New Trends in Differential and Difference Equations and Applications

This is a reprint of articles from the Special Issue published online in the open-access journal Axioms (ISSN 2075-1680) from 2018 to 2019 (available at https://www.mdpi.com/journal/axioms/special issues/differential difference equations)

Deep learning applied to computational mechanics: A comprehensive review, state of the art, and the classics

Three recent breakthroughs due to AI in arts and science serve as motivation: An award winning digital image, protein folding, fast matrix multiplication. Many recent developments in artificial neural networks, particularly deep learning (DL), applied and relevant to computational mechanics (solid, fluids, finite-element technology) are reviewed in detail. Both hybrid and pure machine learning (ML) methods are discussed. Hybrid methods combine traditional PDE discretizations with ML methods either (1) to help model complex nonlinear constitutive relations, (2) to nonlinearly reduce the model order for efficient simulation (turbulence), or (3) to accelerate the simulation by predicting certain components in the traditional integration methods. Here, methods (1) and (2) relied on Long-Short-Term Memory (LSTM) architecture, with method (3) relying on convolutional neural networks. Pure ML methods to solve (nonlinear) PDEs are represented by Physics-Informed Neural network (PINN) methods, which could be combined with attention mechanism to address discontinuous solutions. Both LSTM and attention architectures, together with modern and generalized classic optimizers to include stochasticity for DL networks, are extensively reviewed. Kernel machines, including Gaussian processes, are provided to sufficient depth for more advanced works such as shallow networks with infinite width. Not only addressing experts, readers are assumed familiar with computational mechanics, but not with DL, whose concepts and applications are built up from the basics, aiming at bringing first-time learners quickly to the forefront of research. History and limitations of AI are recounted and discussed, with particular attention at pointing out misstatements or misconceptions of the classics, even in well-known references. Positioning and pointing control of a large-deformable beam is given as an example.Comment: 275 pages, 158 figures. Appeared online on 2023.03.01 at CMES-Computer Modeling in Engineering & Science

A hermite radial basis functions control volume numerical method to simulate transport problems

This thesis presents a Control Volume (CV) method for transient transport problems where the cell surface fluxes are reconstructed using local interpolation functions that besides interpolating the nodal values of the field variable, also satisfies the governing equation at some auxiliary points in the interpolation stencils. The interpolation function relies on a Hermitian Radial Basis Function (HRBF) mesh less collocation approach to find the solution of auxiliary local boundary/initial value problems, which are solved using the same time integration scheme adopted to update the global control volume solution. By the use of interpolation functions that approximate the governing equation, a form of analytical upwinding scheme is achieved without the need of using predefined interpolation stencils according to the magnitude and direction of the local advective velocity. In this way, the interpolation formula retains the desired information about the advective velocity field, allowing the use of centrally defined stencils even in the case of advective dominant problems. This new CV approach, which is referred to as the CV-HRBF method, is applied to a series of transport problems characterised by high Peclet number. This method is also more flexible than the classical CV formulations because the boundary conditions are explicitly imposed in the interpolation formula, without the need for artificial schemes (e.g. utilising dummy cells). The flexibility of the local meshless character of the CVHRBF is shown in the modelling of the saturated zone of the unconfined aquifer where a mesh adapting algorithm is needed to track the phreatic surface (moving boundary). Due to the use of a local RBF interpolation, the dynamic boundary condition can be applied in an arbitrary number of points on the phreatic surface, independently from the mesh element. The robustness of the Hermite interpolation is exploited to formulate a non-overlapping non-iterative multi-domain scheme where physical matching conditions are satisfied locally, i.e. imposing the continuity of the function and flux at the sub-domain interface