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1. Variational Surface Cutting

   Sharp, Nicholas; Crane, Keenan (July 2018, ACM transactions on graphics)

   This paper develops a global variational approach to cutting curved surfaces so that they can be flattened into the plane with low metric distortion. Such cuts are a critical component in a variety of algorithms that seek to parameterize surfaces over flat domains, or fabricate structures from flat materials. Rather than evaluate the quality of a cut solely based on properties of the curve itself (e.g., its length or curvature), we formulate a flow that directly optimizes the distortion induced by cutting and flattening. Notably, we do not have to explicitly parameterize the surface in order to evaluate the cost of a cut, but can instead integrate a simple evolution equation defined on the cut curve itself. We arrive at this flow via a novel application of shape derivatives to the Yamabe equation from conformal geometry. We then develop an Eulerian numerical integrator on triangulated surfaces, which does not restrict cuts to mesh edges and can incorporate user-defined data such as importance or occlusion. The resulting cut curves can be used to drive distortion to arbitrarily low levels, and have a very different character from cuts obtained via purely discrete formulations. We briefly explore potential applications to computational design, as well as connections to space filling curves and the problem of uniform heat distribution. 

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2. Y-Shaped Cutting for the Systematic Characterization of Cutting and Tearing

   https://doi.org/10.1007/s11340-019-00476-5  

   Zhang, B; Shiang, C.-S.; Yang, S. J.; Hutchens, S. B. (January 2019, Experimental mechanics)

   Though they share the similarity of inducing material failure at a crack tip, the cutting and tearing energies of soft materials cannot be quantitatively related to one another. One of the reasons for this lack of understanding comes from additional complications that arise during standard cutting techniques. Decades ago, Lake and Yeoh [Int. J. of Fracture, 1978] described a natural rubber cutting method that uses a `Y-shaped' sample geometry to mitigate several of these challenges, including minimizing friction and controlling the strain energy available to drive fracture. The latter, understood via a fracture mechanics framework, enables relative tuning between a tearing contribution to the cutting energy and a cutting contribution. In this manuscript, we extend Lake and Yeoh's largely unreplicated results to softer, more highly deformable polydimethylsiloxane (PDMS) materials. The range of applicability of this technique to variations in material response, sample geometry, boundary conditions, and cutting rate is large. We utilize this flexibility to describe factors leading to the onset of a material-dependent, stick-slip cutting response, which occurs at low cutting rates and high tearing contributions. Furthermore, variation in cutting blade radius reveals a minimum cutting energy threshold even for blades with radii on the order of a few tens of nanometers. For blunter blades, cutting energy reflects the effects of material strain-stiffening. These results establish the Y-shaped cutting geometry as a useful tool in the study of soft fracture. 

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3. Verifying Cake-Cutting, Faster

   Bertram, Noah; Lai, Tean; Hsu, Justin (July 2024, International Conference on Computer Aided Verification)

   Envy-free cake-cutting protocols procedurally divide an infinitely divisible good among a set of agents so that no agent prefers another's allocation to their own. These protocols are highly complex and difficult to prove correct. Recently, Bertram, Levinson, and Hsu introduced a language called Slice for describing and verifying cake-cutting protocols. Slice programs can be translated to formulas encoding envy-freeness, which are solved by SMT. While Slice works well on smaller protocols, it has difficulty scaling to more complex cake-cutting protocols. We improve Slice in two ways. First, we show any protocol execution in Slice can be replicated using piecewise uniform valuations. We then reduce Slice's constraint formulas to formulas within the theory of linear real arithmetic, showing that verifying envy-freeness is efficiently decidable. Second, we design and implement a linear type system which enforces that no two agents receive the same part of the good. We implement our methods and verify a range of challenging examples, including the first nontrivial four-agent protocol. 

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4. Cutting holes in bistable folds

   https://doi.org/10.1016/j.mechrescom.2021.103700  

   Yu, Tian; Andrade-Silva, Ignacio; Dias, Marcelo A.; Hanna, J.A. (May 2021, Mechanics Research Communications)
5. The long-term case for partial-cutting over clear-cutting in the southern Appalachians USA

   https://doi.org/10.1007/s11056-019-09731-y  

   Elliott, Katherine J.; Miniat, Chelcy F.; Medenblik, Andrea S. (March 2020, New Forests)

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