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FRANCESCO MEZZADRI

Ricercatore t.d. art. 24 c. 3 lett. B
Dipartimento di Ingegneria "Enzo Ferrari"

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Pubblicazioni

2023 - A Framework for Physics-Informed Deep Learning Over Freeform Domains [Articolo su rivista]
Mezzadri, F.; Gasick, J.; Qian, X.
abstract

Deep learning is a popular approach for approximating the solutions to partial differential equations (PDEs) over different material parameters and boundary conditions. However, no work has yet been reported on learning PDE solutions over changing shapes of the underlying domain. We present a framework to train neural networks (NN) and physics-informed neural networks (PINNs) to learn the solutions to PDEs defined over varying freeform domains. This is made possible through our adoption of a parametric non-uniform rational B-Spline (NURBS) representation of the underlying physical shape. Distinct physical domains are mapped to a common parametric space via NURBS parameterization. In our approach, we formulate NNs and PINNs that learn the solutions to PDEs as a function of the shape of the domain itself through shape parameters. Under this formulation, the loss function is based on an unchanging parametric domain that maps to a variable physical domain. Residual computation in PINNs is made possible through the Jacobian of the mapping. Numerical results show that our networks can be trained to predict the solutions to a PDE defined over an entire set of shapes. We focus on the linear elasticity PDE and show how we can build a surrogate model that is able to predict displacements and stresses over a variety of freeform domains. To assess the efficacy of all networks in this work, data efficiency, network accuracy, and the capacity of networks to extrapolate are considered and compared between NNs and PINNs. The comparison includes cases where little training data is available. Transfer learning and applications to shape optimization are analyzed as well. A selection of the used codes and datasets is provided at https://github.com/fmezzadri/shape_parameterized.git.

2023 - Design of innovative self-expandable femoral stents using inverse homogenization topology optimization [Articolo su rivista]
Carbonaro, D.; Mezzadri, F.; Ferro, N.; De Nisco, G.; Audenino, A. L.; Gallo, D.; Chiastra, C.; Morbiducci, U.; Perotto, S.
abstract

In this study, we propose a novel computational framework for designing innovative self-expandable femoral stents. First, a two-dimensional stent unit cell is designed by inverse homogenization topology optimization. In particular, the unit cell is optimized in terms of contact area with the target of matching prescribed mechanical properties. The topology optimization is enriched by an anisotropic mesh adaptation strategy, enabling a time-and cost-effective procedure that promotes original layouts. Successively, the optimized stent unit cell is periodically repeated on a hollow cylindrical surface to construct the corresponding three-dimensional device. Finally, structural mechanics and computational fluid dynamics simulations are carried out to verify the performance of the newly-designed stent. The proposed workflow is being tested through the design of five proof-of-concept stents. These devices are compared through specific performance evaluations, which include the assessments of the minimum requirement for usability, namely the ability to be crimped into a catheter, and the quantification of the radial force, the foreshortening, the structural integrity and the induced blood flow disturbances.& COPY; 2023 The Author(s). Published by Elsevier B.V. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/).

2022 - A generalization of the equivalence relations between modulus-based and projected splitting methods [Articolo su rivista]
Mezzadri, F.; Galligani, E.
abstract

2022 - A modulus-based formulation for the vertical linear complementarity problem [Articolo su rivista]
Mezzadri, F.
abstract

We introduce a modulus-based formulation for vertical linear complementarity problems (VLCPs) with an arbitrary number l of matrices. This formulation can be used to set up a variety of modulus-based solution methods, including, for example, the modulus-based matrix splitting methods for VLCPs that we here introduce. In this context, we particularly analyze the methods for problems with l = 2 (providing also sufficient conditions for their global convergence) and we then generalize the formulation of the methods to any l. Finally, some numerical experiments are solved to evaluate the performance of the proposed methods, which we compare with an existing smoothing Newton method for VLCPs.

2022 - Density gradient‐based adaptive refinement of analysis mesh for efficient multiresolution topology optimization [Articolo su rivista]
Mezzadri, Francesco; Qian, Xiaoping
abstract

2022 - Projected Splitting Methods for Vertical Linear Complementarity Problems [Articolo su rivista]
Mezzadri, Francesco; Galligani, Emanuele
abstract

2021 - A generalization of irreducibility and diagonal dominance with applications to horizontal and vertical linear complementarity problems [Articolo su rivista]
Mezzadri, Francesco; Galligani, Emanuele
abstract

2021 - Analysis of Resonant Soft X-ray Reflectivity of Anisotropic Layered Materials [Articolo su rivista]
Pasquali, Luca; Mahne, Nicola; Giglia, Angelo; Verna, Adriano; Sponza, Lorenzo; Capelli, Raffaella; Bonfatti, Matteo; Mezzadri, Francesco; Galligani, Emanuele; Nannarone, Stefano
abstract

We present here a method for the quantitative prediction of the spectroscopic specular reflectivity line-shape in anisotropic layered media. The method is based on a 4 × 4 matrix formalism and on the simulation from the first principles (through density functional theory—DFT) of the anisotropic absorption cross-section. The approach was used to simulate the reflectivity at the oxygen K-edge of a 3,4,9,10-perylene-tetracarboxylic dianhydride (PTCDA) thin film on Au(111). The effect of film thickness, orientation of the molecules, and grazing incidence angle were considered. The simulation results were compared to the experiment, permitting us to derive information on the film geometry, thickness, and morphology, as well as the electronic structure.

2021 - Modulus-based matrix splitting methods for a class of horizontal nonlinear complementarity problems [Articolo su rivista]
Mezzadri, F.; Galligani, E.
abstract

In this paper, we generalize modulus-based matrix splitting methods to a class of horizontal nonlinear complementarity problems (HNCPs). First, we write the HNCP as an implicit fixed-point equation and we introduce the proposed solution procedures. We then prove the convergence of the methods under some assumptions. We also comment on how the proposed methods and convergence theorems generalize existing results on (standard) linear and nonlinear complementarity problems and on horizontal linear complementarity problems. Finally, numerical experiments are solved to demonstrate the efficiency of the procedures. In this context, the effects of the splitting, of the dimension of the matrices, and of the nonlinear term of the problem are analyzed.

2020 - A second-order measure of boundary oscillations for overhang control in topology optimization [Articolo su rivista]
Mezzadri, Francesco; Qian, Xiaoping
abstract

2020 - Modulus-based matrix splitting methods for horizontal linear complementarity problems [Articolo su rivista]
Mezzadri, Francesco; Galligani, Emanuele
abstract

2020 - Modulus-based synchronous multisplitting methods for solving horizontal linear complementarity problems on parallel computers [Articolo su rivista]
Mezzadri, F.
abstract

In this article, we generalize modulus-based synchronous multisplitting methods to horizontal linear complementarity problems. In particular, first we define the methods of our concern and prove their convergence under suitable smoothness assumptions. Particular attention is devoted also to modulus-based multisplitting accelerated overrelaxation methods. Then, as multisplitting methods are well-suited for parallel computations, we analyze the parallel behavior of the proposed procedures. In particular, we do so by solving various test problems by a parallel implementation of our multisplitting methods. In this context, we carry out parallel computations on GPU with CUDA.

2020 - On the convergence of modulus-based matrix splitting methods for horizontal linear complementarity problems in hydrodynamic lubrication [Articolo su rivista]
Mezzadri, Francesco; Galligani, Emanuele
abstract

2020 - On the solution of general absolute value equations [Articolo su rivista]
Mezzadri, F.
abstract

In this note, we provide necessary and sufficient conditions that ensure the existence and uniqueness of solution of the general form of absolute value equations (AVEs), Ax−B|x|=b. The performed analysis is based on the equivalence between AVEs and horizontal linear complementarity problems (HLCPs). New sufficient conditions are proposed as well. We then compare the proposed conditions with recent results in the literature and we detail how efficient solution methods for HLCPs can be easily applied to the solution of general AVEs. Finally, we provide comments on the solvability of general AVEs under conditions larger than uniqueness of solution.

2019 - A modulus-based nonsmooth Newton’s method for solving horizontal linear complementarity problems [Articolo su rivista]
Mezzadri, F.; Galligani, E.
abstract

2019 - A nonlinearity lagging method for non-steady diffusion equations with nonlinear convection terms [Articolo su rivista]
Mezzadri, F.; Galligani, E.
abstract

We analyze an iterative procedure for solving nonlinear algebraic systems arising from the discretization of nonlinear, non-steady reaction-convection-diffusion equations with non-constant (and, in general, nonlinear) velocity terms. The basic idea underlying the procedure consists in lagging the diffusion and the velocity terms of the discretized system, which is thus partly linearized. After analyzing the discretized system and proving some results on the monotonicity of the operators and on the uniqueness of the solution, we prove sufficient conditions that ensure the convergence of this lagged method. We also describe the inner iteration and show how the weakly nonlinear systems arising at each lagged iteration can be solved efficiently. Finally, we analyze numerically the entire solution process by several numerical experiments.

2019 - On the equivalence between some projected and modulus-based splitting methods for linear complementarity problems [Articolo su rivista]
Mezzadri, F.
abstract

In this paper, we analyze the relationship between projected and (possibly accelerated) modulus-based matrix splitting methods for linear complementarity problems. In particular, first we show that some well-known projected splitting methods are equivalent, iteration by iteration, to some (accelerated) modulus-based matrix splitting methods with a specific choice of the parameter Ω. We then generalize this result to any Ω by formulating new classes of projected splitting methods and also provide a formal projection-based formulation for general (accelerated) modulus-based matrix splitting methods. Finally, we introduce and solve several test problems to evaluate also numerically the equivalence between the analyzed methods.

2019 - Splitting Methods for a Class of Horizontal Linear Complementarity Problems [Articolo su rivista]
Mezzadri, F.; Galligani, E.
abstract

In this paper, we propose two splitting methods for solving horizontal linear complementarity problems characterized by matrices with positive diagonal elements. The proposed procedures are based on the Jacobi and on the Gauss–Seidel iterations and differ from existing techniques in that they act directly and simultaneously on both matrices of the problem. We prove the convergence of the methods under some assumptions on the diagonal dominance of the matrices of the problem. Several numerical experiments, including large-scale problems of practical interest, demonstrate the capabilities of the proposed methods in various situations.

2018 - A lagged diffusivity method for reaction-convection-diffusion equations with Dirichlet boundary conditions [Articolo su rivista]
Mezzadri, Francesco; Galligani, Emanuele
abstract

In this paper we solve a 2D nonlinear, non-steady reaction–convection–diffusion equation subject to Dirichlet boundary conditions by an iterative procedure consisting in lagging the diffusion term. First, we analyze the procedure, which we call Lagged Diffusivity Method. In particular, we provide a proof of the uniqueness of the solution and of the convergence of the lagged iteration when some assumptions are satisfied. We also describe outer and inner solvers, with special regard to how to link the stopping criteria in an efficient way. Numerical experiments are then introduced in order to evaluate the role of different linear solvers and of other components of the solution procedure, considering also the effect of the discretization.

2018 - An inexact Newton method for solving complementarity problems in hydrodynamic lubrication [Articolo su rivista]
Mezzadri, F.; Galligani, E.
abstract

We present an iterative procedure based on a damped inexact Newton iteration for solving linear complementarity problems. We introduce the method in the framework of a popular problem arising in mechanical engineering: the analysis of cavitation in lubricated contacts. In this context, we show how the perturbation and the damping parameter are chosen in our method and we prove the global convergence of the entire procedure. A Fortran implementation of the method is finally analyzed. First, we validate the procedure and analyze all its components, performing also a comparison with a recently proposed technique based on the Fischer–Burmeister–Newton iteration. Then, we solve a 2D problem and provide some insights on an efficient implementation of the method exploiting routines of the Lapack and of the PETSc packages for the solution of inner linear systems.

2018 - Topology optimization of self-supporting support structures for additive manufacturing [Articolo su rivista]
Mezzadri, Francesco; Bouriakov, Vladimir; Qian, Xiaoping
abstract

2017 - On the Lagged Diffusivity Method for the solution of nonlinear finite difference systems [Articolo su rivista]
Mezzadri, Francesco; Galligani, Emanuele
abstract

In this paper, we extend the analysis of the Lagged Diffusivity Method for nonlinear, non-steady reaction-convection-diffusion equations. In particular, we describe how the method can be used to solve the systems arising from different discretization schemes, recalling some results on the convergence of the method itself. Moreover, we also analyze the behavior of the method in case of problems presenting boundary layers or blow-up solutions.

2016 - A Chebyshev technique for the solution of optimal control problems with nonlinear programming methods [Articolo su rivista]
Mezzadri, Francesco; Galligani, Emanuele
abstract

This paper concerns with the solution of optimal control problems transcribed into nonlinear programming (NLP) problems by using approximations based on the Chebyshev series expansion. We first consider the relationship between the necessary conditions of the optimal control problem and the ones of the corresponding NLP problem. We then consider applying the Chebyshev based method to problems depending on both a time and a space variable by handling the space dependency with finite difference discretizations. We eventually write in AMPL language a collection of optimization problems (most of which includes also a space variable), which we solve with a series of solvers, so to analyze the behavior of the method and the influence on the solution of the parameters of the approximation and of the used solver.

2013 - On the numerical solution of elliptic and parabolic optimal control problems [Articolo su rivista]
Mezzadri, Francesco; Galligani, Emanuele
abstract

This paper regards the solution of some optimal control problems, transcribed as constrained optimization problems, with the main purpose of analysing how the solution changes when the number of discretization points is modified. The problems studied, which include both boundary and distributed elliptic optimal control problems as well as parabolic control problems, have in fact been written in AMPL language and solved by changing the size of the discretization mesh, so to analyse the variance of the minimum of the objective as the number of discretization points increases. Moreover, the problems have been solved using different solvers (such as MINOS, LOQO, IPOPT, SNOPT and KNITRO) and different kinds of discretization so to consider the influence of these parameters on the final results.