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 1242 products
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20/2022 - 04/13/2022
Clementi, L.; Gregorio, C; Savarè, L.; Ieva, F; Santambrogio, M.D.; Sangalli, L.M.
A Functional Data Analysis Approach to Left Ventricular Remodeling Assessment | Abstract | | Left ventricular remodeling is a mechanism common to various cardiovascular diseases affecting myocardial morphology. It can be often overlooked in clinical practice since the parameters routinely employed in the diagnostic process (e.g., the ejection fraction) mainly focus on evaluating volumetric aspects. Nevertheless, the integration of a quantitative assessment of structural modifications can be pivotal in the early individuation of this pathology. In this work, we propose an approach based on functional data analysis to evaluate myocardial
contractility. A functional representation of ventricular shape is introduced, and functional principal component analysis and depth measures are used to discriminate healthy subjects from those affected by left ventricular hypertrophy. Our approach enables the integration of higher informative content compared to the traditional clinical parameters, allowing for a synthetic representation of morphological changes in the myocardium, which could be further explored and considered for future clinical practice implementation. |
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19/2022 - 04/11/2022
Lupo Pasini, M.; Perotto, S.
Hierarchical model reduction driven by machine learning for parametric advection-diffusion-reaction problems in the presence of noisy data | Abstract | | We propose a new approach to generate a reliable reduced model for a parametric elliptic problem, in the presence of noisy data. The reference model reduction procedure is the directional HiPOD method, which combines Hierarchical Model reduction with a standard Proper Orthogonal Decomposition, according to an offline/online paradigm.
In this paper we show that directional HiPOD looses in terms of accuracy when problem data are affected by noise. This is due to the interpolation driving the online phase, since it replicates, by definition, the noise trend. To overcome this limit, we replace interpolation with Machine Learning fitting models which better discriminate relevant physical features in the data from irrelevant unstructured noise. The numerical assessment, although preliminary, confirms the potentialities of the new approach. |
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18/2022 - 04/11/2022
Bennati, L; Vergara, C; Giambruno, V; Fumagalli, I; Corno, A.F; Quarteroni, A; Puppini, G; Luciani, G.B
An image-based computational fluid dynamics study of mitral regurgitation in presence of prolapse | Abstract | | Purpose: In this work we performed an imaged-based computational
study of the systolic fluid dynamics in presence of Mitral Valve Regurgitation
(MVR). In particular, we compare healthy and different regurgitant
scenarios with the aim of quantifying different hemodynamic quantities.
Methods: We performed computational fluid dynamic (CFD) simulations
in the left ventricle, left atrium and aortic root, with a resistive
immersed method, a turbulence model, and with imposed systolic wall
motion reconstructed from Cine-MRI images, which allowed us to segment
also the mitral valve. For the regurgitant scenarios we considered
an increase of the heart rate and a dilation of the left ventricle.
Results: Our results highlighted that MVR gave rise to regurgitant
jets through the mitral orifice impinging against the atrial
walls and scratching against the mitral valve leading to high values
of Wall Shear Stresses (WSS) with respect to the healthy case.
Conclusion: CFD with prescribed wall motion and immersed mitral
valve revealed to be an effective tool to quantitatively describe hemodynamics
in the case of MVR and to compare different regurgitant
scenarios. Our findings highlighted in particular the presence of
transition to turbulence in the atrium and allowed us to quantify
some important cardiac indices such as cardiac output and WSS. |
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17/2022 - 04/11/2022
Regazzoni, F.
Stabilization of staggered time discretization schemes for 0D-3D fluid-structure interaction problems | Abstract | | In this paper we analyze the numerical oscillations affecting time-staggered schemes for 0D-3D fluid-structure interaction (FSI) problems, which arise e.g. in the field of cardiovascular modeling, and we propose a novel stabilized scheme that cures this issue. We study two staggered schemes. In the first one, the 0D fluid model prescribes the pressure to the 3D structural mechanics model and receives the flow. In the second one, on the contrary, the fluid model receives the pressure and prescribes the flow. These schemes are respectively known, in the FSI literature, as Dirichlet-Neumann and Neumann-Dirichlet schemes, borrowing these terms from domain decomposition methods, although here a single iteration is performed before moving on to the next time step. Should the fluid be enclosed in a cavity, the Dirichlet-Neumann scheme is affected by non-physical oscillations whose origin lies in the balloon dilemma, for which we provide an algebraic interpretation. Moreover, we show that also the Neumann-Dirichlet scheme can be unstable for a range of parameter choices. Surprisingly, increasing either the viscous dissipation or the inertia of the structure favours the onset of oscillations. Our analysis provides an explanation for this fact and yields sharp stability bounds on the time step size. Inspired by physical considerations on the onset of oscillations, we propose a numerically consistent stabilization term for the Neumann-Dirichlet scheme. We prove that our proposed stabilized scheme is absolutely stable for any choice of time step size. These results are verified by several numerical tests. Finally, we apply the proposed stabilized scheme to an important problem in cardiac electromechanics, namely the coupling between a 3D model and a closed-loop lumped-parameter model of blood circulation. In this setting, our proposed scheme successfully removes the non-physical oscillations that would otherwise affect the numerical solution. |
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16/2022 - 03/17/2022
G. Ciaramella, T. Vanzan
Substructured Two-grid and Multi-grid Domain Decomposition Methods | Abstract | | Two-level Schwarz domain decomposition methods are very powerful techniques for the efficient numerical solution of partial differential equations (PDEs). A two-level domain decomposition method requires two main components: a one-level preconditioner (or its corresponding smoothing iterative method), which is based on domain decomposition techniques, and a coarse correction step, which relies on a coarse space. The coarse space must properly represent the error components that the chosen one-level method is not capable to deal with. In
the literature, most of the works introduced efficient coarse spaces obtained as the span of functions defined on the entire space domain of the considered PDE. Therefore, the corresponding two-level preconditioners and iterative methods are defined in volume.
In this paper, we use the excellent smoothing properties of Schwarz domain decomposition methods to define, for general elliptic problems, a new class of substructured two-level methods, for which both Schwarz smoothers and coarse correction steps are defined on the interfaces (except for the application of the smoother that requires volumetric subdomain solves). This approach has several advantages. On the one hand, the required computational effort is cheaper than the one required by classical volumetric two-level methods. On the other hand,
our approach does not require, like classical multi-grid methods, the explicit construction of coarse spaces, and it permits a multilevel extension, which is desirable when the high dimension of the problem or the scarce quality of the coarse space prevents the efficient numerical solution. Numerical experiments demonstrate the effectiveness of the proposed new numerical framework. |
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15/2022 - 03/17/2022
G. Ciaramella, T. Vanzan
Spectral coarse spaces for the substructured parallel Schwarz method | Abstract | | The parallel Schwarz method (PSM) is an overlapping Domain Decomposition (DD) method to solve partial differential equations (PDEs). Similarly to classical nonoverlapping DD methods, the PSM admits a substructured formulation, that is, it can be formulated as an iteration acting on variables defined exclusively on the interfaces of the overlapping decomposition.
In this manuscript, spectral coarse spaces are considered to improve the convergence and robustness of the substructured PSM. In this framework, the coarse space functions are defined exclusively on the interfaces. This is in contrast to classical two-level volume methods, where the coarse functions are defined in volume, though with local support. The approach presented in this work has several advantages. First, it allows one to use some of the well-known efficient coarse spaces proposed in the literature, and facilitates the numerical construction of efficient coarse spaces. Second, the computational work is comparable or lower than standard volume two-level methods. Third, it opens new interesting perspectives as the analysis of the new two-level substructured method requires the development of a new convergence analysis of general two-level iterative methods. The new analysis casts light on the optimality of coarse spaces: given a fixed dimension m, the spectral coarse space made by the first m dominant eigenvectors is not necessarily the minimizer of the asymptotic convergence factor. Numerical experiments demonstrate the effectiveness of the proposed new numerical framework. |
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14/2022 - 03/17/2022
Zappon, E.; Manzoni, A.; Quarteroni A.
Efficient and certified solution of parametrized one-way coupled problems through DEIM-based data projection across non-conforming interfaces | Abstract | | One of the major challenges of coupled problems is to manage nonconforming meshes at the interface between two models and/or domains, due to different numerical schemes or domains discretizations employed. Moreover, very often complex submodels depend on (e.g., physical or geometrical) parameters. Understanding how outputs of interest are affected by parameter variations thus plays a key role to gain useful insights on the problem's physics; however, expensive repeated solutions of the problem using high-fidelity, full-order models are often unaffordable. In this paper, we propose a parametric reduced order modeling (ROM) technique for parametrized one-way coupled problems made by a first independent model, the master model, and a second model, the slave model, that depends on the master model through Dirichlet interface conditions. We combine a reduced basis (RB) method, applied to each subproblems, with the discretized empirical interpolation method (DEIM) to efficiently interpolate or project Dirichlet data across conforming and non-conforming meshes at the domains interface, building a low-dimensional representation of the overall coupled problem. The proposed technique is then numerically verified by considering a series of test cases involving both steady and unsteady problems, and deriving a-posteriori error estimates on the solution of the coupled problem in both cases. This work arises from the need to solve staggered cardiac electrophysiological models and represents the first step towards the setting of ROM techniques for the more general two-way Dirichlet-Neumann coupled problems solved with domain decomposition sub-structuring methods, when interface non-conformity is involved. |
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13/2022 - 02/26/2022
Grasselli, M.; Parolini, N.; Poiatti, A.; Verani, M.
Non-isothermal non-Newtonian fluids: the stationary case | Abstract | | The stationary Navier-Stokes equations for a non-Newtonian incompressible fluid are coupled with the stationary heat equation and subject to Dirichlet type boundary conditions. The viscosity is supposed to depend on the temperature and the stress depends on the strain through a suit-able power law depending on p in (1,2) (shear thinning case). For this problem we establish the existence of a weak solution as well as we prove some regularity results both for the Navier-Stokes and the Stokes cases.Then, the latter case with the Carreau power law is approximated through a FEM scheme and some error estimates are obtained. Such estimates are then validated through some two-dimensional numerical experiments. |
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