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

Bezier Clippingおよびその計算機支援形状設計への応用

Recently, computer aided geometry design for industrial products has become popular. Geometry design including free form surface is very significant in CAD/CAM system. In geometry designs there are some problems that need to be solved such as curve-curve intersection, surface-surf ace intersection, and curve-surf ace intersection. A display method of desired shapes is also very important. To display curved surfaces we have to solve a hidden line problem and a hidden surface problem. Shading models including shadowing, reflection, and refraction are required to get photo-realistic images of designed shapes. This paper proposes Bezier clipping method which can solve all of the problems mentioned above. Bezier clipping is a robust iterative method which can solve functions expressed by polynomials of a high degree. This paper describes the basic idea of Bezier clipping and its application to computer aided geometry design and computer graphics

Geometric continuity and compatibility conditions for 4-patch surfaces

When considering regularity of surfaces, it is its geometry that is of interest. Thus, the concept of geometric regularity or geometric continuity of a specific order is a relevant concept. In this paper we discuss necessary and sufficient conditions for a 4-patch surface to be geometrically continuous of order one and two or, in other words, being tangent plane continuous and curvature continuous respectively. The focus is on the regularity at the point where the four patches meet and the compatibility conditions that must appear in this case. In this article the compatibility conditions are proved to be independent of the patch parametrization, i.e., the compatibility conditions are universal. In the end of the paper these results are applied to a specific parametrization such as Bezier representation in order to generalize a 4-patch surface result by Sarraga.Comment: 25 pages, 6 figure

The implementation of the vegter yield criterion and a physically based hardening rule in finite elements

A new material description for sheet metal forming using Finite Elements has been developed. The description consists of a yield criterion and a hardening rule. In contrast to most former criteria the new criterion is based on multi-axial stress states. The yield criterion is extended with a physically based hardening rule, in which the flow stress depends on the strain and strain rate. A Limiting Dome Height test is used to examine the material description

Recent developments in finite element simulations of the deep drawing process

Some new developments in a finite element code for the deep drawing process are presented\ud in this paper. First the phenomenon of friction is treated. A Stribeck friction model has been\ud developed which accounts for the dependency of the friction coefficient on the local contact\ud conditions. Secondly a new yield criterion has been developed by Vegter. This Vegter yield\ud criterion is based on multi-axial stress states. Finally attention will be paid to reduce the CPUtime\ud of a simulation when drawbeads are used. An equivalent drawbead model has been\ud developed to avoid an enormous increase in calculation time

Intersection test and blossoming perturbation for disk parametric curves and ball parametric surfaces

Errors in curve and surface representation due to inaccuracies in the data are considered and accounted for by introducing disk parametric curves and ball parametric surfaces. Intersection test algorithms and interval extensions using blossoming are discussed for each of the three cases of Bezier curves, tensor product surfaces, and triangular patches. A stability analysis is also performed for each of the three cases. It is shown that under certain restrictions disk Bezier curves and triangular ball Bezier patches are stable with respect to perturbations of the control disks ( balls); whereas tensor product ball Bezier surfaces are in general not

Algebraic level sets for CAD/CAE integration and moving boundary problems

Boundary representation (B-rep) of CAD models obtained from solid modeling kernels are commonly used in design, and analysis applications outside the CAD systems. Boolean operations between interacting B-rep CAD models as well as analysis of such multi-body systems are fundamental operations on B-rep geometries in CAD/CAE applications. However, the boundary representation of B-rep solids is, in general, not a suitable representation for analysis operations which lead to CAD/CAE integration challenges due to the need for conversion from B-rep to volumetric approximations. The major challenges include intermediate mesh generation step, capturing CAD features and associated behavior exactly and recurring point containment queries for point classification as inside/outside the solid. Thus, an ideal analysis technique for CAD/CAE integration that can enable direct analysis operations on B-rep CAD models while overcoming the associated challenges is desirable. ^ Further, numerical surface intersection operations are typically necessary for boolean operations on B-rep geometries during the CAD and CAE phases. However, for non-linear geometries, surface intersection operations are non-trivial and face the challenge of simultaneously satisfying the three goals of accuracy, efficiency and robustness. In the class of problems involving multi-body interactions, often an implicit knowledge of the boolean operation is sufficient and explicit intersection computation may not be needed. Such implicit boolean operations can be performed by point containment queries on B-rep CAD models. However, for complex non-linear B-rep geometries, the point containment queries may involve numerical iterative point projection operations which are expensive. Thus, there is a need for inexpensive, non-iterative techniques to enable such implicit boolean operations on B-rep geometries. ^ Moreover, in analysis problems with evolving boundaries (ormoving boundary problems), interfaces or cracks, blending functions are used to enrich the underlying domain with the known behavior on the enriching entity. The blending functions are typically dependent on the distance from the evolving boundaries. For boundaries defined by free form curves or surfaces, the distance fields have to be constructed numerically. This may require either a polytope approximation to the boundary and/or an iterative solution to determine the exact distance to the boundary. ^ In this work a purely algebraic, and computationally efficient technique is described for constructing signed distance measures from Non-Uniform Rational B-Splines (NURBS) boundaries that retain the geometric exactness of the boundaries while eliminating the need for iterative and non-robust distance calculation. The proposed technique exploits the NURBS geometry and algebraic tools of implicitization. Such a signed distance measure, also referred to as the Algebraic Level Sets, gives a volumetric representation of the B-rep geometry constructed by purely non-iterative algebraic operations on the geometry. This in turn enables both the implicit boolean operations and analysis operations on B-rep geometries in CAD/CAE applications. Algebraic level sets ensure exactness of geometry while eliminating iterative numerical computations. Further, a geometry-based analysis technique that relies on hierarchical partition of unity field compositions (HPFC) theory and its extension to enriched field modeling is presented. The proposed technique enables direct analysis of complex physical problems without meshing, thus, integrating CAD and CAE. The developed techniques are demonstrated by constructing algebraic level sets for complex geometries, geometry-based analysis of B-rep CAD models and a variety of fracture examples culminating in the analysis of steady state heat conduction in a solid with arbitrary shaped three-dimensional cracks. ^ The proposed techniques are lastly applied to investigate the risk of fracture in the ultra low-k (ULK) dies due to copper (Cu) wirebonding process. Maximum damage induced in the interlayer dielectric (ILD) stack during the process steps is proposed as an indicator of the reliability risk. Numerical techniques based on enriched isogeometric approximations are adopted to model damage in the ULK stacks using a cohesive damage description. A damage analysis procedure is proposed to conduct damage accumulation studies during Cu wirebonding process. Analysis is carried out to identify weak interfaces and potential sites for crack nucleation as well as damage nucleation patterns. Further, the critical process condition is identified by analyzing the damage induced during the impact and ultrasonic excitation stages. Also, representative ILD stack designs with varying Cu percentage are compared for risk of fracture