Quaderni di Dipartimento

Quaderni MOX

• MOX 60 - 20/05/2005
  Di Pietro, Daniele A., Lo Forte, Stefania; Parolini, Nicola
  Mass Preserving Finite Element Implementations of Level Set Methods

  Abstract

  Abstract

  In the last two decades, the level set method has been extensively used for the numerical solution of interface problems in different domains. The basic idea is to embed the interface as the level set of a regular function. In this paper we focus on the numerical solution of hypervolic interface advection equations which appears in free-surface fluid dynamics problems, where naive finite element implementations are unsatisfactory. As a matter of fact, practitioners in fluid dynamics often complain that the mass of each fluid component is not conserved, a phenomenon which is therefore often referred to a mass loss. In this paper we propose and compare two finite element implementations that cure this ill-behaviour without the need to resort to spurius strategies (such as, e.g., particle level set). The first relies on a discontinuous Galerkin discretization, which is known to give very good performance when facing hyperbolic problems the second is a stabilized continuous FEM implementation based on the stabilization method presented in [1], which is free from many of the problems that classical methods exhibit when applied to unsteady problems.

• MOX 59 - 03/05/2005
  Micheletti, Stefano; Perotto, Simona; Verani, Marco
  An Adaptive Uzawa Algorithm for Output Functionals Control

  Abstract

  Abstract

  In this paper we address the approximation of a linear output functional J(u), where u is the solution of an elliptic problem, through suitable duality arguments. In particular, moving from the adaptive Uzawa algorithm proposed by E. Baensch et al. and S. Dahlke et al., we first cast the original problem in a saddle-point formulation and then we provide two new algorithms for computing an approximation $u_h$ to $u,$ such that $|J(u) - (u_h)|$ be below a prescribed tolerance t. Some test cases are included to assess the reliability of the proposed method in the 1D case togheter with some preliminary two dimensional results.

• MOX 58 - 26/04/2005
  Deparis, Simone; Discacciati, Marco; Fourestey, Gilles; Quarteroni, Alfio
  Fluid-structure algorithms based on Steklov-Poincaré operators

  Abstract

  	Abstract
  	In this paper we review some classical algorithms for fluid structure interaction problems and we propose an alternative viewpoint mutuated from the domain decomposition theory. This approach yields preconditioned Richardson iterations on the Steklov-Poincaré nonlinear equation at the fluid-structure interface

• MOX 57 - 19/04/2005
  Di Pietro, Daniele A.; Veneziani, Alessandro
  Expression Templates Implementation of Continuous and Discontinous Galerkin Methods

  Abstract

  Abstract

  Efficiency and flexibility are often mutually exclusive features in a code. This still prompts a large part of the Scientific Computing Community to use traditional procedural language. In the last years, however, new programming techniques have been introduced allowing for a high level of abstraction without loss of performance. In this paper we present an application of the Expression Templates technique introduced in [13] to the assembly step of a finite element computation. We show that a suitable implementation, such that the compiler has the role of parsing abstract operations, allows for user-friendliness and gain in performance with respect to more traditional techniques. Both the cases of confonforming and discontinuous Galerkin finite element discretization are considered. The proposed implementation is finally applied to a number of problems entailing different kind of complications.

• MOX 56 - 24/03/2005
  Martin, Vincent; Clément, F.; Decoene, A.; Gerbeau, J.F.
  Parameter identification for a one-dimensional blood flow model

  Abstract

  Abstract

  The purpose of this work is to use a variational method to identify some of the parameters of one-dimensional models for blood flow in arteries. These parameters can be fit to approach as much as possible some data coming from experimental measurements or from numerical simulations performed using more complex models. A nonlinear least squares approach to parameter estimation was taken, based on the optimization of a cost function. The resolution of such an optimization problem generally requires the efficient and accurate computation of the gradient of the cost function with respect to the parameters. This gradient is computed analytically when the one-dimensional hyperbolic model is discretized with a second order Taylor-Galerkin scheme. An adjoint approach, involving the resolution of an adjoint problem, was used. Some preliminary numerical tests are shown. In these simulation, we mainly focused on detrmining a parameter that is linked to the mechanical properties of the arterial walls, the compliance. The synthetic data we used to estimated the parameter were obtained from a numerical computation performed with a more precise model: a three-dimensional fluid structure interaction model. The first results seem to be promising. In particular, it is woth noticing that the estimated compliance which gives the best fit is qiute different from the values one would have expected a priori.

• MOX55 - 18/02/2005
  Dede', L.; Quarteroni, A.
  Optimal Control and Numerical Adaptivity for Advection-Diffusion Equations

  Abstract

  Abstract

  We propose a general approach for the numerical approximation of optimal control problems governed by a linear advection-diffusion equation, based on a stabilization method applied to the Lagrangian functional, rather than stabilizing the state and adjoint equation separately. This approach yields a coherently stabilized control problem. Besides, it allows a straightforward a posteriori error estimate in which estimates of higher order terms are needless. Our a posteriori estimates stems from splitting the error on the cost functional into the sum of an iteration error plus a discretization error. Once the former is reduced below a given threshold (and therefore the computed solution is near the optimal solution), the adaptive strategy is operated on the discretization error. To prove the effectiveness of the proposed methods, we report some numerical tests, referring to problems in which the control term is the source term of the advection-diffusion equation.

• MOXP4 - 12/02/2005
  Veneziani, Alessandro; Foa, Erika
  Simulazione Numerica di Pompe per Macchine Lavatrici

  Abstract

  	Abstract

• MOX 54 - 21/01/2005
  Agoshkov, Valery; Quarteroni, Alfio; Rozza, Gianluigi
  A Mathematical Approach in the Design of Arterial Bypass Using Unsteady Stokes Equations

  Abstract

  	Abstract
  	In this paper we present an approach for the study of Aorto-Coronaric bypass anastomoses configurations using unsteady Stokes equations. The theory of optimal control based on adjoint formulation is applied in order to optimize the shape of the zone of the incoming branch of the bypass (the toe) into the coronary according to several optimality criteria.