Optimization of materials and structures is a crucial step in the design of man-made mechanical components for a wide field of engineering applications. It also plays a key role in mechanobiology of living systems, being involved by nature across the scales, from single-cell to tissues and organs, as a strategy to minimize metabolic cost and maximize biomechanical performances. The synergy between the continuously increasing development of high-resolution 3D printing technologies and the possibility to predict chemical and physical properties through molecular dynamics-based numerical analyses has recently contributed to boost the use of both design and topology optimization procedures. They are employed in ab initio simulations as key strategies for deciding microstructures to improve mechanical performances and, concretely, to achieve prototypes of new material components. With this in mind, we here propose to abandon the classical approach of using a single scalar objective function employed in the classical design and topology optimization strategies, to introduce multiple quantities to be minimized, identified as the differences between material yield stress and the maximum von Mises stress. After mathematically justifying the well-posedness of this unconventional choice for the case at hand, it is highlighted that the proposed strategy is based on the concept of "equalizing" a proper stress measure at any point of the body and, for this reason, it is baptized as Galilei’s optimization, in honor of the Italian scholar who somehow first wondered about the possibility of changing sizes of beams to have uniform internal forces and, in turn, minimum weight. By exploiting analytical solutions and ad hoc implementing a parametric finite element algorithm to be applied to a wide variety of solids with arbitrary complex structural geometries, including nested or hierarchically organized architectures, it is first demonstrated that the proposed optimization strategy roughly retraces principles invoked by nature to guide growth, remodeling and shaping of biomaterials. More importantly, by means of several benchmark examples, we finally show the proposed procedure might be also helpfully employed to conceive a new class of micro-structured, eventually 3D-printed materials exhibiting surprising post-elastic properties, such as high overall resilience and toughness, in particular obtaining a decrease of stress concentration and a slowing down of crack propagation as direct effects of the optimization, which de facto minimizes stress gradients wherever in the solid domain.

中文翻译：

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设计应力以优化和加固类桁架结构

材料和结构的优化是为广泛的工程应用领域设计人造机械部件的关键步骤。它还在生命系统的机械生物学中发挥着关键作用，自然涉及从单细胞到组织和器官的各个尺度，作为最小化代谢成本和最大化生物力学性能的策略。高分辨率 3D 打印技术的不断发展与通过基于分子动力学的数值分析预测化学和物理特性的可能性之间的协同作用最近促进了设计和拓扑优化程序的使用。它们在 ab initio 模拟中被用作决定微观结构以提高机械性能的关键策略，具体而言，实现新材料组件的原型。考虑到这一点，我们在这里建议放弃在经典设计和拓扑优化策略中使用单个标量目标函数的经典方法，引入多个要最小化的量，确定为材料屈服应力和最大 von 之间的差异。米塞斯压力。在数学上证明了这种非常规选择对手头情况的适定性后，强调所提出的策略是基于在身体的任何一点“均衡”适当的压力测量的概念，因此，它被称为伽利略的优化，以纪念这位意大利学者，他以某种方式首先想知道改变梁的尺寸以获得均匀的内力，进而使重量最小的可能性。通过利用分析解决方案和临时实施参数化有限元算法以应用于具有任意复杂结构几何形状的各种实体，包括嵌套或分层组织的体系结构，首先证明了所提出的优化策略粗略地追溯了自然调用的原则指导生物材料的生长、重塑和成型。更重要的是，通过几个基准示例，我们最终展示了所提出的程序也可能有助于构思一类新的微结构、最终 3D 打印的材料，这些材料表现出令人惊讶的后弹性特性，例如高整体弹性和韧性，特别是作为优化的直接效果，获得应力集中的减少和裂纹扩展的减慢，