Method of asymptotic partial domain decomposition for non-steady problems: heat equation on a thin structure

The non-steady heat equation is considered in thin structures. The asymptotic expansion of the solution is constructed. The error estimates for high order asymptotic approximations are proved. The method of asymptotic partial domain decomposition is justified for the non-steady heat equation

Finite volume implementation of the method of asymptotic partial domain decomposition for the heat equation on a thin structure

International audienceThe non-steady heat equation is considered in thin structures. The asymptotic expansion of the solution constructed earlier is used for evaluation of the partial derivatives of the solution. The method of partial asymptotic domain decomposition is applied to the non-steady heat equation. It reduces the original 2D model to a hybrid dimension one, partially 2D, partially 1D with some special interface conditions between the 2D and 1D parts. The finite volume method is applied for the numerical solution of the hybrid dimension model. The error estimate is proved. The numerical experiment confirms the theoretical error evaluation

Steady state non-Newtonian flow with strain rate dependent viscosity in domains with cylindrical outlets to infinity

The paper deals with a stationary non-Newtonian flow of a viscous fluid in unbounded domains with cylindrical outlets to infinity. The viscosity is assumed to be smoothly dependent on the gradient of the velocity. Applying the generalized Banach fixed point theorem, we prove the existence, uniqueness and high order regularity of solutions stabilizing in the outlets to the prescribed quasi-Poiseuille flows. Varying the limit quasi-Poiseuille flows, we prove the stability of the solution

Multiscale modeling of light absorption in tissues: limitations of classical homogenization approach.

International audienceIn biophotonics, the light absorption in a tissue is usually modeled by the Helmholtz equation with two constant parameters, the scattering coefficient and the absorption coefficient. This classic approximation of "haemoglobin diluted everywhere" (constant absorption coefficient) corresponds to the classical homogenization approach. The paper discusses the limitations of this approach. The scattering coefficient is supposed to be constant (equal to one) while the absorption coefficient is equal to zero everywhere except for a periodic set of thin parallel strips simulating the blood vessels, where it is a large parameter ω. The problem contains two other parameters which are small: ε, the ratio of the distance between the axes of vessels to the characteristic macroscopic size, and δ, the ratio of the thickness of thin vessels and the period. We construct asymptotic expansion in two cases: ε --> 0, ω --> ∞, δ --> 0, ωδ --> ∞, ε2ωδ --> 0 and ε --> 0, ω --> ∞, δ --> 0, ε2ωδ --> ∞, and and prove that in the first case the classical homogenization (averaging) of the differential equation is true while in the second case it is wrong. This result may be applied in the biomedical optics, for instance, in the modeling of the skin and cosmetics

Homogenization for periodic media: from microscale to macroscale

International audienceThe mathematical tools for the up-scaling from micro to macro in periodic media are considered: the homogenization techniques, the multiscale modelling, etc. The main applications of these tools are the mechanics of composite materials, flows in porous media, and discrete models for nanostructure

Parallelization of the algorithm of asymptotic partial domain decompositionin thin tube structures

International audienceThe method of asymptotic partial domain decomposition for thin tube structures (finite unions of thin cylinders) is revisited. Its application to the Newtonian and non-Newtonian flows in great systems of vessels is considered. The possibility of a parallelization of its algorithm is discussed for linear and non-linear models

Homogenization and multicontinuum models for high contrast composites

International audienceAbstract We consider the high contrast periodic composite materials having high ratio ω >>1 of physical constants of the components such as the heat conductivity or Young’s moduli. The ratio of the characteristic microscopic size and the characteristic macroscopic size is a small parameter e. If the component having high constants has the shape of small isolated particles then the homogenized (macroscopic, effective) model demonstrates a loss of the effective wave velocity. If the topology of the component having high constants corresponds to several (at least two) connected periodic sets disconnected one from other and if ε2ω>>1 or ε2ω = const then the macroscopic model is described by a multicontinuum, i.e. several continua in each point

Boundary conditions for the high order homogenized equation: laminated rods, plates and composites

International audienceThe high order homogenization technique generates the so called infinite order homogenized equation. Its coefficients were widely discussed in composite mechanics literature because they are closely related to the so called high order strain gradients theories. However, it was not clear what is the correct mathematical setting for this equation and what are the asymptotically exact boundary conditions. In the present Note we give a variational formulation for the high order homogenized equation by the projection of the initial problem on the "ansatz subspace". This formulation generates the appropriate boundary conditions for the high order homogenized equation. The error estimates for the solution of the original problem and the homogenized one are obtained