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

Isogeometric Analysis on V-reps: first results

Inspired by the introduction of Volumetric Modeling via volumetric representations (V-reps) by Massarwi and Elber in 2016, in this paper we present a novel approach for the construction of isogeometric numerical methods for elliptic PDEs on trimmed geometries, seen as a special class of more general V-reps. We develop tools for approximation and local re-parametrization of trimmed elements for three dimensional problems, and we provide a theoretical framework that fully justify our algorithmic choices. We validate our approach both on two and three dimensional problems, for diffusion and linear elasticity.Comment: 36 pages, 44 figures. Reviewed versio

Two triangulations methods based on edge refinement

In this paper two curvature adaptive methods of surface triangulation are presented. Both methods are based on edge refinement to obtain a triangulation compatible with the curvature requirements. The first method applies an incremental and constrained Delaunay triangulation and uses curvature bounds to determine if an edge of the triangulation is admissible. The second method uses this function also in the edge refinement process, i.e. in the computation of the location of a refining point, and in the re-triangulation needed after the insertion of this refining point. Results are presented, comparing both approachesPostprint (published version

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

A simple approach to the numerical simulation with trimmed CAD surfaces

In this work a novel method for the analysis with trimmed CAD surfaces is presented. The method involves an additional mapping step and the attraction stems from its sim- plicity and ease of implementation into existing Finite Element (FEM) or Boundary Element (BEM) software. The method is first verified with classical test examples in structural mechanics. Then two practical applications are presented one using the FEM, the other the BEM, that show the applicability of the method.Comment: 20 pages and 16 figure

Reverse Engineering Trimmed NURB Surfaces From Laser Scanned Data

A common reverse engineering problem is to convert several hundred thousand points collected from the surface of an object via a digitizing process, into a coherent geometric model that is easily transferred to a CAD software such as a solid modeler for either design improvement or manufacturing and analysis. These data are very dense and make data-set manipulation difficult and tedious. Many commercial solutions exist but involve time consuming interaction to go from points to surface meshes such as BSplines or NURBS (Non Uniform Rational BSplines). Our approach differs from current industry practice in that we produce a mesh with little or no interaction from the user. The user can produce degree 2 and higher BSpline surfaces and can choose the degree and number ofsegments as parameters to the system. The BSpline surface is both compact and curvature continuous. The former property reduces the large storage overhead, and the later implies a smooth can be created from noisy data. In addition, the nature ofthe BSpline allows one to easily and smoothly alter the surface, making re-engineering extremely feasible. The BSpline surface is created using the principle ofhigher orders least squares with smoothing functions at the edges. Both linear and cylindrical data sets are handled using an automated parameterization method. Also, because ofthe BSpline's continuous nature, a multiresolutional-triangulated mesh can quickly be produced. This last fact means that an STL file is simple to generate. STL files can also be easily used as input to the system.Mechanical Engineerin