Smooth quasi-developable surfaces bounded by smooth curves

Computing a quasi-developable strip surface bounded by design curves finds wide industrial applications. Existing methods compute discrete surfaces composed of developable lines connecting sampling points on input curves which are not adequate for generating smooth quasi-developable surfaces. We propose the first method which is capable of exploring the full solution space of continuous input curves to compute a smooth quasi-developable ruled surface with as large developability as possible. The resulting surface is exactly bounded by the input smooth curves and is guaranteed to have no self-intersections. The main contribution is a variational approach to compute a continuous mapping of parameters of input curves by minimizing a function evaluating surface developability. Moreover, we also present an algorithm to represent a resulting surface as a B-spline surface when input curves are B-spline curves.Comment: 18 page

Accessibility for Line-Cutting in Freeform Surfaces

Manufacturing techniques such as hot-wire cutting, wire-EDM, wire-saw cutting, and flank CNC machining all belong to a class of processes called line-cutting where the cutting tool moves tangentially along the reference geometry. From a geometric point of view, line-cutting brings a unique set of challenges in guaranteeing that the process is collision-free. In this work, given a set of cut-paths on a freeform geometry as the input, we propose a conservative algorithm for finding collision-free tangential cutting directions. These directions, if they exist, are guaranteed to be globally accessible for fabricating the geometry by line-cutting. We then demonstrate how this information can be used to generate globally collision-free cut-paths. We apply our algorithm to freeform models of varying complexity.RYC-2017-2264

Volumetric Untrimming: Precise decomposition of trimmed trivariates into tensor products

3D objects, modeled using Computer Aided Geometric Design tools, are traditionally represented using a boundary representation (B-rep), and typically use spline functions to parameterize these boundary surfaces. However, recent development in physical analysis, in isogeometric analysis (IGA) in specific, necessitates a volumetric parametrization of the interior of the object. IGA is performed directly by integrating over the spline spaces of the volumetric spline representation of the object. Typically, tensor-product B-spline trivariates are used to parameterize the volumetric domain. A general 3D object, that can be modeled in contemporary B-rep CAD tools, is typically represented using trimmed B-spline surfaces. In order to capture the generality of the contemporary B-rep modeling space, while supporting IGA needs, Massarwi and Elber (2016) proposed the use of trimmed trivariates volumetric elements. However, the use of trimmed geometry makes the integration process more difficult since integration over trimmed B-spline basis functions is a highly challenging task. In this work, we propose an algorithm that precisely decomposes a trimmed B-spline trivariate into a set of (singular only on the boundary) tensor-product B-spline trivariates, that can be utilized to simplify the integration process in IGA. The trimmed B-spline trivariate is first subdivided into a set of trimmed B\'ezier trivariates, at all its internal knots. Then, each trimmed B\'ezier trivariate, is decomposed into a set of mutually exclusive tensor-product B-spline trivariates, that precisely cover the entire trimmed domain. This process, denoted untrimming, can be performed in either the Euclidean space or the parametric space of the trivariate. We present examples on complex trimmed trivariates' based geometry, and we demonstrate the effectiveness of the method by applying IGA over the (untrimmed) results.Comment: 18 pages, 32 figures. Contribution accepted in International Conference on Geometric Modeling and Processing (GMP 2019

The development of a finite elements based springback compensation tool for sheet metal products

Springback is a major problem in the deep drawing process. When the tools are released after the forming stage, the product springs back due to the action of internal stresses. In many cases the shape deviation is too large and springback compensation is needed: the tools of the deep drawing process are changed so, that the product becomes geometrically accurate after springback. In this paper, two different ways of geometric optimization are presented, the smooth displacement adjustment (SDA) method and the surface controlled overbending (SCO) method. Both methods use results from a finite elements deep drawing simulation for the optimization of the tool shape. The methods are demonstrated on an industrial product. The results are satisfactory, but it is shown that both methods still need to be improved and that the FE simulation needs to become more reliable to allow industrial application

Dev2PQ: Planar Quadrilateral Strip Remeshing of Developable Surfaces

We introduce an algorithm to remesh triangle meshes representing developable surfaces to planar quad dominant meshes. The output of our algorithm consists of planar quadrilateral (PQ) strips that are aligned to principal curvature directions and closely approximate the curved parts of the input developable, and planar polygons representing the flat parts of the input. Developable PQ-strip meshes are useful in many areas of shape modeling, thanks to the simplicity of fabrication from flat sheet material. Unfortunately, they are difficult to model due to their restrictive combinatorics and locking issues. Other representations of developable surfaces, such as arbitrary triangle or quad meshes, are more suitable for interactive freeform modeling, but generally have non-planar faces or are not aligned to principal curvatures. Our method leverages the modeling flexibility of non-ruling based representations of developable surfaces, while still obtaining developable, curvature aligned PQ-strip meshes. Our algorithm optimizes for a scalar function on the input mesh, such that its level sets are extrinsically straight and align well to the locally estimated ruling directions. The condition that guarantees straight level sets is nonlinear of high order and numerically difficult to enforce in a straightforward manner. We devise an alternating optimization method that makes our problem tractable and practical to compute. Our method works automatically on any developable input, including multiple patches and curved folds, without explicit domain decomposition. We demonstrate the effectiveness of our approach on a variety of developable surfaces and show how our remeshing can be used alongside handle based interactive freeform modeling of developable shapes

Mini-Workshop: Analytical and Numerical Methods in Image and Surface Processing

The workshop successfully brought together researchers from mathematical analysis, numerical mathematics, computer graphics and image processing. The focus was on variational methods in image and surface processing such as active contour models, Mumford-Shah type functionals, image and surface denoising based on geometric evolution problems in image and surface fairing, physical modeling of surfaces, the restoration of images and surfaces using higher order variational formulations

Special Curve Patterns for Freeform Architecture

In recent years, freeform shapes are gaining more and more popularity in architecture. Such shapes are often challenging to manufacture, and have motivated an active research field called architectural geometry. In this thesis, we investigate patterns of special curves on surfaces, which find applications in design and realization of freeform architectural shapes. We first consider families of geodesic curves or piecewise geodesic curves on a surface, which are important for panelization of the surface and for interior design. We propose a method to propagate a series of such curves across a surface, starting from a given source curve, so that the distance functions between neighboring curves are close to given target distance functions. We use Jacobi fields as first order approximation of the distance functions from a curve to its neighboring curves, and select a Jacobi field which is closest to the target distance function. A neighboring curve is then computed according to the selected Jacobi field by solving an optimization problem. Using different target distance functions, we can generate different patterns of geodesic/piecewise geodesic curves. Our method provides an intuitive and controllable way to design geodesic patterns on freeform surfaces. We then present a method to compute functional webs, which are three families of curves with regular connectivity, where the curves have given special properties. We consider planar, circular and geodesic properties of the curves, which facilitate the fabrication of curve elements. We discretize a web as a regular triangle mesh, where the curves are represented by edge polylines of the mesh. The shape of the web is determined by optimizing a target functional which penalizes the deviation of the curves from their target properties. Furthermore, for webs where all curves are planar, we also show they can be computed in an exact way using three families of planes. By enabling the design of webs composed of curve elements which are easily manufacturable, our method addresses the challenge in realization of webs which have emerged in recent architectural designs