Integration of geometric modeling and numerical analysis based on Constructive Solid Geometry, Boundary Representation, and the Finite Cell Method Part 2: Extension to flawed geometry and volumetric spline-based models

• Applicant: Professor Dr. Ernst Rank
• Subject Area: Applied Mechanics, Statics and Dynamics
• Term: from 2013 to 2018
• Project identifier: Deutsche Forschungsgemeinschaft (DFG) - Project number 232449854

A better integration of (geometric) modelling and numerical analysishas strongly come into the focus of research in ComputationalMechanics during the last ten years. The need for reducing the effortof transferring geometric designs to analysis-suitable objects is one ofthe reasons for the huge success of Isogeometric Analysis (IGA). Ourproposal addresses the general question from a somewhat differentperspective than IGA does. Whereas most results for IsogeometricAnalysis are available for thin-walled structures or those, which canreadily be mapped from two-dimensional geometry we concentrate onsolid, typically bulky models including those obtained by proceduralCAD-geometry or the newly suggested V-models. As structuralanalysis tool we integrate the Finite Cell Method (FCM), a high orderfictitious domain approach into the CAD-to-analysis chain. Within thescope of the first project phase, it could be shown, that ConstructiveSolid Geometry (CSG) and the Finite Cell Method fit perfectlytogether. The inside-outside test being crucial for FCM could berealized by Boolean operations, and the inherent water-tightness ofCSG models rendered geometry healing unnecessary. Many presentCAD systems, however, apply not only CSG but model solids viadescribing their surfaces. Often even a mixture of both basic modelingparadigms is applied in practice, implying in many cases flawed finiteelement meshes and the necessity for mesh healing. Following thegeneral goal of this project, namely to tightly integrate solid modelingand numerical analysis, the first central objective of the secondproject phase is to avoid geometry and mesh healing by couplingFCM to B-Rep. We will extend the mathematical concept of BoundaryRepresentation models to a class of more general models including alarge part of what is called flawed or dirty geometry in daily CADpractice.For these models, we will develop methods for a generalizedpoint membership test. These tests will then be the basis for a FiniteCell computation and drastically simplify the transition from CAD toanalysis. The second major goal will be an extension of the CADanalysisintegration to solid models which include information on theinterior of a structure itself. A prominent recent example of thisgeneralized type of solids are V-models, where attributes of theinterior of a body are described by fields of NURBS. Whereas wealready have vast experience with solid models discretized bypiecewise constant fields defined on voxels (e.g. CT scans),questions related to the smoothness of attribute functions, toconditions for a consistency of NURBS spaces for these attributesand the approximation spaces of the finite cells, to convergenceproperties for coupled V-model/FCM schemes and to an efficientimplementation shall be investigated.

• DFG Programme: Research Grants
• Co-Investigator: Professor Dr.-Ing. Stefan Kollmannsberger