Coordination Funds

3D实体数字化以及数字化制造的最新发展引发了典型工件复杂性的巨大增长，这些工件可以在自动化CAD/CAE/CAM工作流程中处理。设计不再是一个纯粹的几何任务，而是需要考虑到最终工件的预期功能和可用性。它意味着在过程中包含结构分析的必要性。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已建立的工作流程。为了避免这种情况，出现了新的研究领域，如“等几何分析”和“几何处理”，其中NURBS或多面体网格分别被用作常见的几何参考模型。除了依赖一个模型的优点之外，还有一个缺点是在每个工作步骤中，参考模型必须满足不同的要求。从拓扑约束（如无自交）到结构分析要求（如结构模型类型），再到几何要求（如凸性、细胞的纵横比），这些要求各不相同，并且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近似误差获得精确的表示，同时降低图形应用的计算成本。实现这一目标的一种方法是将最佳拟合区域聚类，从而形成多边形表面网格。对于结构分析，计算力学通常采用有限元（FE）程序，这导致结构的应力和应变状态的物理近似误差。减少这种误差的一种方法是通过细化网格来引入额外的自由度。计算力学的目标是在减少计算量的同时使物理近似误差最小化。虽然存在独立的解决方案，但仍需要将数字3D几何描述的特点与有限元技术相结合。我们的目标是针对模拟的特定要求量身定制的多面体网格生成方法，以及利用多面体网格灵活性的新有限元分析方法。与光滑的NURBS表面或标准FE网格相比，任意形状的单元，从平面切面到自由形状的斑块，在网格生成和适应方面提供了更大的灵活性。因此，有必要将数值数学、几何处理和计算力学的专业知识结合起来，开发出能够同时使用通用多面体网格进行设计和分析的新方法。多面体网格在灵活地适应局部几何特征的同时，具有提供目标几何的高效计算表示（较少的单元）的潜力。此外，这种灵活性为局部调整和完善网格结构开辟了新的可能性，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.