Many applications of density-based topology optimization require a very fine mesh, either to obtain high-resolution designs, or to resolve physics in sufficient detail. Solving the discretized state and adjoint equation in every iteration step then becomes computationally demanding, restricting the applicability of the method. Model Order Reduction (MOR) offers a solution for this computational burden, using a reduced vector basis for simulations and resulting in a higher computational speed and reduced storage requirements. When the geometry is separable and the 3D density field can be expressed as a linear combination of products of lower-coordinate (1D or 2D) basis functions, Proper Generalized Decomposition (PGD) is a promising emerging MOR technique. PGD computes the basis functions for the state field on-the-fly as the reduced problem is solved, making it an method. This paper studies the application of PGD in the context of topology optimization. It is used for the optimization of a 3D ribbed floor for minimum elastic compliance and the optimization of a heat sink device for minimum thermal compliance. The geometry of the designs can be expressed as a sum of products of 2D functions of the in-plane coordinates (representing the rib/fin pattern), and 1D functions of the out-of-plane coordinate (representing the distinction between compression slab and ribs or between baseplate and fins). Numerical results demonstrate that PGD can significantly reduce the computational demand, with computation times 100 to 500 times lower than the full 3D approach. The results show that PGD holds promise for large-scale topology optimization of problems with separable geometries.

中文翻译：

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在最小合规拓扑优化的背景下，针对可分离几何形状的问题进行适当的广义分解


基于密度的拓扑优化的许多应用都需要非常精细的网格，以获得高分辨率的设计，或者足够详细地解决物理问题。在每个迭代步骤中求解离散状态和伴随方程变得计算量很大，限制了该方法的适用性。模型降阶 (MOR) 为这种计算负担提供了一种解决方案，它使用减少的矢量基础进行模拟，从而提高计算速度并减少存储要求。当几何结构可分离并且 3D 密度场可以表示为低坐标（1D 或 2D）基函数乘积的线性组合时，适当广义分解 (PGD) 是一种有前途的新兴 MOR 技术。 PGD​​ 在解决简化问题时实时计算状态场的基函数，使其成为一种方法。本文研究了 PGD 在拓扑优化中的应用。它用于优化 3D 肋状地板以实现最小弹性顺应性，以及优化散热器设备以实现最小热顺应性。设计的几何形状可以表示为面内坐标的 2D 函数（代表肋/鳍图案）和面外坐标的 1D 函数（代表压缩板和板之间的区别）的乘积之和。肋或底板和翅片之间）。数值结果表明，PGD 可以显着降低计算需求，计算时间比完整 3D 方法低 100 到 500 倍。结果表明，PGD 有望用于具有可分离几何形状的问题的大规模拓扑优化。