Quaderni di Dipartimento

Quaderni MOX

• MOX 68 - 07/09/2005
  Micheletti, Stefano; Perotto, Simona
  Nested dual-residual a posteriori error estimators for advection-diffusion-reaction problems

  Abstract

  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.

• MOX 67 - 06/09/2005
  Quarteroni, Alfio; Rozza, Gianluigi
  Tecniche a Basi Ridotte per l Ottimizzazione di Configurazioni di Innesto per Bypass Coronarici

  Abstract

  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.

• MOX 66 - 29/07/2005
  Montano, Antonio; Restelli, Marco; Sacco, Riccardo
  Modeling and Numerical Simulation of Tethered Buoy Dynamics

  Abstract

  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.

• MOX 65 - 26/07/2005
  Paganoni, Anna Maria; Secchi, Piercesare
  A numerical study for comparing two response-adaptive designs for continuous treatment effects

  Abstract

  Abstract

  We study two sequential, response-adaptive randomized designs for clinical trials one has been proposed in Bandy-opadhyay and Biswas (2001) and Biswas and Basu (2001), the other stems from randomly reinforced urn introduced and studied in Muliere et al.(2005). Both designs can be used in clinical trials where the response from each patient is a continuous variable. Comparison is conducted through numerical studies and along a new guideline for the evaluation of a response-adaptive design

• MOX 64 - 24/06/2005
  Formaggia, Luca; Sala, Marzio; Saleri, Fausto
  Domain decomposition techniques

  Abstract

  Abstract

  We introduce some parallel domain decomposition preconditioners for iterative solution of sparse linear systems like those arising from the approximation of partial differential equations by finite elements or finite volumes. We first give an overview of algebraic domain decomposition techniques. We then introduce a preconditioner based on a multilevel approximate Schurcomplement system. Then we present a Schwarz-based preconditioner augmented by an algebraic coarse correction operator. Being the definition of a coarse grid a difficult task on unstructured meshes, we propose a general framework to build a coarse operator by using an agglomeration procedure that operates directly on the matrix entries. Numerical results are presented aimed at assessing and comparing the effectiveness of the two methodologies. The main application will concern computational fluid dynamics (CFD), and in particular the simulation of compressible flow around aeronautical configurations.

• MOX 63 - 01/06/2005
  Restelli, Marco; Bonaventura, Luca; Sacco, Riccardo
  A flux form, semi - Lagrangian method for the scalar advection equation usign Discontinuous Galerkin reconstruction

  Abstract

  Abstract

  A new semi Lagrangian formulation is proposed for the discretization of the scalar advection equation in flux form. The approach combines the accuracy and flexibility of the Discontinuous Galerkin method with the computational efficiency and robustness of Semi-Lagrangian techniques. Unconditional stability of the proposed discretization is proven in the Von Neumann sense for the one dimensional case. A monotonization technique is then introduced, based on the Flux Corrected Transoport approach. This yields a multidimensional monotonic scheme for the piecewise constant component of the computed solution, while reducing the numerical diffusion of monotonization approaches more common in the Discontinuous Galerkin framework. The accuracy and stability of the method are further demonstrated by two dimensional tracer advection tests. The comparison with results obtained by standard semi - Legrangian and Discontinuous Galerkin methods highlights several computational advantages of the new technique.

• MOX 62 - 30/05/2005
  Deparis, Simone; Discacciati, Marco; Fourestey, Gilles; Quarteroni, Alfio
  Heterogeneous Domain Decomposition Methods for Fluid-Structure Interaction Problems

  Abstract

  	Abstract
  	In this note, we propose Steklov-Poincaré iterative algorithms (mutuated from the analogy with heterogeneous domain decomposition) to solve fluid-structure interaction problems. Although our framework is very general, the driving application is concerned with the interaction of blood flow and vessel wall in large arteries.

• MOX 61 - 23/05/2005
  Gervasio, Paola; Saleri, Fausto; Veneziani, Alessandro
  Algebraic fractional step schemes with spectral methods for the incompressible Navier-Stokes equations

  Abstract

  Abstract

  The numerical investigation of a recent family of algebraic fractional-step methods for the solution of the incompressible time-dependent Navier-Stokes equations is presented. These methods are improved verdion of the Yosida method proposed in [29] and [28] and one of them (the Yosida4 method) is proposed in this paper for the first time. They rely on approximate LU block factorization of the matrix obtained after the discretization in time and space of the Navier-Stokes system, yielding a splitting in the velocity and pressure computation. In this paper we analyze the numerical performances of these schemes when the space discretization is carried out with a spectral element method, with the aim of investigating the impact of the splitting on the global accuracy of the computation.