Integrated form finding, simulation and optimization using the generalized force density method based on mixed functionals

该项目的目标是将力密度法(KDM)扩展为一种通用的工具，用于解决形状寻找、优化、结构模拟和切割图形分析等混合问题。KDM将进一步发展成为一种彻底的结构分析工具，它不仅可以处理完全新颖的形状查找任务，而且可以以显著更高的精度处理传统的形状查找任务。方法论上的新方面是将力密度视为变形之外的等价变量。作为力和几何之间的调节剂的力密度的特殊公式补充了对能量共轭力和位移的经典理解。由于力密度的重新参数化，已知(非线性)问题的数学性质发生了显著变化；仿真、找形和裁剪图形分析可以智能而有效地结合在一起处理。它开辟了从力密度到标准静力学的新视角，包括直接和伴随模拟以及灵敏度分析。由于它与连续介质力学的一致集成，该方法在数学和数值上都是可靠的。作为标量的力密度也可以容易和清晰地显示为3D结构，这应该用于开发交互方法和与实践相关的实施。最后但并非最不重要的是，目标是与相关工程部门的合作伙伴一起展示大型、令人印象深刻的例子的实际相关性，这些合作伙伴显著推动了当前的技术水平，并开辟了新的应用领域。方法和模型是在自己的或开源软件(Kratos、Carat++、Teda、Kiwi！3D)中实现的。

The proposed project sets itself the goal of expanding the force density method (KDM) into a universally applicable tool for solving mixed problems of form finding, optimization, structural simulation and cutting pattern analysis. The KDM is to be developed into a further, thorough tool of structural analysis, which allows to handle not only completely novel but also conventional tasks of form finding with significantly better accuracy. The methodically new aspect is to treat the force densities as equivalent variables in addition to the deformations. The classical understanding of energetically conjugated forces and displacements is complemented by the special formulation of the force density as moderator between force and geometry. As a result of the reparameterization by means of force densities, the mathematical properties of known (nonlinear) questions change significantly; simulation, form finding and cutting pattern analysis can be smart and efficiently dealt with together. It opens up new perspectives along the force densities into standard statics, including direct and adjoint simulation and sensitivity analysis. Because of its consistent integration into continuum mechanics, the method will be mathematically and numerically sound. The force densities as scalar quantities can also be easily and clearly visualized for 3D structures, which should be used for the development of interactive methods and practice-relevant implementations. Last but not least, the goal is to demonstrate the practical relevance of large, impressive examples together with cooperation partners from the relevant engineering sector, which significantly advance the current state of the art and open up new fields of application. Methods and models are implemented in own or open source software (KRATOS, Carat ++, TeDA, kiwi!3d).