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Weighted triangulations for geometry processing
"... In this paper, we investigate the use of weighted triangulations as discrete, augmented approximations of surfaces for digital geometry processing. By incorporating a scalar weight per mesh vertex, we introduce a new notion of discrete metric that defines an orthogonal dual structure for arbitrary t ..."
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In this paper, we investigate the use of weighted triangulations as discrete, augmented approximations of surfaces for digital geometry processing. By incorporating a scalar weight per mesh vertex, we introduce a new notion of discrete metric that defines an orthogonal dual structure for arbitrary triangle meshes and thus extends weighted Delaunay triangulations to surface meshes. We also present alternative characterizations of this primaldual structure (through combinations of angles, areas, and lengths) and, in the process, uncover closedform expressions of mesh energies that were previously known in implicit form only. Finally, we demonstrate how weighted triangulations provide a faster and more robust approach to a series of geometry processing applications, including the generation of wellcentered meshes, selfsupporting surfaces, and sphere packing.
Assembling selfsupporting structures
 ACM Transactions on Graphics
, 2014
"... Figure 1: We propose a construction method for selfsupporting structures that uses chains, instead of a dense formwork, to support the blocks during the intermediate construction stages. Our algorithm finds a workminimizing sequence that guides the construction of the structure, indicating which c ..."
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Figure 1: We propose a construction method for selfsupporting structures that uses chains, instead of a dense formwork, to support the blocks during the intermediate construction stages. Our algorithm finds a workminimizing sequence that guides the construction of the structure, indicating which chains are necessary to guarantee stability at each step. From left to right: a selfsupporting structure, an intermediate construction stage with dense formwork, an intermediate construction stage with our method and the assembled model. Selfsupporting 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 selfsupporting 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 divideandconquer 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 selfsupporting puzzles.
Wire Mesh Design
"... Figure 1: Wire mesh design allows creating physical realizations (1st and 5th images) of a given design surface (2nd and 4th images) composed of interwoven material (middle image) in an interactive, optimizationsupported design process. Both the torso and the Igea face are constructed from a single ..."
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Figure 1: Wire mesh design allows creating physical realizations (1st and 5th images) of a given design surface (2nd and 4th images) composed of interwoven material (middle image) in an interactive, optimizationsupported design process. Both the torso and the Igea face are constructed from a single sheet of regular wire mesh. We present a computational approach for designing wire meshes, i.e., freeform surfaces composed of woven wires arranged in a regular grid. To facilitate shape exploration, we map material properties of wire meshes to the geometric model of Chebyshev nets. This abstraction is exploited to build an efficient optimization scheme. While the theory of Chebyshev nets suggests a highly constrained design space, we show that allowing controlled deviations from the underlying surface provides a rich shape space for design exploration. Our algorithm balances globally coupled material constraints with aesthetic and geometric design objectives that can be specified by the user in an interactive design session. In addition to sculptural art, wire meshes represent an innovative medium for industrial applications including composite materials and architectural façades. We demonstrate the effectiveness of our approach using a variety of digital and physical prototypes with a level of shape complexity unobtainable using previous methods.
Statics aware grid shells
 Computer Graphics Forum
, 2015
"... b c Figure 1: We perform a FEM static analysis of the input surface to obtain a stress field, from which we derive a double orthogonal line field (a), an anisotropy scalar field (b) and a density scalar field (c). Then we build a polygonal tessellation whose elements are sized and aligned according ..."
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b c Figure 1: We perform a FEM static analysis of the input surface to obtain a stress field, from which we derive a double orthogonal line field (a), an anisotropy scalar field (b) and a density scalar field (c). Then we build a polygonal tessellation whose elements are sized and aligned according to the stress tensor field; this tessellation is optimized for symmetry and regularity of faces. A small scale model is fabricated to validate the result with load tests. We introduce a framework for the generation of polygonal gridshell architectural structures, whose topology is designed in order to excel in static performances. We start from the analysis of stress on the input surface and we use the resulting tensor field to induce an anisotropic nonEuclidean metric over it. This metric is derived by studying the relation between the stress tensor over a continuous shell and the optimal shape of polygons in a corresponding gridshell. Polygonal meshes with uniform density and isotropic cells under this metric exhibit variable density and anisotropy in Euclidean space, thus achieving a better distribution of the strain energy over their elements. Meshes are further optimized taking into account symmetry and regularity of cells to improve aesthetics. We experiment with quad meshes and hexdominant meshes, demonstrating that our gridshells achieve better static performances than stateoftheart gridshells.
Reciprocal diagrams
"... ∗ • Threedimensional extension of graphic statics using polyhedral form and force diagrams. • Defining the topological and geometrical relationships of 3D reciprocal diagrams. • Design of compression and tensiononly spatial structures with externally applied loads. • Designing complex funicular sp ..."
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∗ • Threedimensional extension of graphic statics using polyhedral form and force diagrams. • Defining the topological and geometrical relationships of 3D reciprocal diagrams. • Design of compression and tensiononly spatial structures with externally applied loads. • Designing complex funicular spatial forms by aggregating convex force polyhedral cells. • CAD implementation to manipulate the geometry of the force and explore its effects on the forms. a r t i c l e i n f o Article history:
Anisotropic Simplicial Meshing Using Local Convex Functions
"... Figure 1: Anisotropic meshes generated by our method. Left: 2D meshing. The anisotropic metric is defined as the Hessian of an analytic function evincing a large range of anisotropy ratios (λ ∈ [1.9, 394.4]). Zoom in on the image to see meshing details. Middle: 3D surface meshing of the Happy Buddha ..."
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Figure 1: Anisotropic meshes generated by our method. Left: 2D meshing. The anisotropic metric is defined as the Hessian of an analytic function evincing a large range of anisotropy ratios (λ ∈ [1.9, 394.4]). Zoom in on the image to see meshing details. Middle: 3D surface meshing of the Happy Buddha from curvature tensors estimated from a highresolution reference mesh (115474 vertices). Our relaxation produces a high quality result (63284 vertices) starting with an initial lowresolution mesh (5000 vertices). Right: volumetric meshing in a 3D cube. Anisotropy changes substantially (λ ∈ [1, 40]) and rapidly over the domain. The lower image shows a crosssection. We present a novel method to generate highquality simplicial meshes with specified anisotropy. Given a surface or volumetric domain equipped with a Riemannian metric that encodes the desired anisotropy, we transform the problem to one of functional approximation. We construct a convex function over each mesh simplex whose Hessian locally matches the Riemannian metric, and iteratively adapt vertex positions and mesh connectivity to minimize the difference between the target convex functions and their piecewiselinear interpolation over the mesh. Our method generalizes optimal Delaunay triangulation and leads to a simple and efficient algorithm. We demonstrate its quality and speed compared to stateoftheart methods on a variety of domains and metrics.
Efficient Construction and Simplification of Delaunay Meshes YongJin Liu∗
"... Figure 1: We present an efficient algorithm to convert an arbitrary manifold triangle meshM to a Delaunay mesh (DM), which has the same geometry of M. Our algorithm can also produce progressive Delaunay meshes, allowing a smooth choice of detail levels. Since DMs are represented using conventional m ..."
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Figure 1: We present an efficient algorithm to convert an arbitrary manifold triangle meshM to a Delaunay mesh (DM), which has the same geometry of M. Our algorithm can also produce progressive Delaunay meshes, allowing a smooth choice of detail levels. Since DMs are represented using conventional mesh data structures, the existing digital geometry processing algorithms can benefit the numerical stability of DM without changing any codes. For example, DMs significantly improve the accuracy of the heat method for computing geodesic distances. Delaunay meshes (DM) are a special type of triangle mesh where the local Delaunay condition holds everywhere. We present an efficient algorithm to convert an arbitrary manifold triangle mesh M into a Delaunay mesh. We show that the constructed DM has O(Kn) vertices, where n is the number of vertices in M and K is a modeldependent constant. We also develop a novel algorithm to simplify Delaunay meshes, allowing a smooth choice of detail levels. Our methods are conceptually simple, theoretically sound and easy to implement. The DM construction algorithm also scales well due to its O(nK logK) time complexity.
Support Substructures: SupportInduced PartLevel Structural Representation
"... Abstract—In this work we explore a supportinduced structural organization of object parts. We introduce the concept of support substructures, which are special subsets of object parts with support and stability. A bottomup approach is proposed to identify such substructures in a support relation g ..."
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Abstract—In this work we explore a supportinduced structural organization of object parts. We introduce the concept of support substructures, which are special subsets of object parts with support and stability. A bottomup approach is proposed to identify such substructures in a support relation graph. We apply the derived highlevel substructures to partbased shape reshuffling between models, resulting in nontrivial functionally plausible model variations that are difficult to achieve with symmetryinduced substructures by the stateoftheart methods. We also show how to automatically or interactively turn a single input model to new functionally plausible shapes by structure rearrangement and synthesis, enabled by support substructures. To the best of our knowledge no single existing method has been designed for all these applications.
ABSTRACT Discrete Differential Geometry of Thin Materials for Computational Mechanics
, 2013
"... Instead of applying numerical methods directly to governing equations, another approach to computation is to discretize the geometric structure specific to the problem first, and then compute with the discrete geometry. This structurerespecting discretedifferentialgeometric (DDG) approach often ..."
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Instead of applying numerical methods directly to governing equations, another approach to computation is to discretize the geometric structure specific to the problem first, and then compute with the discrete geometry. This structurerespecting discretedifferentialgeometric (DDG) approach often leads to new algorithms that more accurately track the physically behavior of the system with less computational effort. Thin objects, such as pieces of cloth, paper, sheet metal, freeform masonry, and steelglass structures are particularly rich in geometric structure and so are wellsuited for DDG. I show how understanding the geometry of time integration and contact leads to new algorithms, with strong correctness guarantees, for simulating thin elastic objects in contact; how the performance of these algorithms can be dramatically improved without harming the geometric structure, and thus the guarantees, of the original formulation; how the geometry of static equilibrium can be used to efficiently solve design problems related to masonry or glass buildings; and how discrete developable surfaces can be used to model thin sheets undergoing isometric deformation.
Johannes Wallner,c Helmut Pottmanna,d
"... Abstract. This paper builds on recent progress in computing with geometric constraints, which is particularly relevant to architectural geometry. Not only do various kinds of meshes with additional properties (like planar faces, or with equilibrium forces in their edges) become available for inte ..."
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Abstract. This paper builds on recent progress in computing with geometric constraints, which is particularly relevant to architectural geometry. Not only do various kinds of meshes with additional properties (like planar faces, or with equilibrium forces in their edges) become available for interactive geometric modeling, but so do other arrangements of geometric primitives, like honeycomb structures. The latter constitute an important class of geometric objects, with relations to “Lobel” meshes, and to freeform polyhedral patterns. Such patterns are particularly interesting and pose research problems which go beyond what is known for meshes, e.g. with regard to their computing, their flexibility, and the assessment of their fairness. 1