Geometric Modeling: Interoperability and New Challenges (Dagstuhl Seminar 21471)

Abstract

This report documents the program and the outcomes of Dagstuhl Seminar 21471 “Geometric Modeling: Interoperability and New Challenges”. This seminar was initially planned on May 2021, and was delayed due to the pandemic. The seminar took place as a hybrid version with on site and remote participants. It provided a great opportunity for exchanges which, as pointed out by participants, were very appreciated in this period where international scientific interactions have been diminished. This report summarizes the seminar communications, first by providing the abstracts of the talks which present recent results in geometric modeling. Moreover, the scientific exchanges during the seminar provided a great basis for scientific discussions that resulted to the included five reports which highlight the new and future challenges in Geometric Modeling. In this talk we present the spatial counterpart of the recently introduced class of planar Pythagorean-Hodograph (PH) B–Spline curves. Spatial Pythagorean-Hodograph B–Spline curves are odd-degree, non-uniform, parametric spatial B–Spline curves whose arc length is a B–Spline function of the curve parameter and can thus be computed explicitly without numerical quadrature. We provide the general construction of these curves using quaternion algebra and formulate the problem of point interpolation by clamped and closed PH B–Spline curves of arbitrary odd degree. In particular, we provide closed form solutions for the cubic and the quintic cases, and discuss how degree-(2 n + 1), C n -continuous PH B–Spline curves can be computed by optimizing several scale-invariant fairness measures with interpolation constraints. Finally, we define Rational B-Spline Euler Rodrigues Frames (RBSERF) for regular PH B-Spline curves as well as rational tensor product B-Spline pipe surfaces. A functional is introduced to minimize the rotation of the RBSERF, and the results are illustrated on the corresponding rational pipe surface. The design and analysis of adaptive isogeometric methods with hierarchical spline constructions has attracted remarkable interest in the last few years. In order to increase the flexibility of the hierarchical approximation framework, while simultaneously preserving the performance of the overall adaptive scheme, particular attention is currently devoted to address the fast formation of system matrices arising from hierarchical discretization as well as to the development of effective multi-patch extensions. The talk will present recent results on these directions. strategies to generate 2-dimensional random auxetic meta-materials. Starting from a dense irregular network, we seek to reduce the Poisson’s ratio, by pruning bonds (edges) based solely on geometric criteria. To this end, we first deduce some prominent geometric features from regular auxetic networks and then introduce a strategy combining a pure geometric pruning algorithm followed by a physics-based testing phase to determine the resulting Poisson’s ratio of our networks. We provide numerical results and statistical validation. We also show physical tests with both laser-cut rubber networks and 3D-printed networks showing auxetic behaviour. shape-signature vector MI 4 . In the case of the PD-based hull, 7 parameters sensitive to MI 4 are also among the 8 parameters sensitive to C w . Interestingly, similar results are obtained for the GMF-based hull, where 6 out of 7 sensitive parameters to C w are also sensitive to MI 4 . Afterwards, two different design spaces are constructed for both hull models, one with sensitive parameters obtained with C w and the other with MI 4 . Shape optimisation is performed in both spaces performed with a meta-heuristic optimisation approach. Final optimisation results showed that the design generated from design space constructed with sensitive parameters of C w and MI 4 for both types of hulls offer similar performance. These results indicate that PSA performed with moments can reasonably estimate parameters’ sensitivity to the design’s physics with considerably reduced computational cost. Industrial X-ray CT scanners have delivered non-destructive evaluation of industrial products with its capability of inspecting even inside the body of products. This paper introduces a new approach to accelerate inspection of a large number of the same mechanical parts by scanning their heap in a bin at once. The scanning result is a CT volumetric image containing all of these parts out of which each part is segmented for inspection. This segmentation is a kind of template matching problem. However, random postures and dense contacts of the binned parts prohibit extracting the parts one-by-one using a traditional template matching due to its high computational complexity. To reduce the computational complexity, we convert both the scanned volumetric images of the template and the binned parts to simpler graph structures, and then, we solve well-studied graph matching problem to distinguish each part. We convert a discrete volume data to a distance field by the distance transform, and then, construct a graph consisting of nodes at extremum points of the distance field based on the Morse theory. The experimental evaluation demonstrates that our method without manual arrangement of the target parts works even for the scan of a heap of 50 binned parts in CT volumes of about 800 3 voxels, and an average processing time is as short as 30 minutes.