Matteo Giacomini

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, 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.