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）扩展为一种普遍适用的工具，用于解决寻形、优化、结构模拟和切割模式分析等混合问题。KDM将发展成为一个进一步的，彻底的结构分析工具，它不仅可以处理完全新颖的，而且可以处理传统的形式查找任务，其精度显着提高。方法上的新方面是将力密度作为变形的等效变量来处理。力密度作为力和几何之间的调节因子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、找形和切割模式分析可以智能、高效地一起处理。它开辟了新的视角，沿着力密度进入标准静力学，包括直接和伴随的模拟和灵敏度分析。由于它与连续介质力学的一致性，该方法在数学上和数值上都是合理的。对于三维结构，力密度作为标量也可以很容易和清晰地可视化，这应该用于开发交互方法和实践相关的实现。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，令人印象深刻的示例的实际相关性，这将大大推进当前的艺术状态并开辟新的应用领域。方法和模型在自己或开源软件（KRATOS, krat++, TeDA, kiwi!3d）中实现。

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).