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 1288 prodotti
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41/2010 - 24/11/2010
Ambrosi, D.; Arioli, G.; Nobile, F.; Quarteroni, A.
Electromechanical coupling in cardiac dynamics: the active strain approach | Abstract | | The coupling between cardiac mechanics and electric signaling is addressed in a non–standard framework in which the electrical potential dictates the active strain (not stress) of the muscle. The physiological and mathematical motivations leading us to this choice are illustrated. The propagation of the electric signal is assumed to be governed by the FitzHugh–
Nagumo equations, rewritten in material coordinates with a deforming substrate; the solution is compared with the rigid case and differences in celerity and width of a pulse are discussed. The role of visco-elasticity is pointed
out. We show that the stretching of coordinates is insufficient to originate electromechanical feedback; nevertheless, it can increase the energy of a perturbation enough to produce a traveling pulse: an energy estimate and numerical evidence are reported. To support these conclusions, numerical simulations in two dimensions show the interplay between electric propagation and mechanical strain. |
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40/2010 - 21/11/2010
D'Angelo, C.; Scotti, A.
A Mixed Finite Element Method for Darcy Flow in Fractured Porous Media with non-matching Grids | Abstract | | We consider an incompressible flow problem in a N-dimensional fractured porous domain(Darcy’s problem). The fracture is represented by a (N − 1)-dimensional interface, exchanging fluid with the surrounding media. In this paper we consider the lowest-order(RT0, P0) Raviart-Thomas mixed finite element method for the approximation of the coupled Darcy’s flows in the porous media and within the fracture, with independent meshes for the respective domains. This is achieved thanks to an enrichment with discontinuous basis functions on triangles crossed by the fracture and a weak imposition of interface conditions.
First, we study the stability and convergence properties of the resulting numerical scheme in the uncoupled case, when the known solution of the fracture problem provides an immersed boundary condition. We detail the implementation issues and discuss the algebraic properties of the associated linear system. Next, we focus on the coupled problem and propose an iterative porous domain / fracture domain iterative method to solve for fluid flow in both the porous media and the fracture and compare the results with those of a traditional monolithic approach.
Numerical results are provided confirming convergence rates and algebraic properties predicted by the theory. In particular, we discuss preconditioning and equilibration techniques to make the condition number of the discrete problem independent of the position of the immersed interface. Finally, two and three dimensional simulations of Darcy’s flow in different configurations (highly and poorly permeable fracture) are analyzed and discussed. |
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39/2010 - 16/11/2010
D'Angelo, C.
Finite Element Approximation of Elliptic Problems with Dirac Measure Terms in Weighted Spaces. Applications to 1D-3D Coupled Problems | Abstract | | In this work we study the stability and the convergence rates of the finite element approximation of elliptic problems involving Dirac measures, using weighted Sobolev spaces and weighted discrete norms. Our approach handles
both the cases where the measure is simply a right hand side or it represents an additional term, i.e. solution-dependent, in the formulation of the problem.
The main motivation of this study is to provide a methodological tool to treat elliptic problems in fractured domains, where the coupling terms are seen as Dirac measures concentrated on the fractures. We first establish a decomposition lemma, which is our fundamental tool for the analysis of the considered problems in the non-standard setting of weighted spaces. Then, we consider the stability of the Galerkin approxima-
tion with finite elements in weighted norms, with uniform and graded meshes.
We introduce a discrete decomposition lemma that extends the continuous one and allows to derive discrete inf-sup conditions in weighted norms. Then, we focus on the convergence of the finite element method. Due to the lack of regularity, the convergence rates are suboptimal for uniform meshes; we show that using graded meshes optimal rates are recovered. Our theoretical results are supported by several numerical experiments. Finally, we show how our theoretical results apply to certain coupled problems involving
fluid flow in porous three-dimensional media with one-dimensional fractures, that are found in the analysis of microvascular flows.
Keywords: elliptic problems, measure, Dirac measure, weighted spaces, nite element method, graded
mesh, error estimates, reduced models, multiscale models, microcirculation. |
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38/2010 - 15/11/2010
Flournoy, N.; May, C.; Secchi, P.
Response-adaptive designs in clinical trials for targeting the best treatment: an overview | Abstract | | Response-adaptive designs are increasingly being implemented in clinical trials, particularly early phase trials, and they have increasingly stimulated the work of researchers. This paper reviews a particular class of response-adaptive designs, which have a di fferent property from the most adaptive designs in literature. These
are response-adaptive designs targeting asymptotically the superior response, that is, treating with the superior treatment with
probability converging to one. The model underlying such designs is a randomly reinforced urn. In the context of clinical trials, this prop-
erty is particularly attractive from an ethical point of view. This overview starts from the early paper of [8] until the recent work by [9].
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37/2010 - 12/11/2010
Discacciati, Marco; Quarteroni, Alfio; Quinodoz, Samuel
Numerical approximation of internal discontinuity interface problems | Abstract | | This work focuses on the finite element discretization of boundary value problems whose solution presents either a discontinuity and/or a discontinuous conormal derivative across an interface inside the computational domain.
The interface is characterized via a level-set function. The discontinuities are accounted for using suitable extension operators whose numerical
implementation requires a very low computational
e ffort. Numerical results to validate our approach are presented in one, two and three dimensions. |
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36/2010 - 11/11/2010
Arioli, G.; Koch, H.
Non-Symmetric low-index solutions for a symmetric boundary value problem | Abstract | | We consider the equation -Laplacian(u)=w*u^3 on a square domain in R^2, with Dirichlet boundary conditions, where w is a given positive function that is invariant under all (Euclidean) symmetries of the square. This equation is shown to have a solution u, with Morse index 2, that is neither symmetric nor antisymmetric with respect to any nontrivial symmetry of the square. Part of our proof is computer-assisted. An analogous result is proved for index 1.
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35/2010 - 10/11/2010
Arioli, G.; Koch, H.
Integration of dissipative PDEs: a case study | Abstract | | We develop a computer-assisted technique to construct and analyze orbits of dissipative evolution equations. As a case study, the methods are applied to the Kuramoto-Sivashinski equation. We prove the existence of a hyperbolic periodic orbit.
Keywords: Kuramoto-Sivashinski equation, hyperbolicity, periodic orbit, computer-assisted proof
AMS Subject Classification: 37L05, 37L45, 35K35 |
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34/2010 - 09/11/2010
Abba', A.; Bonaventura, L.
A mimetic finite difference method for Large Eddy Simulation of incompressible flow | Abstract | | A finite difference discretization of the three-dimensional, incompressible Navier-Stokes equations is presented, based on finite difference operators that satisfy discrete analogs of some basic calculus identities. These
mimetic properties yield a numerical method for which a discrete form of the vorticity equation can be derived naturally from the discrete momentum equation, by application of the mimetic rotation operator. As a result,a discrete approximation of vorticity is exactly preserved, for inviscid flows, independently of the mesh size. The vorticity preservation property guarantees that no spurious vorticity is generated by the nonlinear advective terms in absence of viscosity. A mimetic discretization of the viscous terms and an appropriate treatment for rigid wall boundary conditions are also
proposed. The relationship of this approach to other similar techniques is discussed. The proposed method is validated on several idealized test cases for laminar incompressible flow, in which it is compared to a widely used finite difference discretization. The method is then applied to Large Eddy Simulation of incompressible flow, demonstrating the advantages of the inherentconservation properties in a comparison with experimental data and DNS results especially when strong vorticity production takes place at the boundaries. |
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