Quaderni di Dipartimento

Quaderni MOX

• MOX 34 - 30/03/2004
  Bottasso, Carlo L.; Causin, Paola; Sacco, Riccardo
  Flux-Upwind Stabilization of the Discontinuous Petrov-Galerkin Formulation with Lagrange Multipliers for Advection-Diffusion Problems

  Abstract

  Abstract

  In this work we considerthe dual-primal Discontinuous Petrov-Galerkin (DPG) method for the advection-diffusion model problem. Since in the DPG method both mixed internal variables are discontinuous, a static condensation procedure can be carried out, leading to a single-field nonconforming discretization scheme. For this latter formulation, we propose a flux-upwind stabilization technique to deal with the advection-dominated case. The resulting scheme is conservative and satisfies a discrete maximum principle under standard geometrical assumptions on the computational grid. A convergence analysis is developed, proving first-order accuracy of the method in a discrete $H^1-norm$, anf the numerical performance of the scheme is validated on benchmark problems with sharp internal and boundary layers.

• MOX 33 - 25/03/2004
  Abba', A.; Saleri, F.; D'Angelo, C.
  A 3D Shape Optimization Problem in Heat Transfer: Analysis and Approximation via BEM

  Abstract

  Abstract

  In this paper an optimal shape control problem dealing with heat transfer in enclosures is studied. We model and enclosure heated by a flame surface (taking account of radiation, conductor and convection effects), and we try to find an optimal flame shape which minimizes some cost functional defined on the temperature field. This kind of problem arises in industrial furnaces optimization, being temperature uniformity one of the most important aspects in industrial plant analysis and design. Analytical results (smoothness of the control-to-state mapping, existence of an optimal shape in a certain admissible class) as well as numerical optimization results by the boundary element method are obtained we employ the gradient method to optimize the flame shape, exploiting the adjoint equation associated with the state equation and the cost function.

• MOX 32 - 08/03/2004
  Carstensen, Carsten; Causin, Paola; Sacco, Riccardo
  A Posteriori Dual-Mixed (Hybrid) Adaptive Finite Element Error Control for Lamé and Stokes Equations

  Abstract

  Abstract

  A unified and robust mathematical model for compressible and incompressible elasticity can be obtained by rephrasing the Hermann formulation within the Hellinger-Reissner principle. The quasi-optimally converging extension of PEERS (Plane Elasticity Element with Reduced Symmetry) is called Dual-Mixed Hybrid formulation (DMH). Explicit residual-based a posteriori error estimates for DMII are introduced and are mathematically shown to be locking-free, reliable, and efficient. The estimator serves as a refinement indicator in an adaptive algorithm for effective automatic mesh generation. Numerical evidence supports that the adaptive scheme leads to optimal convergence for Lamé and Stokes benchmark problems with singularities.

• MOX 31 - 27/02/2004
  Formaggia, Luca; Lamponi, Daniele; Tuveri, Massimiliano; Veneziani, Alessandro
  Numerical Modeling of 1D Arterial Networks Coupled with a Lumped Parameters Description of the Heart

  Abstract

  Abstract

  The investigation on the pressure wave propagation along the arterial network and its relationships with vascular physiopatologies can be supported nowadays by numerical simulation (see e.g. [25]). One dimensional (1D) mathematical models, based on systems of two partial differential equations for each arterial segment suitably matched at bifucations, can be simulated with low computationsl costs and provide useful insights into the role of wave reflections. For instance, those induced by the stiffening of the arterial walls or a vascular endoprothesis, and their influence on the cardiac work. Some recent works have indeed moved in this direction ([19,6,25,24,33]). The specific contribution of the present paper is to illustrate a 1D numerical model in which there is an effective coupling between the heart action and the 1D system. Often, the action of the heart on the arterial system is modelled as a boundary condition at the entrance of the aorta. However, it is well known that the left ventricle and the vascular network are strongly coupled single mechanical system (see [15,25]). This coupling can be relevant in the numerical description of pressure waves propagation particularly when dealing with patological situation. In this work we propose a simple lumped parameter model for the heart and show how it can be coupled numerically with a 1D model for the arteries. Numerical results actually confirm the relevant impact of the heart-arteries coupling in realistic simulations.

• MOX 30 - 10/12/2003
  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

  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.

• MOXD01 - 01/11/2003
  Paganoni, Anna Maria; Pontiggia, Laura
  Laboratorio informatico di Statistica Matematica su supporto Excel

  Abstract

  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.

• MOX 29 - 21/10/2003
  Micheletti, Stefano; Perotto, Simona
  Anisotropic mesh adaptivity in CFD

  Abstract

  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.

• MOX28 - 10/10/2003
  Secchi, Piercesare
  A game between two Bayesians for estimating the mean of a Gaussian distribution

  Abstract

  	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$.