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 1287 prodotti
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03/2020 - 10/01/2020
Ferro, N.; Micheletti, S.; Perotto, S.
Compliance-stress constrained mass minimization for topology optimization on anisotropic meshes | Abstract | | In this paper, we generalize the SIMPATY algorithm, which combines the SIMP method with anisotropic mesh adaptation to solve the minimum compliance problem with a mass costraint. In particular, the mass of the final layout is now minimized and both a maximum compliance and a maximum stress can be enforced as either mono- or multi-constraints. The new algorithm, named MSC-SIMPATY, is able to sharply detect the material-void interface, thanks to the anisotropic mesh adaptation. The presented test cases deal with three different scenarios, with a focus on the effect of the constraints on the final layouts and on the performance of the algorithm. |
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02/2020 - 10/01/2020
Fresca, S.; Dede', L.; Manzoni, A.
A comprehensive deep learning-based approach to reduced order modeling of nonlinear time-dependent parametrized PDEs | Abstract | | Traditional reduced order modeling techniques such as the reduced basis (RB) method (relying, e.g., on proper orthogonal decomposition (POD)) suffer from severe limitations when dealing with nonlinear time-dependent parametrized PDEs, because of the fundamental assumption of linear superimposition of modes they are based on. For this reason, in the case of problems featuring coherent structures that propagate over time such as transport, wave, or convection-dominated phenomena, the RB method usually yields inefficient reduced order models (ROMs) if one aims at obtaining reduced order approximations sufficiently accurate compared to the high-fidelity, full order model (FOM) solution. To overcome these limitations, in this work, we propose a new nonlinear approach to set reduced order models by exploiting deep learning (DL) algorithms. In the resulting nonlinear ROM, which we refer to as DL-ROM, both the nonlinear trial manifold (corresponding to the set of basis functions in a linear ROM) as well as the nonlinear reduced dynamics (corresponding to the projection stage in a linear ROM) are learned in a non-intrusive way by relying on DL algorithms; the latter are trained on a set of FOM solutions obtained for different parameter values. In this paper, we show how to construct a DL-ROM for both linear and nonlinear time-dependent parametrized PDEs; moreover, we assess its accuracy on test cases featuring different parametrized PDE problems. Numerical results indicate that DL-ROMs whose dimension is equal to the intrinsic dimensionality of the PDE solutions manifold are able to approximate the solution of parametrized PDEs in situations where a huge number of POD modes would be necessary to achieve the same degree of accuracy. |
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01/2020 - 05/01/2020
Pozzi, S.; Vergara, C.
Mathematical and numerical models of atherosclerotic plaque progression in carotid arteries | Abstract | | We propose a mathematical model for the description of plaque progression in carotid arteries.
This is based on the coupling of a fluid-structure interaction problem, arising between blood and vessel wall,
and differential problems for the cellular evolution. A numerical model is also proposed. This is based on
the splitting of the coupled problem based on a suitable strategy to manage the multiscale-in-time
nature of the problem. We present some preliminary numerical results both in ideal and real scenarios. |
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60/2019 - 30/12/2019
Ieva, F; Paganoni, A.M.; Romo, J.; Tarabelloni, N.
roahd Package: Robust Analysis of High Dimensional Data | Abstract | | The focus of this paper is on the open-source R package roahd (RObust Analysis
of High dimensional Data), see Tarabelloni et al. (2017). roahd has been developed to gather
recently proposed statistical methods that deal with the robust inferential analysis of univariate
and multivariate functional data. In particular, efficient methods for outlier detection and related
graphical tools, methods to represent and simulate functional data, as well as inferential tools for
testing differences and dependency among families of curves will be discussed, and the associated
functions of the package will be described in details. |
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58/2019 - 30/12/2019
Antonietti, P.F; Manzini, G.; Mourad, H.M.; Verani, M.
The virtual element method for linear elastodynamics models. Design, analysis, and implementation | Abstract | | We design the conforming virtual element method for the numerical simulation of two dimensional time-dependent elastodynamics problems. We investigate the performance of the method both theoretically and numerically. We prove the stability and the convergence of the semi-discrete approximation in the energy norm and derive optimal error estimates. We also show the convergence in the $L^2$ norm.
The performance of the virtual element method is assessed on a set
of different computational meshes, including non-convex cells up to order four in the $h$-refinement setting. Exponential convergence is also experimentally seen in the $p$-refinement setting. |
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57/2019 - 30/12/2019
Antonietti, P.F.; Bertoluzza, S.; Prada, D.; Verani M.
The Virtual Element Method for a Minimal Surface Problem | Abstract | | In this paper we consider the Virtual Element discretization of a minimal surface problem, a quasi-linear elliptic partial differential equation modeling the problem of minimizing the area of a surface subject to a prescribed boundary condition. We derive optimal error estimate and present several numerical tests assessing the validity of the theoretical results. |
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56/2019 - 30/12/2019
Antonietti, P.F.; Berrone, S.; Borio A.; D'Auria A.; Verani, M.; Weisser, S.
Anisotropic a posteriori error estimate for the Virtual Element Method | Abstract | | We derive an anisotropic a posteriori error estimate for the adaptive conforming Virtual Element approximation of a paradigmatic two-dimensional elliptic problem. In particular, we introduce a quasi-interpolant operator and exploit its approximation results to prove the reliability of the error indicator. We design and implement the corresponding adaptive polygonal anisotropic algorithm. Several numerical tests assess the superiority of the proposed algorithm in comparison with standard polygonal isotropic mesh refinement schemes. |
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55/2019 - 18/12/2019
Agosti, A.; Ciarletta, P.; Garcke, H.; Hinze, M.
Learning patient-specific parameters for a diffuse interface glioblastoma model from neuroimaging data | Abstract | | Parameters in mathematical models for glioblastoma multiforme (GBM) tumour growth
are highly patient specific. Here we aim to estimate parameters in a Cahn-Hilliard type
diffuse interface model in an optimised way using model order reduction (MOR) based on
proper orthogonal decomposition (POD). Based on snapshots derived from finite element
simulations for the full order model (FOM) we use POD for dimension reduction and solve
the parameter estimation for the reduced order model (ROM). Neuroimaging data are used
to define the highly inhomogeneous diffusion tensors as well as to define a target functional in
a patient specific manner. The reduced order model heavily relies on the discrete empirical
interpolation method (DEIM) which has to be appropriately adapted in order to deal with
the highly nonlinear and degenerate parabolic PDEs. A feature of the approach is that
we iterate between full order solves with new parameters to compute a POD basis function
and sensitivity based parameter estimation for the ROM problems. The algorithm is applied
using neuroimaging data for two clinical test cases and we can demonstrate that the reduced
order approach drastically decreases the computational effort. |
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