\section{Research Background}  In the last few decades, the advances in digital fabrication technologies and the use of computational design methods have provided the opportunity to explore the construction of complex building shapes with high accuracy, even constructing a complete structure in one-go by means of 3D printing, as in Contour Crafting \cite{khoshnevis2004automated}. Recently there has been a movement towards mass customization procedures due to the easy access we have to CNC and robotic fabrication \cite{Frick2016}. However, such processes often require sophisticated technologies, which would be typically unavailable/or costly to set up in regions in need of shelters. The Super-Adobe method (see \cite{khalili1999earthquake}) allows for a low-tech construction using in-situ materials. Constructing shelters using this method, however, still requires some on-site training, and also the use of molds. Factors such as cost and skilled workers become an issue that need to be addressed when trying to implement modern technological construction methods \cite{Ramage2010}.Therefore, we explore the possibility of constructing reliable earthy shelters only relying on low-tech construction methods using merely simple hand tools, ideally to be built without large molds.  \section{Problem Statement}  With the added criterion of structurally validating the approximated polyhedral shell structure, the goal is to design a process that generates habitable shelters by using computational design methods in producing various adobe brick geometries given the constraint of using simple low-tech fabrication techniques. We seek to propose a shell construction procedure that is potentially build-able by the local communities seeking to inhabit them. We formulate three problems to solve for designing polyhedral approximated adobe shells.  \begin{enumerate}  \item designing an optimal adobe shell structure: given a manifold roof surface on the ground(to be made a funicular shell), desired an optimal surface shape for a thin shell structure;  \item obtaining a sound polyhedral approximation of the shell: given a surface, desired a topologically correct polyhedral approximation and the finite-element mesh of the polyhedralized structure;  \item finding an optimal shape for a polyhedral brick: given different polyhedral primitives/bricks and their relative geometric lattices, desired the one that results in the best (validated) compression-only structure.  \end{enumerate}  \section{Hypothesis and Proposition}  We propose that an optimal shape for a thin shell structure can be approximated by polyhedral bricks in order to simplify and speed-up construction by local labor forces. The hypothesis is that a non-cubic polyhedral brick (from the family of space filling polyhedrons) might perform better than a cubic/traditional adobe brick; in that, when aggregated, such bricks can potentially better conduct the forces through themselves as illustrated in Figure \ref{fig:Aggregation}. The assumption is that aggregating rectilinear bricks can result either in redundancies or would require expert knowledge and/or intricate scaffolds in complex curvilinear brick laying, especially when dealing with compression only shell structures.  The major challenge in building the proposed polyhedral shells is not only reliably validating the structure of the global shell, but also providing a proposition for how the structure can be stabilized during the construction process without the use of complex scaffolding techniques that might not be available or be difficult to manufacture without sophisticated technologies. In this method, due to the space-filling nature of the proposed polyhedrons, we propose to use the unused bricks as temporary scaffolding during the construction process.
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 \subsection{Topological Polyhedralization}  Given a surface (in this case the surface representing the geometry of the optimized thin shell structure), we want to find a sound/well-connected polyhedral approximation of this shape. We initially take the bounding box of the surface in question, enlarge it with a buffer, no larger than the largest dimension of our bricks, to ensure complete coverage. Then, based on an arbitrary space-filling polyhedron, we construct a full polyhedral tessellation of the space inside the bounding box. From the polyhedrons inside the tessellation, we need to pick and choose the ones required to minimally (and sufficiently/properly) approximate the shape of the surface in question. Based on the ideas of Topological Voxelization, proposed by \cite{lainel} and implemented by \citep{nourian} and by virtue of the Poincaré Duality Theorem [Henry Poincaré 1854-1912] (inspired by \cite{pigot} and \cite{lee}; originally stated in \cite{poincare1904cinquieme}) we propose that the minimally sufficient topological polyhedralization of the surface to be done by intersecting the 1D edges of the candidate polyhedrons with the 2D surface in question. Poincaré Duality theorem states that there exists a pairing between k-dimensional features and their intersecting dual features of dimension $(n-k)$, where $n$ denotes the dimension of the space within which the features are embedded. If any edge of a space-filling polyhedron, from the complete lattice, intersects with the input surface, it means that the polyhedron in question must be included in the output approximation.  In addition, again by using the Poincaré duality theorem, we construct the Finite-Element-Model of the approximate structure based on the adjacency graph of the polyhedral bodies. If two polyhedral 3D bodies have a 2D face in common, that means, by virtue of Poincaré duality, there exists a dual 1D edge, connecting their dual 0D vertexes. Below is a summary of possible dualities in a 3D world, from which we have respectively referred to the cases stated in the last two rows, i.e. the 2D faces of the input surface have dual 1D edges belonging to the space filling polyhedrons, and that the polyhedral 3D bodies have dual 0D vertexes and 1D edges connecting these vertexes dual to the shared 2D faces of the polyhedrons:  \subsubsection{Space Filling Polyhedrons}  A Plesiohedron \cite{plesiohedron} is a space filling polyhedron. Such polyhedrons can be used to generate a volumetric tessellation. The most intuitive and simple space filling geometry that we are all aware of is the cube. Due to its geometry, by aggregating it in every way, as long as the faces of the cubes are aligned, one can represent any form in a space-filling manner. There are other space filling polyhedrons that we seek to investigate as potentially better candidates for brick construction of a doubly curved shell structure. We specifically look at three different space filling polyhedrons with the intention of investigating these geometries as potentially more optimal and flexible brick elements to be applied to shell structures.
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 \subsubsection{Topological Polyhedralization}  After tessellating the space we need to polyhedralize the funicular surface within the tessellation. By intersecting the edges of the polyhedral tessellation with the funicular surface, we can identify the modules that approximate the input surface. Since the funicular surface is the ideal shape to redirect the forces to ground, we need to make sure that the rasterization is topologically and geometrically similar to the original surface.  \subsection{Finite Element Analysis}
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 Based on the polyhedral tessellated lattice mentioned above, a new mesh is generated based on the topology of the tessellated structure in order to more easily test the overall performance in a finite element solver. The newly generated mesh is representative of our bricked surface where each vertex of the mesh is representative of one brick, i.e. one element in the finite element model. The edges of the latices, connecting all the vertexes of the mesh, represent the springs for the finite element analyses. Given that the elements in this case are adobe bricks, the assumption is that the stiffness modulus of these elements are high enough to be considered as a solid \cite{Illampas2011}.  \subsubsection{Generating a Finite Element Model}  We have made a simplified Finite Element Model of our polyhedral model that is a dual graph with finite 1-dimensional elements abstracted as springs representing the mortar between polyhedral bricks. The polyhedral bricks have been idealized and abstracted as rigid bodies, therefore they are reduced to vertexes in this model. The Finite Element Model therefore only contains 0-dimensional vertexes representing the rigid bodies and their weights, and 1-dimensional edges representing spring elements. We have analyzed this model using the toolkit Karamba \cite{preisinger} as a beam model, whose beam/linear elements are modeled as springs. Note that in our idealization there are no 2-dimensional elements, therefore, effectively the Finite Element Model is abstracted as a spatial structure, not a shell structure anymore.  Using the Karamba plug-in for Rhinoceros \& Grasshopper, we used the resultant  finite element model \ref{fig:Karamba1} as an input for our finite element analysis. The results are limited due to the fact that, given the input model, we had to treat the shell analysis as a grid shell. The results indicate that the majority of the structure is in equilibrium with some areas being in compression and only a few areas in tension. This may result in issues due to the fact that the adobe bricks and the lime based mortar that we propose for construction, should be completely in compression. Remedying this issue would involve altering the input mesh to produce a more suitable funicular surfaces that may result in more compression forces rather than tension. Regardless of this, the finite element analysis done in Karamba should be supplemented with more sophisticated finite element analysis software that can take the original brick geometry as an input rather than the abstracted lattice that we have used.
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 \subsubsection{Comparison of FEA Results}  As indicated in Figure \ref{fig:Karamba1}, we have structurally analyzed the lattices of each of the three polyhedral bricks exhibited in Figure \ref{fig:Poincaré}. The colors indicate the different utilization forces applied to the elements; red and blue being the two extremes, and white indicating the elements in equilibrium. On the left we have represented the lattice of each geometry, the middle figures show case the plan view of the structure. Most importantly, on the right, we have visualized all the elements that are considered to be in tension. Therefore, by evaluating the amount of tension elements in each lattice, we can better understand the performance of each polyhedral brick in the global structure. On observation, it seems that the second polyhedron we tested results in the least amount of tension elements. We can conclude from this that the second polyhedron and its accompanying lattice structure results in a shell structure that is mostly in compression, and therefore potentially the most optimal candidate for a adobe brick from the geometries that we have tested. These findings indicate to us that there is merit in further research of these space filling polyhedrons where more optimal geometries might prove to be better candidates for compression only shell structures.  \section{Conclusion and Discussion}  This preliminary model still needs to be tested and validated with actual material properties of the mortar that is to be used in between adobe bricks. The numerical results, therefore, could only be considered as indicative. Although the range of values calculated for compression/tension stresses are unreliable due to the unknown material properties, the model and the results can be used to qualitatively compare different polyhedral tessellations and their effectiveness in reducing tensile stresses. \cmmnt {Please here add some comparative results from different lattices dual to different polyhedral tessellations.}  \section{Limitations and Future Work}  This project is essentially a work in progress. The biggest challenge in this process is to carefully design a manufacturing procedure that is low cost and most importantly, teachable to local communities with no access to technologically sophisticated equipment. Indeed the procedure of creating molds for the polyhedral bricks would be the best option, however a certain level of rigorous quality control needs to be considered to deal with errors and imperfections. It should be stated that the method in this research is not suggesting the ultimate solution to low-tech fabrication of complex geometries, there should be a learning and trial phase to truly evaluate how this methodology might perform under real life circumstances both physically and socially. In addition to this, the structural validation of these shells should be done with multiple finite element tools in order to check whether the results across various platforms are consistent to ensure the safe assemblage of the resultant adobe shells. Moreover, we also need to validate every state throughout the construction process, using the unused adobe bricks to create a temporary scaffolding to support the unfinished shell as it is being assembled. In addition, further research is necessary to re-compute the shell geometries to exclude any and all tension stresses which the current model cannot do entirely. Furthermore, given the prospect of using untrained workers, the revised work flow could include automatically generated construction drawings for laying bricks layer by layer. In any case, these adobe shells can be categorized as shelters for temporary use rather than proper long term dwellings. A factor that needs to be taken into account is the role that the lime-based mortar would play in terms of the overall mass and resultant forces on the shell. The next step would be to investigate whether this is the best solution to bond all the polyhedral adobe bricks together or whether a dry connection system could be designed which could simplify the construction process further.
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