Geometry-Analysis Integration Extended: The Power of Immersed Boundary Methods


Isogeometric analysis (IGA) has been in the focus of an increasingly large number of researchers since its event in 2007. Constructed by spline spaces with higher order continuity the attraction of this extension to finite element methods is based on two major pillars: Use of the same spaces for description of the geometry and Ansatz functions can drastically reduce the effort for transition from CAD models to analysis, and the higher continuous approximation spaces themselves often offer significant advantages w.r.t. computational effort and accuracy.

Surface models or thin-walled structures are very often described by NURBS or similar spline-based functions and therefore ideally fit to the IGA paradigm. Yet, in engineering practice many more types of geometric models are used. The long list starts with models applying Constructive Solid Geometry (CSG), which is frequently combined with surface models and extended to feature-based designs. Completely different model types result from tomographic methods, where a body is defined only by a discretized density distribution, or from photographic images yielding point clouds to approximate the surface of a structure. Furthermore, problems in the transition process from (any) geometric to an analysis suitable model frequently result from flawed designs with imprecise or mathematically inconsistent, ‘dirty’ geometry.

In this presentation it will be shown how immersed boundary or embedded domain methods significantly extend opportunities for a geometry-analysis integration beyond isogeometric analysis. We will use low- and high-order finite element spaces as well as spline-based, ‘IGA-like’ spaces and demonstrate that classical techniques like local mesh adaptation can be applied in a completely analogous way to boundary fitting methods. Applications to various types of geometric models including questions on the robustness w.r.t. ‘dirty’ geometry will be discussed. We will address an extension of the method to fracture problems and present an example for a camera-to-analysis workflow, where a point cloud of an object is obtained from photos of a cell-phone camera and directly transferred, without surface reconstruction and mesh generation, to a three-dimensional structural analysis.