result in a shellular funicular form diagram (Figure 1). The paper concludes by evaluating the mechanical performance of continuous shellular structures compared to their discrete counterparts. 

Special Issue: Machine Hallucinations: Architecture and Artificial Intelligence Nature has always been the master of design skills to which humans only aspire, but new approaches bring that aspiration closer to our reach than ever before. Through 4.5 billion years of iterations, nature has shown us its extraordinary craftsmanship, breeding a variety of species whose body structures have gradually evolved to adapt to natural phenomena and make full use of their unique characteristics. The dragonfly wing, among body structures, is an extreme example of efficient use of materials and minimal weight while remaining strong enough to withstand the tremendous forces of flight. It has long been the object of scientific research examining its structural advantages to apply its principles to fabricated designs.1 We can imitate its form and create duplicates, but thoroughly understanding the dragonfly wing’s mechanism, behavior, and design logic is no trivial task. 

The Auxetic materials are structural systems with a negative Poisson’s ratio. Such materials show unexpected behavior when subjected to uni-axial compression or tension forces. For instance, they expand perpendicular to the direction of an applied compressive force. This behavior is the result of their internal structural geometry. These materials, with their unique behavior, have recently found many applications in the fields of sensors, medical devices, sport wears, and aerospace. Thus, there is a lot of relevant research in the artificial design of auxetic metamaterials and the prediction of their behavior [2]. Since the behavior of these materials heavily relies on the geometry of their internal structure, the geometry-based methods of structural design, known as graphic statics, are very well suited to derive their geometry or describe their behavior. Nevertheless, graphic statics has never been used in the design of such materials. For the first time, this paper provides an introduction to the use of graphic statics in the design and form-finding of auxetic metamaterials. The paper explains multiple equilibrium states of various auxetic structures using algebraic formulations of 2d/3d graphic statics [1, 3]. Moreover, it sheds light on the geometric behavior of auxetic materials by changing the force diagram of graphic statics. Therefore, it suggests a novel approach in predicting the changes in the geometry of the material under various loading conditions by controlling the force equilibrium geometrically. 

This paper introduces a web-based interactive educational platform for 3D/polyhedral graphic statics (PGS) [1]. The Block Research Group (BRG) at ETH Zürich developed a dynamic learning and teaching platform for structural design. This tool is based on traditional graphic statics. It uses interactive 2D drawings to help designers and engineers with all skill levels to understand and utilize the methods [2]. However, polyhedral graphic statics is not easy to learn because of its characteristics in three-dimensional. All the existing computational design tools are heavily dependent on the modeling software such as Rhino or the Python-based computational framework like Compass [3]. In this research, we start with the procedural approach, developing libraries using JavaScript, Three.js, and WebGL to facilitate the construction by making it independent from any software. This framework is developed based on the mathematical and computational algorithms deriving the global equilibrium of the structure, optimizing the balanced relationship between the external magnitudes and the internal forces, visualizing the dynamic reciprocal polyhedral diagrams with corresponding topological data. This instant open-source application and the visualization interface provide a more operative platform for students, educators, practicers, and designers in an interactive manner, allowing them to learn not only the topological relationship but also to deepen their knowledge and understanding of structures in the steps for the construction of the form and force diagrams and analyze it. In the simplified single-node example, the multi-step geometric procedures intuitively illustrate 3D structural reciprocity concepts. With the intuitive control panel, the user can move the constraint point’s location through the inserted gumball function, the force direction of the form diagram will be dynamically changed from compression-only to tension and compression combined. Users can also explore and design innovative, efficient spatial structures with changeable boundary conditions and constraints through real-time manipulating both force distribution and geometric form, such as adding the number of supports or subdividing the global equilibrium in the force diagram. Eventually, there is an option to export the satisfying geometry as a suitable format to share with other fabrication tools. As the online educational environment with different types of geometric examples, it is valuable to use graphical approaches to teach the structural form in an exploratory manner. 

This research investigates the design of structurally performant, lightweight architectural elements produced through concrete 3D printing (C3DP). Traditionally, concrete requires dense and sturdy formwork, whose production adds significantly to the total cost and results in massive and heavy parts after demolding. C3DP offers the unique opportunity to both eliminate the need for formwork and to create lighter parts by introducing internal voids and cavities. The advent of additive manufacturing in a broad range of scales, materials, industries, and applications, led to increased interest and intense research into different types of porous structures, their geometry, and structural performance under various boundary conditions. Precise control over the sparse distribution of material allows not only for parts with similar strength at reduced mass but even for modifications of mechanical properties, like turning brittle materials into elastic or shock-absorbent ones. While with powder-based additive manufacturing processes like metal 3D printing, truss-based lattices have become very popular for the light-weighting of parts or to provide tissue growth scaffolds for medical implants, their geometry – a sparse space frame resulting in numerous individual contour islands and accentuated overhangs – cannot as easily be produced by C3DP, which is based on a continuous material extrusion. Alternative types of micro-structures, so-called triply periodic minimal surfaces (TPMS), are better suited for this process as they are, as their name suggests, consisting of one continuous surface dividing space into two separate but interwoven subspaces. TPMS are therefore very popular for the efficient design of heat exchangers. We develop and present a continuous and integrated workflow, in which the architectural elements and their structural requirements are designed through transitioning back and forth between the force and the form diagram using 3D graphic statics [1]. The members and their topology from the abstract graph of the conceptual form diagram are seamlessly connected to the volumetric modeling (VM) framework, responsible for the definition of the part geometry [2]. VM represents form assigned distance functions (SDF) and can easily handle complex topologies and flawless Boolean operations of not only the outer shell geometry but also the internal micro-structural infill patterns (Fig. 1, a). In an iterative feedback loop, the infill can be further optimized to leave the material only along certain internal stress trajectories (force flows). This functional grading controlling the relative density is done based on the FE analysis results. The stress distribution is thereby defined as a three-dimensional field (Fig. 1, b). Its values can factor into the SDF equation and be used to modify the wavelength (periodicity) of the TPMS, the local thickness of the surface shell, the solid to void fraction by shifting the threshold iso-value or even the alignment and orientation of the unit cells (Fig. 1, c). They can be arranged in an orthogonal, polar- or even spherical coordinate system to optimally adapt to structural necessities. The TPMS pattern can also gradually transition from one type into another type along the gradient of a spatial function. 

This research investigates the use of graphic statics in analyzing the structural geometry of a natural phenomenon to understand their performance and their relevant design parameters. Nature has always been inspiring for designers, engineers, and scientists. Structural systems in nature are constantly evolving to optimize themselves with their boundary conditions and the applied loads. Such phenomena follow certain design rules that are quite challenging for humans to formulate or even comprehend. A dragonfly wing is an instance of a high-performance, lightweight structure that has intrigued many researchers to investigate its geometry and its performance as one of the most light-weight structures designed by nature [1]. There are extensive geometrical and analytical studies on the pattern of the wing, but the driving design logic is not clear. The geometry of the internal members of the dragonfly wings mainly consists of convex cells which may, in turn, represent a compression-only network on a 2D plane. However, this phenomenon has never been geometrically analyzed from this perspective to confirm this hypothesis. In this research, we use the methods of 2D graphic statics to construct the force diagram from the given structural geometry of the wing. We use algebraic and geometric graphic statics to unfold the topological and geometric properties of the form and force diagrams such as the degrees of indeterminacies of the network [2]. We then reconstruct the compression-only network of the wing for more than 300 cases for the same boundary conditions and the edge lengths of the independent edges of the network. Comparing the magnitude of the internal forces of the reconstructed network with the actual structure of the wing using the edge length of the force diagram will shed light on the performance of the structure. Multiple analytical studies will be provided to compare the results in both synthetic and natural networks and drive solid conclusions. The success in predicting the internal force flow in the natural structural pattern using graphic statics will expand the use of these powerful methods in reproducing the exact geometry of the natural structural system for use in many engineering and scientific problems. It will also ultimately help us understand the design parameters and boundary conditions for which nature produces its masterpieces.