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Altenhofen, Christian; Ewald, Tobias; Stork, André; Fellner, Dieter W.

Analyzing and Improving the Parameterization Quality of Catmull-Clark Solids for Isogeometric Analysis

2021

IEEE Computer Graphics and Applications

In the field of physically based simulation, high quality of the simulation model is crucial for the correctness of the simulation results and the performance of the simulation algorithm. When working with spline or subdivision models in the context of isogeometric analysis, the quality of the parameterization has to be considered in addition to the geometric quality of the control mesh. Following Cohen et al.’s concept of model quality in addition to mesh quality, we present a parameterization quality metric tailored for Catmull–Clark (CC) solids. It measures the quality of the limit volume based on a quality measure for conformal mappings, revealing local distortions and singularities. We present topological operations that resolve these singularities by splitting certain types of boundary cells that typically occur in interactively designed CC-solid models. The improved models provide higher parameterization quality that positively affects the simulation results without additional computational costs for the solver.

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Ewald, Tobias; Reif, Ulrich [Referent]; Hormann, Kai [Korreferent]

Analyse geometrischer univariater Subdivisionsalgorithmen

2020

Darmstadt, TU, Diss., 2020

Subdivisionsalgorithmen generieren Freiformgeometrien durch iteratives Verfeinern polygonaler Daten, bspw. Polygonzüge bei univariater Subdivision. Dabei kann die Frage "Konvergiert die Polygonzugfolge gegen eine Grenzkurve und wie glatt ist diese?" im klassischen Fall linearer Algorithmen mit einer systematischen Regularitätstheorie beantwortet werden. Für nichtlineare Verfahren im Euklidischen, die unvermeidliche Nachteile linearer Algorithmen umgehen, gibt es nur Einzeluntersuchungen oder numerische Experimente. Diese Arbeit führt die große Klasse der geometrischen Subdivisionsschemata (GLUED-Schema) ein, zeigt für sie eine universelle $C^{2,\\alpha}$-Regularitätstheorie und gibt erstmalig rigorose Glattheitsnachweise für prominente Beispiele an. Besagte Klasse erweitert sich im Nichtstationären auf die GLUGs-Schemata, für die eine Konvergenztheorie angegeben ist. Letztlich vereinheitlicht eine allgemeingültige Proximitätstheorie für beliebige Algorithmen und beliebige $C^k$-Glattheit, genannt PAS-Theorie, die GLUED-, GLUGs- und lineare Theorien.

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Luu, Thu Huong; Altenhofen, Christian; Ewald, Tobias; Stork, André; Fellner, Dieter W.

Efficient Slicing of Catmull–Clark Solids for 3D Printed Objects with Functionally Graded Material

2019

Computers & Graphics

In the competition for the volumetric representation most suitable for functionally graded materials in additively manufactured (AM) objects, volumetric subdivision schemes, such as Catmull-Clark (CC) solids, are widely neglected. Although they show appealing properties, e_cient implementations of some fundamental algorithms are still missing. In this paper, we present a fast algorithm for direct slicing of CC-solids generating bitmaps printable by multi-material AMmachines. Our method optimizes runtime by exploiting constant time limit evaluation and other structural characteristics of CCsolids. We compare our algorithm with the state of the art in trivariate trimmed spline representations and show that our algorithm has similar runtime behavior as slicing trivariate splines, fully supporting the benefits of CC-solids.