Coordination Funds

3D实体数字化和数字化制造的最新发展引发了可在自动化CAD/CAE/CAM工作流程中处理的典型零件的复杂性的巨大增长。设计不再是一项纯粹的几何任务，而是需要考虑最终工件的所需功能和适用性。这意味着有必要将结构分析纳入这一过程。因此，对底层数字3D模型进行繁琐且容易出错的转换是工业中的既定工作流程。为了避免这种情况，出现了新的研究领域，如“等距分析”和“几何处理”，它们分别使用NURBS或多边形网格作为通用的几何参考模型。除了依赖于一个模型的优点外，缺点是在每个工作步骤中参考模型必须满足不同的要求。这些要求从拓扑约束(例如无自交)到结构分析要求(例如结构模型的类型)到几何要求(例如单元的凸度、纵横比)，并源于特定工作步骤的特定目标。几何处理的目标是以较小的几何逼近误差获得精确的表示，同时降低图形应用的计算成本。实现此目标的一种方法是聚集最佳拟合区域，从而生成多边形曲面网格。对于结构分析，计算力学采用典型的有限元程序，这会导致结构应力和应变状态的物理逼近误差。减少此误差的一种方法是通过细化网格来引入更多自由度。计算力学的目的是在减少计算量的同时最小化物理逼近误差。虽然存在独立的解决方案，但需要将数字3D几何描述的特征与有限元技术相结合。我们的目标是针对模拟的特定要求而定制的多边形网格生成方法，以及利用多边形网格的灵活性的新的有限元分析方法。与光滑NURBS曲面或标准有限元网格相比，从平面面片到自由曲面片的任意形状单元在网格生成和自适应方面提供了更大的灵活性。因此，有必要结合数值数学、几何处理和计算力学的专业知识，开发新的方法，使一般多边形网格能够同时用于设计和分析。多边形网格可以在灵活适应局部几何特征的同时，为目标几何图形提供高效的计算表示(只需很少的单元)。此外，这种灵活性为局部调整和优化网格结构提供了新的可能性，从而允许网格生成和模拟的紧密交错操作。

Recent developments in the digitization of 3D bodies as well as in digital fabrication have triggered a tremendous growth in complexity of typical workpieces that can be handled in automated CAD/CAE/CAM workflows. The design is no longer a purely geometric task but needs to take the desired functionality and serviceability of the final workpiece into account. It implies the necessity to include structural analysis into the process. Therefore, a tedious and error-prone conversion of the underlying digital 3D models is an established workflow in industry. To avoid this, novel research areas have emerged like "Isogeometric Analysis" and "Geometry Processing" where NURBS or polytope meshes, respectively, are used as a common geometric reference model. Besides the advantages of relying on one model, there is a disadvantage that in each work step the reference model has to satisfy different requirements. These vary from topological constraints (e.g. self-intersection free) to structural analysis requirements (e.g. type of structural model) up to geometric requirements (e.g. convexity, aspect ratio of cells) and stem from the particular aims of the specific work steps. Geometry processing aims for an accurate representation with a small geometric approximation error while reducing the computational cost for graphics applications. A way to reach this goal is to cluster best-fitting regions resulting in a polygonal surface mesh. For the structural analysis computational mechanics employs typically finite element (FE) programs, which lead to a physical approximation error of the stress and strain state of the structure. One way to reduce this error is to introduce additional degrees of freedom by refining the mesh. The aim of computational mechanics is to minimize the physical approximation error while reducing the computational effort. While stand-alone solutions exist, there is a need to combine the features of digital 3D geometry descriptions with finite element techniques. We aim at polytope mesh generation methods that are tailored to the specific requirements of the simulation, as well as new FE analysis methods that exploit the flexibility of polytopic meshes. Compared to smooth NURBS surfaces or standard FE meshes, arbitrarily shaped elements, ranging from planar facets to freeform patches, offer more flexibility in mesh generation and adaptation. Therefore, it is necessary to combine the expertise from numerical mathematics, geometry processing and computational mechanics to develop novel methods that enable the usage of general polytope meshes for design and analysis simultaneously. Polytope meshes have the potential to provide a computationally efficient representation (with few cells) of the target geometry while adapting flexibly to local geometric features. Moreover, this flexibility opens up new possibilities to locally adjust and refine the mesh structure allowing for a tightly interleaved operation of mesh generation and simulation.