Locally periodic unfolding method and two-scale convergence on surfaces of locally periodic microstructures

In this paper we generalize the periodic unfolding method and the notion of two-scale convergence on surfaces of periodic microstructures to locally periodic situations. The methods that we introduce allow us to consider a wide range of non-periodic microstructures, especially to derive macroscopic equations for problems posed in domains with perforations distributed non-periodically. Using the methods of locally periodic two-scale convergence (l-t-s) on oscillating surfaces and the locally periodic (l-p) boundary unfolding operator, we are able to analyze differential equations defined on boundaries of non-periodic microstructures and consider non-homogeneous Neumann conditions on the boundaries of perforations, distributed non-periodically

Multiscale reaction-diffusion systems describing concrete corrosion : modeling and analysis

This thesis deals with the modeling and multiscale analysis of reaction-diffusion systems describing concrete corrosion processes due to the aggressive chemical reactions occurring in concrete. We develop a mathematical framework that can be useful in forecasting the service life of sewer pipes. We aim at identifying reliable and easy-to-use multiscale models able to forecast the penetration of sulfuric acid into sewer pipes walls. For modeling of corrosion processes, we take into account balance equations expressing physico-chemical processes that take place in the microstructures (pores) of the partially saturated concrete. We consider two dierent modeling strategies: (1) we propose microscopic reaction-diusion systems to delineate the corrosion processes at the pore level and (2) we consider a distributed microstructure model containing information from two separated spatial scales (micro and macro). All systems of dierential equations are semi-linear, weakly coupled, and partially diusive. Since the precise microstructure of the material is far too complex to be described accurately, we consider two approximations, namely uniformly-periodic and locally-periodic array of microstructures, which are tractable by using averaging mathematical tools. We use different homogenization techniques to obtain the effective behavior of the microscopically oscillating quantities. For the formal derivation of our multiscale models, we apply the asymptotic expansion method to the microscopic reaction-diffusion systems defined in locally-periodic domains for two special choices of scaling in ¿ of the diffusion coefficients. We end up with (i) upscaled systems and (ii) distributed-microstructure systems. As far as rigorous derivations are concerned, we apply the notion of two-scale convergence to the PDE system defined in the uniformly periodic domain. To deal with the non-diffusive object, i.e. the ordinary dierential equation tracking the damage-by-reaction, we combine the two-scale convergence idea with the periodic-boundary-unfolding technique. Additionally, we use the periodic unfolding techniques to obtain corrector estimates assessing the quality of the averaging method. These estimates are convergence rates measuring the error contribution produced while approximating macroscopic solutions by microscopic ones. We derive these estimates under minimal regularity assumptions on the solutions to the microscopic and macroscopic systems, microstructure boundaries, and to the corresponding auxiliary cell problems. We prove the well-posedness of a distributed-microstructure reaction-diusion system which includes transport (diusion) and reaction effects emerging from two separated spatial scales. We perform this analysis by incorporating a variational inequality requiring minimal regularity assumptions on the initial data. We ensure basic estimates like positivity and L8-bounds on the solution to the system. Then we prove the global-in-time existence and uniqueness of a suitable class of positive and bounded solutions. To predict the position of the corrosion front penetrating the concrete, we use our distributed-microstructure model to perform simulations at macroscopic length scales while taking into account transport and reactions occurring at small length scales. Using an ad hoc logarithmic expression, we approximate numerically macroscopic pH proles dropping down with the onset of corrosion. We extract from the gypsum proles the approximate position of the corrosion front penetrating the uncorroded concrete. We illustrate numerically that as the macroscopic mass-transfer Biot number BiM -> 8, BiM naturally connects two different multiscale reaction-diusion scenarios: the solution of the distributed-microstructure system having the Henry's law acting as micro-macro transmission condition converges to the solution of the matched distributed-microstructure system