Search

This paper proposes localized subspace iteration (LSI) methods to construct generalized finite element basis functions for elliptic problems with multiscale coefficients. The key components of the proposed method consist of the localization of the original differential operator and the subspace iteration of the corresponding local spectral problems, where the localization is conducted by enforcing the local homogeneous Dirichlet condition and the partition of the unity functions. From a novel perspective, some multiscale methods can be regarded as one iteration step under approximating the eigenspace of the corresponding local spectral problems. Vice versa, new multiscale methods can be designed through subspaces of spectral problem algorithms. Then, we propose the efficient localized standard subspace iteration (LSSI) method and the localized Krylov subspace iteration (LKSI) method based on the standard subspace and Krylov subspace, respectively. Convergence analysis is carried out for the proposed method. Various numerical examples demonstrate the effectiveness of our methods. In addition, the proposed methods show significant superiority in treating long-channel cases over other well-known multiscale methods., Comment: 23 pages

In this paper, we propose and analyze a new semi-implicit stochastic multiscale method for the radiative heat transfer problem with additive noise fluctuation in composite materials. In the proposed method, the strong nonlinearity term induced by heat radiation is first approximated, by a semi-implicit predictor-corrected numerical scheme, for each fixed time step, resulting in a spatially random multiscale heat transfer equation. Then, the infinite-dimensional stochastic processes are modeled and truncated using a complete orthogonal system, facilitating the reduction of the model's dimensionality in the random space. The resulting low-rank random multiscale heat transfer equation is approximated and computed by using efficient spatial basis functions based multiscale method. The main advantage of the proposed method is that it separates the computational difficulty caused by the spatial multiscale properties, the high-dimensional randomness and the strong nonlinearity of the solution, so they can be overcome separately using different strategies. The convergence analysis is carried out, and the optimal rate of convergence is also obtained for the proposed semi-implicit stochastic multiscale method. Numerical experiments on several test problems for composite materials with various microstructures are also presented to gauge the efficiency and accuracy of the proposed semi-implicit stochastic multiscale method., Comment: 30 pages

This paper discusses the theory and numerical method of two-scale analysis for the multiscale Landau-Lifshitz-Gilbert equation in composite ferromagnetic materials. The novelty of this work can be summarized in three aspects: Firstly, the more realistic and complex model is considered, including the effects of the exchange field, anisotropy field, stray field, and external magnetic field. The explicit convergence orders in the $H^1$ norm between the classical solution and the two-scale solution are obtained. Secondly, we propose a robust numerical framework, which is employed in several comprehensive experiments to validate the convergence results for the Periodic and Neumann problems. Thirdly, we design an improved implicit numerical scheme to reduce the required number of iterations and relaxes the constraints on the time step size, which can significantly improve computational efficiency. Specifically, the projection and the expansion methods are given to overcome the inherent non-consistency in the initial data between the multiscale problem and homogenized problem.

In the present work, we consider multi-scale computation and convergence for nonlinear time-dependent thermo-mechanical equations of inhomogeneous shells possessing temperature-dependent material properties and orthogonal periodic configurations. The first contribution is that a novel higher-order macro-micro coupled computational model is rigorously devised via multi-scale asymptotic technique and Taylor series approach for high-accuracy simulation of heterogeneous shells. Benefitting from the higher-order corrected terms, the higher-order multi-scale computational model keeps the conservation of local energy and momentum for nonlinear thermo-mechanical simulation. Moreover, a global error estimation with explicit rate of higher-order multi-scale solutions is first derived in the energy norm sense. Furthermore, an efficient space-time numerical algorithm with off-line and on-line stages is presented in detail. Adequate numerical experiments are conducted to confirm the competitive advantages of the presented multi-scale approach, exhibiting not only the exceptional numerical accuracy, but also the less computational expense for heterogeneous shells.

Designing efficient and high-accuracy numerical methods for complex dynamic incompressible magnetohydrodynamics (MHD) equations remains a challenging problem in various analysis and design tasks. This is mainly due to the nonlinear coupling of the magnetic and velocity fields occurring with convection and Lorentz forces, and multiple physical constraints, which will lead to the limitations of numerical computation. In this paper, we develop the MHDnet as a physics-preserving learning approach to solve MHD problems, where three different mathematical formulations are considered and named $B$ formulation, $A_1$ formulation, and $A_2$ formulation. Then the formulations are embedded into the MHDnet that can preserve the underlying physical properties and divergence-free condition. Moreover, MHDnet is designed by the multi-modes feature merging with multiscale neural network architecture, which can accelerate the convergence of the neural networks (NN) by alleviating the interaction of magnetic fluid coupling across different frequency modes. Furthermore, the pressure fields of three formulations, as the hidden state, can be obtained without extra data and computational cost. Several numerical experiments are presented to demonstrate the performance of the proposed MHDnet compared with different NN architectures and numerical formulations.

In this paper, we introduce an innovative approach for addressing Bayesian inverse problems through the utilization of physics-informed invertible neural networks (PI-INN). The PI-INN framework encompasses two sub-networks: an invertible neural network (INN) and a neural basis network (NB-Net). The primary role of the NB-Net lies in modeling the spatial basis functions characterizing the solution to the forward problem dictated by the underlying partial differential equation. Simultaneously, the INN is designed to partition the parameter vector linked to the input physical field into two distinct components: the expansion coefficients representing the forward problem solution and the Gaussian latent noise. If the forward mapping is precisely estimated, and the statistical independence between expansion coefficients and latent noise is well-maintained, the PI-INN offers a precise and efficient generative model for Bayesian inverse problems, yielding tractable posterior density estimates. As a particular physics-informed deep learning model, the primary training challenge for PI-INN centers on enforcing the independence constraint, which we tackle by introducing a novel independence loss based on estimated density. We support the efficacy and precision of the proposed PI-INN through a series of numerical experiments, including inverse kinematics, 1-dimensional and 2-dimensional diffusion equations, and seismic traveltime tomography. Specifically, our experimental results showcase the superior performance of the proposed independence loss in comparison to the commonly used but computationally demanding kernel-based maximum mean discrepancy loss.

The coupling effects in multiphysics processes are often neglected in designing multiscale methods. The coupling may be described by a non-positive definite operator, which in turn brings significant challenges in multiscale simulations. In the paper, we develop a regularized coupling multiscale method based on the generalized multiscale finite element method (GMsFEM) to solve coupled thermomechanical problems, and it is referred to as the coupling generalized multiscale finite element method (CGMsFEM). The method consists of defining the coupling multiscale basis functions through local regularized coupling spectral problems in each coarse-grid block, which can be implemented by a novel design of two relaxation parameters. Compared to the standard GMsFEM, the proposed method can not only accurately capture the multiscale coupling correlation effects of multiphysics problems but also greatly improve computational efficiency with fewer multiscale basis functions. In addition, the convergence analysis is also established, and the optimal error estimates are derived, where the upper bound of errors is independent of the magnitude of the relaxation coefficient. Several numerical examples for periodic, random microstructure, and random material coefficients are presented to validate the theoretical analysis. The numerical results show that the CGMsFEM shows better robustness and efficiency than uncoupled GMsFEM., Comment: 26 pages

In this paper, we propose and analyze a new stochastic homogenization method for diffusion equations with random and fast oscillatory coefficients. In the proposed method, the homogenized solutions are sought through a two-stage procedure. In the first stage, the original oscillatory diffusion equation is approximated, for each fixed random sample w, by a spatially homogenized diffusion equation with piecewise constant coefficients, resulting a random diffusion equation. In the second stage, the resulted random diffusion equation is approximated and computed by using an efficient multi-modes Monte Carlo method which only requires to solve a diffusion equation with a constant diffusion coefficient and a random right-hand side. The main advantage of the proposed method is that it separates the computational difficulty caused by the spatial fast oscillation of the solution and that caused by the randomness of the solution, so they can be overcome separately using different strategies. The convergence of the solution of the spatially homogenized equation (from the first stage) to the solution of the original random diffusion equation is established and the optimal rate of convergence is also obtained for the proposed multi-modes Monte Carlo method. Numerical experiments on some benchmark test problems for random composite materials are also presented to gauge the efficiency and accuracy of the proposed two-stage stochastic homogenization method.

Background and objective: Combined evaluation of lumbosacral structures (e.g. nerves, bone) on multimodal radiographic images is routinely conducted prior to spinal surgery and interventional procedures. Generally, magnetic resonance imaging is conducted to differentiate nerves, while computed tomography (CT) is used to observe bony structures. The aim of this study is to investigate the feasibility of automatically segmenting lumbosacral structures (e.g. nerves & bone) on non-contrast CT with deep learning. Methods: a total of 50 cases with spinal CT were manually labeled for lumbosacral nerves and bone with Slicer 4.8. The ratio of training: validation: testing is 32:8:10. A 3D-Unet is adopted to build the model SPINECT for automatically segmenting lumbosacral structures. Pixel accuracy, IoU, and Dice score are used to assess the segmentation performance of lumbosacral structures. Results: the testing results reveals successful segmentation of lumbosacral bone and nerve on CT. The average pixel accuracy is 0.940 for bone and 0.918 for nerve. The average IoU is 0.897 for bone and 0.827 for nerve. The dice score is 0.945 for bone and 0.905 for nerve. Conclusions: this pilot study indicated that automatic segmenting lumbosacral structures (nerves and bone) on non-contrast CT is feasible and may have utility for planning and navigating spinal interventions and surgery., Comment: 11 pages, 5 figures