Analysis of a fully discretized FDM-FEM scheme for solving thermo-elastic-damage coupled nonlinear PDE systems

In this paper, we consider a nonlinear PDE system governed by a parabolic heat equation coupled in a nonlinear way with a hyperbolic momentum equation describing the behavior of a displacement field coupled with a nonlinear elliptic equation based on an internal damage variable. We present a numerical scheme based on a low-order Galerkin finite element method (FEM) for the space discretization of the time-dependent nonlinear PDE system and an implicit finite difference method (FDM) to discretize in the direction of the time variable. Moreover, we present a priori estimates for the exact and discrete solutions for the pointwise-in-time $L^2$-norm. Based on the a priori estimates, we rigorously prove the convergence of the solutions of the fully discretized system to the exact solutions. Denoting the properties of the internal parameters, we find the order of convergence concerning the discretization parameters

Global-Local Forward Models within Bayesian Inversion for Large Strain Fracturing in Porous Media

In this work, Bayesian inversion with global-local forwards models is used to identify the parameters based on hydraulic fractures in porous media. It is well-known that using Bayesian inversion to identify material parameters is computationally expensive. Although each sampling may take more than one hour, thousands of samples are required to capture the target density. Thus, instead of using fine-scale high-fidelity simulations, we use a non-intrusive global-local (GL) approach for the forward model. We further extend prior work to a large deformation setting based on the Neo-Hookean strain energy function. The resulting framework is described in detail and substantiated with some numerical tests

Hierarchical LU preconditioning for the time-harmonic Maxwell equations

The time-harmonic Maxwell equations are used to study the effect of electric and magnetic fields on each other. Although the linear systems resulting from solving this system using FEMs are sparse, direct solvers cannot reach the linear complexity. In fact, due to the indefinite system matrix, iterative solvers suffer from slow convergence. In this work, we study the effect of using the inverse of $\mathcal{H}$-matrix approximations of the Galerkin matrices arising from N\'ed\'elec's edge FEM discretization to solve the linear system directly. We also investigate the impact of applying an $\mathcal{H}-LU$ factorization as a preconditioner and we study the number of iterations to solve the linear system using iterative solvers

Using layer-wise training for Road Semantic Segmentation in Autonomous Cars

A recently developed application of computer vision is pathfinding in self-driving cars. Semantic scene understanding and semantic segmentation, as subfields of computer vision, are widely used in autonomous driving. Semantic segmentation for pathfinding uses deep learning methods and various large sample datasets to train a proper model. Due to the importance of this task, accurate and robust models should be trained to perform properly in different lighting and weather conditions and in the presence of noisy input data. In this paper, we propose a novel learning method for semantic segmentation called layer-wise training and evaluate it on a light efficient structure called an efficient neural network (ENet). The results of the proposed learning method are compared with the classic learning approaches, including mIoU performance, network robustness to noise, and the possibility of reducing the size of the structure on two RGB image datasets on the road (CamVid) and off-road (Freiburg Forest) paths. Using this method partially eliminates the need for Transfer Learning. It also improves network performance when input is noisy

Bayesian Inversion with Open-Source Codes for Various One-Dimensional Model Problems in Computational Mechanics

The complexity of many problems in computational mechanics calls for reliable programming codes and accurate simulation systems. Typically, simulation responses strongly depend on material and model parameters, where one distinguishes between backward and forward models. Providing reliable information for the material/model parameters, enables us to calibrate the forward model (e.g., a system of PDEs). Markov chain Monte Carlo methods are efficient computational techniques to estimate the posterior density of the parameters. In the present study, we employ Bayesian inversion for several mechanical problems and study its applicability to enhance the model accuracy. Seven different boundary value problems in coupled multi-field (and multi-physics) systems are presented. To provide a comprehensive study, both rate-dependent and rate-independent equations are considered. Moreover, open source codes (https://doi.org/10.5281/zenodo.6451942) are provided, constituting a convenient platform for future developments for, e.g., multi-field coupled problems. The developed package is written in MATLAB and provides useful information about mechanical model problems and the backward Bayesian inversion setting. © 2022, The Author(s)