Quaderni di Dipartimento

Quaderni MOX

• MOX 36 - 16/04/2004
  Miglio, Edie; Perotto, Simona; Saleri, Fausto
  Strategies of model coupling: application to free surface flow problems

  Abstract

  Abstract

  The motion of water in a complex hydrodynamic configuration is characterized by a wide spectrum of space and time scales, due to coexistence of physical phenomena of different nature. Consequently, the numerical simulation of a hydrodynamic system of this type is characterized by a large computational cost. In this paper, after introducing a quite general setting for model coupling, we discuss two techniques to reduce such a computational effort by suitably coupling different hydrodynamic models. The first approach is based on a dimensionally heterogeneous-physically homogeneous coupling strategy driven a priori physical considerations. As second strategy we suggest a dimensionally homogeneous-physically heterogeneous coupling. This time the subdomain-to-model correspondence is identified automatically thanks to a suitable a posteriori modeling error estimator. The range of applicability of both the approaches is finally examined on some test cases.

• MOX 37 - 16/04/2004
  Micheletti, Stefano; Perotto, Simona
  Reliability and efficiency of an anisotropic Zienkiewicz-Zhu error estimator

  Abstract

  Abstract

  In this paper we study the efficiency and the reliability of an anisotropic a posteriori error estimator in the case of the Poisson problem supplied with mixed boundary conditions. The error estimator may be classified as a residual-basedone, but its novelty is twofold: firstly, it employs anisotropic estimates of the interpolation error for linear triangular finite elements and, secondly, it makes use of the Zienkiewicz-Zhu recovery procedure to approximate the gradient of the exact solution. Finally, we describe the adaptive procedure used to obtain a numerical solution satisfying a given accuracy, and we include some numerical test cases to assess the robustness of the proposed numerical algorithm.

• MOX 35 - 08/04/2004
  Veneziani, Alessandro; Vergara, Christian
  Flow rate defective boundary conditions in haemodynamics simulations

  Abstract

  Abstract

  In the numerical simulation of blood flow problems it might happen that the only available boundary conditions prescribe the flow rate incoming/outcoming the vascular district at hand. In order to have a well posed Navier-Stokes problem, these conditions need to be completed. In the bioengineering community, this problem is usually faced by choosing a priori a velocity profile on the inflow/outflow sections, to be fitted with the assigned flow rates. This approach strongly influences the accuracy of the numerical solutions. A less perturbative strategy is based on the so-called do-nothing approach, advocated in Heywood, Rannacher, turek, Int. J. Num. Fl, 1996. An equivalent approach, but easier from the numerical discretization viewpoint, has been proposed in Formaggia, Gerbeau, Nobile, Quarteroni, SIAM J Num An, 2002. It is based on an augmented formulation of the problem, in which the conditions on the flow rates are prescribed in a weak sense by means of Lagrangian multipliers. In this paper we analyze the unsteady augmented Navier-Stokes problem, proving a well posedness result. Moreover, we present some numerical methods for solving the augmented problem, based on a splitting of the computation of velocity and pressure on one side and the Lagrangian multiplier on the other one. In this way, we show how it is possible to solve the augmented problem resorting to available Navier-Stokes solvers.

• 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.