Proper generalized decomposition in the context of minimum compliance topology optimization for problems with separable geometries

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 a priori 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.