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 1237 products
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12/2013 - 03/09/2013
Antonietti, P.F.; Beirao da Veiga, L.; Bigoni, N.; Verani, M.
Mimetic finite differences for nonlinear and control problems | Abstract | | In this paper we review some recent applications of the mimetic finite difference method to nonlinear problems (variational inequalities and quasilinear elliptic equations) and optimal control problems governed by linear elliptic partial differential equations. Several numerical examples show the effectiveness of mimetic finite differences in building accurate and robust numerical approximations. Finally, we draw some conclusions highlighting possible further applications of the mimetic finite difference method to nonlinear Stokes equations and shape optimization problems. |
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11/2013 - 03/01/2013
Discacciati, M.; Gervasio, P.; Quarteroni, A.
The Interface Control Domain Decomposition (ICDD) Method for Elliptic Problems | Abstract | | Interface controls are unknown functions used as Dirichlet or Robin boundary data on the interfaces of an overlapping decomposition designed for solving second order elliptic boundary value problems. The controls are computed through an optimal control problem with either distributed or interface observation. Numerical results show that, when interface observation is considered, the resulting Interface Control Domain Decomposition (ICDD) method is robust with respect to coefficients variations; it can exploit non-conforming meshes and provides optimal convergence with respect to the discretization parameters; finally it can be easily used to face heterogeneous advection - advection/diffusion couplings. |
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10/2013 - 02/28/2013
Antonietti, P.F.; Beirao da Veiga, L.; Mora, D.; Verani, M.
A stream virtual element formulation of the Stokes problem on polygonal meshes | Abstract | | In this paper we propose and analyze a novel stream formulation
of the Virtual Element Method (VEM) for the solution of the
Stokes problem. The new formulation hinges upon the introduction
of a suitable stream function space (characterizing the divergence
free subspace of discrete velocities) and it is equivalent to the
velocity-pressure (inf-sup stable) mimetic scheme presented in [L. Beirao da Veiga, V. Gyrya, K. Lipnikov and G. Manzini, Mimetic finite difference method for the Stokes problem on polygonal meshes, J. Comput. Phys. (2009)]
(up to a suitable reformulation into the VEM framework). Both schemes are
thus stable and linearly convergent but the new method results to be more
desirable as it employs much less degrees of freedom and it is based on a
positive definite algebraic problem. Several numerical experiments assess
the convergence properties of the new method and show its computational
advantages with respect to the mimetic one.
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09/2013 - 02/23/2013
Vergara, C.; Palamara, S.; Catanzariti, D.; Pangrazzi, C.; Nobile, F.; Centonze, M.; Faggiano, E.; Maines, M.; Quarteroni, A.; Vergara, G.
Patient-specific computational generation of the Purkinje network driven by clinical measuraments | Abstract | | Rationale: The propagation of the electrical signal in the Purkinje network is the starting point of the activation of the muscular cells in the ventricle and of the contraction of the heart. Anomalous propagation in such a network can cause disorders such as ventricular fibrillation. In the computational models describing the electrical activity of the ventricle is therefore important to account for the Purkinje fibers.
Objective: Aim of this work is to apply to real cases a method for the generation of the Purkinje network driven by patient-specific measures of the activation on the endocardium, to compute the activation maps in the ventricle and to compare the accuracy with that of other strategies proposed so far in the literature.
Methods and Results: We consider MRI images of two patients and data of the activation times on the endocardium acquired by means of the EnSite NavX system. To generate the Purkinje network we use a fractal law driven by the measures. Our results show that for a normal activation our algorithm is able to reduce considerably the errors (19.9±5.3% with our algorithm vs 33.9±6.8% with the best of the other strategies for patient 1, and 28.6±6.0% vs 63.9±8.6% for patient 2).
Conclusions: In this work we showed the reliability of the proposed method to generate a patient-specific Purkinje network. This allowed to improve the accuracy of computational models for the description of the electrical activation in the ventricle. |
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08/2013 - 02/22/2013
Chen, P.; Quarteroni, A.; Rozza, G.
A Weighted Reduced Basis Method for Elliptic Partial Differential Equations with Random Input Data | Abstract | | In this work we propose and analyze a weighted reduced basis method to solve elliptic partial differential equation (PDE) with random input data. The PDE is first transformed into a weighted parametric elliptic problem depending on a finite number of parameters. Distinctive importance at different values of the parameters are taken into account by assigning different weight to the samples in the greedy sampling procedure. A priori convergence analysis is carried out by constructive approximation of the exact solution with respect to the weighted parameters. Numerical examples are provided for the assessment of the advantages of the proposed method over the reduced basis method and stochastic collocation method in both univariate and multivariate stochastic problems. Keywords: weighted reduced basis method, stochastic partial differential equation, uncertainty quantication, stochastic collocation method, Kolmogorov N-width, exponential convergence |
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07/2013 - 02/21/2013
Chen, P.; Quarteroni, A.; Rozza, G.
A Weighted Empirical Interpolation Method: A-priori Convergence Analysis and Applications | Abstract | | We extend the conventional empirical interpolation method to a weighted empirical interpolation method in order to approximate nonlinear parametric functions with weighted parameters, e.g. random variables obeying various probability distributions. A priori convergence analysis is provided for the proposed method and the error bound by Kolmogorov N-width is improved from the recent work. We apply our method to geometric Brownian motion, exponential Karhunen-Loeve expansion and reduced basis approximation of non-affine stochastic elliptic equations. We demonstrate its improved accuracy and efficiency over the empirical interpolation method, as well as sparse grid stochastic collocation method. Keywords: empirical interpolation method, a priori convergence analysis, greedy algorithm, Kolmogorov N-width, geometric Brownian motion, Karhunen-Loeve expansion, reduced basis method |
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06/2013 - 02/18/2013
Dedè, L.; Quarteroni, A.
Isogeometric Analysis for second order Partial Differential Equations on surfaces | Abstract | | We consider the numerical solution of second order Partial Differential Equations (PDEs) on lower dimensional manifolds, specifically on surfaces
in three dimensional spaces. For the spatial approximation, we consider Isogeometric Analysis which facilitates the encapsulation of the exact
geometrical description of the manifold in the analysis when this is represented by B–splines or NURBS. Our analysis addresses linear, nonlinear,
time dependent, and eigenvalues problems involving the Laplace–Beltrami operator on surfaces. Moreover, we propose a priori error estimates
under h–refinement in the general case of second order PDEs on the lower dimensional manifolds. We highlight the accuracy and efficiency of Isogeometric Analysis with respect to the exactness of the geometrical representations of the surfaces. |
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05/2013 - 02/16/2013
Caputo, M.; Chiastra, C.; Cianciolo, C.; Cutri , E.; Dubini, G.; Gunn, J.; Keller, B.; Zunino, P.;
Simulation of oxygen transfer in stented arteries and correlation with in-stent restenosis | Abstract | | Computational models are used to study the combined effect of biomechanical and biochemical factors on coronary in-stent restenosis, which is a post-operative remodeling and regrowth of the stented artery tissue. More precisely, we address numerical simulations based on Navier-Stokes and mass transport equations to study the role of perturbed wall shear stresses and reduced oxygen concentration in a geometrical model reconstructed from a real porcine artery treated with a stent. Joining emph{in vivo} and emph{in silico} tools of investigation has multiple benefits in this case. On one hand, the geometry of the arterial wall and of the stent closely correspond to a real implanted configuration. On the other hand, the inspection of histological tissue samples informs us on the location and intensity of in-stent restenosis. As a result of that, we are able to correlate geometrical factors, such as the axial variation of the artery diameter and its curvature, the numerical quantification of biochemical stimuli, such as wall shear stresses, and the availability of oxygen to the inner layers of the artery, with the appearance of in-stent restenosis. This study shows that the perturbation of the vessel curvature could induce hemodynamic conditions that stimulate an undesired arterial remodeling. |
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