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A class of Nonlocal Matching Boundary Condition (NMBC) is presented for multiscale analysis that employs non-ordinary peridynamics (NOPD) with correspondence material model. The NMBC is cast in the form of parameterized expressions involving displacements and their higher-order time derivatives of the peridynamic (PD) nodes at the numerical interface. The corresponding parameters are solved by zeroing the associated residual and its higher order derivatives that are functions of the dispersion relation at the particular wave length of interest. After deriving the dispersion relations for both the standard and stabilized versions of NOPD with the correspondence material model, NMBCs are established for both 1D and 2D and the robustness is demonstrated in numerical examples. It is shown that stabilized NOPD effectively reduces the effects of the zero energy modes in the standard NOPD. Furthermore, nonlocal implementation is essential in eliminating the edge effects at the interface. [ABSTRACT FROM AUTHOR]

In this paper we propose a class of central difference schemes for resolving the Caputo fractional derivative. The accuracy may reach any selected integer order. More precisely, the Caputo fractional derivative operator is decomposed into symmetric and antisymmetric components. Starting from difference schemes of lower order accuracy for each component, we enhance the accuracy by a weighted average of shifted differences. The weights are calculated by matching the symbols of the scheme and the operators. We further illustrate the application of the proposed schemes to a fractional advection–diffusion equation. Together with the Crank–Nicolson algorithm, it reaches designed accuracy order, and is unconditionally stable. Numerical tests are presented to demonstrate the nice features. [ABSTRACT FROM AUTHOR]

This paper presents an in-depth numerical analysis of spatial fractional advection–diffusion equations (FADE) utilizing the finite element method (FEM). A traditional Galerkin finite element formulation of the pure fractional diffusion equation without advection may yield numerical oscillations in the solution depending on the fractional derivative order. These oscillations are similar to those that may arise in the integer-order advection–diffusion equation when using the Galerkin FEM. In a Galerkin formulation of a FADE, these oscillations are further compounded by the presence of the advection term, which we show can be characterized by a fractional element Peclet number that takes into account the fractional order of the diffusion term. To address this oscillatory behavior, a Petrov–Galerkin method is formulated using a fractional stabilization parameter to eliminate the oscillatory behavior arising from both the fractional diffusion and advection terms. A compact formula for an optimal fractional stabilization parameter is developed through a minimization of the residual of the nodal solution. Steady state and transient one-dimensional cases of the pure fractional diffusion and fractional advection–diffusion equations are implemented to demonstrate the effectiveness and accuracy of the proposed formulation. [ABSTRACT FROM AUTHOR]

This paper presents a tensor decomposition (TD) based reduced-order model of the hierarchical deep-learning neural networks (HiDeNN). The proposed HiDeNN-TD method keeps advantages of both HiDeNN and TD methods. The automatic mesh adaptivity makes the HiDeNN-TD more accurate than the finite element method (FEM) and conventional proper generalized decomposition (PGD) and TD, using a fraction of the FEM degrees of freedom. This work focuses on the theoretical foundation of the method. Hence, the accuracy and convergence of the method have been studied theoretically and numerically, with a comparison to different methods, including FEM, PGD, TD, HiDeNN and Deep Neural Networks. In addition, we have theoretically shown that the PGD/TD converges to FEM at increasing modes, and the PGD/TD solution error is a summation of the mesh discretization error and the mode reduction error. The proposed HiDeNN-TD shows a high accuracy with orders of magnitude fewer degrees of freedom than FEM, and hence a high potential to achieve fast computations with a high level of accuracy for large-size engineering and scientific problems. As a trade-off between accuracy and efficiency, we propose a highly efficient solution strategy called HiDeNN-PGD. Although the solution is less accurate than HiDeNN-TD, HiDeNN-PGD still provides a higher accuracy than PGD/TD and FEM with only a small amount of additional cost to PGD. • A novel reduced order deep learning interpolant, HiDeNN-TD is proposed. • HiDeNN-TD allows r-adaptivity in each separated spatial dimension. • HiDeNN-TD achieves a high accuracy with orders of magnitude fewer DoFs than FEM. • PGD/TD solution errors consist of mesh discretization and mode reduction errors. • The proposed variant HiDeNN-PGD has a comparable computational cost to PGD. [ABSTRACT FROM AUTHOR]

Virtual clustering analysis (VCA) has been developed for numerical homogenization of heterogeneous material. The integral form of the material system is the Lippmann–Schwinger equation, which imposes boundary condition at infinity for fictitious surrounding homogeneous reference material. The artificially chosen reference stiffness induces a distribution for traction on the material boundary. The deviation from a uniform loading traction boundary condition degrades the accuracy in predicting the average stiffness of the material under consideration. In this work, we suggest that the induced traction should be within one standard deviation from the loading traction, and propose an adaptive strategy to update the reference stiffness. Numerical tests for inclusion problems with elasto-plasticity compositions verify the effectiveness of the proposed strategy. [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]