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 1349 products
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MOX 72 - 12/12/2005
Dogan, G.; Morin, P.; Nochetto, R.H., Verani, Marco
Finite Element Methods for Shape Optimization and Applications | Abstract | | We present a variational framework for shape optimization problems that establishes and clarifies explicit connections among the continuous formulation, its full discretization and the resulting linear algebraic systems.
Our approach hinges on the following essential features: shape differential calculus, a semi-implicit time discretization and a finite element method for space discretization. We use shape differential calculus to express variations
of bulk and surface energies with respect to domain changes. The semi-implicit time discretization allows us to track the domain boundary without an explicit parametrization, and has the exibility to choose different descent directions by varying the scalar product used for the computation of normal velocity. We propose a Schur complement approach to solve the resulting linear systems e_ciently. We discuss applications of this framework to image segmentation, optimal shape design for PDE, and surface diffusion, along with the choice of suitable scalar products in each case. We illustrate the method with several numerical experiments, some developing pinch-off and topological changes in finite time.
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MOX 71 - 11/24/2005
Babuska, I.; Nobile, Fabio; Tempone, Raul
A stochastic collocation method for elliptic partial differential equations with random input data | Abstract | | In this paper we propose and analyze a Stochastic-Collocation method to solve elliptic Partial Differential Equations with random coefficients and forcing terms (input data of the model). The input data are assumed to depend on a fi nite number of random variables. The method consists in a Galerkin approximation in space and a collocation in the zeros of suitable tensor product orthogonal polynomials (Gauss points) in the probability space and naturally leads to the solution of uncoupled deterministic prob¬lems as in the Monte Carlo approach. It can be seen as a generalization of the Stochastic Galerkin method proposed in [Babuˇ ska -Tempone-Zouraris, SIAM J. Num. Anal. 42(2004)] and allows one to treat easily a wider range of situations, such as: input data that depend non-linearly on the random variables, diffusivity coefficients with unbounded second moments , random variables that are correlated or have unbounded support. We provide a rigorous convergence analysis and demonstrate exponential con¬vergence of the “probability error” with respect of the number of Gauss points in each direction in the probability space, under some regularity assumptions on the random input data. Numerical examples show the effectiveness of the method. |
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MOX 70 - 10/28/2005
Veneziani, Alessandro; Vergara, Christian
An approximate method for solving incompressible Navier-Stokes problem with flow rate conditions | Abstract | | We consider the incompressible Navier-Stokes problem with flow rate boundary conditions. This problem has been investigated in [2] and [12], following a Langrage multiplier approach. This approach has the drawback of high computational costs. In this paper, we propose an approximate formulation of the problem, yielding a strong reduction of the computational cost. The error analysis shows that the error introduced by this approximate formulation is confined in a smoll region of the boundary. This is a confirmed by the numerical simulations |
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MOXP5 - 10/20/2005
Olgiati, Emanuela; Paglieri, Luca; Salvati, Simonetta; Secchi, Piercesare
Storia di un Caso: Intervalli di Confidenza per una Proporzione per la regolazione della qualità del servizio nel settore energetico nazionale | Abstract | | Si descrive il processo di costruzione di un metodo di controllo campionario della qualita' dei servizi di distribuzione misura e vendita dell'energia elettrica e del gas attualmente adottato dall'Autorita' per l'energia elettrica e il gas italiana |
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MOX 69 - 09/27/2005
Quarteroni, A.; Rozza, G.; Dede', L.; Quaini, A.
Numerical Approximation of a Control Problem for Advection-Diffusion Processes | Abstract | | Two different approaches are proposed to enhance the efficiency of the numerical resolution of optimal control problems governed by a linear advection-diffusion equation. In the framework of Galerkin-Finite Element (FE) method, we adopt a novel a posteriori error estimate of the discretization error in the cost functional this estimated is used in the course of a numerical adaptive strategy for the generation of efficient grids for the resolution of the optimal control problem. Moreover, we propose to solve the control problem by adopting a reduced basis (RB) technique, hence ensuring rapid, reliable and repeated evaluations of input-output relationship. Our numerical tests show that by this technique a substantial saving of computational costs can be achieved. |
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MOX 68 - 09/07/2005
Micheletti, Stefano; Perotto, Simona
Nested dual-residual a posteriori error estimators for advection-diffusion-reaction problems | Abstract | | In this work we introduce a fully computable
dual-based a posteriori error estimator for
standard scalar advection-diffusion-reaction
problems. In particular, such an estimator does not depend
on neither the primal nor the dual exact solution, but only on the
corresponding Galerkin finite element approximations.
This new approach merges the main advantages of the dual-based and of the
residual-based error analysis, being devised as a residual-based estimator
nested in a dual-based one.
This allows us to explicitly approximate suitable functionals of
the solution, in the spirit of a classical goal-oriented analysis,
at the same cost as a dual-based strategy, the solution of two
differential problems being involved.
The related issue of optimal mesh adaptivity is also addressed.
Several two-dimensional numerical test cases validate the proposed theory
as well as the employed adaptive procedure. |
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MOX 67 - 09/06/2005
Quarteroni, Alfio; Rozza, Gianluigi
Tecniche a Basi Ridotte per l Ottimizzazione di Configurazioni di Innesto per Bypass Coronarici | Abstract | | Viene applicato un metodo a basi ridotte per equazioni alle derivate parziali su domini parametrizzati allo scopo di approssimare il flusso del sangue attraverso un bypass aorto-coronarico. L obiettivo del lavoro è quello di fornire (a) un analisi di sensività per grandezze geometriche rilevanti nella caratterizzazione della configurazione di innesto di un bypass e (b) una rapida e affidabile previsione del valore di certi funzionali integrali, denominati outputs (quali, per esempio, indici legati a grandezze fluidodinamiche). Le linee guida del lavoro sono finalizzate a (i) ottenere valide indicazioni circa le procedure di progetto e innesto chirurgico di un bypass, nella prospettiva futura dello sviluppo e dell uso di protesi bioartificiali, (ii) sviluppare metodi numerici per l ottimizzazione e la progettazione biomeccanica e, (iii) fornire una relazione di tipo input-output retta da modelli con complessità matematica e costi computazionali inferiori rispetto a quelli che si avrebbero risolvendo le equazioni della fluidodinamica mediante il metodo classico degli elementi finiti. |
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MOX 66 - 07/29/2005
Montano, Antonio; Restelli, Marco; Sacco, Riccardo
Modeling and Numerical Simulation of Tethered Buoy Dynamics | Abstract | | In this article we deal with the numerical simulation of the dynamics of a tethered buoy, which is a mechanical system for marine applications consisting of a rigid floating bonding (buoy) connected by an elastic cable to the bottom of the fluid environment. A novel mixed finite element formulation is proposed for the spatial numerical approximation of the equations governing the dynamics of the elastic cable. This is done to allow a robust modeling of the cable, event in the limit of an infinite value of the Young modulus, in a way that is similar to mixed formulations for incompressible fluid-mechanics. The dynamics of the floating bodyis described by the classical Euler equations of motion, written using quaternion variables to end up with a numerically robust algorithm in presence of large rotations. For the time discretization of the resulting coupled system of nonlinear differential equations, the Backward Euler implicit method is adopted due to the stability requirements of the problem at hand, while a damped Newton method is used for linearization. Finally, the accuracy and robustness of the proposed numerical procedure are validated in the simulation of the tethered buoy system under various static and dynamic working conditions. |
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