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 1251 prodotti
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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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33/2010 - 08/11/2010
Migliorati, Giovanni; Quarteroni, Alfio
Multilevel Schwarz Methods for Elliptic Partial Differential Equations | Abstract | | We investigate multilevel Schwarz domain decomposition preconditioners, to efficiently solve linear systems arising from numerical discretizations of elliptic Partial Di fferential Equations by the fiite element method. In our
analysis we deal with unstructured mesh partitions and with subdomain boundaries resulting from using the mesh partitioner. We start from two-level preconditioners with either aggregative or interpolative coarse level
components, then we focus on a strategy to increase the number of levels. For all preconditioners, we consider the additive residual update and its multiplicative variants within and between levels. Moreover, we compare the preconditioners behaviour, regarding scalability and rate of convergence. Numerical results are provided for elliptic boundary-value problems, including a convection-di ffusion problem when suitable stabilization
becomes necessary. |
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32/2010 - 07/11/2010
Malossi, A. Cristiano I.; Blanco, Pablo J.; Deparis, Simome; Quarteroni, Alfio
Algorithms for the partitioned solution of weakly coupled fluid models for cardiovascular flows | Abstract | | The main goal of the present work is to devise robust iterative strategies to partition the solution of the Navier–Stokes equations in a three-dimensional(3D) computational domain, into non overlapping 3D subdomains,which communicate through the exchange of integrated quantities
across the interfaces. The novel aspect of the present approach is that at coupling boundaries the conservation of flow rates and of the associated dual variables is imposed, entailing a weak physical coupling. For the solution
of the non-linear problem, written in terms of interfaces variables, two strategies are compared: relaxed fixed point iterations and Newton iterations. The algorithm is tested in several configurations for problems which involve more than two components at each coupling interface. In such cases it is shown that relaxed fixed point methods are not convergent, whereas the Newton method leads in all the tested cases to convergent schemes. One of the appealing aspects of the strategy proposed here is the
flexibility in the setting of boundary conditions at branching points, where no hierarchy is established a priori, unlike classical Gauss–Seidel methods.
Such an approach can be applied in two other different contexts: (i) when coupling dimensionally-heterogeneous models, just by replacing some of the 3D models by one-dimensional (or zero-dimensional) condensed ones, and (ii) as a preconditioner method for domain decomposition methods for the
Navier–Stokes equations. These two issues are also addressed in the present work. Finally, several examples of application are presented, ranging from academic examples to some related to the computational hemodynamics field. |
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31/2010 - 06/11/2010
Manzoni, Andrea; Quarteroni, Alfio; Rozza, Gianluigi
Shape optimization for viscous flows by reduced basis methods and free-form deformation | Abstract | | In this paper we present a new approach for shape optimization that combines two different types of model reduction: a suitable low-dimensional
parametrization of the geometry (yielding a geometrical reduction) combined with reduced basis methods (yielding a reduction of computational complexity). More precisely, free-form deformation techniques are introduced
for the geometry description and its parametrization, while reduced basis methods are used upon a finite element discretization to solve systems of parametrized partial differential equations. This allows an efficient flow field computation and cost functional evaluation during the iterative optimization
procedure, resulting in effective computational savings with respect to usual shape optimization strategies. This approach is very general
and can be applied for a broad variety of problems. To prove its effectivity, in this paper we apply it to find the optimal shape of aorto-coronaric bypass anastomoses based on vorticity minimization in the down-field region.
Stokes equations are used to model blood flow in the coronary arteries. |
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30/2010 - 05/11/2010
Blanco, Pablo J.; Discacciati, Marco; Quarteroni, Alfio
Modeling dimensionally-heterogeneous problems: analysis, approximation and applications | Abstract | | In the present work a general theoretical framework for coupled dimensionally-heterogeneous partial differential equations is developed. This isdone by recasting the variational formulation in terms of coupling interface variables. In such a general setting we analyze existence and uniqueness of solutions for both the continuous problem and its finite dimensional approximation. This approach also allows the development of different iterative substructuring solution methodologies involving dimensionallyhomogeneous
subproblems. Numerical experiments are carried out to test our theoretical results.
Keywords: Multiphysics, Heterogeneous PDE models, Augmented formulation,Domain decomposition, Finite elements. |
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29/2010 - 04/11/2010
Lesinigo, Matteo; D'Angelo, Carlo; Quarteroni, Alfio
A multiscale Darcy-Brinkman model for fluid flow in fractured porous media | Abstract | | The aim of this work is to present a reduced mathematical model for describing fluid flow in porous media featuring open channels or fractures.
The Darcy s law is assumed in the porous domain while the Stokes-Brinkman equations are considered in the fractures. We address the case of fractures whose thickness is very small compared to the characteristic diameter of the
computational domain, and describe the fracture as if it were an interface between porous regions. We derive the corresponding interface model governing the fluid flow in the fracture and in the porous media, and establish the
well-posedness of the coupled problem. Further, we introduce a finite element scheme for the approximation of the coupled problem, and discuss solution strategies. We conclude by showing the numerical results related to several test cases and compare the accuracy of the reduced model compared with the non-reduced one. |
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