Watertightization of Trimmed Surfaces at Intersection Boundary

This paper introduces a watertight technique to deal with the boundary representation of surface-surface intersection in CAD. Surfaces play an important role in today's geometric design. The mathematical model of non-uniform rational B-spline surfaces (NURBS) is the mainstream and ISO standard. In the situation of surface-surface intersection, things are a little complicated, for some parts of surfaces may be cut-off, so called trimmed surfaces occur, which is the central topic in the past decades in CAD community of both academia and industry. The main problem is that the parametric domain of the trimmed surface generally is not the standard square or rectangle, and rather, typically, bounded by curves, based on point inverse of the intersection points and interpolated. The existence of gaps or overlaps at the intersection boundary makes hard the preprocessing of CAE and other downstream applications. The NURBS are in this case hard to keep a closed form. In common, a special data structure of intersection curves must be affiliated to support downstream applications, while the data structure of the whole CAD system is not unified, and the calculation is not efficient. In terms of Bezier surface, a special case of NURBS, this paper designs a reparameterization or normalization to transform the trimmed surface into a group of Bezier surface patches in standard parametric domain [0,1]X[0,1]. And then the boundary curve of normalized Bezier surface patch can be replaced by the intersection curve to realize watertight along the boundary. In this way, the trimmed surface is wiped out, the "gap" between CAD and CAE is closed.Comment: 10 pages,6 figure

T-spline based unifying registration procedure for free-form surface workpieces in intelligent CMM

With the development of the modern manufacturing industry, the free-form surface is widely used in various fields, and the automatic detection of a free-form surface is an important function of future intelligent three-coordinate measuring machines (CMMs). To improve the intelligence of CMMs, a new visual system is designed based on the characteristics of CMMs. A unified model of the free-form surface is proposed based on T-splines. A discretization method of the T-spline surface formula model is proposed. Under this discretization, the position and orientation of the workpiece would be recognized by point cloud registration. A high accuracy evaluation method is proposed between the measured point cloud and the T-spline surface formula. The experimental results demonstrate that the proposed method has the potential to realize the automatic detection of different free-form surfaces and improve the intelligence of CMMs

B-Spline meshing for high-order finite element analyses of multi-physics problems

Multi-physics problems often involve differential equations of higher-order, which cannot be solved with standard finiteelement methods. B-splines as finite element basis functions provide the required continuity and smoothness. However, the meshgeneration for arbitrarily shaped domains is non-intuitively and traditional techniques often lead to distorted elements.Here a strategy is presented to design isoparametric B-spline based meshes for curves, surfaces, and volumes. The error of thehomeomorphic transformation into curved boundaries is estimated. For selected two and three-dimensional shapes, the knotvectors and the control points are calculated.Exemplarily, a finite element analysis of a helical structure subjected to a chemo-mechanical deformation with phase decompositionis performed

A generalized finite element formulation for arbitrary basis functions : from isogeometric analysis to XFEM

Many of the formulations of cm-rent research interest, including iosogeometric methods and the extended finite element method, use nontraditional basis functions. Some, such as subdivision surfaces, may not have convenient analytical representations. The concept of an element, if appropriate at all, no longer coincides with the traditional definition. Developing a new software for each new class of basis functions is a large research burden, especially, if the problems involve large deformations, non-linear materials, and contact. The objective of this paper is to present a method that separates as much as possible the generation and evaluation of the basis functions from the analysis, resulting in a formulation that can be implemented within the traditional structure of a finite clement program but that permits the use of arbitrary sets of basis functions that are defined only through the input file. Elements ranging from a traditional linear four-node tetrahedron through a higher-order element combining XFEM and isogeometric analysis may be specified entirely through an input file without any additional programming. Examples of this framework to applications with Lagrange elements, isogeometric elements, and XFEM basis functions for fracture are presented

Conversion of trimmed NURBS surfaces to Catmull-Clark subdivision surfaces

This paper introduces a novel method to convert trimmed NURBS surfaces to untrimmed subdivision surfaces with Bézier edge conditions. We take a NURBS surface and its trimming curves as input, from this we automatically compute a base mesh, the limit surface of which fits the trimmed NURBS surface to a specified tolerance. We first construct the topology of the base mesh by performing a cross-field based decomposition in parameter space. The number and positions of extraordinary vertices required to represent the trimmed shape can be automatically identified by smoothing a cross field bounded by the parametric trimming curves. After the topology construction, the control point positions in the base mesh are calculated based on the limit stencils of the subdivision scheme and constraints to achieve tangential continuity across the boundary. Our method provides the user with either an editable base mesh or a fine mesh whose limit surface approximates the input within a certain tolerance. By integrating the trimming curve as part of the desired limit surface boundary, our conversion can produce gap-free models. Moreover, since we use tangential continuity across the boundary between adjacent surfaces as constraints, the converted surfaces join with G1 continuity. © 2014 The Authors.EPSRC, Chinese Government (PhD studentship) and Cambridge Trust

Subdivide and Conquer: Adapting Non-Manifold Subdivision Surfaces to Surface-Based Representation and Reconstruction of Complex Geological Structures

Methods from the field of computer graphics are the foundation for the representation of geological structures in the form of geological models. However, as many of these methods have been developed for other types of applications, some of the requirements for the representation of geological features may not be considered, and the capacities and limitations of different algorithms are not always evident. In this work, we therefore review surface-based geological modelling methods from both a geological and computer graphics perspective. Specifically, we investigate the use of NURBS (non-uniform rational B-splines) and subdivision surfaces, as two main parametric surface-based modelling methods, and compare the strengths and weaknesses of the two approaches. Although NURBS surfaces have been used in geological modelling, subdivision surfaces as a standard method in the animation and gaming industries have so far received little attention—even if subdivision surfaces support arbitrary topologies and watertight boundary representation, two aspects that make them an appealing choice for complex geological modelling. It is worth mentioning that watertight models are an important basis for subsequent process simulations. Many complex geological structures require a combination of smooth and sharp edges. Investigating subdivision schemes with semi-sharp creases is therefore an important part of this paper, as semi-sharp creases characterise the resistance of a mesh structure to the subdivision procedure. Moreover, non-manifold topologies, as a challenging concept in complex geological and reservoir modelling, are explored, and the subdivision surface method, which is compatible with non-manifold topology, is described. Finally, solving inverse problems by fitting the smooth surfaces to complex geological structures is investigated with a case study. The fitted surfaces are watertight, controllable with control points, and topologically similar to the main geological structure. Also, the fitted model can reduce the cost of modelling and simulation by using a reduced number of vertices in comparison with the complex geological structure

Conversion of B-rep CAD models into globally G¹ triangular splines

Existing techniques that convert B-rep (boundary representation) patches into Clough-Tocher splines guarantee watertight, that is C0, conversion results across B-rep edges. In contrast, our approach ensures global tangent-plane, that is G1, continuity of the converted B-rep CAD models. We achieve this by careful boundary curve and normal vector management, and by converting the input models into Shirman-Séquin macro-elements near their (trimmed) B-rep edges. We propose several different variants and compare them with respect to their locality, visual quality, and difference with the input B-rep CAD model. Although the same global G1 continuity can also be achieved by conversion techniques based on subdivision surfaces, our approach uses triangular splines and thus enjoys full compatibility with CAD