Quaderni MOX
Pubblicazioni
del Laboratorio di Modellistica e Calcolo Scientifico MOX. I lavori riguardano prevalentemente il campo dell'analisi numerica, della statistica e della modellistica matematica applicata a problemi di interesse ingegneristico. Il sito del Laboratorio MOX è raggiungibile
all'indirizzo mox.polimi.it
Trovati 1237 prodotti
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58/2024 - 03/09/2024
Ciaramella, G.; Vanzan, T.
Variable reduction as a nonlinear preconditioning approach for optimization problems | Abstract | | When considering an unconstrained minimization problem, a standard approach is to solve the optimality system with a Newton method possibly preconditioned by, e.g., nonlinear elimination. In this contribution, we argue that nonlinear elimination could be used to reduce the number of optimization variables by artificially constraining them to satisfy a subset of the optimality conditions. Consequently, a reduced objective function is derived which can now be minimized with any optimization algorithm. By choosing suitable variables to eliminate, the conditioning of the reduced optimization problem is largely improved. We here focus in particular on a right preconditioned gradient descent and show theoretical and numerical results supporting the validity of the presented approach. |
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56/2024 - 02/09/2024
Parolini, N.; Covello, V.; Della Rocca, A.; Verani, M.,
Design of a checkerboard counterflow heat exchanger for industrial applications | Abstract | | This work is devoted to the design of a checkerboard air-gas heat exchanger suitable for industrial applications. The design of the heat exchanger is optimized in order to obtain the maximum increase of the outlet air temperature, considering different geometrical design parameters and including manufacturing constraints. The heat exchanger efficiency has been assessed by means of the $epsilon$-NTU method. The perfomances are compared with traditional finned recuperators and appreciable enhancement of the exchanger efficiency has been observed adopting the new design. |
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57/2024 - 02/09/2024
Negrini, G.: Parolini, N.; Verani, M.
An Immersed Boundary Method for Polymeric Continuous Mixing | Abstract | | We introduce a new implementation of the Immersed Boundary method in the finite-volume library OpenFOAM. The implementation is tailored to the simulation of temperature-dependent non-Newtonian polymeric flows in complex moving geometries, such as those characterizing the most popular polymeric mixing technologies. |
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55/2024 - 01/09/2024
Artoni, A.; Ciaramella, G.; Gander, M.J.; Mazzieri, I.
Schwarz Waveform Relaxation and the Unmapped Tent-Pitching Method in 3D | Abstract | | Several Parallel-in-Time (PinT) algorithms, especially multilevel methods like Parareal and MGRIT, struggle when applied to hyperbolic partial differential equations. There are however also very effective PinT methods for hyperbolic problems which use the hyperbolic nature of the problem to their advantage. Typical examples are Schwarz Waveform Relaxation methods, and the Mapped and Unmapped Tent Pitching methods. We present and study here for the first time the Unmapped Tent Pitching method in three spatial dimensions, applied to a second order wave equation. We give a general equivalence result with the Mapped Tent Pitching algorithm using Schwarz Waveform Relaxation to build the link, and also characterize in detail the resulting 4D space-time tents generated implicitly by the Unmapped Tent Pitching method. This leads to a complete convergence analysis of the method in 3D. |
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54/2024 - 14/08/2024
Antonietti, P.F.; Botti, M.; Cancrini, A.; Mazzieri, I.
A polytopal discontinuous Galerkin method for the pseudo-stress formulation of the unsteady Stokes problem | Abstract | | This work aims to construct and analyze a discontinuous Galerkin method on polytopal grids (PolydG) to solve the pseudo-stress formulation of the unsteady Stokes problem. The pseudo-stress variable is introduced due to the growing interest in non-Newtonian flows and coupled interface problems, where stress assumes a fundamental role.
The space-time discretization of the problem is achieved by combining the PolydG approach with the implicit theta-method time integration scheme. For both the semi- and fully-discrete problems we present a detailed stability analysis. Moreover, we derive convergence estimates for the fully discrete space-time discretization. A set of verification tests is presented to verify the theoretical estimates and the application of the method to cases of engineering interest. |
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53/2024 - 12/08/2024
Caldana, M.; Hesthaven, J. S.
Neural ordinary differential equations for model order reduction of stiff systems | Abstract | | Neural Ordinary Differential Equations (ODEs) represent a significant advancement at the intersection of machine learning and dynamical systems, offering a continuous-time analog to discrete neural networks. Despite their promise, deploying neural ODEs in practical applications often encounters the challenge of stiffness, a condition where rapid variations in some components of the solution demand prohibitively small time steps for explicit solvers. This work addresses the stiffness issue when employing neural ODEs for model order reduction by introducing a suitable reparametrization in time. The considered map is data-driven and it is induced by the adaptive time-stepping of an implicit solver on a reference solution. We show the map produces a nonstiff system that can be cheaply solved with an explicit time integration scheme. The original, stiff, time dynamic is recovered by means of a map learnt by a neural network that connects the state space to the time reparametrization. We validate our method through extensive experiments, demonstrating improvements in efficiency for the neural ODE inference while maintaining robustness and accuracy. The neural network model also showcases good generalization properties for times beyond the training data. |
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52/2024 - 02/08/2024
Botti, M.; Mascotto, L.
A Necas-Lions inequality with symmetric gradients on star-shaped domains based on a first order Babuska-Aziz inequality | Abstract | | We prove a Necas-Lions inequality with symmetric gradients on two and three dimensional domains that are star-shaped with respect to a ball B; the constants in the inequality are explicit with respect to the diameter and the radius of B. Crucial tools in deriving this inequality are a first order Babuska-Aziz inequality based on Bogovskii's construction of a right-inverse of the divergence and Fourier transform techniques proposed by Duran. As a byproduct, we derive arbitrary order estimates in arbitrary dimension for that operator. |
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51/2024 - 23/07/2024
Antonietti, P.F.; Corti, M.; Lorenzon, G.
A discontinuous Galerkin method for the three-dimensional heterodimer model with application to prion-like proteins’ dynamics | Abstract | | Neurocognitive disorders, such as Alzheimer's and Parkinson's, have a wide social impact. These proteinopathies involve misfolded proteins accumulating into neurotoxic aggregates. Mathematical and computational models describing the prion-like dynamics offer an analytical basis to study the diseases' evolution and a computational framework for exploring potential therapies. This work focuses on the heterodimer model in a three-dimensional setting, a reactive-diffusive system of nonlinear partial differential equations describing the evolution of both healthy and misfolded proteins. We investigate traveling wave solutions and diffusion-driven instabilities as a mechanism of neurotoxic pattern formation. For the considered mathematical model, we propose a space discretization, relying on the Discontinuous Galerkin method on polytopal/polyhedral grids, allowing high-order accuracy and flexible handling of the complicated brain’s geometry. Further, we present a-priori error estimates for the semi-discrete formulation and we perform convergence tests to verify the theoretical results. Finally, we conduct simulations using realistic data on a three-dimensional brain mesh reconstructed from medical images. |
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