Vibeke Skytt

Locally refined spline surfaces for representation of terrain data

In this paper we describe the use of a novel representation, LR B-spline surfaces, and apply this representation in the treatment of geographical data. These data sets are typically very large and LR B-spline surfaces offer a compact representation of overall smooth data with local details. We briefly describe the properties of the LR B-spline representation, and discuss the details of two approximation methods adapted for LR B-splines: least squares approximation and multilevel B-spline approximation (MBA). The described techniques are demonstrated on several examples of terrain data in the form of point clouds. Graphical abstractAn LR B-spline surface approximating a point cloud representing Gaustatoppen in Norway. The point cloud covers 7kmi?7km and orth is pointing upwards to the right. The photograph has a slightly different view.Display Omitted HighlightsLR B-splines is a new surface representation for local refinement of spline spaces.Least squares approximation is adapted for LR B-splines.Multilevel B-spline approximation is adapted for LR B-splines.The methods are applied to approximate terrain data.

Spline volume fairing

The use of trivariate NURBS in isogeometric analysis has put the quality of parametrization of NURBS volumes on the agenda. Sometimes a NURBS volume needs a better parametrization to meet requirements regarding smoothness, approximation or periodicity. In this paper we generalize various smoothing methods that already exist for bivariate parametric spline surfaces to trivariate parametric spline volumes. We will also address how rational and polynomial spline volumes create different challenges and solutions in the algorithms.

Isogeometric Representation and Analysis - Bridging the Gap between CAD and Analysis

In 2005 Prof. Thomas J. R. Hughes proposed isogeometric analysis using volumetric NURBS elements rather than traditional finite elements for analysis. NURBS is the standard approach for representation of free form curves and sculptured surfaces in Computer Aided Design, and can represent elementary shapes such as sphere, cylinders, and torus exactly. Used in analysis NURBS consequently offers exact geometry representation, simplified design optimization and tighter integration of analysis and CAD. In this paper we address different aspects of isogeometric representation and analysis with a main focus on the relation to CAD, and how CAD can change to improve analysis by incorporating isogeometric representation.

Intersection Algorithms and CAGD

An approach for calculating intersections between parametric surfaces based on long experience in developing intersection algorithms for industrial use, is presented. In addition two novel methods that help improve quality and performance of intersection algorithms are described: An initial assessment of the intersection complexity to identify most transversal intersections, and to identify surface regions with possible complex intersection topology. To find regions where the surfaces possibly intersect, and regions where surface normals possibly are parallel, the computational power of multi-core CPUs and programmable graphics processors (GPUs) is used for subdivision of the surfaces and their normal fields. Approximate implicitization of surface regions to help analyse singular and near singular intersections.

Challenges in Surface-Surface Intersections

Tangential and singular situations are still challenges in a system for surface-surface intersections. This paper presents several real world examples of hard intersection problems, and proposes methods on how to deal with them. In particular, solutions which use the possibility of representing a parametric surface as an algebraic surface through the use of approximate implicitization, are in focus. This allows us to transform an intersection between two parametric surfaces to the problem of finding zeroes of a function of two parameters.

Deconfliction and Surface Generation from Bathymetry Data Using LR B-splines

A set of bathymetry point clouds acquired by different measurement techniques at different times, having different accuracy and varying patterns of points, are approximated by an LR B-spline surface. The aim is to represent the sea bottom with good accuracy and at the same time reduce the data size considerably. In this process the point clouds must be cleaned by selecting the “best” points for surface generation. This cleaning process is called deconfliction, and we use a rough approximation of the combined point clouds as a reference surface to select a consistent set of points. The reference surface is updated using only the selected points to create an accurate approximation. LR B-splines is the selected surface format due to its suitability for adaptive refinement and approximation, and its ability to represent local detail without a global increase in the data size of the surface.

Models for Isogeometric Analysis from CAD

In isogeometric analysis splines are used for describing both the geometric shape and the solution fields of the analysis. Although splines are used in boundary structure CAD models, the CAD representation cannot be used directly in isogeometric analysis. We focus on how to build block structured models suited for isogeometric analysis from CAD-models. The blocks are quadrilateral or hexahedral depending on the dimension of the space. A CAD solid represented by its outer and inner hulls is replaced by a trivariate structure where each block is a spline volume and the blocks meet with at least C0 continuity. To avoid continuity conditions involving several coefficients at block boundaries when representing the blocks by NURBS we do not allow T-joints, i.e., the blocks have to meet in a corner-to-corner configuration. No such conditions are necessary for LR B-splines as LR B-splines are constructed on domains with T-junctions. The local refinement properties of LR B-splines facilitate adapting the structure of the spline space to the local variations of both the shape model and the analysis model. Although the data structure for LR B-splines is more complex than for NURBS, the data volume needed for representing a model using LR B-splines will in most cases be much smaller for LR B-splines than for NURBS.

Trivariate spline representations for computer aided design and additive manufacturing

Abstract Digital representations targeting design and simulation for Additive Manufacturing (AM) are addressed from the perspective of Computer Aided Geometric Design. We discuss the feasibility for multi-material AM for B-rep based CAD, STL, sculptured triangles as well as trimmed and block-structured trivariate locally refined spline representations. The trivariate spline representations support Isogeometric Analysis (IGA), and topology structures supporting these for CAD, IGA and AM are outlined. The ideas of (Truncated) Hierarchical B-splines, T-splines and LR B-splines are outlined and the approaches are compared. An example from the EC H2020 Factories of the Future Research and Innovation Actions CAxMan illustrates both trimmed and block-structured spline representations for IGA and AM.