Scientific Reports

MOX Reports

• MOX 86 - 07/10/2006
  Amadori, Debora; Ferrari, Stefania; Formaggia, Luca
  Derivation and analysis of a fluid-dynamical model in thin and long elastic vessels

  Abstract

  Abstract

  Starting from the three-dimensional Newtonian and incompressible Navier-Stokes equations in a compliant straight vessel, we derive a reduced one-dimensional model by an averaging procedure which takes into consideration the elastic properties of the wall structure. In particular, we neglect terms of the first order with respect to the ratio between the vessel radius and length. Furthermore, we consider that the viscous effects are negligible with respect to the propagative phenomena. The result is a one-dimensional nonlinear hyperbolic system of two equations in one space dimension, which describes the mean longitudinal velocity of flow and the radial wall displacement. The modelling technique here applied to straight cylindrical vessels may be generalized to account for curvature and torsion. An analysis of well posedness is presented which demonstrates, under reasonable hypothesis, the global in time existence of regular solutions.

• MOX 85 - 06/22/2006
  Nobile, Fabio; Tempone, Raul; Webster, Clayton
  A sparse grid stochastic collocation method for elliptic partial differential equations with random input data

  Abstract

  Abstract

  This work proposes and analyzes a sparse grid stochastic collocation method for solving elliptic partial differential equations with random coefficients and forcing terms (input data of the model). This method can be viewed as an extension of the Stochastic Collocation method proposed in [Babuska-Nobile-Tempone, Technical report, MOX, Dipartimento di Matematica, 2005] which consists of a Galerkin approximation in space and a collocation at the zeros of suitable tensor product orthogonal polynomials in probability space and naturally leads to the solution of uncoupled deterministic problems as in the Monte Carlo method. The full tensor product spaces suffer from the curse of dimensionality since the dimension of the approximating space grows exponentially fast in the number of random variables. If the number of random variables is moderately large, this work proposes the use of sparse tensor product spaces utilizing either Clenshaw-Curtis or Gaussian interpolants. For both situations this work provides rigorous convergence analysis of the fully discrete problem and demonstrates: (sub)-exponential convergence of the “probability error” in the asymptotic regime and algebraic convergence of the “probability error” in the pre-asymptotic regime, with respect to the total number of collocation points. The problem setting in which this procedure is recommended as well as suggestions for future enhancements to the method are discussed. Numerical examples exemplify the theoretical results and show the effectiveness of the method.

• MOX 84 - 06/12/2006
  Dede', L.
  Optimal flow control for Navier-Stokes equations: drag minimization

  Abstract

  Abstract

  Optimal control and shape optimization techniques have an increasing role in Fluid Dynamics problems governed by Partial Differential Equations (PDEs). In this paper we consider the problem of drag minimization for a body in relative motion in a fluid by controlling the velocity through the body boundary. With this aim we handle with an optimal control approach applied to the steady incompressible Navier-Stokes equations. We use the Lagrangian functional approach and we adopt the Lagrangian multiplier method for the treatment of the Dirichlet boundary conditions, which include the control function itself. Moreover we express the drag coefficient, which is the functional to be minimized, through the variational form of the Navier-Stokes equations. In this way we can derive, in a straightforward manner, the adjoint and sensitivity equations associated with the optimal control problem, even in presence of Dirichlet control functions. The problem is solved numerically by an iterative optimization procedure applied to state and adjoint PDEs which we approximate by the finite element method.

• MOX 83 - 05/26/2006
  Abba', A.; Bonaventura, L.
  A vorticity preserving finite difference discretization for the incompressible Navier-Stokes equations

  Abstract

  Abstract

  A finite difference discretization of the three-dimensional, incompressible Navier Stokes equations is introduced, based on ideas that have been applied successfully to geophysical flows over the last four decades. The proposed spatial discretization is mass conservative and vorticity preserving, in the sense that a discrete form of the vorticity equation is derived naturally from the discrete momentum equation by application of a discrete rotation operator. A vorticity preserving discretization of the viscous terms and an appropriate treatment for rigid wall boundary conditions are also proposed. The relationship of this approach to other similar techniques is discussed. The results are compared to those of a standard finite difference discretization approach in a number of relevant test cases, which demonstrate the advantages of the proposed method, especially when strong vorticity production takes place at the boundaries.

• MOX 82 - 04/10/2006
  Corno, A.F.; Prosi, M.; Fridez, P.; Zunino, P.; Quarteroni, A.;von Segesser, L.K.
  The non-circular shape of FloWatch®-PAB prevents the need for pulmonary

  Abstract

  Abstract

  Objective: To evaluate the differences between non-circular shape of FloWatch®-PAB and conventional pulmonary artery (PA) banding.Methods: Geometrical analysis. Conventional banding and FloWatch®-PAB perimeters were plotted against cross sections.Computational Fluid Dynamics (CFD) model. CFD compared non-circular FloWatch®-PAB cross sections with conventional banding regarding pressure gradients.Clinical data. Seven children, median age 2months (7days-3years), median weight 4.2kg (3.2-9.8kg) with complex congenital heart defects underwent PA banding with FloWatch®-PAB implantation.Results. Geometrical analysis. Conventional banding: progressive reduction of cross sections was accompanied by progressive reduction of PA perimeters. FloWatch®-PAB: with equal reduction of cross sections the PA perimeter remained constant.CFD model. Non-circular and circular banding provided same trans-banding pressure gradients for same cross sections at any given flow.Clinical data. Mean PA internal diameter at banding was 13.3±4.5mm. After mean interval of 5.9±3.7 months, all children underwent intra-cardiac repair and simple FloWatch®-PAB removal without PA reconstruction. Mean PA internal diameter with FloWatch®-PAB removal increased from 3.0±0.8mm to 12.4±4.5mm (normal mean internal diameter for the age = 9.9±1.6). No residual pressure gradient was recorded in correspondence of the site of the previous FloWatch®-PAB implantation in 6/7 patients, 10mmHg peak and 5mmHg mean gradient in 1/7.Conclusions. The non-circular shape of FloWatch®-PAB can replace conventional circular banding with the following advantages:a)the pressure gradient will remain essentially the same as for conventional circular banding for any given cross section, but with significantly smaller reduction of PA perimeter b)PA reconstruction at the time of de-banding for intra-cardiac repair can be avoided.

• MOX 81 - 03/06/2006
  Balossino, R., Pennati, G.; Migliavacca, F.; Formaggia, L.; Veneziani, A.; Tuveri, M.; Dubini, G.
  Influence of boundary conditions on fluid dynamics in models of the cardiovascular system: a multiscale approach applied to the carotid bifurcation

  Abstract

  Abstract

  Background: This work aims at addressing an important problem in the simulation of detailed 3D hemodynamic models of vascular districts with complex anatomy. Namely, to define appropriate boundary conditions accounting for both local as well as global effects. Approach: The approach devised in this work is based on a multiscale model, where the Navier-Stokes equations for the district of interest are coupled to a non-linear system of ordinary differential equations which describes the global circulatory system as a lumped parameter network. The multiscale approach is applied to three 3D models of a carotid bifurcation which differ only in the severity of a stenosis in the internal carotid artery. The results of the multiscale simulations are compared to those obtained by two stand-alone models of the carotid bifurcation, which differ in the adopted strategy in prescribing the boundary conditions. Results: Significant differences are found in the results between the multiscale and the stand-alone models in terms of flows, pressures and wall shear stresses distribution in the 3D domain. Conclusions: The capability to numerically predict the hemodynamic changes due to the presence of a stenosis is highly dependent on the availability of correct boundary conditions. The geometrical multiscale approach offers a logical and proper alternative to the use of measured data to prescribe realistic boundary conditions and predict new hemodynamic scenarios.

• MOX 80 - 01/31/2006
  Ponzini, R.; Vergara, Christian; Redaelli, A.; Veneziani, Alessandro
  Reliable CFD-based estimation of flow rate in haemodynamics measures

  Abstract

  Abstract

  Physical useful measures in current clinical practice refer often to the blood ow rate, that is related to the mean velocity. However, the direct measure of the latter is currently not possible using a Doppler velocimetry technique. Therefore, the usual approach to calculate the ow rate with this technique consists in measuring the maximum velocity and in estimating the mean velocity, making the hypothesis of parabolic profile, that in realistic situations brings to strongly inaccurate estimates. In this paper, we propose a different way for estimating the ow rate regarded as a function of maximum velocity and Womersley number. This relation is obtained by fixing a parametrized representation and by evaluating the parameters by means of a least square approach working on the numerical results of CFD simulations (about 200). Numerical simulations are carried out by prescribing the ow rate, not the velocity profile. In this way, no bias are implicitly induced in prescribing boundary conditions. Validation tests based on numerical simulations show that the proposed relation improves the flow rate estimation.

• MOX 79 - 01/30/2006
  Quarteroni, Alfio; Rozza, Gianluigi; Quaini, Annalisa
  Reduced basis methods for optimal control of advection-diffusion problems

  Abstract

  Abstract

  The reduced basis (RB) method is proposed for the approximation of multiparametrized equations governing an optimal control problem. The idea behind the RB method is to project the solution onto a space of small dimension, specifically designed on the problem at hand, and to decouple the generation and projection stages (off-line/on-line computational procedures) of the approximation process in order to solve parametrized equations in a rapid and not expensive way. The application that we investigate is an air pollution control problem: we aim at regulating the emissions of industrial chimneys in order to keep the pollutant concentration below a certain threshold over an observation area, like a town. Adopting the RB method for both state and adjoint equations of the optimal control problem leads to important computational savings with respect to the use of the Galerkin-finite element method. We consider different parametrization (control, physical and geometrical input parameters) so that we are able to solve the control problem from a global and decisional point of view.