基于相场法的结构拓扑optimization混合变分公式
Mixed variational formulations for structural topology optimization based on the phase-field approach
摘要：
我们提出了一个变分原理相场泛函结构拓扑优化与混合（三场）Hu-Washizu功能，然后直接包括在制定平衡，本构和协调方程。然后将所得混合变分泛函专门化以导出经典拓扑优化公式（其中待分配的材料的量是充当问题的全局约束的\emph{a priori}分配量）以及新颖拓扑优化公式（其中待分配的材料的量被最小化，因此没有对问题的预先施加的约束）。这两种制剂的数值求解通过实施混合有限元计划，与第二种方法避免了引入的全球约束，因此尊重方便的局部性质的有限元离散。此外，在所提出的方法中，可以获得用于设置相场相关模拟参数的适当值的指导方针，并且由于相场和Hu-Washizu原理的结合，可以容易地采用单片算法解决方案。一个有见地的和广泛的数值调查结果在一个详细的收敛性研究和讨论所获得的最终设计。数值结果清楚地突出了两种配方之间的差异，以及与整体解决方案策略相关的优势;数值研究解决了二维和三维应用。
英文摘要：
We propose a variational principle combining a phase-field functional for structural topology optimization with a mixed (three-field) Hu-Washizu functional, then including directly in the formulation equilibrium, constitutive, and compatibility equations. The resulting mixed variational functional is then specialized to derive a classical topology optimization formulation (where the amount of material to be distributed is an \emph{a priori} assigned quantity acting as a global constraint for the problem) as well as a novel topology optimization formulation (where the amount of material to be distributed is minimized, hence with no pre imposed constraint for the problem). Both formulations are numerically solved by implementing a mixed finite element scheme, with the second approach avoiding the introduction of a global constraint, hence respecting the convenient local nature of the finite element discretization. Furthermore, within the proposed approach it is possible to obtain guidelines for settings proper values of phase-field-related simulation parameters and, thanks to the combined phase-field and Hu-Washizu rationale, a monolithic algorithm solution scheme can be easily adopted. An insightful and extensive numerical investigation results in a detailed convergence study and a discussion on the obtained final designs. The numerical results clearly highlight differences between the two formulations as well as advantages related to the monolithic solution strategy; numerical investigations address both two-dimensional and three-dimensional applications.
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