三维空间单轴载荷下柔度最小化的最优细尺度结构
Optimal fine-scale structures in compliance minimization for a uniaxial load in three space dimensions
摘要：
我们考虑的形状和拓扑最优化问题设计的结构，最大限度地减少材料消耗和（线性）弹性顺应性在一个固定的给定的边界载荷的加权和。众所周知，这个问题通常不是适定的，因为它的解决方案通常需要使用无限精细的微观结构。因此，我们研究奇摄动问题的影响，增加了结构周长的成本。对于二维空间中的单轴和剪切载荷，文献中已经推导出相应的能量标度律。这项工作现在推导出在三维空间中的单轴载荷的情况下的标度律，这可以被认为是最简单的三维设置。从本质上讲，它是预期（并确认在这篇文章中），对于单轴负载的遵守行为几乎像一个标量通量问题的耗散，使下界从模式分析超导体可以直接应用。然而，上界需要对从超导体已知的结构进行非平凡的修改。这些是必要的，因为在弹性力学中，人们有力矩平衡的附加约束。
英文摘要：
We consider the shape and topology optimization problem to design a structure that minimizes a weighted sum of material consumption and (linearly) elastic compliance under a fixed given boundary load. As is well-known, this problem is in general not well-posed since its solution typically requires the use of infinitesimally fine microstructure. Therefore we examine the effect of singularly perturbing the problem by adding the structure perimeter to the cost. For a uniaxial and a shear load in two space dimensions, corresponding energy scaling laws were already derived in the literature. This work now derives the scaling law for the case of a uniaxial load in three space dimensions, which can be considered the simplest three-dimensional setting. In essence, it is expected (and confirmed in this article) that for a uniaxial load the compliance behaves almost like the dissipation in a scalar flux problem so that lower bounds from pattern analysis in superconductors can directly be applied. The upper bounds though require nontrivial modifications of the constructions known from superconductors. Those become necessary since in elasticity one has the additional constraint of torque balance.
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