Convolution Hierarchical Deep-Learning Neural Network Tensor Decomposition (C-HiDeNN-TD) for high-resolution topology optimization

Deep learning in computational mechanics: a review

“…23 The dilation parameter depends on x and thus introduces additional degrees of freedom throughout the domain. 24 After close examination and exchanges with the authors of [506,508,[510][511][512], we have concluded that the current speed-up is mainly attributable to the simultaneously employed reduced order models. Minor improvements in accuracy are possible through the employed r -adaptivity and convolutions, however, accompanied by an increase in computational effort.…”

“…A different approach are the hierarchical deep-learning NNs (HiDeNNs) [506] with extensions in [507][508][509][510][511][512]. Here, shape functions are treated as NNs constructed from basic building blocks.…”

“…Special care has to be taken to avoid element inversion, which is handled by an additional loss term. Inverse problems can similarly be solved by using learnable input parameters, as presented for topology optimization [512].…”

“…The enhanced space has been incorporated in the HiDeNN framework. While an independent confirmation is still missing, the authors promise a speedup of several orders of magnitude compared to traditional finite element solvers [512] 24 . 23 The dilation parameter depends on x and thus introduces additional degrees of freedom throughout the domain.…”

“…Minor improvements in accuracy are possible through the employed r -adaptivity and convolutions, however, accompanied by an increase in computational effort. Furthermore, we have been informed that the hiDeNN methodology is implemented most efficiently in the standard finite element way, i.e., without NNs, unlike the descriptions in [506,508,[510][511][512]. Thus the shape functions N e i (x) are implemented in a straightforward manner within automatic differentiation frameworks in order to obtain the derivatives of the solution u with respect to x, and the loss function with respect to the nodal positions x e i and degrees of freedom u e .…”