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 1239 products
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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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12/2022 - 02/25/2022
Antonietti, P.F.; Dassi, F.; Manuzzi, E.
Machine Learning based refinement strategies for polyhedral grids with applications to Virtual Element and polyhedral Discontinuous Galerkin methods | Abstract | | We propose two new strategies based on Machine Learning techniques to handle polyhedral grid refinement, to be possibly employed within an adaptive framework. The first one employs the k-means clustering algorithm to partition the points of the polyhedron to be refined. This strategy is a variation of the well known Centroidal Voronoi Tessellation. The second one employs Convolutional Neural Networks to classify the “shape” of an element so that “ad-hoc” refinement criteria can be defined. This strategy can be used to enhance existing refinement strategies, including the k-means strategy, at a low online computational cost. We test the proposed algorithms considering two families of finite element methods that support arbitrarily shaped polyhedral elements, namely the Virtual Element Method (VEM) and the Polygonal Discontinuous Galerkin (PolyDG) method. We demonstrate that these strategies do preserve the structure and the quality of the underlaying grids, reducing the overall computational cost and mesh complexity. |
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11/2022 - 02/22/2022
Sampaoli, S.; Agosti, A.; Pozzi, G.; Ciarletta,P.
A toy model of misfolded protein aggregation and neural damage propagation in neudegenerative diseases | Abstract | | Neurodegenerative diseases (NDs) result from the transformation and accumulation of misfolded proteins within the nervous system. They have common features, like the chronic nature and the progressive destruction of neurons in specific areas of the brain. Several mathematical models have been proposed to investigate the biological processes underlying NDs, focusing on the kinetics of polymerization and fragmentation at the microscale and on the spread of neural damage at a macroscopic level. The aim of this work is to bridge the gap between microscopic and macroscopic approaches proposing a toy partial differential model able to take into account both for the short-time dynamics of the misfolded proteins aggregating in plaques and the long-term evolution of tissue damage. Using the theoretical framework of mixtures theory, we considered the brain as a biphasic material made of misfolded protein aggregates and of healthy tissue. The resulting Cahn-Hilliard type equation for the misfolded proteins contains a growth term depending on the local availability of precursor proteins, that follow a reaction-diffusion equation. The misfolded proteins also posses a chemotactic mass flux driven by gradients of neural damage, that is caused by local accumulation of misfolded protein and that evolves slowly according to an Allen-Cahn equation. The partial differential model is solved numerically using the finite element method in a simple two-dimensional domain, evaluating the effects of the mobility of the misfolded protein and the diffusion of the neural damage. We considered both isotropic and anisotropic mobility coefficients, highlighting that the spreading front of the neural damage follows the direction of the largest eigenvalue of the mobility tensor. In both cases, we computed two biomarkers for quantifying the aggregation in plaques and the evolution of neural damage, that are in qualitative agreement with the characteristic Jack curves for many NDs. |
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10/2022 - 02/22/2022
Fresca, S.; Manzoni, A.
Real-time simulation of parameter-dependent fluid flows through deep learning-based reduced order models | Abstract | | Simulating fluid flows in different virtual scenarios is of key importance in engineering applications. However, high-fidelity, full-order models relying, e.g., on the finite element method, are unaffordable whenever fluid flows must be simulated in almost real-time. Reduced order models (ROMs) relying, e.g., on proper orthogonal decomposition (POD) provide reliable approximations to parameter-dependent fluid dynamics problems in rapid times. However, they might require expensive hyper-reduction strategies for handling parameterized nonlinear terms, and enriched reduced spaces (or Petrov-Galerkin projections) if a mixed velocity-pressure formulation is considered, possibly hampering the evaluation of reliable solutions in real-time. Dealing with fluid-structure interactions entails even higher difficulties. The proposed deep learning (DL)-based ROMs overcome all these limitations by learning in a non-intrusive way both the nonlinear trial manifold and the reduced dynamics. To do so, they rely on deep neural networks, after performing a former dimensionality reduction through POD enhancing their training times substantially. The resulting POD-DL ROMs are shown to provide accurate results in almost real-time for the flow around a cylinder benchmark, the fluid-structure interaction between an elastic beam attached to a fixed, rigid block and a laminar incompressible flow, and the blood flow in a cerebral aneurysm. |
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