Quaderni di Dipartimento

Quaderni MOX

• 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.

• 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.