基于变密度法的周期性拓扑优化

Periodic Topology Optimization Using Variable Density Method

板条状结构的设计域具有较大的长宽比,常规的拓扑优化方法无法获得清晰的、周期性的拓扑形式或求解困难。以结构的最小柔度为目标函数,单元相对密度为设计变量,构建周期性拓扑优化问题的数学模型。提出一种基于变密度理论固体各向同性微结构材料惩罚模型法(Solid isotropic microstructures with penalization,SIMP)的周期性拓扑优化的方法。在数学模型中设置额外的约束条件,保证优化结构可以得到周期性的拓扑形式。通过优化准则法推导出虚拟子域设计变量的迭代公式,利用体积约束计算出拉格朗日乘子。引入过滤函数解决拓扑优化容易出现数值计算不稳定,导致棋盘格、网格依赖性等问题。利用所提出的方法,通过平面矩形悬臂梁结构算例,获得平面矩形悬臂梁结构的周期性拓扑形式。结果表明,在优化过程中,出现周期性的孔洞。随着迭代次数的增加,孔洞数目没有增加,说明该方法具有较强的稳健性。子域数目取值不同时,均可以得到具有周期性的拓扑形式,且具有良好的一致性。

The design domain of lath-shaped structure has a large length-width ratio, so it is difficult to obtain a solution or a clear and periodic topology configuration using the conventional algorithm. The mathematical model for periodic topology optimization is built; in which mean compliance is taken as objective function and relative densities of elements are taken as design variables. A method for periodic topology optimization is presented using variable density method solid isotropic microstructures with penalization （SIMP）. An additional constraint condition is taken part in the mathematical model to ensure a topological structure which possesses periodicity. The iterative formula of virtual sub-domain design variables is deduced taking advantage of optimality criteria method and Lagrange multiplier is calculated using volume constraint. A filtering function is imported in order to solve checkerboard and mesh-independent. Results show that periodic holes are appeared in the process of optimization. The numbers of holes do not change as the iterative number increasing, which shows that the proposed method has stronger robustness. Periodic topology configuration which has a good consistency is achieved when the number of sub-domain is different.