Quaderni MOX
Pubblicazioni
del Laboratorio di Modellistica e Calcolo Scientifico MOX. I lavori riguardano prevalentemente il campo dell'analisi numerica, della statistica e della modellistica matematica applicata a problemi di interesse ingegneristico. Il sito del Laboratorio MOX è raggiungibile
all'indirizzo mox.polimi.it
Trovati 1249 prodotti
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53/2017 - 23/10/2017
Bertagna, L.; Deparis, S.;Formaggia, L.; Forti, D.; Veneziani A.
The LifeV library: engineering mathematics beyond the proof of concept | Abstract | | LifeV is a library for the finite element (FE) solution of partial differential equations in one, two, and three dimensions. It is written in C++ and designed to run on diverse parallel architectures, including cloud and high performance computing facilities. In spite of its academic research nature, meaning a library for the development and testing of new methods, one distinguishing feature of LifeV is its use on real world problems and it is intended to provide a tool for many engineering applications. It has been actually used in computational hemodynamics, including cardiac mechanics and fluid-structure interaction problems, in porous media, ice sheets dynamics for both forward and inverse problems. In this paper we give a short overview of the features of LifeV and its coding paradigms on simple problems. The main focus is on the parallel environment which is mainly driven by domain decomposition methods and based on external libraries such as MPI, the Trilinos project, HDF5 and ParMetis.
(Dedicated to the memory of Fausto Saleri) |
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54/2017 - 23/10/2017
Dede', L; Quarteroni, A.
Isogeometric Analysis of a Phase Field Model for Darcy Flows with Discontinuous Data | Abstract | | We consider a phase field model for Darcy flows with discontinuous data in porous media; specifically, we adopt the Hele-Shaw-Cahn-Hillard equations of [Lee, Lowengrub, Goodman, Physics of Fluids, 2002] to model flows in the Hele-Shaw cell through a phase field formulation which incorporates discontinuities of physical data, namely density and viscosity, across interfaces. For the spatial approximation of the problem, we use NURBS-based Isogeometric Analysis in the framework of the Galerkin method, a computational framework which is particularly advantageous for the solution of high order Partial Differential Equations and phase field problems which exhibit sharp but smooth interfaces. In this paper, we verify through numerical tests the sharp interface limit of the phase field model which in fact leads to an internal discontinuity interface problem; finally, we show the efficiency of Isogeometric Analysis for the numerical approximation of the model by solving a benchmark problem, the so-called "rising bubble" problem. |
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55/2017 - 23/10/2017
Agosti, A.; Cattaneo, C.; Giverso, C.; Ambrosi, D.; Ciarletta, P.
A computational platform for the personalized clinical treatment of glioblastoma multiforme | Abstract | | In this work, we develop a computational tool to predict the patient-specific evolution of a highly malignant
brain tumour, the glioblastoma multiforme (GBM), and its response to therapy. A diffuse-interface mathematical
model based on mixture theory is fed by clinical neuroimaging data that provide the anatomical and microstructural
characteristics of the patient brain. The model is numerically solved using the finite element method, on the basis
of suitable numerical techniques to deal with the resulting Cahn-Hilliard type equation with degenerate mobility and
single-well potential. We report the results of simulations performed on the real geometry of a patient brain, proving
how the tumour expansion is actually dependent on the local tissue structure. We also report a sensitivity analysis
concerning the effects of the different therapeutic strategies employed in the clinical Stupp protocol. The simulated
results are in qualitative agreement with the observed evolution of GBM during growth, recurrence and response to
treatment. Taken as a proof-of-concept, these results open the way to a novel personalized approach of mathematical
tools in clinical oncology.
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52/2017 - 16/10/2017
Beretta, E.; Ratti, L.; Verani, M.
A phase-field approach for the interface reconstruction in a nonlinear elliptic problem arising from cardiac electrophysiology | Abstract | | In this work we tackle the reconstruction of discontinuous coefficients in a semilinear elliptic equation from the knowledge of the solution on the boundary of the domain, an inverse problem motivated by biological application in cardiac electrophysiology.
We formulate a constraint minimization problem involving a quadratic mismatch functional enhanced with a regularization term which penalizes the perimeter of the inclusion to be identified. We introduce a phase-field relaxation of the problem, replacing the perimeter term with a Ginzburg-Landau-type energy. We prove the Gamma-convergence of the relaxed functional to the original one (which implies the convergence of the minimizers), we compute the optimality conditions of the phase-field problem and define a reconstruction algorithm based on the use of the Frechet derivative of the functional. After introducing a discrete version of the problem we implement an iterative algorithm and prove convergence properties. Several numerical results are reported, assessing the effectiveness and the robustness of the algorithm in identifying arbitrarily-shaped inclusions.
Finally, we compare our approach to a shape derivative based technique, both from a theoretical point of view (computing the sharp interface limit of the optimality conditions) and from a numerical one. |
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51/2017 - 16/10/2017
Gerbi, A.; Dede', L.; Quarteroni, A.
A monolithic algorithm for the simulation of cardiac electromechanics in the human left ventricle | Abstract | | In this paper, we propose a monolithic algorithm for the numerical solution of an electromechanics model of the left ventricle in the human heart. We consider the monodomain equation together with the Bueno-Orovio minimal ionic model for the description of the electrophysiology and the Holzapfel-Ogden strain energy function within the active strain framework for the mechanics of the myocardium. For the latter, we use for the first time in the context of electromechanics a transmurally variable active strain formulation. The Finite Element Method is used for the space discretization, while Backward Differentiation Formulas are used for the time discretization. Both implicit and semi-implicit schemes are addressed in this paper: the Newton method is used to solve the nonlinear system arising in the implicit scheme, while the semi-implicit scheme (corresponding to extrapolation of nonlinear terms from previous timesteps) yields a linear problem at each timestep. In the latter case, stability constraints may pose limitations in the timestep size. Much emphasis is laid into on the preconditioning strategy, which is based on the factorization of a block Gauss-Seidel preconditioner combined with the use of parallel preconditioners for each of the single core models composing the full electromechanics model. This monolithic preconditioner can be easily extended to cases where other ionic models are adopted and,
besides heart models, to other integrated problems arising in different multiphysics applications in engineering and applied sciences. Several numerical simulations are carried out in a high performance computing framework for both idealized and patient-specific left ventricle geometries. The latter are obtained from medical MRI images through suitable segmentation procedures to generate the computational mesh. Personalized pressure-volume loops are produced by means of the computational procedure and used to synthetically interpret and analyze the outputs of the electromechanics model. |
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50/2017 - 16/10/2017
Formaggia, F.; Vergara, C.
Defective boundary conditions for PDEs with applications in haemodynamics | Abstract | | This works gives an overview of the mathematical treatment of state-of-the-art techniques for partial differential problems where boundary data are provided only in
terms of averaged quantities. A condition normally indicated as ``defective boundary condition''. We present and analyze several procedures by which this type of problems can be handled |
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49/2017 - 13/09/2017
Antonietti, P. F.,; Pennesi, G.
V-cycle multigrid algorithms for discontinuous Galerkin methods on non-nested polytopic meshes | Abstract | | In this paper we analyse the convergence properties of V-cycle multigrid algorithms for the numerical solution of the linear system of equations arising from discontinuous Galerkin discretization of second-order elliptic partial differential equations on polytopal meshes. Here, the sequence of spaces that stands at the basis of the multigrid scheme is possibly non nested and is obtained based on employing agglomeration with possible edge/face coarsening. We prove that the method converges uniformly with respect to the granularity of the grid and the polynomial approximation degree p, provided that the number of smoothing steps, which depends on p, is chosen sufficiently large. |
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48/2017 - 30/08/2017
Regazzoni, F.; Dedè, L.; Quarteroni, A.
Active contraction of cardiac cells: a reduced model for sarcomere dynamics with cooperative interactions | Abstract | | We propose a reduced ODE model for the mechanical activation of cardiac myofilaments, which is based on explicit spatial representation of nearest-neighbour interactions. Our model is derived from the cooperative Markov Chain model of Washio et al. 2012, under the assumption of conditional independence of specific sets of events. This physically motivated assumption allows to drastically reduce the number of degrees of freedom, thus resulting in a significantly large computational saving. Indeed, the original Markov Chain model involves a huge number of degrees of freedom (order of 10^21) and is solved by means of the Monte Carlo method, which notoriously reaches statistical convergence in a slow fashion. With our reduced model, instead, numerical simulations can be carried out by solving a system of ODEs, reducing the computational time by more than 10000 times. Moreover, the reduced model is accurate with respect to the original Markov Chain model. We show that the reduced model is capable of reproducing physiological steady-state force-calcium and force-length relationships with the observed asymmetry in apparent cooperativity near the calcium level producing half activation. Finally, we also report good qualitative and quantitative agreement with experimental measurements under dynamic conditions. |
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