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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MOX 30 - 12/10/2003
Discacciati, Marco; Quarteroni, Alfio
Convergence Analysis of a Subdomain Iterative Method for the Finite Element Approximation of the Coupling of Stokes and Darcy Equations | Abstract | | We consider a Galerkin Finite Element approximation of the Stokes-Darcy problem which models the coupling between surface and groundwater flows. Then we propose an iterative subdomain method for its solution, inspired to the domain decomposition theory. The convergence analysis that we develop is based on the properties of the discrete Steklov-Poincaré operators associated to the given coupled problem. An optimal preconditioner for Krylov methods is proposed and analyzez. |
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MOXD01 - 11/01/2003
Paganoni, Anna Maria; Pontiggia, Laura
Laboratorio informatico di Statistica Matematica su supporto Excel | Abstract | | Lo scopo di queste dispense è quello di introdurre il lettore all'uso del software Excel nell'analisi statistica di dati, ed esse nascono da esperienze didattiche avute in differenti corsi di Laboratorio di Statistica. Ogni tematica statistica affrontata viene trattata partendo da un esempio concreto sul quale si esegue un'analisi guidata, descritta passaggio per passaggio. Le nozioni di probabilità e statistica necessarie sono presupposte note, e vengono richiamate solo per uniformare le notazioni. Nota: disponibile in rete solo l'introduzione e il sommario. |
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MOX 29 - 10/21/2003
Micheletti, Stefano; Perotto, Simona
Anisotropic mesh adaptivity in CFD | Abstract | | Why anisotropy? The straightforward answer to this question could be: because anisotropy is everywhere! Actually, when numerically solving a problem in Computational Fluid Dynamics (CFD), or in some other areas, there are many instances where the solution shows directional features such as great variations along certain directions with less significant changes along other ones, e.g. boundary and internal layers, singularities or shocks. A correct orientation and deformation of the mesh elements (longest edges parallel to, e.g. the boundary layers) yields a great reduction of the number of triangles. Moreover, in the anisotropic case the layers are captured more sharply.
The leitmotiv of an anisotropic analysis can be stated as: for a fixed solution accuracy, reduced the number of degrees of freedom involved in the approximation of the problem at hand by better orienting the mesh elements according to some suitable features of the solution, or vice versa, given a contraint on the number of elements, find the mesh maximizing the accuracy of the numerical solution.
However, things may not be so straightforward. While anisotropy is proved to superior in terms of effectiveness for the most accurate computations in many cases, yet, there are some instances where a structured Adaptive Mesh Refinement (AMR) procedure turns out to be more simple to carry out, especially in view of an implementation in a parallel environment. Moreover, in the unstructured case, the main drawback of the anisotropic approach compared to the anisotropic one, is the more complex analysis required to fully describe the element dimensions and orientation. Though, this heavier burden is the strenght of the method.
The outline of the article is the following. In the Sect. 2 we introduce the anisotropic framework by recalling some anisotropic interpolation error estimates, representing the main tool used in the a posteriori error analysis addressed in Sect. 3. This analysis is discussed in the case of a general differential operator, moving from the adjoint theory for goal-oriented error control, and it is then detailed for the advection-diffusion-reaction and the Stokes problems. Finally, in Sect. 4 the effectiveness of the anisotropic philosophy is assessed on some numerical test cases. |
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MOX28 - 10/10/2003
Secchi, Piercesare
A game between two Bayesians for estimating the mean of a Gaussian distribution | Abstract | | Two Bayesian players are engaged in a multi-stage competition where the final goal for each of them is to estimate the mean $ mu$ of a Normal distribution $ mathcal{N}$ with variance equal to 1, minimizing the total costs due to sampling and variance of the posterior distribution of $ mu$. |
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MOX 27 - 10/02/2003
Rozza, Gianluigi
Reduced Basis Methods for Elliptic Equations in sub-domains with A-Posteriori Error Bounds and Adaptivity | Abstract | | We present an application in multi-parametrized subdomains based on a technique for the rapid and reliable prediction of linear-functional output of elliptic coercive partial differential equations with affine parameter dependence (reduced basis methods). The essential components are (i) (provably) rapidly convergent global reduce-basis approximation - Galerkin projection onto a space W_N spanned by solutions of the governing partial differential equation at N selected points in parameter space (ii) a posteriori error estimation-relaxations of the error-residual equation that provide inexpensive bounds for the error in the outputs of interest and (iii) off-line/on-line computational procedures-methods wich decouple the generation and projection stages of the approximation process. The operation count for the on-line stage-in Which, given a new parameter value, we calculate the output of interest and associated error bound-depends only on N (typically very small) and the parametric complexity of the problem the method is thus ideally suited for the repeated and rapid evaluations required in the context of parameter estimation, design, optimization, and real-time control.
In [11] a rigorous a posteriori error bound framework for reduced-basis approximations of elliptic coercive equations is developed. The resulting error estimates are, in some cases, quite sharp: the ratio of the estimated error in the output to the true error in the output, or effectivity, is close to (but always greater than) unity. We use a posteriori bound error estimator applied also to an adaptive procedure in choosing the approximation space and its dimension, minimizing the estimated erro of the effectivity [23].
The application is based on a parametrized geometry, divided in subdomains, each of them depending by geometrical quantities that can be useful for future haemodynamics applications [15], such as the bypass configuration problem (stenosis lenght, graft angle, artery diameter, incoming bypass diameter and outflow lenght). For future development guidelines we suggest to see [19] and [16]. |
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MOX 26 - 09/29/2003
Saleri, Fausto; Veneziani, Alessandro
Pressure-correction algebraic splitting methods for the incompressible navier-stokes equations | Abstract | | In this paper we present a new family of methods for the effective numerical solution of the incompressible unsteady Navier-Stokes equations. These methods resort to an algebraic splitting of the discretized problem based on inexact LU block-factorizations of the corresponding matrix (following [21]). In particular, we will start from inexact algebraic factorizations of algebraic Chorin-Temam and Yosida type and introduce a pressure correction step aiming at improving the time accuracy. one of the schemes obtained in this way (the Algebraic Chorin-Temam Pressure Correction Method) resembles a method previously introduced in the framework of differential projection schemes (see [24], [19]). The stability and the dependence of splitting error on the time step of the new methods is investigated and tested on several numerical cases. |
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MOXP1 - 09/10/2003
Formaggia, Luca; Micheletti, Stefano; Savini, Barbara
Modellazione Bidimensionale e Simulazione Numerica del Funzionamento di una Candeletta di Preriscaldo
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MOX 25 - 07/29/2003
Ballestra, Luca V.; Sacco, Riccardo
Numerical Problems in Semiconductor Simulation Using the Hydrodynamical Model: a Second-Order Finite Difference Scheme | Abstract | | In this paper a second-order Total Variation Diminuishing (TVD) finite diffence scheme of upwind type is employed for the numerical approximation of the classical hydrodynamic model for semiconductors proposed byB. and Baccarani-Wordeman. In particular, the high-order hyperbolic fluxes are evaluated by a suitable extrapolation on adjacent cell of the first-order fluxes of Roe, while total variation diminuishing is achieved by limiting th slopes of the discrete Riemann invariants using the so-called Flux Corrected Transport approach.
Extensive numerical simulations are performed on a submicron n^+ - n - n^+ ballistic diode. The numerical experiments show that the spurious oscillations arising in the electron current are not completely suppressed by the TVD scheme, and can lead to serious numerical instabilities when the solution of the hydrodynamic model is non-smooth and the computational mesh is coarse.
The accuracy of the numerical method is investigated in terms of conservation of the steady electron current. The Obtained results show that the second-order scheme does not behave much better than a corresponding first-order one due to a poor performance of the slope limiters caused by the presence of local extrema of the Riemann invariant associated with the hyperbolic system. |
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