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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55/2017 - 10/23/2017
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 - 10/16/2017
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 - 10/16/2017
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 - 10/16/2017
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 - 09/13/2017
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 - 08/30/2017
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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47/2017 - 08/29/2017
Menghini, F.; Dede, L.; Forti, D.; Quarteroni, A.
Hemodynamics in a left atrium based on a Variational Multiscale-LES numerical model | Abstract | | Standard studies of the cardiovascular system are based on advanced experimental and imaging techniques, however, in the past few years, they are being complemented by computational fluid dynamics simulations of blood
flows with increasing level of details. The vast majority of works dealing with the heart hemodynamics focus on the left ventricle, both in patient-specific and idealized geometries, while the fluid dynamics of the left atrium is much less investigated. In this work we propose a computational model of a left atrium suitable to provide physically meaningful fluid dynamics indications and other outputs as the velocity profile at the mitral valve. A Variational Multiscale model is used to obtain a stable formulation of the Navier-Stokes equations discretized by means of the Finite Element method and to account for turbulence modeling within the framework of Large Eddy Simulation (LES). We present and discuss numerical results regarding the fluid dynamics of the left atrium with the focus on possible transitions to turbulence. We also provide a comparison with the results obtained using a SUPG formulation. |
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46/2017 - 08/08/2017
Agosti, A.; Gower, A.L.; Ciarletta, P.
The constitutive relations of initially stressed incompressible Mooney-Rivlin materials | Abstract | | Initial stresses originate in soft materials by the occurrence of misfits in the undeformed microstruc-
ture. Since the reference configuration is not stress-free, the effects of initial stresses on the hyperelastic
behavior must be constitutively addressed. Notably, the free energy of an initially stressed material
may not possess the same symmetry group as the one of the same material deforming from a naturally
unstressed configuration. This work assumes an explicit dependence of the hyperelastic strain energy
density on both the deformation gradient and the initial stress tensor, taking into account for their inde-
pendent invariants. Using this theoretical framework, a constitutive equation is derived for an initially
stressed body that naturally behaves as an incompressible Mooney-Rivlin material. The strain energy
densities for initially stressed neo-Hookean and Mooney materials are derived as special sub–cases. By
assuming the existence of a virtual state that is naturally stress-free, the resulting strain energy functions
are proved to fulfill the required frame–independence constraints. In the case of plane strain condition,
great simplifications arise in the expression of the constitutive relations. Finally, the resulting constitu-
tive relations prove useful guidelines for designing non-destructive methods for the quantification of the
underlying initial stresses in naturally isotropic materials.
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