Matteo Giacomini

Guillaume Dollé, Omar Duran, Nelson Feyeux et al.

In this paper, we propose a mathematical model to describe the functioning of a bioreactor landfill, that is a waste management facility in which biodegradable waste is used to generate methane. The simulation of a bioreactor landfill is a very complex multiphysics problem in which bacteria catalyze a chemical reaction that starting from organic carbon leads to the production of methane, carbon dioxide and water. The resulting model features a heat equation coupled with a non-linear reaction equation describing the chemical phenomena under analysis and several advection and advection-diffusion equations modeling multiphase flows inside a porous environment representing the biodegradable waste. A framework for the approximation of the model is implemented using Feel++, a C++ open-source library to solve Partial Differential Equations. Some heuristic considerations on the quantitative values of the parameters in the model are discussed and preliminary numerical simulations are presented.

Matteo Giacomini, Olivier Pantz, Karim Trabelsi

In this paper we introduce a novel certified shape optimization strategy - named Certified Descent Algorithm (CDA) - to account for the numerical error introduced by the Finite Element approximation of the shape gradient. We present a goal-oriented procedure to derive a certified upper bound of the error in the shape gradient and we construct a fully-computable, constant-free a posteriori error estimator inspired by the complementary energy principle. The resulting CDA is able to identify a genuine descent direction at each iteration and features a reliable stopping criterion. After validating the error estimator, some numerical simulations of the resulting certified shape optimization strategy are presented for the well-known inverse identification problem of Electrical Impedance Tomography.

Matteo Giacomini, Olivier Pantz, Karim Trabelsi

In this article, we consider the problem of optimal design of a compliant structure under a volume constraint, within the framework of linear elasticity. We introduce the pure displacement and the dual mixed formulations of the linear elasticity problem and we compute the volumetric expressions of the shape gradient of the compliance by means of the velocity method. A preliminary qualitative comparison of the two expressions of the shape gradient is performed through some numerical simulations using the Boundary Variation Algorithm.

Matteo Giacomini, Pedro Díez

Surrogate models provide compact relations between user-defined input parameters and output quantities of interest, enabling the efficient evaluation of complex parametric systems in many-query settings. Such capabilities are essential in a wide range of applications, including optimisation, control, data assimilation, uncertainty quantification, and emerging digital twin technologies in various fields such as manufacturing, personalised healthcare, smart cities, and sustainability. This article reviews established methodologies for constructing surrogate models exploiting either knowledge of the governing laws and the dynamical structure of the system (physics-based) or experimental observations (data-driven), as well as hybrid approaches combining these two paradigms. By revisiting the design of a surrogate model as a functional approximation problem, existing methodologies are reviewed in terms of the choice of (i) a reduced basis and (ii) a suitable approximation criterion. The paper reviews methodologies pertaining to the field of Scientific Machine Learning, and it aims at synthesising established knowledge, recent advances, and new perspectives on: dimensionality reduction, physics-based, and data-driven surrogate modelling based on proper orthogonal decomposition, proper generalised decomposition, and artificial neural networks; multi-fidelity methods to exploit information from sources with different fidelities; adaptive sampling, enrichment, and data augmentation techniques to enhance the quality of surrogate models.

Matteo Giacomini, Antonio Huerta

A surrogate-based topology optimisation algorithm for linear elastic structures under parametric loads and boundary conditions is proposed. Instead of learning the parametric solution of the state (and adjoint) problems or the optimisation trajectory as a function of the iterations, the proposed approach devises a surrogate version of the entire optimisation pipeline. First, the method predicts a quasi-optimal topology for a given problem configuration as a surrogate model of high-fidelity topologies optimised with the homogenisation method. This is achieved by means of a feed-forward net learning the mapping between the input parameters characterising the system setup and a latent space determined by encoder/decoder blocks reducing the dimensionality of the parametric topology optimisation problem and reconstructing a high-dimensional representation of the topology. Then, the predicted topology is used as an educated initial guess for a computationally efficient algorithm penalising the intermediate values of the design variable, while enforcing the governing equations of the system. This step allows the method to correct potential errors introduced by the surrogate model, eliminate artifacts, and refine the design in order to produce topologies consistent with the underlying physics. Different architectures are proposed and the approximation and generalisation capabilities of the resulting models are numerically evaluated. The quasi-optimal topologies allow to outperform the high-fidelity optimiser by reducing the average number of optimisation iterations by $53\%$ while achieving discrepancies below $4\%$ in the optimal value of the objective functional, even in the challenging scenario of testing the model to extrapolate beyond the training and validation domain.

Marco Discacciati, Ben J. Evans, Matteo Giacomini

A strategy to construct physics-based local surrogate models for parametric Stokes flows and coupled Stokes-Darcy systems is presented. The methodology relies on the proper generalized decomposition (PGD) method to reduce the dimensionality of the parametric flow fields and on an overlapping domain decomposition (DD) paradigm to reduce the number of globally coupled degrees of freedom in space. The DD-PGD approach provides a non-intrusive framework in which end-users only need access to the matrices arising from the (finite element) discretization of the full-order problems in the subdomains. The traces of the finite element functions used for the discretization within the subdomains are employed to impose arbitrary Dirichlet boundary conditions at the interface, without introducing auxiliary basis functions. The methodology is seamless to the choice of the discretization schemes in space, being compatible with both LBB-compliant finite element pairs and stabilized formulations, and the DD-PGD paradigm is transparent to the employed overlapping DD approach. The local surrogate models are glued together in the online phase by solving a parametric interface system to impose continuity of the subdomain solutions at the interfaces, without introducing Lagrange multipliers to enforce the continuity in the entire overlap and without solving any additional physical problem in the reduced space. Numerical results are presented for parametric single-physics (Stokes-Stokes) and multi-physics (Stokes-Darcy) systems, showcasing the accuracy, robustness, and computational efficiency of DD-PGD, and its capability to outperform DD methods based on high-fidelity finite element solvers in terms of computing times.

Matteo Giacomini, Simona Perotto

A finite element-based image segmentation strategy enhanced by an anisotropic mesh adaptation procedure is presented. The methodology relies on a split Bregman algorithm for the minimisation of a region-based energy functional and on an anisotropic recovery-based error estimate to drive mesh adaptation. More precisely, a Bayesian energy functional is considered to account for image spatial information, ensuring that the methodology is able to identify inhomogeneous spatial patterns in complex images. In addition, the anisotropic mesh adaptation guarantees a sharp detection of the interface between background and foreground of the image, with a reduced number of degrees of freedom. The resulting split-adapt Bregman algorithm is tested on a set of real images showing the accuracy and robustness of the method, even in the presence of Gaussian, salt and pepper and speckle noise.

Matteo Giacomini

The certified descent algorithm (CDA) is a gradient-based method for shape optimization which certifies that the direction computed using the shape gradient is a genuine descent direction for the objective functional under analysis. It relies on the computation of an upper bound of the error introduced by the finite element approximation of the shape gradient. In this paper, we present a goal-oriented error estimator which depends solely on local quantities and is fully-computable. By means of the equilibrated fluxes approach, we construct a unified strategy valid for both conforming finite element approximations and discontinuous Galerkin discretizations. The new variant of the CDA is tested on the inverse identification problem of electrical impedance tomography: both its ability to identify a genuine descent direction at each iteration and its reliable stopping criterion are confirmed.