Objective multiscale analysis of random heterogeneous materials

The multiscale framework presented in [1, 2] is assessed in this contribution for a study of random heterogeneous materials. Results are compared to direct numerical simulations (DNS) and the sensitivity to user-defined parameters such as the domain decomposition type and initial coarse scale resolution is reported. The parallel performance of the implementation is studied for different domain decompositions

Matlab parallel codes for 3D slope stability benchmarks

This contribution is focused on a description of implementation details for solver related to the slope stability benchmarks in 3D. Such problems are formulated by the standard elastoplastic models containing the Mohr-Coulomb yield criterion and by the limit analysis of collapse states. The implicit Euler method and higher order finite elements are used for discretization. The discretized problem is solved by non-smooth Newton-like methods in combination with incremental methods of limit load analysis. In this standard approach, we propose several innovative techniques. Firstly, we use recently developed sub-differential based constitutive solution schemes. Such an approach is suitable for non-smooth yield criteria, and leads better return-mapping algorithms. For example, a priori decision criteria for each return-type or simplified construction of consistent tangent operators are applied. The parallel codes are developed in MATLAB using Parallel Computing Toolbox. For parallel implementation of linear systems, we use the TFETI domain decomposition method. It is a non-overlapping method where the Lagrange multipliers are used to enforce continuity on the subdomain interfaces and satisfaction of the Dirichlet boundary conditions

General treatment of essential boundary conditions in reduced order models for non-linear problems

Inhomogeneous essential boundary conditions must be carefully treated in the formulation of Reduced Order Models (ROMs) for non-linear problems. In order to investigate this issue, two methods are analysed: one in which the boundary conditions are imposed in an strong way, and a second one in which a weak imposition of boundary conditions is made. The ideas presented in this work apply to the big realm of a posteriori ROMs. Nevertheless, an a posteriori hyper-reduction method is specifically considered in order to deal with the cost associated to the non-linearity of the problems. Applications to nonlinear transient heat conduction problems with temperature dependent thermophysical properties and time dependent essential boundary conditions are studied. However, the strategies introduced in this work are of general application.Fil: Cosimo, Alejandro. Consejo Nacional de Investigaciones Científicas y Técnicas. Centro Científico Tecnológico Conicet - Santa Fe. Centro de Investigaciones en Métodos Computacionales. Universidad Nacional del Litoral. Centro de Investigaciones en Métodos Computacionales; ArgentinaFil: Cardona, Alberto. Consejo Nacional de Investigaciones Científicas y Técnicas. Centro Científico Tecnológico Conicet - Santa Fe. Centro de Investigaciones en Métodos Computacionales. Universidad Nacional del Litoral. Centro de Investigaciones en Métodos Computacionales; ArgentinaFil: Idelsohn, Sergio Rodolfo. Consejo Nacional de Investigaciones Científicas y Técnicas. Centro Científico Tecnológico Conicet - Santa Fe. Centro de Investigaciones en Métodos Computacionales. Universidad Nacional del Litoral. Centro de Investigaciones en Métodos Computacionales; Argentina. Institució Catalana de Recerca i Estudis Avancats; España. International Center for Numerical Methods in Engineering; Españ

A Generalized Schwarz-type Non-overlapping Domain Decomposition Method using Physics-constrained Neural Networks

We present a meshless Schwarz-type non-overlapping domain decomposition method based on artificial neural networks for solving forward and inverse problems involving partial differential equations (PDEs). To ensure the consistency of solutions across neighboring subdomains, we adopt a generalized Robin-type interface condition, assigning unique Robin parameters to each subdomain. These subdomain-specific Robin parameters are learned to minimize the mismatch on the Robin interface condition, facilitating efficient information exchange during training. Our method is applicable to both the Laplace's and Helmholtz equations. It represents local solutions by an independent neural network model which is trained to minimize the loss on the governing PDE while strictly enforcing boundary and interface conditions through an augmented Lagrangian formalism. A key strength of our method lies in its ability to learn a Robin parameter for each subdomain, thereby enhancing information exchange with its neighboring subdomains. We observe that the learned Robin parameters adapt to the local behavior of the solution, domain partitioning and subdomain location relative to the overall domain. Extensive experiments on forward and inverse problems, including one-way and two-way decompositions with crosspoints, demonstrate the versatility and performance of our proposed approach