Optimal fine-scale structures in compliance minimization for a uniaxial load
Optimal fine-scale structures in compliance minimization for a uniaxial load
复制标题
单轴载荷柔度最小化的最佳精细结构
DOI：
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发表时间：
2014
期刊：
影响因子：
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通讯作者：
B. Wirth
中科院分区：
文献类型：
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作者：
R. Kohn;B. Wirth
我们考虑对承受固定边界载荷的弹性结构 O⊂R2 的拓扑和几何形状进行优化，即我们的目标是最小化材料体积 Vol(O)、结构周长 Per(O) 和结构柔度 Comp(O)（这是载荷所做的功）的加权和。作为第一个简单且有启发性的案例，本文讨论了二维施加均匀单轴拉伸载荷的情况。如果周长的权重 ϵ 很小，则最佳几何形状会表现出非常精细的结构，无法通过数值优化来解决。相反，我们证明了 ε 中的最小能量如何缩放，这涉及构建一系列接近最优的几何形状，从而提供定性见解。该构造基于经典的分支程序，具有合规性最小化所特有的一些功能。能量标度的证明还需要一个与 ansatz 无关的下界，我们通过经典的凸对偶论证（仅限于二维和单轴载荷）导出一次，并通过复合材料有效弹性模量的 Hashin-Shtrikman 边界的基于傅立叶的细化导出一次。我们还强调了与标量偏微分方程约束的形状优化的密切关系和区别，以及与在 I 型超导体中间态中观察到的图案形成的联系。
We consider the optimization of the topology and geometry of an elastic structure O⊂R2 subjected to a fixed boundary load, i.e. we aim to minimize a weighted sum of material volume Vol(O), structure perimeter Per(O) and structure compliance Comp(O) (which is the work done by the load). As a first simple and instructive case, this paper treats the situation of an imposed uniform uniaxial tension load in two dimensions. If the weight ϵ of the perimeter is small, optimal geometries exhibit very fine-scale structure which cannot be resolved by numerical optimization. Instead, we prove how the minimum energy scales in ϵ, which involves the construction of a family of near-optimal geometries and thus provides qualitative insights. The construction is based on a classical branching procedure with some features unique to compliance minimization. The proof of the energy scaling also requires an ansatz-independent lower bound, which we derive once via a classical convex duality argument (which is restricted to two dimensions and the uniaxial load) and once via a Fourier-based refinement of the Hashin–Shtrikman bounds for the effective elastic moduli of composite materials. We also highlight the close relation to and the differences from shape optimization with a scalar PDE-constraint and a link to the pattern formation observed in intermediate states of type-I superconductors.
影响因子：
3
作者：
H. Knüpfer;R. V. Kohn;F. Otto
通讯作者：
F. Otto