Quaderni di Dipartimento

Quaderni MOX

• MOX 20 - 26/05/2003
  Bottasso, Carlo L.; Micheletti, Stefano; Sacco, Riccardo
  A multiscale formulation of the discontinuous Petrov-Galerkin method for advective-diffusive problems

  Abstract

  Abstract

  We consider the Discontinuous Petrov-Galerkin method for the advection-diffusion model problem and we investigate the application of the variational multiscale method to this formulation. We show the exact modeling of the fine scale modes at the element level for the linear case, and we discuss the approximate modeling both in the linear and in the non-linear cases. Furthermore, we highlight the existing link between this multiscale formulation and the $p$ version of the finite element method. Numerical examples illustrate the behavior of the proposed scheme.

• MOX 19 - 07/05/2003
  Causin, Paola; Sacco, Riccardo
  A discontinuous Petrov-Galerkin method with Lagrangian multipliers for second order elliptic problems

  Abstract

  Abstract

  We present a discontinuous Petrov-Galerkin method (DPG) for finite element discretization scheme of second order elliptic boundary value problems. The novel approach emanates from a one-element weak formulation of the differential problem (that is typical of Discontinuous Galerkin methods (DG)) which is based on introducing variables defined in the interior and on the boundary of the element. The interface variables are suitable Lagrangian multipliers that enforce interelement continuity of the solution and of its normal derivate, thus providing the proper connection between neighboring elements. The internal variables can be eliminated in favor of the interface variables using static condensation to end up with a system of reduced size having as unknowns the Lagrangian multipliers. A stability and convergence analysis of the novel formulation is carried out and its connection with mixed-hybrid and DG method is explored. Numerical tests on several benchmark problems are included to validate the convergence perfomance and the flux-conservation properties of the DPG method.

• MOX 18 - 16/04/2003
  Causin, Paola; Restelli, Marco; Sacco, Riccardo
  A simulation system based on mixed-hybrid finite elements for thermal oxidation in semiconductor technology

  Abstract

  Abstract

  In this work we deal with the numerical simulation of thermal oxidation in silicon device technology. This application involves the coupled solution of a diffusion-reaction problem and of a fluid-structure interaction problem. These two problems are mutually dependent through the exchange of stresses and fluxes that are typically post-processed fields in standard finite element approaches, and as such they may suffer form a lack of accuracy and from physical inconsistencies. In this article, we propose a novel approach to the simulation of the thermal oxidation process, that is characterized by the use of mixed and hybrid finite elements. the main advantage of such formulations is that stresses and fluxes are directly computed quantities, rather than obtained from post-processing techniques. We also address the procedures and the techniques that must be devised for handling the coupled interaction problem and the presence of a computational grid moving in time. The numerical approach we propose is eventually validated on a realistic example of the thermal oxidation process in a local oxidation structure (LOCOS)

• MOX 17 - 05/04/2003
  Abba', A.; Cercignani, C.; Valdettaro, L.
  Modelli e simulazione LES di correnti turbolente e applicazione alla combustione

  Abstract

  Abstract

  Le finalità della presente ricerca sono l analisi e sviluppo di modelli per la simulazione numerica di correnti turbolente incomprimibili e di fluidi turbolenti reagenti, mediante la tecnica della Large Eddy Simulation (LES). La prima parte dell articolo è dedicata all analisi, sulla base di test a priori, di diversi modelli di turbolenza per correnti incomprimibili in assenza di reazioni chimiche. I modelli vengono valutati mediante test a priori avvalendoci di risultati provenienti da simulazioni numeriche dirette di turbolenza omogenea isotropa a Reynolds 1000 e di uno strato limite turbolento tridimensionale. Nella seconda parte si troveranno invece le descrizioni delle equazioni e del metodo numerico utilizzato per la simulazione di correnti turbolente in presenza di reazioni chimiche secondo differenti approcci: il metodo dello scalare conservato, per reazioni chimiche veloci in fiamme di diffusione l approccio a fronte di fiamma per flussi premiscelati equazioni di conservazione delle specie chimiche per reazioni complesse, a più passi e non in equilibrio. Per ognuno di questi vengono formulati nuovi modelli basati sui modelli dinamici presentati nella prima parte, e vengono presentati i risultati delle simulazioni numeriche.

• MOX 15 - 05/03/2003
  Formaggia, Luca; Micheletti, Stefano; Perotto, Simona
  Anisotropic mesh adaption in Computational Fluid Dynamics: application to the advection-diffusion-reaction and the Stokes problems

  Abstract

  Abstract

  In this work we develop an anisotropic a posteriori error analysis of the advection-diffusion-reaction and the Stokes problems. This is the first step towards the study of more complex situation, such as the Oseen and Navier-Stokes equations, which are very common in Computational Fluid Dynamic (CFD) applications. The leading idea of our analysis consists in combining the anisotropic interpolation error estimates for affine triangular finite element provided in [14,15] with a posteriori error analysis based on a dual problem associated with the problem at hand [6,34]. Anisotropic interpolation estimates take into account more in detail the geometry of the triangular elements, i.e. not just their diameter but also their aspect ratio and orientation. Ont he other hand, the introduction of the dual problem allows us to control suitable functionals of the discretization error, e.g. the lift and drag around bodies in external flows, mean and local values, etc. The combined use of both approaches yelds an adaptive algorithm which, via an iterative process, can be used for designing the optimal mesh for the problem at hand

• MOX 16 - 05/03/2003
  Canuto, Claudio; Quarteroni, Alfio
  Spectral Methods

  Abstract

  Abstract

  Spectral methods represent a family of methods for the numerical approximation of partial differential equations. Their common denominator is to rely on high order polynomial expansions, notably trigonometric polynomials for periodic problems, orthogonal Jacobi polynomials for non periodic boundary value problems. They have the potential of providing high rate of convergence when applied to problems with regular data. They can be regarded as members of the broad family of (generalized) Galerkin methods with munerical evaluation of integrals based on Gaussian nodes. In a first part we introduce the methods on a computational domain of simple shape, analyze their approximation properties as well as their algorithmic features. Next we address the issue of how these methods can be extended to more complex geometrical domains be retaining their distinctive approximation properties.

• MOX 13 - 03/02/2003
  Deparis, Simone; Fernandez, Miguel A.; Formaggia, Luca
  Acceleration of a fixed point algorithm for fluid-structure interaction using transpiration conditions

  Abstract

  Abstract

  In this work, we address the numerical solution of fluid-structure interaction problems. This issue is particularly difficulty to face when the fluid and the solid densities are of the same order, for instance as it happens in hemodynamic applications, since fully implicit coupling schemes are required to ensure stability of the resulting method. Thus, at each time step, we have to solve a highly non-linear coupled system, since the fluid domain depends on the unknown displacement of the structure. Standar strategies for solving this non-linear problems, are fixed point based methods such as Block-Gauss-Seidel (BGS) iteractions. Unfortunately, these methods are very CPU time consuming and usually show slow convergence. We propose a modified fixed-point algorithm which combines the standard BGS iteraction with transpiration formulation. Numerical experiments show the great improvement in computing time with respect to the standard BGS method.

• MOX 14 - 03/02/2003
  Sherwin, S.J.; Formaggia, Luca; Peiro, Jaume; Franke, V.
  Computational Modelling of 1d blood flow with variable mechanical properties and its application to the simulation of wave propagation in the human arterial system

  Abstract

  Abstract

  In this paper we numerically investigate a one-dimensional model of blood flow in human arteries using both discontinuous Galerkin and a Taylor-Galerkin formulation. The derivation of the model and the numerical schemes are detailed and applied to two model numerical experiments. We first study the effect of an intervenction, such the implantantion of a vascular prothesis (e.g. a stent), which leadsto an abrupt variation of the mechanical characteristics of an artery. We then discuss the simulation of the propagation of the pressure and velocity waveforms in the human arterial tree using a simplified model consisting of the 55 main arteries.