Convolution hierarchical deep-learning neural network (C-HiDeNN) with graphics processing unit (GPU) acceleration

We propose the Convolution Hierarchical Deep-learning Neural Network (C-HiDeNN) that can be tuned to have superior accuracy, higher smoothness, and faster convergence rates like higher order finite element methods (FEM) while using only linear element’s degrees of freedom. This is based on our newly developed convolution interpolation theory (Lu et al. in Comput Mech, 2023) and this article focuses on the deep-learning interpretation of C-HiDeNN with graphics processing unit (GPU) programming using JAX library in Python. Instead of increasing the degrees of freedom like higher order FEM, C-HiDeNN takes advantage of neighboring elements to construct the so-called convolution patch functions. The computational overhead of C-HiDeNN is reduced by GPU programming and the total solution time is brought down to the same order as commercial FEM software running on a CPU, however, with orders of magnitude better accuracy and faster convergence rates. C-HiDeNN is locking-free regardless of element types (even with 3-node triangular elements or 4-node tetrahedral elements). C-HiDeNN is also capable of r-h-p-mesh adaptivity like its predecessor HiDeNN (Zhang et al. in Comput Mech 67:207–230, 2021) with additional “a” (dilation parameter) adaptivity that stems from the convolution patch function and “p” adaptivity with higher accuracy and with the same degrees of freedom as that of the linear finite elements. C-HiDeNN potentially has myriad future applications in multiscale analysis, additive and advanced manufacturing process simulations, and high-resolution topology optimization. Details on these applications can be found in the companion papers (Lu et al. 2023; Saha et al. in Comput Mech, 2023; Li et al. in Comput Mech, 2023) published in this special issue.