Theory and applications of bijective barycentric mappings

Barycentric coordinates provide a convenient way to represent a point inside a triangle as a convex combination of the triangle's vertices, and to linearly interpolate data given at these vertices. Due to their favourable properties, they are commonly applied in geometric modelling, finite element methods, computer graphics, and many other fields. In some of these applications it is desirable to extend the concept of barycentric coordinates from triangles to polygons. Several variants of such generalized barycentric coordinates have been proposed in recent years. An important application of barycentric coordinates consists of barycentric mappings, which allow to naturally warp a source polygon to a corresponding target polygon, or more generally, to create mappings between closed curves or polyhedra. The principal practical application is image warping, which takes as input a control polygon drawn around an image and smoothly warps the image by moving the polygon vertices. A required property of image warping is to avoid fold-overs in the resulting image. The problem of fold-overs is a manifestation of a larger problem related to the lack of bijectivity of the barycentric mapping. Unfortunately, bijectivity of such barycentric mappings can only be guaranteed for the special case of warping between convex polygons or by triangulating the domain and hence renouncing smoothness. In fact, for any barycentric coordinates, it is always possible to construct a pair of polygons such that the barycentric mapping is not bijective. In the first part of this thesis we illustrate three methods to achieve bijective mappings. The first method is based on the intuition that, if two polygons are sufficiently close, then the mapping is close to the identity and hence bijective. This suggests to ``split'' the mapping into several intermediate mappings and to create a composite barycentric mapping which is guaranteed to be bijective between arbitrary polygons, polyhedra, or closed planar curves. We provide theoretical bounds on the bijectivity of the composite mapping related to the norm of the gradient of the coordinates. The fact that the bound depends on the gradient implies that these bounds exist only if the gradient of the coordinates is bounded. We focus on mean value coordinates and analyse the behaviour of their directional derivatives and gradient at the vertices of a polygon. The composition of barycentric mappings for closed planar curves leads to the problem of blending between two planar curves. We suggest to solve it by linearly interpolating the signed curvature and then reconstructing the intermediate curve from the interpolated curvature values. However, when both input curves are closed, this strategy can lead to open intermediate curves. We present a new algorithm for solving this problem, which finds the closed curve whose curvature is closest to the interpolated values. Our method relies on the definition of a suitable metric for measuring the distance between two planar curves and an appropriate discretization of the signed curvature functions. The second method to construct smooth bijective mappings with prescribed behaviour along the domain boundary exploits the properties of harmonic maps. These maps can be approximated in different ways, and we discuss their respective advantages and disadvantages. We further present a simple procedure for reducing their distortion and demonstrate the effectiveness of our approach by providing examples. The last method relies on a reformulation of complex barycentric mappings, which allows us to modify the ``speed'' along the edges to create complex bijective mappings. We provide some initial results and an optimization procedure which creates complex bijective maps. In the second part we provide two main applications of bijective mapping. The first one is in the context of finite elements simulations, where the discretization of the computational domain plays a central role. In the standard discretization, the domain is triangulated with a mesh and its boundary is approximated by a polygon. We present an approach which combines parametric finite elements with smooth bijective mappings, leaving the choice of approximation spaces free. This approach allows to represent arbitrarily complex geometries on coarse meshes with curved edges, regardless of the domain boundary complexity. The main idea is to use a bijective mapping for automatically warping the volume of a simple parametrization domain to the complex computational domain, thus creating a curved mesh of the latter. The second application addresses the meshing problem and the possibility to solve finite element simulations on polygonal meshes. In this context we present several methods to discretize the bijective mapping to create polygonal and piece-wise polynomial meshes

Re-surface: the novel use of deployable and actively-bent gridshells as reusable, reconfigurable and intuitive concrete shell formwork

Following a well-documented rise in the popularity of concrete shell application in the 20th century, thin concrete shells have experienced a global decline despite their potential as efficient structures with an economy of material use with aesthetics benefits. This phenomenon is subject to geographically determined socio-economic conditions and competition from other building solutions as a result of technological advancement in alternative construction systems. Importantly, their decline was attributed to limitations inherent to concrete shell formwork and construction methods. Being able to produce efficient shaping did not ensure that this method of construction is most cost efficient as it still remains difficult to construct double curved surfaces. The thesis addresses the limitations associated with past and present concrete shell building by proposing the use of actively-bent gridshells as re-configurable and reusable formwork for concrete shells to be designed and built. The hypothesis uses deployable scissor-jointed actively-bent gridshells as re-configurable and reusable formwork for concrete shell construction. This was developed from a series of Flash research (Benjamin, 2012) as student construction workshops to investigate the design and creation of actively-bent gridshells held between December 2008 and March 2011 in Sheffield. In this study, to understand this new system, scaled models of actively-bent gridshells were used as preliminary design aid. Deployed into three dimensional forms from a flexible flat grid mat, the structures were rigidized by bracing through triangulation restraints. The temporary rigid structure was subsequently enveloped with fabric onto which concrete was applied to create the concrete shell, thus acting as formwork. This formwork was then removed following the curing of the concrete cast to be reused repeatedly, or reconfigured into another concrete shell form. Hence, the thesis draws on the concepts, principles and ideas pertaining to three key architectural technologies: 1. concrete shell, 2. actively-bent gridshells and 3.fabric formwork. The thesis then presents a series of four prototype concrete shells constructed from different materials spanning between 1.3 meters and 2.45 meters in the workshops at the University of Edinburgh built between August 2014 and September 2015. For each experimental construction, the process of gridshell construction, fabric formwork preparation, concrete casting, gridshell formwork decentring and different design elements of openings, edges and anchorage abutments were analysed and discussed under the themes of construction, architectural tectonics and structure. The tectonic of process and material is understood and discussed based on the idea of stereogeneity (Manelius, 2012). Specifically, the relationship between gridshell as formwork and the concreting process was studied, analysed and assimilated in concrete shells built with progressive sophistication and elegance, culminating in a doubly-curved concrete shell that demonstrated both synclastic and anticlastic geometries, with further abutment simplification, edge leaning and physical openings incorporation. The study concludes with a physical concrete shell model formed by applying concrete onto fabric formwork to cover the Weald and Downland Jerwood gridshell. In the 1:20 scaled model, the proposed method is speculatively applied onto fabric stretched between pre-determined curvatures of the as-built gridshell. This formwork was subsequently removed for reuse, re-deployed and reconfigured. Using finite element analysis, the structural behaviour of the gridshell made of glass-fibre reinforced tubes and structural characteristics of the resultant concrete shell was checked. The interaction between the three technologies are discussed architectonically and structurally to inform guidelines for potential life-scale application. The thesis evidences the feasibility of the proposed system. It re-purposes a scaled model of a deployable gridshell as a physical modelling tool to facilitate concrete shell design, for both pure compression shells and "improper" shells, demonstrating its adaptability. It also promotes and reinvigorates concrete shells as possible architectural systems serving to instigate future research to revive concrete shell construction as an intelligent and intuitive way of creating structures with material economy, structural efficiency and visual elegance