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Finite Elements / Éléments finis
(Org: Roger Pierre)

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FRANCOIS BERTRAND, Department of Chemical Engineering, École Polytechnique de Montréal, Montreal, Quebec  H4R 2V3
A local refinement based fictitious domain method for the simulation of fluid flow in complex geometries [PDF]

The simulation of fluid flow in industrial processes often involves geometries that may contain mobile internal parts. The use of classical finite element (or finite volume) methods to tackle such problems is far from trivial since a new mesh is needed at each time iteration owing to the motion of these internal parts. The objective of this work is to combine a fictitious domain method with a mesh refinement technique that relies upon one single reference mesh. The method will be discussed in detail and two-dimensional and three-dimensional applications will be presented. In particular, it will be shown that the proposed strategy is quite efficient for the simulation of fluid flow in geometries with moving parts and small gaps.

YVES BOURGAULT, Universite d'Ottawa, Ottawa, Ontario
A mortar element for coupling hyperbolic and parabolic problems [PDF]

The mortar element method is now very popular to decompose elliptic problems on multiple sub-domains. The main feature of this method is its ability to deal with nonmatching grids on sub-domain interfaces without loosing any accuracy of the global solution, while allowing the parallel computing of the solution. As far as we know, the mortar method has been introduced for elliptic or parabolic PDEs only. Its extension to hyperbolic problems (such as the Euler equations for inviscid flows) or mixed-type equations (such as the Navier-Stokes equations for compressible flows) would be a definite asset.

The present work is an initial step into the development of an ``all-at-once'' mortar methods that works for all type of equations, first concentrating on its development for the linear advection equation. The proposed mortar method works for hyperbolic equations, through a combination of streamline-diffusion up-winding, discontinuous and mortar finite element terms in the Galerkin formulation. A weak flux continuity condition at the sub-domain interface is enforced by means of Lagrange multipliers which yields a solution with optimal accuracy even with non-matching grids at sub-domain interfaces. The method can be consistently applied to the advection-diffusion equation. The method has been implemented using MPI and numerical results will be shown for the pure advection as well as the advection-diffusion equations.

ALAIN CHARBONNEAU, Université du Québec en Outaouais, Saint-Jean-Bosco, Hull, Québec  J8Y 3G5
Une méthode adaptative d'éléments finis permettant le calcul des modes propres d'un guide d'ondes optiques [PDF]

Les guides d'ondes optiques (GOO) sont des composants fondamentaux de certains dispositifs de transmission de données par ondes lumineuses utilisés dans les secteurs des télécommunications et du génie (instruments de mesure de paramètres physiques tels la température, la pression, ...).

Dans cet exposé, nous nous intéressons au calcul des modes propres des GOO qui présentent un axe longitudinal d'invariance. Puisque nous cherchons aussi à modéliser la biréfringence de certains types de GOO, nous sommes conduits à résoudre un problème aux valeurs propres issu des équations de Maxwell, dans le cadre de l'optique guidée, dites pleinement vectorielles.

C'est dans ce contexte que nous présentons une méthode adaptative d'éléments finis qui permet de calculer de façon précise les composantes des modes propres d'un champ électromagnétique se propageant dans un GOO composé de matériaux diélectriques isotropes ou anisotropes. Des applications de cette méthode de calcul seront présentées.

MICHEL DELFOUR, Centre de Recherches Mathématiques et Département de Mathématiques et de Statistique, Université de Montréal
Approximation of the dose for thin coated stents in interventional cardiology [PDF]

Stents are used in interventional cardiology to keep a diseased vessel open. New stents are coated with a medicinal agent to prevent early reclosure due to the proliferation of smooth muscle cells. It is the dose of the agent which effectively acts on the cells in the wall of the vessel. This paper gives mathematical models of the dose for a periodic stent and an asymptotic stent. It studies the effect of the number of struts and the ratio between the area of the coated struts and the targeted area of the vessel. Theoretical and numerical results are presented with emphasis on the critical choice of finite element approximations for diffusion-transport equations in the presence of the stent which behaves as a Neumann sieve at the interface between the lumen and the wall of the vessel. (joint paper with A. Garon (Ecole Polytechnique and Vito Longo (Université de Montreal)

KOKOU DOSSOU, Département de mathématiques et de statistique, Université Laval, Québec  G1K 7P4
Higher order vector edge finite element analysis of optical waveguides [PDF]

We will present some applications of vector edge finite element methods to the analysis of optical fiber such as the computation of the propagation constant and propagation mode and that of the birefringence. To address the need for a more accurate finite element approximation we develop a higher order vector edge finite element model. It is well known that the use of standard nodal finite element methods does not work well for electromagnetic problems. Although edge elements appear to be reliable, some care must be taken in order to avoid spurious modes. We will discuss some observations and mathematical properties which ensure that the higher order vector finite element converges and is free of spurious modes.

ROSS ETHIER, University of Toronto, Toronto, Ontario  M5S 3G8
Finite element modelling of coronary artery hemodynamics [PDF]

The coronary arteries are responsible for supplying blood to the heart muscle and are a common site of arterial disease, which leads in its end stages to heart attack. Development of arterial disease in these arteries appears to be strongly influenced by biomechanical factors, including blood flow (hemodynamic) features. To better understand the disease process we therefore desire to model flow patterns in the coronary arteries. The modelling challenges in these arteries include very large deformations of the artery over the cardiac cycle, complex 3D geometries, and significant flow unsteadiness. Here we review some of the techniques used to overcome these challenges. In brief, coronary artery geometries are determined based on post-mortem casts and movies of beating hearts (cineangiograms). Flow modelling uses the Arbitrary Lagrangian-Eulerian (ALE) approach; mesh updating is based on a spring analogy model modified to preserve element quality during complex 3D motions. Flow unsteadiness is based on intra-operative measurements of blood flow wave forms in the affected arteries. An overview of our results will be given, demonstrating the primary effects of arterial geometry (particularly complex, compound curvature), with smaller effects due to flow pulsation and arterial motion.

ANDRÉ FORTIN, Université Laval, Québec  G1K 7P4
Reconstruction géométrique, estimation d'erreur et remaillage adaptatif: application à la mise en forme des polymères [PDF]

Dans cet exposé, nous présenterons brièvement les outils de reconstruction géométrique, d'estimation d'erreurs et de remaillage adaptatif développés au GIREF au cours des dernières années. La reconstruction géométrique consiste, à partir uniquement d'un maillage donné, à identifier les frontières et à recréer la géométrie du problème de manière à être totalement indépendant de quelque logiciel de CAD que ce soit. On peut ensuite résoudre le problème par une méthode numérique quelconque et estimer l'erreur commise. Le maillage est ensuite rafiné ou dérafiné suivant l'importance de l'erreur estimée de maniére à respecter la géométrie reconstruite au préalable.

Nous présenterons par la suite quelques applications à la mise ne forme des polyméres: écoulement dans une contraction de rapport 18 à 1, écoulement dans des mélangeurs statiques, etc.

ROBERT GUENETTE, Université Laval, Québec  G1K 7P4
Méthodes de dualité convexe pour la résolution par éléments finis de problèmes de contact en mécanique des solides [PDF]

De nombreux problèmes industriels exigent de tenir compte du contact mécanique et/ou thermique entre divers matériaux. Le présent exposé est motivé par des applications dans le secteur de l'aluminium et celui de la conception de moteurs d'avion. La résolution de problèmes de contact pose des défis de taille pour le numéricien. Ceci est principalement dû à la nature non différentiable des lois de contact conduisant à des inéquations variationnelles. De plus, les méthodes classiques de résolution du contact ne sont pas efficaces pour les problèmes de grande taille visés dans les applications.

On posera le problème dans le contexte général de l'élasticité en grande déformation incluant le frottement mécanique entre les différents corps élastiques. On utilisera les méthodes de dualité convexe pour le traitement de la non différentiabilité des lois de contact. On proposera une linéarisation des inéquations non linéaires et une discrétisation par éléments finis. Pour les problèmes de contact sans frottement, le système discret sera résolu par un algorithme de gradient conjugué projeté appliqué au problème dual. Des résultats numériques seront présentés pour le calcul approché des déplacements de deux corps élastiques discrétisés par des maillages incompatibles à l'interface de contact.

DANIEL LEROUX, Université Laval, Québec  G1K 7P4
An appropriate finite-element pair to simulate inertia-gravity waves [PDF]

Most of atmospheric, oceanic and hydrological models typically employ gridpoint, finite and spectral-element techniques. For all these numerical methods the coupling between the momentum and continuity equations usually leads to spurious solutions in the representation of inertia-gravity waves. The spurius modes have a wide range of characteristics and may take the form of pure inertia oscillations, Coriolis modes and pressure modes. The spurius modes are small-scale artifacts which are trapped within the model grid, and can cause aliasing and an accumulation of energy in the smallest-resolvable scale, leading to noisy solutions. Their appearance is mainly due to an inappropriate placement of variables on the grid and/or a bad choice of approximation function spaces. We present a triangular finite-element pair candidate, which `properly' models the dispersion of the inertia-gravity waves. In particular, the discrete frequency increases monotonically with wavenumbers as in the continuum case, contrarily to most of other finite-element pairs (if not all). It will also be shown that, like for most other pairs, this finite element candidate should be employed when a precise calculation of the Rossby modes is not an issue. Results of test problems to simulate the propagation of inertia-gravity waves with the proposed finite-element pair are presented and they are compared with results of other grids. They illustrate the promise of the proposed approach.

P.D. MINEV, Department of Mathematics, Statistics and Sciences, University of Alberta, Edmonton, Alberta
Analysis of a projection/characteristic scheme for incompressible flow [PDF]

The paper presents the convergence analysis of a characteristic/projection scheme for the incompressible Navier-Stokes equations. This scheme is a modification of the scheme analyzed in [1] which does not eliminate the projected velocity field from the system but rather uses it as the advecting field in the explicit characteristic advection. This field has a zero (generalized) divergence and is therefore more suitable for this purpose. It appears that this scheme has the same convergence rate as the one in [1] but on a given grid seems to produce more accurate results. The computational cost is not significantly higher since it requires only one extra inversion of the mass matrix which can be done relatively efficiently. We present numerical results which illustrate the properties of the scheme.

References

[1]

Y. Achdou and J.-L. Guermond, Convergence analysis of a finite element projection/Lagrange-Galerkin method for the incompressible Navier-Stokes equations. SIAM J. Numer. Anal. (3) 37(2000), 799-826.

DOMINIQUE PELLETIER, École Polytechnique de Montréal, Montréal, Ontario
Sensitivity and uncertainty analysis in CFD [PDF]

In this talk we present the sensitivity equation method (SEM) to perform sensitivity and uncertainty analysis of CFD model. A formulation of teh SEM is presented that unifies both value and shape parameters. The SEM is used to cascade uncertainty in the input through a CFD code to obtain uncertainty estimates on the outputs of the CFD simulation. Examples will be presented for flows with strongly temperature dependent properties. Application to turbulent flows will also be discussed.

BRUCE SIMPSON, School of Computer Science, University of Waterloo, Waterloo, Ontario  N2L 3G1
Computing the deltas; efficiency-accuracy trade offs in solving Black Scholes equations [PDF]

Pricing functions for financial options are routinely computed as numerical solutions of partial differential equations of Black Scholes type. The risk associated with issuing an option can be reduced by various hedging strategies for portfolio management. In theory, a zero-risk strategy is possible, which requires continuously modifying the portfolio. These modifications depend on the derivatives of the dynamically changing price function, i.e. the so-called delta hedging parameters. In practice, the ideal hedging strategy may be approximately followed which results in the issuer incurring some risk.

We will look at finite element computation of pricing functions V(S₁,S₂,t) that depend on two underlying assets, and estimation of the the gradient from the numerical solution for hedging parameters. The goal is to determine

a)  a level of accuracy that incurs an acceptable risk

b)  techniques of meshing and gradient estimation which can efficiently meet the accuracy requirement of a).

LEILA SLIMANE, GIREF, Laval
Méthodes mixtes pour la résolution des inéquations variationnelles [PDF]

La méthode des éléments finis mixtes permet de remedier de façon efficace aux phénomènes de verrouillage numérique pouvant apparaître dans la résolution numérique d'équations variationnelles dépendant d'un petit paramètre. Dans le présent exposé nous étendons le champ d'application de cette méthode aux inéquations variationnelles, tout particulièrement au problème de transmission raide avec des conditions aux limites de type Signorini, et au problème de contact unilatéral en élasticité presque incompressible.

Nous commençons par dégager les propriétés communes aux formulations mixtes de ces derniers problèmes. Ensuite, nous nous plaçons dans un cadre abstrait, regroupant les propriétés des exemples précédents, et dans lequel nous établissons des résultats d'existence, d'unicité et de stabilité. Nous donnons aussi des résultats de convergence et des estimations d'erreur dans le cadre d'approximation du problème. Finalement, nous appliquons cette étude à l'approximen élasticité presque incompressible, où nous obtenons des résultats de convergence uniforme pour ces schémas.

AZZEDINE SOULAIMANI, École de technologie supérieure
On the solution of free surface flows with the SPH and related methods [PDF]

SPH (Smoothed Particle Hydrodynamics) is a Lagrangian mesh free method used since the end of the seventeen in the simulation of astrophysics problems. Monaghan proposed extensions to gas dynamics and free surface problems. Original SPH regain more popularity in the nineties, especially for impact and large deformation mechanical problems. In the first part of the talk, a state of the art on SPH will be given. A relationship between SPH and the finite element-finite volume method will be emphasized with application to the solution of the Shallow-water equations. This formulation gives the possibility to introduce well-known finite elements or finite volume stabilization techniques for high speed flows. The problem of dam break in various two-dimensional configurations is used as the benchmark test. The result obtained depends on the concerned problem. For the cases of standard and circular dams, the results are quiet encouraging. The capture of shocks and the shape of the waves were successfully revealed. The success of the SPH in the solution of free surface flows depends on the optimisation of its parameters, a smart choice of particles number, type of the kernel and the smoothing of irregularities in the geometry. In case of irregular boundaries, some difficulties are encountered for imposing proper boundary conditions, and require additional investigations.