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

近十年来，几何建模与数值分析的结合已成为计算力学研究的热点。等几何分析（伊加）之所以取得巨大成功，主要原因之一是需要减少将几何设计转换为适合分析的对象的工作量。我们的建议从一个与伊加稍有不同的角度来解决这个一般性问题。虽然IsogeometricAnalysis的大多数结果可用于薄壁结构或那些可以容易地从二维几何形状映射的结构，但我们专注于实体，通常是庞大的模型，包括通过过程CAD几何形状或新建议的V模型获得的模型。作为结构分析工具，我们将有限单元法（FCM），一种高阶虚拟域方法集成到CAD到分析链中。在第一个项目阶段的范围内，可以证明，构造实体几何（CSG）和有限单元法完美地结合在一起。对于FCM来说，内外检验是关键，可以通过布尔运算来实现，CSG模型固有的水密性使得几何愈合变得不必要。然而，许多现有的CAD系统不仅应用CSG，而且通过描述其表面来建模实体。通常，甚至两种基本建模范式的混合也会应用于实践中，这意味着在许多情况下有限元网格存在缺陷，并且需要进行网格修复。遵循这个项目的总体目标，即紧密结合实体建模和数值分析，第二个项目阶段的第一个中心目标是通过将FCM耦合到B-Rep来避免几何和网格愈合。我们将边界表示模型的数学概念扩展到一类更一般的模型，包括日常CAD实践中称为有缺陷或脏几何的大部分。对于这些模型，we will develop发展methods方法for a generalizedpoint广义point点membership成员test测试.这些测试将成为计算单元计算的基础，并大大简化从CAD到分析的过渡。第二个主要目标是将CAD分析集成扩展到实体模型，其中包括结构本身内部的信息。最近一个突出的例子，这种广义类型的固体是V模型，其中属性的内部的一个机构描述的领域的NURBS。虽然我们已经有了大量的经验与实体模型离散bypecewise常数场定义的体素（如CT扫描），问题有关的光滑性的属性功能，到条件的一致性NURBS空间为这些属性和近似空间的有限细胞，收敛性能耦合V-模型/FCM计划和一个有效的实现将进行调查。

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.