Courses

Multi-physics finite elements modeling and Deep-learning for the metal forming industry. 
 

Introduction to multigrid methods. Application of Schwarz domain decomposition methods as smoothers. 

Mixed stabilised finite element methods.

 

Reduced-order models: linear and nonlinear approaches.

Domain decomposition for local surrogate models of parametric systems. 

Matteo Giacomini, Universitat Politècnica de Catalunya (Spain). 

Hybridisable discontinuous Galerkin methods.

 

Title: Multi-physics finite elements modeling and Deep-learning for the metal forming industry

Abstract: In this talk, we'll go through the mathematical, physical and numerical notions required in multi-physics simulations for complex manufacturing processes. A review of fundamental principles and their interaction at the macroscopic scale of process manufacturing will be given. Followed by an overview of the numerical ingredients usually required for transforming such first principle into algebraic problems. Within these ingredients, the finite element method plays a foundational role as the framework enabling combining the different sources of non-linearities within architectures for which convergency can be strongly or weakly stablish through proof or experimentation. Several applications in the realm of electromagnetically-driven processes will be covered to explore how industry already profits from the combined power of such approaches. A last section will be dedicated to deep-learning and the rise of physics-driven learning architectures which can be used to bring to live the massive amount of data generated by numerical simulations, accelerating the engineering cycle, as well as bridging gaps with complex material modeling. 

 

Title: Mixed stabilised finite element methods.

Abstract: Mixed problems are boundary value problems involving two or more unknowns that belong to different functional spaces. Such problems are pervasive in computational mechanics, where they arise in the modeling of phenomena including solid mechanics, fluid dynamics, and electromagnetism. The well-posedness of these problems depends on the satisfaction of a general inf-sup (or stability) condition for the combined space of all unknowns, which itself stems from the inf-sup conditions linking the individual functional spaces.  When considering their finite element approximation, well-posedness requires that the discrete finite element spaces preserve the inf-sup condition of the continuous problem. Ensuring this property can be highly challenging, and in some cases, practically infeasible. An alternative approach is to modify the discrete variational formulation of the problem. One effective strategy is to design a stabilized finite element formulation based on the Variational Multiscale (VMS) concept.  This course presents the general theory of mixed problems and their finite element formulations, with a brief introduction to hybrid formulations. Rather than focusing primarily on inf-sup stable approximations, the emphasis will be on the design of finite element methods grounded in the VMS framework. Both the stability and convergence properties of the resulting schemes will be analyzed. Throughout the course, key examples—including Darcy’s problem, the Stokes problem, Maxwell’s equations, and mixed formulations of elasticity—will serve as guiding applications.

 

Click here to download the course materials.

 

 

 

Title: Hybridisable discontinuous Galerkin methods.

 

Abstract: This short course provides an overview of hybridisable discontinuous Galerkin (HDG) formulations and their application to a range of physical problems, including thermal phenomena, incompressible flows, and compressible flows. The course begins by introducing the fundamental concepts underlying HDG methods, such as the mixed form of discontinuous Galerkin (DG) approximations, the principle of hybridisation that enables static condensation to reduce the number of globally-coupled degrees of freedom, and the formulation of numerical fluxes inspired by Riemann solvers to ensure stability and accuracy in convection-dominated regimes. The course explores the derivation and implementation of HDG schemes for different classes of partial differential equations, highlighting their advantages in achieving high-order accuracy, local conservation, and efficient parallelisation. Special attention is given to the role of stabilisation parameters, boundary conditions, and post-processing techniques to achieve super-convergence of the solution. Practical implementation aspects are discussed using the open-source software HDGlab, a MATLAB-based library designed for rapid prototyping and educational exploration of HDG methods. 
 

Click here to download the course materials.

Click here to download the course materials.

 

 

Title: Manifold learning for model order reduction in engineering

Abstract: We propose a general framework for projection-based model order reduction assisted by deep neural networks. For parametric elliptic equations this approach is theoretically based on Céa's Lemma. The proposed methodology, called ROM-net, consists in using deep learning techniques to adapt the reduced-order model to a stochastic input tensor whose nonparametrized variabilities strongly influence the quantities of interest for a given physics problem. In particular, we introduce the concept of dictionary-based ROM-nets, where deep neural networks recommend a suitable local reduced-order model from a dictionary. The dictionary of local reduced-order models is constructed from a clustering of vector subspaces in a Grassmann manifold. It enables the identification of the local subspace in which the solutions evolve for different input tensors. This methodology is applied to an anisothermal elastoplastic problem in structural mechanics coupled to a stochastic thermal field. When using deep neural networks, the selection of the best reduced-order model for a given thermal loading is 60 times faster than when following the clustering procedure used in the training phase.  The implementation of local hyper-reduction schemes using a dictionary-based ROM-net is straightforward. The extension to variational inequalities will be addressed at the end of the lecture.