论文标题
通过最佳质量分布的方法设置自由材料设计问题
Setting the Free Material Design problem through the methods of optimal mass distribution
论文作者
论文摘要
该论文介绍了旨在从弹性材料中构建最不合规的结构的自由材料设计（FMD）问题，其本构磁场以张量值的量度$λ$的形式扮演设计变量的作用，该量在设计域中支持。从点角度来看，本构张量被转介给给定各向异性类$ \ mathscr {h} $，而成本$ c（λ）$的积分从上面有限。构成量张量将凸$ p $ - 均匀的弹性潜力$ j $进行了参数化。这项工作提出了存在的结果，并表明可以将原始问题简化为G.Bouchitté和G. Buttazzo的最佳质量分布理论的线性约束问题（LCP）。 （FMD）和（LCP）的定理链接解决方案允许有效解决原始问题。开发的理论涵盖了文献中已知的几种最佳各向异性设计问题，并解锁了新的优化问题，包括设计弹性响应在张力和压缩中不对称的材料制成的结构。通过采用明确得出的最佳条件，我们给出了几个最佳设计的分析示例。
The paper deals with the Free Material Design (FMD) problem aimed at constructing the least compliant structures from an elastic material the constitutive field of which play the role of the design variable in the form of a tensor valued measure $λ$ supported in the design domain. Point-wise the constitutive tensor is referred to a given anisotropy class $\mathscr{H}$ while the integral of a cost $c(λ)$ is bounded from above. The convex $p$-homogeneous elastic potential $j$ is parameterized by the constitutive tensor. The work puts forward the existence result and shows that the original problem can be reduced to the Linear Constrained Problem (LCP) known from the theory of optimal mass distribution by G. Bouchitté and G. Buttazzo. A theorem linking solutions of (FMD) and (LCP) allows to effectively solve the original problem. The developed theory encompasses several optimal anisotropy design problems known in the literature as well as it unlocks new optimization problems including design of structures made of a material whose elastic response is dissymmetric in tension and compression. By employing the explicitly derived optimality conditions we give several analytical examples of optimal designs.
