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On the existence of optimal shapes in architecture

机译：建筑中最佳形状存在

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摘要

We consider shape optimization problems for elasticity systems in architecture. A typical objective in this context is to identify a structure of maximal stability that is close to an initially proposed one. For structures without external forces on varying parts, classical methods allow proving the existence of optimal shapes within well-known classes of bounded uniformly Lipschitz domains. We discuss this for maximally stable roof structures. We then introduce a more general framework that includes external forces on varying parts (for instance, caused by loads of snow on roofs) and prove the existence of optimal shapes, now in a subclass of bounded uniformly Lipschitz domains, endowed with generalized surface measures on their boundaries. These optimal shapes realize the infi-mum of the corresponding energy of the system. Generalizing further to yet another, very new framework, now involving classes of bounded uniform domains with fractal measures on their boundaries, we finally prove the existence of optimal architectural shapes that actually realize the minimum of the energy. As a by-product we establish the well-posedness of the elasticity system on such domains. In an auxiliary result we show the convergence of energy functionals along a sequence of suitably converging domains. This result is helpful for an efficient approximation of an optimal shape by shapes that can be constructed in practice.
机译：我们考虑架构中弹性系统的形状优化问题。在这种情况下典型的目的是识别接近最初提出的稳定性的结构的结构。对于在不同部件上没有外部力的结构，经典方法允许在众所周知的均匀嘴唇尖端域中证明存在最佳形状的存在。我们讨论了最大稳定的屋顶结构。然后我们引入了更一般的框架，包括在不同部件上的外力（例如，由屋顶上的雪负荷引起）并证明了最佳形状的存在，现在在有界均匀嘴唇尖端域的子类中，赋予广义表面措施他们的界限。这些最佳形状实现了系统的相应能量的Infi-Mum。概括到另一个非常新的框架，现在涉及具有分形措施的有界统一结构域的阶级，我们最终证明存在最佳的架构形状，实际上实现了能量的最小值。作为副产品，我们在这种域中建立弹性系统的良好姿势。在辅助结果中，我们沿一系列适当的会聚域显示能量函数的收敛。该结果有助于通过可以在实践中构造的形状有效近似的最佳形状。

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