Mathématiques industrielles
Jeudi, 20 juin
Salle George V
Modelling Mass Transfer in a Rotating Disk Reaction Vessel
Bohun, Sean C.
University of Ontario Institute of Technology

Presented is research focussing on the process of mass transfer in a rotating disk apparatus, formulated as a Stefan problem driven by the dynamics of the chemical reactions in the bulk solution. Calcite (calcium carbonate) is chosen as the material undergoing dissolution because of the central role it plays in geological and man-made structures. The wide range in the rates of the various reactions allows for a natural decoupling of the problem into a simplified set of weakly coupled convective-reaction-diffusion equations for the slowly evolving species, connected with a set of algebraic relations for the species that react rapidly. Further simplification is possible by identifying the dominant reaction as determined by the acidity. Numerical solutions of the model are compared to the results of an asymptotic analysis and predicted dissolution rates are contrasted with experimentally obtained results.

Applications of the finite element method to wood and wood-based products
Cloutier, Alain (1), André Fortin (2), Jean Deteix (2), Zanin Kavazovic (1), Costel Barbuta (3) and Pierre Blanchet (3)
(1) Département des sciences du bois et de la forêt, Université Laval, (2) GIREF, Département de mathématiques et de statistique, Université Laval, (3) FPInnovations

The use of the finite element method to wood and wood-based products is fairly recent. Wood is a biomaterial. This involves high variability in its material properties and structure. Moreover, wood is strongly hygroscopic, which impacts its manufacturing processes and its behavior in service. This also involves that most of its material properties vary strongly with moisture content. In this presentation, a few finite element model case studies will be showcased including wood drying and dimensional stability, the hot pressing of medium density fiberboard and the dimensional stability of engineered wood flooring under cyclic relative humidity conditions. The industrial problems, models used and solutions obtained will be presented. The challenges involved in modeling a variable material such as wood will be outlined.

A preconditioned GCR solver for an active set strategy in frictional contact
Fortin, Michel, Patrick Lacasse and Jean Deteix
GIREF, Département de mathématiques et de statistique, Université Laval

In this presentation, we consider elasticity problems where  frictional deformable-deformable contact (Coulomb's law) is discretised by a mortar (Lagrange multiplier) method. We present an algorithm which allies an active set strategy with a preconditioned GCR method to obtain a fast and accurate solver, Lagrangian multipliers are indeed a natural way to discretise the problem (Alart and Curnier1991), (Wriggers and Zavarise 1993) and (deSaxce and Feng 1998).

Modelling fuel cells in start-up and reactant starvation conditions
Wetton, Brian R.
University of British Columbia

Material degradation is currently one of the main technical issues in the development of polymer electrolyte membrane fuel cells (PEMFC). This degradation includes carbon corrosion of the catalyst support and catalyst migration that can occur at high electrochemical potentials. These high potentials can occur at start-up, shut-down or reactant starvation. This work reviews the known conditions that lead to high potentials in PEMFC and develops models that describe the phenomena concisely. Computations describing these cells in four steady starvation situations are presented: anode starvation; cathode starvation; and two types of local anode starvation. Reverse reactions, oxygen and hydrogen evolution, must be included to correctly identify the behaviour of fuel cells in these situations. Using logarithmic scaling of concentrations and adaptive grid refinement in the flow direction, computational implementations of the model can robustly handle the depletion of reactants in the flow and corresponding changes in potentials. The model is extended to time dependent computations using simpler descriptions of channel flow but including capacitance and stack-level electrical coupling effects. Simulations of fuel cell start-up are presented. In all cases, the results of the simulations are compared to literature and new experimental results. Strong qualitative agreement is obtained.

An updated Lagrangian method with error estimation and adaptive remeshing for very large deformation elasticity problems
Léger, Sophie, André Fortin, Cristian Tibirna and Michel Fortin
GIREF, Département de mathématiques et de statistique, Université Laval

Accurate simulations of large deformation hyperelastic materials by the finite element method is still a challenging problem. In a total Lagrangian formulation, even when using a very fine initial mesh, the simulation can break down due to severe mesh distortion.  Error estimation and adaptive remeshing on the initial geometry are helpful and can provide more accurate solutions (with a smaller number of degrees of freedom) but are not sufficient to attain very large deformations. The updated Lagrangian formulation where the geometry is periodically updated is then preferred. Remeshing may still be necessary to control the quality of the elements and to avoid too severe mesh distortion.  It then requires frequent data transfer from the old mesh to the new one and this is a very delicate issue. If these transfers are not done appropriately, accuracy can be severely affected. In this paper, we present an updated Lagrangian formulation where the error on the finite element solution is estimated and adaptive remeshing is performed in order  to concentrate the elements of the  mesh where the error is large, to coarsen the mesh where the error is small and at the same time to control mesh distortion. In this way, we can reach high level of deformations while preserving the accuracy of the solution. Special attention is given to data transfer methods and a very accurate cubic Lagrange projection method is introduced. A very efficient continuation method is used to automatically pilot the complete algorithm including load increase, error estimation, adaptive remeshing and data transfer. A number of examples will be presented and analyzed.