Macroscopic corrosion front computations of sulfate attack in sewer pipes based on a micro-macro reaction-diffusion model

We consider a two-scale reaction diffusion system able to capture the corrosion of concrete with sulfates. Our aim here is to define and compute two macroscopic corrosion indicators: typical pH drop and gypsum profiles. Mathematically, the system is coupled, endowed with micro-macro transmission conditions, and posed on two different spatially-separated scales: one microscopic (pore scale) and one macroscopic (sewer pipe scale). We use a logarithmic expression to compute values of pH from the volume averaged concentration of sulfuric acid which is obtained by resolving numerically the two-scale system (microscopic equations with direct feedback with the macroscopic diffusion of one of the reactants). Furthermore, we also evaluate the content of the main sulfatation reaction (corrosion) product –the gypsum– and point out numerically a persistent kink in gypsum's concentration profile. Finally, we illustrate numerically the position of the free boundary separating corroded from not-yet-corroded regions

Homogenization Method and Multiscale Modeling

This mini-course addresses graduate students and young researchers in mathematics and engineering sciences interested in applying both formal and rigorous averaging methods to real-life problems described by means of partial differential equations (PDEs) posed in heterogeneous media. As a background application scenario we choose to look at the interplay between reaction, diffusion and flow in periodic porous materials, but broadly speaking, a similar procedure would apply for, e.g., acoustic and/or electromagnetic wave propagation phenomena in composite (periodic) media as well. We start off with the study of oscillatory elliptic PDEs formulated firstly in fixed and, afterwards, in periodically-perforated domains. We remove the oscillations by means of a (formal) asymptotic homogenization method. The output of this procedure consists of a “guessed” averaged model equations and explicit rules (based on cell problems) for computing the effective coefficients. As second step, we introduce the concept of two-scale convergence (and correspondingly, the two-scale compactness) in the sense of Allaire and Nguetseng and derive rigorously the averaged PDE models and coefficients obtained previously. This step uses the framework of Sobolev and Bochner spaces and relies on basic tools like weak convergence methods, compact embeddings as well as extension theorems in Sobolev spaces. We particularly emphasize the role the choice of microstructures (pores, perforations, subgrids, etc.) plays in performing the overall averaging procedure. Finally, we focus our attention on a two-scale partly dissipative reaction-diffusion system with periodically distributed microstructure modeling chemical attack on concrete structures. We present a two-scale finite difference scheme able to approximate the unique weak solution to the two-scale system and prove its convergence. We illustrate numerically the typical micro-macro behavior of the active concentrations involved in the corrosion process and give details on how a two-scale FD scheme can be implemented in C. The main objective of the course is to endow the audience with a rather flexible mathematical homogenization tool so that he/she can quickly start applying this averaging methodology to other PDEs scenarios describing physico-chemical processes in media with microstructures

Multiscale sulfate attack on sewer pipes : numerical study of a fast micro-macro mass transfer limit

We present two multiscale reaction-diffusion (RD) systems modeling sulfate attack in concrete structures (here: sewer pipes). The systems are posed on two different spatially separated scales. The only difference between them is the choice of the micro-macro transmission condition. We explore numerically the way in which the macroscopic Biot number $Bi^M$ connects the two reaction-diffusion scenarios. We indicate connections between the solution of the "regularized" system (with moderate size of $Bi^M$) and the solution to the "matched" system (with blowing up size of $Bi^M$), where Henry’s law plays the role of the micro-macro transmission condition

Multiscale sulfate attack on sewer pipes: Numerical study of a fast micro-macro mass transfer limit

MI: Global COE Program Education-and-Research Hub for Mathematics-for-IndustryグローバルCOEプログラム「マス･フォア･インダストリ教育研究拠点」We present two multiscale reaction-diffusion (RD) systems modeling sulfate attack in concrete structures (here: sewer pipes). The systems are posed on two different spatially separated scales. The only difference between them is the choice of the micro-macro transmission condition. We explore numerically the way in which the macroscopic Biot number Bi^M connects the two reaction-diffusion scenarios. We indicate connections between the solution of the \u22regularized\u22 system (with moderate size of Bi^M) and the solution to the \u22matched\u22 system (with blowing up size of Bi^M), where Henry\u27s law plays the role of the micro-macro transmission condition

Multiscale sulfate attack on sewer pipes: Numerical study of a fast micro-macro mass transfer limit

We present two multiscale reaction-diffusion (RD) systems modeling sulfate attack in concrete structures (here: sewer pipes). The systems are posed on two different spatially separated scales. The only difference between them is the choice of the micro-macro transmission condition. We explore numerically the way in which the macroscopic Biot number Bi^M connects the two reaction-diffusion scenarios. We indicate connections between the solution of the "regularized" system (with moderate size of Bi^M) and the solution to the "matched" system (with blowing up size of Bi^M), where Henry's law plays the role of the micro-macro transmission condition.MI: Global COE Program Education-and-Research Hub for Mathematics-for-IndustryグローバルCOEプログラム「マス･フォア･インダストリ教育研究拠点