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This paper presents a general Convolution Hierarchical Deep-learning Neural Networks (C-HiDeNN) computational framework for solving partial differential equations. This is the first paper of a series of papers devoted to C-HiDeNN. We focus on the theoretical foundation and formulation of the method. The C-HiDeNN framework provides a flexible way to construct high-order C n approximation with arbitrary convergence rates and automatic mesh adaptivity. By constraining the C-HiDeNN to build certain functions, it can be degenerated to a specification, the so-called convolution finite element method (C-FEM). The C-FEM will be presented in detail and used to study the numerical performance of the convolution approximation. The C-FEM combines the standard C 0 FE shape function and the meshfree-type radial basis interpolation. It has been demonstrated that the C-FEM can achieve arbitrary orders of smoothness and convergence rates by adjusting the different controlling parameters, such as the patch function dilation parameter and polynomial order, without increasing the degrees of freedom of the discretized systems, compared to FEM. We will also present the convolution tensor decomposition method under the reduced-order modeling setup. The proposed methods are expected to provide highly efficient solutions for extra-large scale problems while maintaining superior accuracy. The applications to transient heat transfer problems in additive manufacturing, topology optimization, GPU-based parallelization, and convolution isogeometric analysis have been discussed. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

Computationally modeling the laser powder bed fusion process can be expensive despite the fact that the laser covers a small fraction of the entire domain. Hence, we developed GO-MELT, a multilevel approach inspired by the variational multiscale method. GO-MELT calculates the full temperature solution using three coupled thermal solvers, each solving one of three overlapping domains, subsequently referred to as levels. Level 1's solver obtains a global, coarse-scale solution. Level 2's solver computes a meso-scale thermal field spanning the melt pool region. Level 3's solver obtains a fine-scale temperature field spanning the laser's immediate region. Coarser solutions provide boundary conditions for finer solvers. Finer solutions compute subgrid scales that influence coarser solvers through additional source terms. Being independently meshed, Levels 2 and 3 can track a laser along its tool path without remeshing. Moreover, fixed-sized, structured meshes allow for GPU acceleration using Google's JAX library with just-in-time compilation. Five case studies were conducted to demonstrate proper convergence of GO-MELT, to show that GO-MELT can reach the same accuracy as a uniform mesh with fewer degrees of freedom, and to quantify the computational load from each level's solver. A simulated production run averaged 1.64 ms per time step after taking 9.36 h to complete 20.5 million time steps, which is approximately 678 × faster than a uniform mesh solver with identical resolution and GPU acceleration. Future work will implement different solvers into GO-MELT to improve fidelity and speed. [Display omitted] • Developed multiscale framework inspired by VMS to combine separate thermal solvers. • Integrated GPU acceleration in Python using Google's JAX library with JIT compilation. • Simulated LPBF printing a 10 mm cube (20.5 million time steps) in 9.36 h. • Achieved 678x speedup compared to a uniform mesh of the same resolution. [ABSTRACT FROM AUTHOR]

Representative volume element (RVE)-based analysis plays a central role in understanding the response of heterogeneous materials to properties and geometry of the constituents. However, the accuracy of RVE analysis on real-life materials requires extra effort on the identification of material constituents and characterization of imperfections (e.g., voids and cracks) introduced in the fabrication process. For these reasons, together with the multiscale and spatially varying nature of heterogeneities, analysis of heterogeneous materials can be prohibitively time-consuming. In this work, a fully convolutional network (FCN)-based framework called RVE-net is proposed to take advantage of the state-of-art use of FCNs in image segmentation and feedforward neural networks in universal approximation to accelerate multiscale analysis, identify microscale material properties, and automatically characterize defects in materials. In contrast with standard numerical methods (e.g., the finite element method), which depend heavily on domain discretization and local interpolations, the RVE-net takes microstructure images — parameterized by a coupled Heaviside and level-set field representation — and loading conditions as inputs. The aim is to directly learn the nonlinear interaction between the microstructures and their local responses in a hierarchical manner. This avoids burdensome discretization and interpolations, makes it possible to transfer the learned structure-response from one microstructure to another, and thus significantly accelerates the modeling of heterogeneous materials. Several numerical examples are performed to examine the performance of the proposed RVE-net. It is demonstrated that the RVE-net can leverage the power of graphics processing units (GPUs) in RVE analysis, inverse derivation of material constituents, and characterization of defects. • Fully convolutional neural network, RVE-net, developed for feature-driven analysis. • RVE-net takes microstructure images and loading conditions as inputs. • Method allows simultaneous parametric study on both geometry and load conditions. • RVE-net can discover both quantities and fields of interest from external data. • RVE-net enables knowledge transfer from existing to new instances. [ABSTRACT FROM AUTHOR]

We are witnessing a rapid transition from Software 1.0 to 2.0. Software 1.0 focuses on manually designed algorithms, while Software 2.0 leverages data and machine learning algorithms (or artificial intelligence) for optimized, fast, and accurate solutions. For the past few years, we have been developing Convolution Hierarchical Deep-learning Neural Network Artificial Intelligence (C-HiDeNN-AI), which enables the realization of Engineering Software 2.0 by opening the next-generation neural network-based computational tools that can simultaneously train data and solve mechanistic equations. This paper focuses on solving partial differential equations with C-HiDeNN. Still, the same neural network can be used for training and calibration with experimental data, which will be discussed in a separate paper. This paper presents a computational framework combining the C-HiDeNN theory with isogeometric analysis (IGA), called Convolution IGA (C-IGA). C-IGA has five key features that advance IGA: (1) arbitrarily high-order smoothness and convergence rates without increasing degrees of freedom; (2) a Kronecker delta property that enables direct imposition of Dirichlet boundary conditions; (3) automatic and flexible global/local mesh-adaptivity with built-in length scale control and adjustable radial basis functions; (4) ability to handle irregular meshes and triangular/tetrahedral elements; and (5) GPU implementation that speeds up the program as fast as finite element method (FEM). Mathematically, we prove that both IGA and C-IGA mappings are equivalent, and by taking a special design and modified anchors as nodes, C-IGA degenerates to IGA. We demonstrate the accuracy, convergence rates, mesh-adaptivity, and performance of C-IGA with several 1D, 2D, and 3D numerical examples. The future applications of C-IGA from topology optimization to product manufacturing with multi-GPU programming are discussed. [ABSTRACT FROM AUTHOR]