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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09/2007 - 04/21/2007
May, Caterina; Flournoy, Nancy
Asymptotics in response-adaptive designs generated by a two-colors, randomly reinforced urn | Abstract | | This paper illustrates asymptotic properties for a response-adaptive
design generated by a two-color, randomly reinforced urn model. The
design considered is optimal in the sense that it assigns patients to
the best treatment with probability converging to one. An approach
to show the joint asymptotic normality of the estimators of the mean
responses to the treatments is provided in spite of the fact that allocation
proportions converge to zero and one. Results on the rate of
convergence of the number of patients assigned to each treatment are
also obtained. Finally, we study the asymptotic behavior of a suitable
test statistic.
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08/2007 - 04/14/2007
Formaggia, Luca; Miglio, Edie; Mola, Andrea; Parolini, Nicola
Fluid-structure interaction problems in free surface flows: application to boat dynamics | Abstract | | We present some recent studies on fluid-structure
interaction problems in the presence of free surface flow. We consider
the dynamics of boats simulated as rigid bodies. Several hydrodynamic
models are presented, ranging from full Reynolds Averaged
Navier-Stokes equations down to reduced models based on potential flow
theory.
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07/2007 - 03/22/2007
Ern, Alexandre; Stephansen, Annette F.; Zunino, Paolo
A Discontinuous Galerkin method with weighted averages for advection-diffusion equations with locally vanishing and anisotropic diffusivity | Abstract | | We consider Discontinuous Galerkin approximations of advection-diffusion equations with anisotropic and discontinuous diffusivity, and propose the symmetric weighted interior penalty (SWIP) method for better coping with locally vanishing diffusivity. The analysis yields convergence results for the natural energy norm that are optimal (with respect to mesh-size) and robust (fully independent of the diffusivity). The convergence results for the advective derivative are optimal with respect to mesh-size and robust for isotropic diffusivity, as well as for anisotropic diffusivity in the dominant advection regime. In the dominant diffusivity regime, an optimal convergence result for the the $L^2$-norm is also recovered.
Numerical results are presented to illustrate the performance of the scheme.
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06/2007 - 02/20/2007
Deponti, Alberto; Bonaventura, Luca; Fraccarollo, Luigi; Miglio, Edie; Rosatti, Giorgio
Analysis of Hyperbolic Systems for Mobile--Bed, Free--Surface Flow Modelling in Arbitrary Cross Sections | Abstract | | A model for mobile--bed river hydraulics based on the conservation equations of liquid mass, solid sediment mass and momentum is presented and analysed. The equations are reduced to one dimension by averaging over the cross section. These equations differ from the classical one--dimensional equations for mobile--bed, free--surface flows, since they take into account the effect of non--uniformities in velocity distribution along cross sections of arbitrary shape. By using appropriate closure formulae for sediment transport and for bottom friction, a system of three non--linear hyperbolic equations is obtained. Its eigenstructure is studied and its dependency on the closure formulae and on the non--dimensional parameters determining the flow and transport regimes is investigated. The existence and uniqueness of classical solutions of this hyperbolic system are discussed and an energy inequality for the frozen coefficient problem is derived.
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05/2007 - 02/18/2007
Alì, G.; Culpo, M.; Micheletti, S.
Domain decomposition techniques for microelectronic modeling | Abstract | | This paper is meant to be the continuation of the previous work [1] where
a coupled ODE/PDE method for the simulation of semiconductor devices
was introduced. From a strictly mathematical viewpoint, analytical results
on coupled PDE/ODE systems (as arising in integrated circuit simulation)
can be found in [2]. In particular, in the present paper, we numerically
investigate an algorithm of Domain Decomposition type for the simulation
of circuits containing distributed devices (x 1) as well as semiconductors
in which some part is modeled with lumped parameters (x 2). It is worth
noticing the original employment of the Domain Decomposition technique
within the con_nes of a heterogeneous PDE/ODE coupling, versus its
typical use in a homogeneous full-PDE context. The results presented
here have been studied in the seminal work [3], while a more thorough
analysis is ongoing [4].
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04/2007 - 02/15/2007
Nobile, Fabio; Tempone, Raul; Clayton G., Webster
An anisotropic sparse grid stochastic collocation method for elliptic partial differential equations with random input data | Abstract | | This work proposes and analyzes an anisotropic sparse grid stochastic collocation method for solving elliptic partial differential equations with random coefficients and forcing terms (input data of the model). The method consists of a Galerkin approximation in the space variables and
a collocation, in probability space, on sparse tensor product grids utilizing either Clenshaw-Curtis or Gaussian knots. Even in the presence of nonlinearities, the collocation approach leads to the solution of uncoupled
deterministic problems, just as in the Monte Carlo method. This work includes a priori and a posteriori procedures to adapt the anisotropy of
the sparse grids to each given problem. These procedures seem to be very effective for the problems under study. The proposed method combines
the advantages of isotropic sparse collocation with those of anisotropic full tensor product collocation: the first approach is effective for problems depending on random variables which weigh approximately equally in the solution, while the benefits of the latter approach become apparent when solving highly anisotropic problems depending on a relatively small number of random variables, as in the case where input random variables are
Karhunen-Loève truncations of “smooth” random fields. This work also provides a rigorous convergence analysis of the fully discrete problem and demonstrates: (sub)-exponential convergence in the asymptotic regime and algebraic convergence in the pre-asymptotic regime, with respect to the total number of collocation points. Numerical examples illustrate the theoretical results and are used to compare this approach with several others, including the standard Monte Carlo. In particular, for moderately large dimensional problems, the sparse grid approach with a properly chosen anisotropy seems to be very efficient and superior to all examined methods.
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03/2007 - 01/23/2007
Santiago, Badia; Quaini, Annalisa; Quarteroni, Alfio
Splitting methods based on algebraic factorization for fluid-structure interaction | Abstract | | We discuss in this paper the numerical approximation of fluid-structure interaction (FSI) problems dealing with strong added-mass effect. We propose new semi-implicit algorithms based on inexact block-LU factorization of the linear system obtained after the space-time discretization and linearization of the FSI problem. As a result, at each iteration the fluid velocity is computed separately from the coupled pressure-structure velocity system, reducing the computational cost.We investigate explicit-implicit decomposition through algebraic splitting techniques originally designed for the FSI problem. This approach leads to two different families of methods which extend to FSI the algebraic pressure correction method and the Yosida method, two schemes that were previously adopted for pure fluid problems. Furthermore, we have considered the inexact factorization of the fluid-structure system as a preconditioner. The numerical properties of these methods have been tested on a model problem representing a blood-vessel system.
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02/2007 - 01/18/2007
Nochetto, Ricardo H.; Veeser, Andreas; Verani, Marco
A safeguarded dual weighted residual method | Abstract | | The dual weighted residual (DWR) method yields reliable a posteriori error bounds for linear output functionals provided that the error incurred by the numerical approximation of the dual solution is negligible. In that case its performance is generally superior than that of global energy norm
error estimators which are
unconditionally reliable. We present a simple numerical example for which neglecting the approximation error leads to severe underestimation of the functional error, thus showing that the DWR method may be unreliable. We propose a remedy that preserves the original performance, namely a DWR method safeguarded by additional asymptotically higher order a posteriori terms. In particular, the enhanced estimator is unconditionally reliable and asymptotically coincides with the original DWR method. These properties are illustrated via the aforementioned example. |
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