Christian Geindreau

Homogenization of Coupled Phenomena in Heterogenous Media

Description: Both naturally–occurring and man–made materials are often heterogeneous materials formed of various constituents with different properties and behaviours. Studies are usually carried out on volumes of materials that contain a large number of heterogeneities. Describing these media by using appropriate mathematical models to describe each constituent turns out to be an intractable problem. Instead they are generally investigated by using an equivalent macroscopic description – relative to the microscopic heterogeneity scale – which describes the overall behaviour of the media. Fundamental questions then arise: Is such an equivalent macroscopic description possible? What is the domain of validity of this macroscopic description? The homogenization technique provides complete and rigorous answers to these questions. This book aims to summarize the homogenization technique and its contribution to engineering sciences. Researchers, graduate students and engineers will find here a unified and concise presentation. The book is divided into four parts whose main topics are Introduction to the homogenization technique for periodic or random media, with emphasis on the physics involved in the mathematical process and the applications to real materials. Heat and mass transfers in porous media Newtonian fluid flow in rigid porous media under different regimes Quasi–statics and dynamics of saturated deformable porous media Each part is illustrated by numerical or analytical applications as well as comparison with the self–consistent approach.

Computational comparison of the bending behavior of aortic stent-grafts

Secondary interventions after endovascular repair of abdominal aortic aneurysms are frequent because stent-graft (SG) related complications may occur (mainly endoleak and SG thrombosis). Complications have been related to insufficient SG flexibility, especially when devices are deployed in tortuous arteries. Little is known on the relationship between SG design and flexibility. Therefore, the aim of this study was to simulate numerically the bending of two manufactured SGs (Aorfix--Lombard Medical (A) and Zenith--Cook Medical Europe (Z)) using finite element analysis (FEA). Global SG behavior was studied by assessing stent spacing variation and cross-section deformation. Four criteria were defined to compare flexibility of SGs: maximal luminal reduction rate, torque required for bending, maximal membrane strains in graft and maximal Von Mises stress in stents. For angulation greater than 60°, values of these four criteria were lower with A-SG, compared to Z-SG. In conclusion, A-SG was more flexible than Z-SG according to FEA. A-SG may decrease the incidence of complications in the setting of tortuous aorto-iliac aneurysms. Our numerical model could be used to assess flexibility of further manufactured as well as newly designed SGs.

Transport properties of heterogeneous materials. Combining computerised X-ray micro-tomography and direct numerical simulations

Feasibility of a method for finding flow permeability of porous materials, based on combining computerised X-ray micro-tomography and numerical simulations, is assessed. The permeability is found by solving fluid flow through the complex 3D pore structures obtained by tomography for actual material samples. We estimate overall accuracy of the method and compare numerical and experimental results. Factors contributing to uncertainty of the method include numerical error arising from the finite resolution of tomographic images and the rather small sample size available with the present tomographic techniques. The total uncertainty of computed values of permeability is, however, not essentially larger than that of experimental results. We conclude that the method provides a feasible alternative for finding fluid flow properties of the kind of materials studied. It can be used to estimate all components of permeability tensor and is useful in cases where direct measurements are not achievable. Analogous methods can be applied to other modes of transport, such as diffusion and heat conduction.

Study of a temperature gradient metamorphism of snow from 3-D images: time evolution of microstructures, physical properties and their associated anisotropy

CONCLUSIONS The temperature gradient metamorphism is a frequent process which affects the snowpack structure leading often to weak layers (faceted crystals and depth hoar). One of its main features is the development of vertical structures of ice but its impact on physical properties has been investigated by few studies. In this poster, we study the time evolution of several properties of a snow slab subjected to a controlled temperature gradient. For this purpose we use 3D images of snow samples obtained by X-ray micro-tomography. Some properties are computed in the x-, yand z-directions of the samples so that we can determine their anisotropy coefficient. Finally, we present two analytical models based on ellipsoidal inclusions as ways to estimate the effective thermal conductivity and permeability of snow.

Microstructural effects on the flow law of power-law fluids through fibrous media

In this work, the flow of power-law fluids through anisotropic fibrous media is revisited, upscaling the fluid flow at the pore scale with the homogenization method of multiple scale expansions for periodic structures. This upscaling technique permits a quantitative study of the seepage law by performing numerical simulation with simple two-dimensional periodic arrays of circular solid inclusions. The significant role of the solid fraction, the fluid rheology and the porous media anisotropy on the resulting macroscopic flow law is underlined from the simulation.

Macroscopic Modeling for Heat and Water Vapor Transfer in Dry Snow by Homogenization

Dry snow metamorphism, involved in several topics related to cryospheric sciences, is mainly linked to heat and water vapor transfers through snow including sublimation and deposition at the ice-pore interface. In this paper, the macroscopic equivalent modeling of heat and water vapor transfers through a snow layer was derived from the physics at the pore scale using the homogenization of multiple scale expansions. The microscopic phenomena under consideration are heat conduction, vapor diffusion, sublimation, and deposition. The obtained macroscopic equivalent model is described by two coupled transient diffusion equations including a source term arising from phase change at the pore scale. By dimensional analysis, it was shown that the influence of such source terms on the overall transfers can generally not be neglected, except typically under small temperature gradients. The precision and the robustness of the proposed macroscopic modeling were illustrated through 2D numerical simulations. Finally, the effective vapor diffusion tensor arising in the macroscopic modeling was computed on 3D images of snow. The self-consistent formula offers a good estimate of the effective diffusion coefficient with respect to the snow density, within an average relative error of 10%. Our results confirm recent work that the effective vapor diffusion is not enhanced in snow.

Severe Bending of Two Aortic Stent-Grafts: An Experimental and Numerical Mechanical Analysis

Stent-grafts (SGs) are commonly used for treating abdominal aortic aneurysms (AAAs) and numerical models tend to be developed for predicting the biomechanical behavior of these devices. However, due to the complexity of SGs, it is important to validate the models. In this work, a validation of the numerical model developed in Demanget et al. (J. Mech. Behav. Biomed. Mater. 5:272–282, 2012) is presented. Two commercially available SGs were subjected to severe bending tests and their 3D geometries in undeformed and bent configurations were imaged from X-ray microtomography. Dedicated image processing subroutines were used in order to extract the stent centerlines from the 3D images. These skeletons in the undeformed configurations were used to set up SG numerical models that are subjected to the boundary conditions measured experimentally. Skeletons of imaged and deformed stents were then quantitatively compared to the numerical simulations. A good agreement is found between experiments and simulations. This validation offers promising perspectives to implementing the numerical models in a computer-aided tool and simulating the endovascular treatments.

Mechanical behaviour of a fibrous scaffold for ligament tissue engineering: Finite elements analysis vs. X-ray tomography imaging

The use of biodegradable scaffolds seeded with cells in order to regenerate functional tissue-engineered substitutes offers interesting alternative to common medical approaches for ligament repair. Particularly, finite element (FE) method enables the ability to predict and optimise both the macroscopic behaviour of these scaffolds and the local mechanic signals that control the cell activity. In this study, we investigate the ability of a dedicated FE code to predict the geometrical evolution of a new braided and biodegradable polymer scaffold for ligament tissue engineering by comparing scaffold geometries issued from FE simulations and from X-ray tomographic imaging during a tensile test. Moreover, we compare two types of FE simulations the initial geometries of which are issued either from X-ray imaging or from a computed idealised configuration. We report that the dedicated FE simulations from an idealised reference configuration can be reasonably used in the future to predict the global and local mechanical behaviour of the braided scaffold. A valuable and original dialog between the fields of experimental and numerical characterisation of such fibrous media is thus achieved. In the future, this approach should enable to improve accurate characterisation of local and global behaviour of tissue-engineering scaffolds.

Coriolis Effects on Filtration Law in Rotating Porous Media

We investigate the filtration law of incompressible viscous Newtonian fluids in rigid non-inertial porous media, for example, rotating porous media. The filtration law is obtained by upscaling the flow at the pore scale. We use the method of multiple scale expansions which gives rigorously the macroscopic behaviour without any prerequisite on the form of the macroscopic equations. For finite Ekman numbers the filtration law is shown to resemble a Darcys law, but with a non-symmetric permeability tensor which depends on the angular velocity of the porous matrix. We obtain the filtration analog of the Hall effect. For large Ekman numbers the filtration law is a small correction to the classical Darcys law. The corrector is antisymmetric. In this case we recover a structure of law which is similar to phenomenological laws introduced in the literature, but with a dissimilar effective coefficient.

NUMERICAL ANALYSIS OF THE WALL STRESS IN ABDOMINAL AORTIC ANEURYSM: INFLUENCE OF THE MATERIAL MODEL NEAR-INCOMPRESSIBILITY

Recently, hyperelastic mechanical models were proposed to well capture the aneurismal arterial wall anisotropic and nonlinear features experimentally observed. These models were formulated assuming the material incompressibility. However in numerical analysis, a nearly incompressible approach, i.e., a mixed formulation pressure-displacement, is usually adopted to perform finite element stress analysis of abdominal aortic aneurysm (AAA). Therefore, volume variations of the material are controlled through the volumetric energy which depends on the initial bulk modulus κ. In this paper, an analytical analysis of the influence of κ on the mechanical response of two invariant-based anisotropic models is first performed in the case of an equibiaxial tensile test. This analysis shows that for the strongly nonlinear anisotropic model, even in a restricted range of deformations, large values of κ are necessary to ensure the incompressibility condition, in order to estimate the wall stress with a reasonable precision. Finite element simulations on idealized AAA geometries are then performed. Results from these simulations show that the maximum stress in the AAA wall is underestimated in previous works, committed errors vary from 26% to 58% depending on the geometrical model complexity. In addition to affect the magnitude of the maximum stress in the aneurysm, we found that too small value of κ may also affect the location of this stress.