1,478 research outputs found
Robustness of Boolean operations on subdivision-surface models
This work was presented in two parts at Dagstuhl seminar 08021.
The two presentations described work in
progress, including a ``backward bound\u27\u27 for a combined backward/forward
error analysis for the problem mentioned in the title.
We seek rigorous proofs that representations of computed sets, produced by
algorithms to compute Boolean operations, are well formed, and that the
algorithms are correct. Such proofs should eventually take account of the use of
finite-precision arithmetic, although the proofs presented here do not.
The representations studied are based on subdivision surfaces. Such
representations are being used more and more frequently in place of trimmed
NURBS representations, and the robustness analysis for these new representations
is simpler than for trimmed NURBS.
The particular subdivision-surface representation used is based on the Loop
subdivision scheme. The analysis is broken into three parts. First, it is
established that the input operands are well-formed two-dimensional manifolds
without boundary. This can be done with existing methods.
Secondly, we introduce the so-called ``limit mesh\u27\u27, and view the
limit meshes corresponding to the input sets as defining an approximate problem
in the sense of a backward error analysis. The presentations mentioned above
described a proof of the corresponding error bound. The third part of the
analysis corresponds to the ``forward bound\u27\u27: this remains to be done
QuickCSG: Fast Arbitrary Boolean Combinations of N Solids
QuickCSG computes the result for general N-polyhedron boolean expressions
without an intermediate tree of solids. We propose a vertex-centric view of the
problem, which simplifies the identification of final geometric contributions,
and facilitates its spatial decomposition. The problem is then cast in a single
KD-tree exploration, geared toward the result by early pruning of any region of
space not contributing to the final surface. We assume strong regularity
properties on the input meshes and that they are in general position. This
simplifying assumption, in combination with our vertex-centric approach,
improves the speed of the approach. Complemented with a task-stealing
parallelization, the algorithm achieves breakthrough performance, one to two
orders of magnitude speedups with respect to state-of-the-art CPU algorithms,
on boolean operations over two to dozens of polyhedra. The algorithm also
outperforms GPU implementations with approximate discretizations, while
producing an output without redundant facets. Despite the restrictive
assumptions on the input, we show the usefulness of QuickCSG for applications
with large CSG problems and strong temporal constraints, e.g. modeling for 3D
printers, reconstruction from visual hulls and collision detection
QuickCSG: Fast Arbitrary Boolean Combinations of N Solids
QuickCSG computes the result for general N-polyhedron boolean expressions
without an intermediate tree of solids. We propose a vertex-centric view of the
problem, which simplifies the identification of final geometric contributions,
and facilitates its spatial decomposition. The problem is then cast in a single
KD-tree exploration, geared toward the result by early pruning of any region of
space not contributing to the final surface. We assume strong regularity
properties on the input meshes and that they are in general position. This
simplifying assumption, in combination with our vertex-centric approach,
improves the speed of the approach. Complemented with a task-stealing
parallelization, the algorithm achieves breakthrough performance, one to two
orders of magnitude speedups with respect to state-of-the-art CPU algorithms,
on boolean operations over two to dozens of polyhedra. The algorithm also
outperforms GPU implementations with approximate discretizations, while
producing an output without redundant facets. Despite the restrictive
assumptions on the input, we show the usefulness of QuickCSG for applications
with large CSG problems and strong temporal constraints, e.g. modeling for 3D
printers, reconstruction from visual hulls and collision detection
Geometric Modeling of Cellular Materials for Additive Manufacturing in Biomedical Field: A Review
Advances in additive manufacturing technologies facilitate the fabrication of cellular materials that have tailored functional characteristics. The application of solid freeform fabrication techniques is especially exploited in designing scaffolds for tissue engineering. In this review, firstly, a classification of cellular materials from a geometric point of view is proposed; then, the main approaches on geometric modeling of cellular materials are discussed. Finally, an investigation on porous scaffolds fabricated by additive manufacturing technologies is pointed out. Perspectives in geometric modeling of scaffolds for tissue engineering are also proposed
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
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
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