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 1268 products
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21/2021 - 04/10/2021
Torti, A.; Galvani, M.; Menafoglio, A.; Secchi, P.; Vantini S.
A General Bi-clustering Algorithm for Hilbert Data: Analysis of the Lombardy Railway Service | Abstract | | A general and flexible bi-clustering algorithm for the analysis of Hilbert data is presented in the Object Oriented Data Analysis framework. The algorithm, called HC2 (i.e. Hilbert Cheng and Church), is a non-parametric method to bi-cluster Hilbert data indexed in a matrix structure.
The Cheng and Church approach is here extended to the general case of data embedded in a Hilbert space and then applied to the analysis of the regional railway service in the Lombardy region with the aim of identifying recurrent patterns in the passengers' daily access to trains and/or stations. The analysed data, modelled as multivariate functional data and time series, allows to measure both overcrowding and travel demand, providing useful insights to best handle the service. |
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20/2021 - 04/10/2021
Pasquale, A.; Ammar, A.; Falcó, A.; Perotto, S.; Cueto, E.; Duval, J.-L.; Chinesta, F.
A separated representation involving multiple time scales within the Proper Generalized Decomposition framework | Abstract | | Solutions of partial differential equations can exhibit multiple time scales. Standard discretization techniques are constrained to capture the finest scale to accurately predict the response of the system. In this paper, we provide an alternative route to circumvent prohibitive meshes arising from the necessity of capturing fine-scale behaviors. The proposed methodology is based on a time-separated representation within the standard Proper Generalized Decomposition, where the time coordinate is transformed into a multi-dimensional time through new separated coordinates, each representing one scale, while continuity is ensured in the scale coupling. For instance, when considering two different time scales, the governing Partial Differential Equation is commuted into a nonlinear system that iterates between the so-called microtime and macrotime, so that the time coordinate can be viewed as a 2D time. The macroscale effects are taken into account by means of a finite element-based macro-discretization, whereas the microscale effects are handled with unidimensional parent spaces that are replicated throughout the time domain. The resulting separated representation allows us a very fine time discretization without impacting the computational efficiency. The proposed formulation is explored and numerically verified on thermal and elastodynamic problems. |
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19/2021 - 03/31/2021
Gillard, M.; Benacchio, T.
FT-GCR: a fault-tolerant generalized conjugate residual elliptic solver | Abstract | | With the steady advance of high performance computing systems
featuring smaller and smaller hardware components, the systems and
algorithms used for numerical simulations increasingly contend with
disruptions caused by hardware failures and bit-levels misrepresenta-
tions of computing data. In numerical frameworks exploiting massive
processing power, the solution of linear systems often represents the
most computationally intensive component. Given the large amount
of repeated operations involved, iterative solvers are particularly vulnerable to bit-flips.
A new method named FT-GCR is proposed here that supplies the
preconditioned Generalized Conjugate Residual Krylov solver with
detection of, and recovery from, soft faults. The algorithm tests on the monotonic decrease of the residual norm and, upon failure, restarts
the iteration within the local Krylov space. Numerical experiments
on the solution of an elliptic problem arising from a stationary flow
over an isolated hill on the sphere show the skill of the method in
addressing bit-flips on a range of grid sizes and data loss scenarios,
with best returns and detection rates obtained for larger corruption
events. The simplicity of the method makes it easily extendable to
other solvers and an ideal candidate for algorithmic fault tolerance
within integrated model resilience strategies. |
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18/2021 - 03/31/2021
Gigante, G.; Vergara, C.
On the choice of interface parameters in Robin-Robin loosely coupled schemes for fluid-structure interaction | Abstract | | We consider two loosely-coupled schemes for the solution of the fluid-structure interaction problem in presence of large added mass effect. In particular, we introduce the Robin-Robin and Robin-Neumann explicit schemes where suitable interface conditions of Robin type are used. For the estimate of interface Robin parameters which guarantee stability of the numerical solution, we propose to optimize the reduction factor of the corresponding strongly-coupled (implicit) scheme, by means of the Optimized Schwarz method. To check the suitabilty of our proposals, we show numerical results both in an ideal cylindrical domain and in a real human carotid. |
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17/2021 - 03/31/2021
Chew, R.; Benacchio, T.; Hastermann, G.; Klein, R.
Balanced data assimilation with a blended numerical model | Abstract | | A challenge arising from the local Bayesian assimilation of data in an atmospheric flow simulation is the imbalances it may introduce. Fast-mode imbalances of the order of the slower dynamics can be negated by employing a blended numerical model with seamless access to the compressible and the soundproof pseudo-incompressible dynamics. Here, the blended modelling strategy by Benacchio et al. (2014) is upgraded in an advanced numerical framework and extended with a Bayesian local ensemble data assimilation method. Upon assimilation of data, the model configuration is switched to the pseudo-incompressible regime for one time-step. After that, the model configuration is switched back to the compressible model for the duration of the assimilation window. The switching between model regimes is repeated for each subsequent assimilation window. An improved blending strategy ensures that a single time-step in the pseudo-incompressible regime is sufficient to filter imbalances. This improvement is based on three innovations: (i) the association of pressure fields computed at different stages of the numerical integration with actual time levels; (ii) a conversion of pressure-related variables between the model regimes derived from low Mach number asymptotics; and (iii) a judicious selection of the pressure variables used in converting numerical model states when a switch of models occurs. Travelling vortex and bubble convection experiments show that the imbalance arising from assimilation of the momentum fields can be eliminated by using this blended model, thereby achieving balanced analysis fields. The leftover imbalance in the thermodynamics can be quantified by scale analysis. |
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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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