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

在过去的十年中，几何建模和数值分析的更好的结合已经成为计算力学研究的焦点。需要减少将几何设计转移到适合分析的对象的努力是等几何分析（IGA）取得巨大成功的原因之一。我们的提案从与IGA不同的角度来解决一般问题。虽然等几何分析的大多数结果可用于薄壁结构或那些可以很容易地从二维几何中映射的结构，但我们专注于固体，通常是笨重的模型，包括那些由程序cad几何或新建议的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，可以看到，构造立体几何（CSG）和有限单元法完美地结合在一起。对于FCM至关重要的内外测试可以通过布尔运算来实现，并且csg模型固有的水密性使得几何修复变得不必要。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应用模型实体。在实践中，通常甚至是两种基本建模范式的混合应用，这意味着在许多情况下有缺陷的有限元网格和网格修复的必要性。遵循该项目的总体目标，即紧密集成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。我们将把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实践中所谓的有缺陷或脏几何的很大一部分。对于这些模型，我们将开发一般化点隶属度测试的方法。然后，这些测试将成为FiniteCell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。这种广义类型的一个突出例子是v模型，其中物体内部的属性由NURBS域描述。虽然我们已经在实体模型离散方面有了丰富的经验，通过定义在体素上的常数场（例如CT扫描），但与属性函数的平滑性，这些属性的NURBS空间和有限单元的近似空间的一致性条件，耦合v模型/FCM方案的收敛性以及有效实现相关的问题将被研究。

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.