Partial Differential Equations (PDE) are fundamental to model different phenomena in science and engineering mathematically. Solving them is a crucial step towards a precise knowledge of the behaviour of natural and engineered systems. In general, in order to solve PDEs that represent real systems to an acceptable degree, analytical methods are usually not enough. One has to resort to discretization methods. For engineering problems, probably the best known option is the finite element method (FEM). However, powerful alternatives such as mesh-free methods and Isogeometric Analysis (IGA) are also available. The fundamental idea is to approximate the solution of the PDE by means of functions specifically built to have some desirable properties. In this contribution, we explore Deep Neural Networks (DNNs) as an option for approximation. They have shown impressive results in areas such as visual recognition. DNNs are regarded here as function approximation machines. There is great flexibility to define their structure and important advances in the architecture and the efficiency of the algorithms to implement them make DNNs a very interesting alternative to approximate the solution of a PDE. We concentrate in applications that have an interest for Computational Mechanics. Most contributions that have decided to explore this possibility have adopted a collocation strategy. In this contribution, we concentrate in mechanical problems and analyze the energetic format of the PDE. The energy of a mechanical system seems to be the natural loss function for a machine learning method to approach a mechanical problem. As proofs of concept, we deal with several problems and explore the capabilities of the method for applications in engineering.

Some new aspects of the continuum strong discontinuity approach (CSDA) to model material failure in geomaterials are addressed. A new global algorithm, for tracking multiple crack lines/surfaces in 2D/3D cases is proposed. It is based on solving a simple heat conduction‐like problem accompanying the standard mechanical algorithm. A viscous perturbation method on the crack surface is also proposed to remedy the instabilities caused by mutual interactions of multiple developing cracks. A simple procedure to compute the critical time step that ensures algorithmic uniqueness is then provided. Numerical simulations of two‐ and three‐dimensional problems displaying a multi‐crack pattern are finally presented. Copyright © 2004 John Wiley & Sons, Ltd.

Mountain regions worldwide present a pronounced spatiotemporal precipitation variability, which added to scarce monitoring networks limits our understanding of the generation processes involved. To improve our understanding of clouds and precipitation dynamics and cross-scale generation processes in mountain regions, we analyzed spatiotemporal rainfall patterns using satellite cloud products (SCP) in the Paute basin (900–4200 m a.s.l. and 6481 km2) in the Andes of Ecuador. Precipitation models, using SCP and GIS data, reveal the spatial extension of three regimes: a three-modal (TM) regime present across the basin, a bimodal (BM) regime, along sheltered valleys, and a unimodal (UM) regime at windward slopes of the eastern cordillera. Subsequently, the spatiotemporal analysis using synoptic information shows that the dry season of the BM regime during boreal summer is caused by strong subsidence inhibiting convective clouds formation. Meanwhile, in UM regions, low advective shallow cap clouds mainly cause precipitation, influenced by water vapor from the Amazon and enhanced easterlies during boreal summer. TM regions are transition zones from UM to BM and zones on the windward slopes of the western cordillera. These results highlight the suitability of satellite and GIS data-driven statistical models to study spatiotemporal rainfall seasonality and generation processes in complex terrain, as the Andes.

SUMMARYA homogenization method is proposed for sandwich structures consisting of two plates interlaced with beams and shells in a periodic, lattice structure. The proposed method is a quasi-continuum approach where the constitutive response is obtained from the generalized forces of the interlacing elements. Buckling is studied as part of this model. Comparison of the homogenized model with fully discrete models show reasonable to very good agreement.