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 1253 products
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11/2016 - 03/01/2016
Zhu, S.; Dedè, L.; Quarteroni, A.
Isogeometric Analysis and proper orthogonal decomposition for the acoustic wave equation | Abstract | | Isogeometric Analysis (IGA) is used in combination with proper orthogonal decomposition (POD) for model order reduction of the time parameterized acoustic wave equations. We propose a fully discrete IGA-Newmark-POD approximation and we analyze the associated numerical error, which features three components due to spatial discretization by IGA, time discretization with the Newmark scheme, and modes truncation by POD. We prove stability and convergence. Numerical examples are presented to show the accuracy and efficiency of IGA-based POD techniques for the model order reduction of the
acoustic wave equation.
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12/2016 - 03/01/2016
Bartezzaghi, A.; Dedè, L.; Quarteroni, A.
Isogeometric Analysis of Geometric Partial Differential Equations | Abstract | | We consider the numerical approximation of geometric Partial Differential Equations (PDEs) defined on surfaces in the 3D space. In particular, we focus on the geometric PDEs deriving from the minimization of an energy functional by L2 -gradient flow. We analyze two energy functionals: the area, which leads to the mean curvature flow, a nonlinear second order PDE, and the Willmore energy, leading to the Willmore flow, a nonlinear fourth order PDE. We consider surfaces represented by single-patch NURBS and discretize the PDEs by means of NURBS-based Isogeometric Analysis in the framework of the Galerkin method. To approximate the high order geometric PDEs we use high order continuous NURBS basis functions. Instead, for the time discretization of the nonlinear geometric PDEs, we use Backward Differentiation Formulas (BDF) with extrapolation of the geometric quantities involved in the weak formulation of the problem; in this manner, we solve a linear problem at each time step. We report numerical results concerning the mean curvature and Willmore flows on different geometries of interest and we show the accuracy and efficiency of the proposed approximation scheme. Keywords: Geometric Partial Differential Equation, Surface, High Order, Isogeometric Analysis, Mean Curvature Flow, Willmore Flow |
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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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