MOX Reports
The preprint collection of the Laboratory for Modeling and Scientific Computation MOX. It mainly contains works on numerical
analysis and mathematical modeling applied to engineering problems. MOX web site is mox.polimi.it
Found 1152 products
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MOX 91 - 10/24/2006
Burman, Erik; Zunino, Paolo
A domain decomposition method based on weighted interior penalties for advection-diffusion-reaction problems | Abstract | | We propose a domain decomposition method for advection-diffusion-reaction equations based on Nitsche s transmission conditions.
The advection dominated case is stabilized using a continuous interior penalty approach based on the jumps in the gradient over element boundaries. We prove the convergence of the finite element solutions of the discrete problem to the exact solution and we propose a parallelizable iterative method.
The convergence of the resulting domain decomposition method is proved and this result holds true uniformly with respect to the diffusion parameter. The numerical scheme that we propose here can thus be applied straightforwardly to diffusion dominated, advection dominated and hyperbolic problems. Some numerical examples are presented in different flow regimes showing the influence of the stabilization parameter on the performance of the iterative method and we compare with some other domain decomposition techniques for advection--diffusion equations.
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MOX 90 - 10/23/2006
Quaini, Annalisa; Quarteroni, Alfio
A semi-implicit approach for fluid-structure interaction based on an algebraic fractional step method | Abstract | | We address the numerical simulation of fluid-structure interaction problems dealing with an incompressible fluid whose density is
close to the structure density. We propose a semi-implicit coupling scheme based on an algebraic fractional-step method. The
basic idea of a semi-implicit scheme consists in coupling implicitly the added-mass effect, while the other terms (dissipation, convection and geometrical non-linearities) are treated explicitly. Thanks to this kind of explicit-implicit splitting, computational costs can be reduced (in comparison to
fully implicit coupling algorithms) and the scheme remains stable for a wide range of discretization parameters. In this paper we propose to derive this kind of splitting from the algebraic
formulation of the coupled fluid-structure problem (after finite-element space discretization). This approach extends for the first time to fluid-structure problems the algebraic
fractional step methodology that was previously advocated to treat the pure fluid problem in a fixed domain. More particularly, for
the specific semi-implicit method presented in this report we adapt the Yosida scheme to the case of a coupled fluid-structure problem. This scheme relies on an approximate LU block factorization of the matrix obtained after the discretization in time and space of the fluid-structure
system.
We analyze the numerical performances of this scheme on 2D fluid-structure simulations performed with a simple 1D structure model. |
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MOX 89 - 10/17/2006
Formaggia, Luca; Veneziani, Alessandro; Vergara, Christian
A new approach to numerical solution of defective boundary problems in incompressible fluid dynamics | Abstract | | We consider the incompressible Navier-Stokes equations with flow rate and mean pressure boundary conditions. There are basically two strategies for solving these defective boundary problems: the variational approach (see J. Heywood, R. Rannacher, S. Turek, Int J Num Meth Fluids 22 (1996), pp. 325-352) and the augmented formulation (see L. Formaggia, J. F. Gerbeau, F. Nobile, A. Quarteroni, SIAM J Num Anal, 40-1 (2002), pp. 376--401, and A. Veneziani, C. Vergara, Int J Num Meth Fluids, 47 (2005), pp. 803--816).
However, these approaches present some drawbacks.
The former, for the flow rate problem, resorts to non standard functional spaces, which are quite difficult to discretize. On the other hand, for the mean pressure problem, it yealds exact solutions only in very specific cases. The latter is applicable only to the flow rate problem, since for the mean pressure problem it provides unfeasible boundary conditions.
In this paper, we propose a new strategy, based on a control reformulation of the problems at hand. This approach allows to treat the two problems successfully within the same framework. We carry out the well-posedness analysis of the problems obtained with this approach and we propose some algorithms for their numerical solution. Several numerical results are presented supporting the effectiveness of our approach.
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MOX 88 - 09/24/2006
Quarteroni, Alfio; Rozza, Gianluigi
Numerical solution of parametrized Navier-Stokes equations by reduced basis methods | Abstract | | We apply the reduced basis method to solve Navier-Stokes
equations in parametrized domains. Special attention is devoted to
the treatment of the parametrized non-linear transport term in
the reduced basis framework, including the case of non-affine
parametric dependence that is treated by an empirical
interpolation method. This method features (1) a rapid
global convergence owing to the property of the Galerkin
projection onto a space W_N spanned by solutions of the
governing partial differential equation at N (optimally)
selected points in the parameter space, and (2) the
off-line/on-line computational procedures which decouple the
generation and projection stages of the approximation process.
This method is well suited for the repeated and rapid evaluations
required in the context of parameter estimation, design,
optimization, and real-time control. Our analysis focuses on:
(i) the pressure treatment of incompressible Navier-Stokes
problem; (ii) the fulfillment of an equivalent inf-sup condition to
guarantee the stability of the reduced basis solutions. The applications that we consider involve parametrized geometries, like e.g. a channel with curved upper wall or an arterial bypass configuration.
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MOX 87 - 09/15/2006
Peiro, J.; Formaggia, L.; Radaelli,A.; Rigamonti, V.
Shape reconstruction from medical images and quality mesh generation via implicit surfaces | Abstract | | The ability of automatically reconstructing physiological shapes, of generating computational meshes, and of calculating flow solutions from medical images is enabling the introduction of computational fluid dynamics (CFD) techniques as an additional tool to aid clinical practice.
This article presents a set of procedures for the shape reconstruction and triangulation of geometries derived from a set of medical images representing planar cross sections of the object. The reconstruction of the shape of the boundary is based on the interpolation of an implicit function through a set of points obtained from the segmentation of the images. This approach is favoured for its ability of smoothly interpolating between sections of different topology. The boundary of the object is an iso-surface of the implicit function that is approximated by a triangulation extracted by the method of marching cubes. The quality of this triangulation is often neither suitable for mesh generation nor for flow solution. We discuss the use of mesh enhancement techniques to maximize the quality of the triangulation together with curvature adaption to optimize mesh resolution.
The proposed methodology is applied to the reconstruction and discretization of two physiological geometries: a femoral by-pass graft and a nasal cavity.
Keywords: Mesh generation; shape reconstruction; mesh enhancement; radial basis functions; implicit surfaces; medical image processing.
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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 | | 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. |
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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 | | 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.
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MOX 84 - 06/12/2006
Dede', L.
Optimal flow control for Navier-Stokes equations: drag minimization | 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.
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