Keywords: coupled heat and moisture transport, Künzel model, parallel computing.

The coupled heat and moisture transport is required in connection with analyses of historical buildings and structures which are usually made of masonry. Detailed models of bricks and plaster can be used only for analyses of small segments of masonry structures. If the whole masonry structure has to be analysed, a suitable homogenization technique has to be used because of the strong material heterogeneity. The classical first order homogenization in a spatial domain in the framework of the two-step multiscale computational scheme was proposed in [1]. A significant dependence of the homogenized properties on actual moisture gradients and corresponding values of both macroscopic temperature and relative humidity was observed.

The Künzel material model [2] of coupled heat and moisture transport was selected for both levels of the analysis. The model is based on the mass and heat balance equations where the temperature and relative humidity are unknown functions. The finite elements in the analysis on the macro-level, which is also called the macro-problem, use the matrices of material parameters which are obtained from the homogenization. It means that the appropriate material matrices are assembled as a result of solution of the corresponding meso-level problems. The enormous computational demand stems from the fact that the material matrix has to be determined for each integration point in the macro-problem. If a standard mesh with thousands or tens of thousands of finite elements is used on the macro-problem, the number of integration points is in the order of tens or hundreds of thousands. Therefore, the meso-level problem has to be solved many times. Although the meso-level problem usually contains only hundreds of elements, the computational requirements are huge.

Fortunately, the solution of many meso-level problems is an excellent example of computation suitable for processor farming [3]. The master processor executes the macro-problem while all other processors, called the slave processors, deal with meso-level problems. The master sends to each slave only the actual temperature, relative humidity and their gradients (8 values in total). On the other hand, the slaves send to the master the material matrix and the coefficient of capacity (40 values in total).

The access to a parallel computer with thousands of processors is not common. Therefore, several meso-level problems are assigned to each slave which leads to the significantly reduced number of processors needed. From the physical point of view, it is also acceptable to use homogenized material parameters in more than one integration point of the macro-problem. Clearly, the material matrices do not change dramatically in the nearby integration points in the macro-problem.

First order homogenization in a spatial domain in the framework of the two-step multiscale computational scheme is applied to real world problems in two and three spatial dimensions. Analyses are performed on a cluster of PC.

1

J. Sýkora, M. Šejnoha, J. Šejnoha, "Homogenization of coupled heat and moisture transport in masonry structures including interfaces", Applied Mathematics and Computation, 2011. doi:10.1016/j.amc.2011.02.050

2

H.M. Künzel, "Simultaneous Heat and Moisture Transport in Building Components", Fraunhofer Institute of Building Physics, Fraunhofer IRB Verlag Stutgart, 1995.

3

B.H.V. Topping, A.I. Khan, "Parallel Finite Element Computations", Saxe-Coburg Publications, Edinburgh, UK, 1996.

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