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 1251 products
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10/2016 - 02/23/2016
Flemisch, B.; Fumagalli, A.; Scotti, A.
A review of the XFEM-based approximation of flow in fractured porous media | Abstract | | This paper presents a review of the available mathematical models and corresponding non-conforming numerical approximations which describe single-phase fluid flow in a fractured porous medium. One focus is on the geometrical difficulties that may arise in realistic simulations such as intersecting and immersed fractures. Another important aspect is the choice of the approximation spaces for the discrete problem: in mixed formulations, both the Darcy velocity and the pressure are considered as unknowns, while in classical primal formulations, a richer space for the pressure is considered and the Darcy velocity is computed a posteriori. In both cases, the extended finite element method is used, which allows for a complete geometrical decoupling among the fractures and rock matrix grids. The fracture geometries can thus be independent of the underlying grid thanks to suitable enrichments of the spaces that are able to represent possible jumps of the solution across the fractures. Finally, due to the dimensional reduction, a better approximation of the resulting boundary conditions for the fractures is addressed. |
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07/2016 - 02/19/2016
Pacciarini, P.; Gervasio, P.; Quarteroni, A.
Spectral Based Discontinuous Galerkin Reduced Basis Element Method for Parametrized Stokes Problems | Abstract | | In this work we extend to the Stokes problem the Discontinuous Galerkin Reduced Basis Element (DGRBE) method introduced in [1]. By this method we aim at reducing the computational cost for the approximation of a parametrized Stokes problem on a domain partitioned into subdomains. During an offline stage, expensive but performed only once, a low-dimensional approximation space is built on each subdomain. For any new value of the parameter, the rapid evaluation of the solution takes place during the online stage and consists in a Galerkin projection onto the low-dimensional subspaces computed offline. The high-fidelity discretization on each subdomain, used to build the local low-dimensional subspaces, is based on spectral element methods. The continuity of both the velocity and the normal component of the Cauchy stress tensor at subdomain interfaces is weakly enforced by a discontinuous Galerkin approach. |
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09/2016 - 02/19/2016
Rizzo, C.B.; de Barros, F.P.J.; Perotto, S.; Oldani, L.; Guadagnini, A.
Relative impact of advective and dispersive processes on the efficiency of POD-based model reduction for solute transport in porous media | Abstract | | We study the applicability of a model order reduction technique to the cost-effective solution of transport of passive scalars in porous media. Transport dynamics is modeled through the advection-dispersion equation (ADE) and we employ Proper Orthogonal Decomposition (POD) as a strategy to reduce the computational burden associated with the numerical solution of the ADE. Our application of POD relies on solving the governing ADE for selected time intervals, termed snapshots. The latter are then employed to achieve the desired model order reduction. The problem dynamics require alternating, over diverse time scales, between the solution of the full numerical transport model, as expressed by the ADE, and its reduced counterpart, constructed through the selected snapshots. We explore the way the selection of these time scales is linked to the Péclet number characterizing transport under steady-state flow conditions taking place in two-dimensional homogeneous and heterogeneous porous media. We find that the length of the time scale within which the POD-based reduced model solution provides accurate results tends to increase with decreasing Péclet number. This suggests that the effects of local scale dispersive processes facilitate the POD method to capture the salient features of the system dynamics embedded in the selected snapshots. Since the dimension of the reduced model is much lower than that of the full numerical model, the methodology we propose enables one to accurately simulate transport at a markedly reduced computational cost. |
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08/2016 - 02/19/2016
Dassi, F.; Perotto, S.; Si, H.; Streckenbach, T.
A priori anisotropic mesh adaptation driven by a higher dimensional embedding | Abstract | | We generalize the higher embedding approach proposed in B. Levy, N. Bonnell, 2012,
to generate an adapted mesh matching the intrinsic directionalities of an assigned function.
In more detail, the original embedding map between the physical (lower dimensional) and the embedded (higher dimensional) setting
is modified to include information associated with the function and with its gradient. Then, we
set an adaptive procedure driven by the embedded metric but performed in the lower dimensional setting
which results into an anisotropic adapted mesh of the physical domain.
The effectiveness of the proposed procedure is extensively investigated on several two-dimensional test cases,
involving both analytical functions and finite element approximations of differential problems.
The preliminary verification in three dimensions corroborates the robustness of the method. |
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06/2016 - 02/16/2016
Micheletti, S.; Perotto, S.; Signorini, M.
Anisotropic mesh adaptation for the generalized Ambrosio-Tortorelli functional with application to brittle fracture | Abstract | | Quasi-static crack propagation in brittle materials is modeled via the
Ambrosio-Tortorelli approximation [7]. The crack is modeled by a smooth
phase-field, defined on the whole computational domain. Since the crack
is confined to a thin layer, the employment of anisotropic adapted grids is
shown to be a really effective tool in containing computational costs. We
extend the error analysis in [3, 4, 5] to the generalized Ambrosio-Tortorelli functional introduced in [8], where a unified framework for several elasticity laws is dealt with as well as a non-convex fracture energy can be accommodated. After deriving an anisotropic a posteriori error estimator, we devise an algorithm which alternates optimization and mesh adaptation. Both anti-plane and plane-strain configurations are numerically checked.
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05/2016 - 02/16/2016
Alfio Quarteroni, A.; Lassila, T.; Rossi, S.; Ruiz-Baier, R.
Integrated Heart - Coupling multiscale and multiphysics models for the simulation of the cardiac function | Abstract | | Mathematical modelling of the human heart and its function can expand our understanding of various cardiac diseases, which remain the most common cause of death in the developed world. Like other physiological systems, the heart can be understood as a complex multiscale system involving interacting phenomena at the molecular, cellular, tissue, and organ levels. This article addresses the numerical modelling of many aspects of heart function, including the interaction of the cardiac electrophysiology system with contractile muscle tissue, the sub-cellular activation-contraction mechanisms, as well as the haemodynamics inside the heart chambers. Resolution of each of these sub-systems requires separate mathematical analysis and specially developed numerical algorithms, which we review in detail. By using specific sub-systems as examples, we also look at systemic stability, and explain for example how physiological concepts such as microscopic force generation in cardiac muscle cells, translate to coupled systems of differential equations, and how their stability properties influence the choice of numerical coupling algorithms. Several numerical examples illustrate three fundamental challenges of developing multiphysics and multiscale numerical models for simulating heart function, namely: (i) the correct upscaling from single-cell models to the entire cardiac muscle, (ii) the proper coupling of electrophysiology and tissue mechanics to simulate electromechanical feedback, and (iii) the stable simulation of ventricular haemodynamics during rapid valve opening and closure.
Key words: Integration of cardiac function, Coupling of multiphysics and multiscale models, Electrophysiology, Nonlinear elasticity, Navier-Stokes equations, Reaction-diffusion systems, Finite element methods, Stability analysis, Numerical simulation |
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04/2016 - 01/01/2016
Pettinati, V.; Ambrosi, D; Ciarletta P.; Pezzuto S.
Finite element simulations of the active stress in the imaginal disc of the Drosophila Melanogaster | Abstract | | During the larval stages of development, the imaginal disc of Drosphila Melanogaster is composed by a monolayer of epithelial cells, which undergo a strain actively produced by the cells themselves. The well organized collective contraction produces a stress field that seemingly has a double morphogenetic role: it orchestrates the cellular organization towards
the macroscopic shape emergence while simultaneously providing a local information on the organ size. Here we perform numerical simulations of such a mechanical control on morphogenesis at a continuum level, using a three-dimensional finite model that accounts for the active cell contraction. The numerical model is able to reproduce the (few) known
qualitative characteristics of the tensional patterns within the imaginal disc of the fruit fly. The computed stress components slightly deviate from planarity, thus confirming the previous theoretical assumptions of a nonlinear elastic analytical model, and enforcing the hypothesis that the mechanical stress may act as a size regulating signal that locally scales
with the global dimension of the domain. |
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03/2016 - 01/01/2016
Tarabelloni, N.; Ieva, F.
On Data Robustification in Functional Data Analysis | Abstract | | The problem of outlier detection in high dimensional settings is nowadays a crucial point for a number of statistical analysis. Outliers are often considered as an error or noise, instead, they may carry important information on the phenomenon under study. If not properly identified, they may lead to model misspecification, biased parameter estimation and incorrect results, especially in those contexts where the number of available statistical units is lower than the number of parameters (for example, Functional Data Analysis).
In this paper we introduce a robustly adjusted version of the functional boxplot, which is the most common tool adopted to perform outlier detection in Functional Data Analysis. A crucial element of the functional boxplot is the inflation factor of the fences, controlling the proportion of observations flagged as outlier. After an overview of the methods and tools currently available in the literature, we will describe a robust method to compute a data-driven value for such inflation factor. In doing so, we will make use of robust estimators of variance-covariance operators and the corresponding eigenvalues and eigenfunctions.
Two simulation studies are proposed to give direct insights into the use of the proposed functional boxplot, and test both the robustness and accuracy of robust variance-covariance estimators, together with the performances of the functional boxplot in recognising truly outlying observations. |
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