MOX Reports
The preprint collection of the Laboratory for Modeling and Scientific Computation MOX. It mainly contains works on numerical
analysis and mathematical modeling applied to engineering problems. MOX web site is mox.polimi.it
Found 1287 products
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16/2021 - 03/31/2021
Salvador, M.; Dede', L.; Manzoni, A.
Non intrusive reduced order modeling of parametrized PDEs by kernel POD and neural networks | Abstract | | We propose a nonlinear reduced basis method for the efficient approximation of parametrized partial differential equations (PDEs), exploiting kernel proper orthogonal decomposition (KPOD) for the generation of a reduced-order space and neural networks for the evaluation of the reduced-order approximation. In particular, we use KPOD in place of the more classical POD, on a set of high-fidelity solutions of the problem at hand to extract a reduced basis. This method provides a more accurate approximation of the snapshots' set featuring a lower dimension, while maintaining the same efficiency as POD. A neural network (NN) is then used to find the coefficients of the reduced basis by following a supervised learning paradigm and shown to be effective in learning the map between the time/parameter values and the projection of the high-fidelity snapshots onto the reduced space. In this NN, both the number of hidden layers and the number of neurons vary according to the intrinsic dimension of the differential problem at hand and the size of the reduced space. This adaptively built NN attains good performances in both the learning and the testing phases. Our approach is then tested on two benchmark problems, a one-dimensional wave equation and a two-dimensional nonlinear lid-driven cavity problem. We finally compare the proposed KPOD-NN technique with a POD-NN strategy, showing that KPOD allows a reduction of the number of modes that must be retained to reach a given accuracy in the reduced basis approximation. For this reason, the NN built to find the coefficients of the KPOD expansion is smaller, easier and less computationally demanding to train than the one used in the POD-NN strategy. |
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15/2021 - 03/10/2021
Fumagalli, A.; Patacchini, F.S.
Model adaptation in a discrete fracture network: existence of solutions and numerical strategies | Abstract | | Fractures are normally present in the underground and are, for some physical processes, of paramount importance. Their accurate description is fundamental to obtain reliable numerical outcomes useful, e.g., for energy management. Depending on the physical and geometrical properties of the fractures, fluid flow can behave differently, going from a slow Darcian regime to more complicated Brinkman or even Forchheimer regimes for high velocity. The main problem is to determine where in the fractures one regime is more adequate than others. In order to determine these low-speed and high-speed regions, this work proposes an adaptive strategy which is based on selecting the appropriate constitutive law linking velocity and pressure according to a threshold criterion on the magnitude of the fluid velocity itself. Both theoretical and numerical aspects are considered and investigated, showing the potentiality of the proposed approach. From the analytical viewpoint, we show existence of weak solutions to such model under reasonable hypotheses on the constitutive laws. To this end, we use a variational approach identifying solutions with minimizers of an underlying energy functional. From the numerical viewpoint, we propose a one-dimensional algorithm which tracks the interface between the low- and high-speed regions. By running numerical experiments using this algorithm, we illustrate some interesting behaviors of our adaptive model on a single fracture and small networks of intersecting fractures. |
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14/2021 - 03/10/2021
Peli, R.; Menafoglio, A.; Cervino, M.; Dovera, L.; Secchi, P;
Physics-based Residual Kriging for dynamically evolving functional random fields | Abstract | | We present a novel approach named Physics-based Residual Kriging for the statistical prediction of spatially dependent functional data. It incorporates a physical model - expressed by a partial differential equation - within a Universal Kriging setting through a geostatistical modelization of the residuals with respect to the physical model.
The approach is extended to deal with sequential problems, where samples of functional data become available along consecutive time intervals, in a context where the physical and stochastic processes generating them evolve, as time intervals succeed one another.
An incremental modeling is used to account for both these dynamics and the misfit between previous predictions and actual observations.We apply Physics-based Residual Kriging to forecast production rates of wells operating in a mature reservoir according to a given drilling schedule. We evaluate the predictive errors of the method in two different case studies. The first deals with a single-phase reservoir where production is supported by fluid injection, while the second considers again a single-phase reservoir but the production is driven by rock compaction. |
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13/2021 - 03/10/2021
Ferro, N.; Perotto, S.; Cangiani, A.
An anisotropic recovery-based error estimator for adaptive discontinuous Galerkin methods | Abstract | | We present a new recovery-based anisotropic error estimator for discontinuous Galerkin finite element approximations of advection-diffusion problems. We propose a metric-based algorithm for mesh adaptation which is driven by this error estimator. Numerical verification on several test cases, both in the steady and in the unsteady setting, shows the effectiveness of the algorithm in capturing the intrinsic directionalities of the solution. |
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12/2021 - 03/02/2021
di Cristofaro, D.; Galimberti, C.; Bianchi, D.; Ferrante, R.; Ferro, N.; Mannisi, M.; Perotto, S
Adaptive topology optimization for innovative 3D printed metamaterials | Abstract | | An adaptive method for designing the infill pattern of 3D printed objects is proposed. In particular, new unit cells for metamaterials are designed in order to match prescribed mechanical specifications. To this aim, we resort to topology optimization at the microscale driven by an inverse homogenization to guarantee the desired properties at the macroscale. The whole procedure is additionally enriched with an anisotropic adaptive generation of the computational mesh. The proposed algorithm is first numerically verified both in a mono- and in a multi-objective context. Then, a mechanical validation and 3D manufacturing through fused-model-deposition are carried out to assess the feasibility of the proposed design workflow. |
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11/2021 - 03/02/2021
Antonietti,P.F.; Manzini, G.; Mazzieri, I.; Scacchi, S.; Verani, M.
The conforming virtual element method for polyharmonic and elastodynamics problems: a review | Abstract | | In this paper we review recent results on the conforming virtual element approximation of polyharmonic and elastodynamics problems. The structure and the content of this review is motivated by three paradigmatic examples of applications: classical and anisotropic Cahn-Hilliard equation and phase field models for brittle fracture, that are briefly discussed in the first part of the paper. We present and discuss the mathematical details of the conforming virtual element approximation of linear polyharmonic problems, the classical Cahn-Hilliard equation and linear elastodynamics problems. |
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10/2021 - 03/02/2021
Di Michele, F.; May, J.; Pera, D.; Kastelic, V.; Carafa, M.; Smerzini, C.; Mazzieri, I.; Rubino, B.; Antonietti, P.F.; Quarteroni, A.; Aloisio, R.; Marcati, P.
Spectral elements numerical simulation of the 2009 L’Aquila earthquake on a detailed reconstructed domain | Abstract | | In this paper we simulate the earthquake that hit the city of L’Aquila on the 6th of April 2009 using the SPEED code (SPectral Elements in Elastodynamics with Discontinuous Galerkin), an open source library able to simulate the propagation of seismic waves in complex three dimensional (3D) domains. Our model includes an accurate 3D reconstruction of the Quaternary deposits, according to the most up-to-date data obtained from the Microzonation studies in Central Italy and a detailed reconstruction of the topography reported using a newly developed tool [61].
The sensitivity of our results with respect to different kinematic seis- mic sources is investigated. The results obtained are in good agreement with the recordings at all available seismic stations at epicentral dis- tances within 20 km range. Finally, a blind source prediction scenario shows that an acceptable agreement between simulations and recordings can be obtained by simulating stochastic rupture realizations with basic input data. A similar approach can be used to model future and past earthquakes for which little information is generally available, thus allowing an enhancement of the associated risk assessment. |
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09/2021 - 02/13/2021
Riccobelli, D.; Noselli, G.; DeSimone, A.
Rods coiling about a rigid constraint: Helices and perversions | Abstract | | Mechanical instabilities can be exploited to design innovative structures, able to change their shape in the presence of external stimuli. In this work, we derive a mathematical model of an elastic beam subjected to an axial force and constrained to smoothly slide along a rigid support, where the distance between the rod midline and the constraint is fixed and finite.
Using both theoretical and computational techniques, we characterize the bifurcations of such a mechanical system, in which the axial force and the natural curvature of the beam are used as control parameters. We show that, in the presence of a straight support, the rod can deform into shapes exhibiting helices and perversions, namely transition zones connecting together two helices with opposite chirality. The mathematical predictions of the proposed model are also compared with some experiments, showing a good quantitative agreement. In particular, we find that the buckled configurations may exhibit multiple perversions and we propose a possible explanation for this phenomenon based on the energy landscape of the mechanical system. |
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