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 1249 products
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16/2016 - 04/06/2016
Agosti, A.; Antonietti, P.F.; Ciarletta, P.; Grasselli, M.; Verani, M.
A Cahn-Hilliard type equation with degenerate mobility and single-well potential. Part I: convergence analysis of a continuous Galerkin finite element discretization. | Abstract | | We consider a Cahn-Hilliard type equation with degenerate mobility
and single-well potential of Lennard-Jones type. This equation models the
evolution and growth of biological cells such as solid tumors. The degeneracy set of the mobility and the singularity set of the cellular potential do not coincide, and the absence of cells is an unstable equilibrium configuration of the potential. This feature introduces a nontrivial difference with respect to the Cahn-Hilliard equation analyzed in the literature. We formulate a continuous finite element approximation of the problem, where the positivity of the solution is enforced through a discrete variational inequality. We prove the existence and uniqueness of the discrete solution together with the convergence to the weak solution. We present simulation results in one and two space dimensions. We also study the dynamics of the spinodal decomposition and the growth and scaling laws of phase ordering dynamics.
In this case we find similar results to the ones obtained in standard phase
ordering dynamics and we highlight the fact that the asymptotic behavior
of the solution is dominated by the mechanism of growth by bulk diffusion.
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15/2016 - 04/06/2016
Ieva, F.; Paganoni, A.M.
A taxonomy of outlier detection methods for robust classification in multivariate functional data | Abstract | | We propose a new method for robust classification of multivariate functional data. We exploit the joint use of two different depth measures, generalizing the outliergram to the multivariate functional framework, aiming at detecting and discarding both shape and magnitude outliers in order to robustify the reference samples of data, composed by G different known groups. We asses by means of a simulation study method’s performance in comparison with different outlier detection methods. Finally we consider
a real dataset: we classify a data minimizing a suitable distance from the
center of reference groups. We compare performance of supervised classification on test sets training the algorithm on original dataset and on the robustified one, respectively. |
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14/2016 - 03/18/2016
Bonomi, D.; Manzoni, A.; Quarteroni, A.
A matrix discrete empirical interpolation method for the efficient model reduction of parametrized nonlinear PDEs: application to nonlinear elasticity problems | Abstract | | When using Newton iterations to solve nonlinear parametrized PDEs in the context of Reduced Basis (RB) methods, the assembling of the RB arrays in the online stage depends in principle on the high-fidelity approximation.
This task is even more challenging when dealing with fully nonlinear problems, for which the global Jacobian matrix has to be entirely reassembled at each Newton step.
In this paper the Discrete Empirical Interpolation Method (DEIM) and its matrix version MDEIM are exploited to perform system approximation at a purely algebraic level, in order to evaluate both the residual vector and the Jacobian matrix very efficiently. We compare different ways to combine solution-space reduction and system approximation, and we derive a posteriori error estimates on the solution accounting for the contribution of DEIM/MDEIM errors.
The capability of the proposed approach to generate accurate and efficient reduced-order models is demonstrated on the solution of two nonlinear elasticity problems. |
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13/2016 - 03/09/2016
Guerciotti, B; Vergara, C; Ippolito, S; Quarteroni, A; Antona, C; Scrofani, R.
Computational study of the risk of restenosis in coronary bypasses | Abstract | | Coronary artery disease, caused by the build-up of atherosclerotic plaques in the coronary vessel wall, is one of the leading causes of death in the world. For high-risk patients, coronary artery bypass graft is the preferred treatment. Despite overall excellent patency rates, bypasses may
fail due to restenosis. In this context, the purpose of this work is to perform a parametric computational study of the fuid-dynamics in patient-specific geometries with the aim of investigating a possible relationship
between coronary stenosis degree and risk of graft failure. Firstly, we propose a strategy to prescribe realistic boundary conditions in absence of measured data, based on an extension of Murray's law to provide the flow division at bifurcations in case of stenotic vessels and non-Newtonian
blood rheology. Then, we carry out numerical simulations in three patients
affected by severe coronary stenosis and treated with a graft, in which
the stenosis degree is virtually varied in order to compare the resulting fluid-dynamics in terms of hemodynamic indices potentially involved in
restenosis development. Our findings suggest that low degrees of coronary
stenosis produce a more disturbed fluid-dynamics in the graft, resulting
in hemodynamic conditions that may promote a higher risk of graft failure. |
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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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