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18the definition of the local snapshot space that takes all possible flows on the interface between coarse cells
into account. In order to reduce the size of the snapshot space, we solve a local spectral problem. We present
a convergence analysis of the presented multiscale method. Numerical results are presented for two-
dimensional problems in three testing geometries along with the errors associated to different number of the
multiscale basis functions used for velocity field. Numerical investigations are conducted for problems with
homogeneous and heterogeneous properties respectively.
Multiscale mathematical modeling of the seepage into the soil under cryolithozone conditions
S. Stepanov, D. Nikiforov and Al. Grigorev
M. K. Ammosov North-Eastern Federal University, Yakutsk
Email: [email protected]
DOI 10.24412/cl-35065-2021-1-02-84
In this work, the numerical modelling of fluid seepage in the presence of permafrost in heterogeneous
soils is considered. The multiphysics model consists of the coupled Richards� equation and the Stefan problem.
These problems often contain heterogeneities due to variations of soil properties. In the paper, we design a
multiscale simulation method based on Generalized Multiscale Finite Element Method (GMsFEM). For this rea-
son, we design coarse-grid spaces for this multiphysics problem and design algorithms for solving the overall
problem. Numerical simulations are carried out on two-dimensional and three-dimensional model problems.
For the case of a three-dimensional, somewhat realistic geometry with a complex surface structure is consid-
ered. We demonstrate the efficiency and accuracy of the proposed method using several representative nu-
merical results.
This work was (partially) supported by the Foundation mega-grant of the Russian 230 Federation Government
N14.Y26.31.0013.
Generalized Multiscale Finite Element Method for the poroelasticity problem in multicontinuum media
A. A. Tyrylgin1, M. V. Vasilyeva2, D. A. Spiridonov1, E. T. Chung3
1 M. K. Ammosov North-Eastern Federal University, Yakutsk
2Institute for Scientific Computation, Texas A&M University, College Station, TX 77843-3368 & Department of
Computational Technologies
3Department of Mathematics, The Chinese University of Hong Kong (CUHK), Hong Kong SAR
Email: [email protected]
DOI 10.24412/cl-35065-2021-1-02-85
In this paper, we consider a poroelasticity problem in heterogeneous multicontinuum media that is widely
used in simulations of the unconventional hydrocarbon reservoirs and geothermal fields. Mathematical model
contains a coupled system of equations for pressures in each continuum and effective equation for displace-
ment with volume force sources that are proportional to the sum of the pressure gradients for each continu-
um. To illustrate the idea of our approach, we consider a dual continuum background model with discrete
fracture networks that can be generalized to a multicontinuum model for poroelasticity problem in complex
heterogeneous media. We present a fine grid approximation based on the finite element method and Discrete
Fracture Model (DFM) approach for two and three-dimensional formulations. The coarse grid approximation is
constructed using the Generalized Multiscale Finite Element Method (GMsFEM), where we solve local spectral
problems for construction of the multiscale basis functions for displacement and pressures in multicontinuum
media. We present numerical results for the two and three dimensional model problems in heterogeneous
fractured porous media. We investigate relative errors between reference fine grid solution and presented
coarse grid approximation using GMsFEM with different numbers of multiscale basis functions. Our results in-
dicate that the proposed method is able to give accurate solutions with few degrees of freedoms.
A generalized multiscale finite element method for neutron transport problems in SP3 approximation
A. O. Vasilev1, D. A. Spiridonov1, A. V. Avvakumov2
1M. K. Ammosov North-Eastern Federal University, Yakutsk
2National Research Center �Kurchatov Institute�, Moscow
Email: [email protected]
DOI 10.24412/cl-35065-2021-1-02-86
The SP3 approximation of the neutron transport equation allows improving the accuracy for both static
and transient simulations for reactor core analysis compared with the neutron diffusion theory. Besides, the
SP3 calculation costs are much less than higher order transport methods (SN or PN). Another advantage of the
SP3 approximation is a similar structure of equations that is used in the diffusion method. Therefore, there is
no difficulty to implement the SP3 solution option to the multi-group neutron diffusion codes.
In this paper, we attempt to employ a model reduction technique based on the multiscale method for
neutron transport equation in SP3 approximation. The proposed method is based on the use of a generalized
multiscale finite element method (GmsFEM). The main idea is to create multiscale basis functions that can be
used to effectively solve on a coarse grid. From calculation results, we obtain that multiscale basis functions
can properly take into account the small-scale characteristics of the medium and provide accurate solutions.
The application of the SP3 methodology based on solution of the lambda-spectral problems has been tested
for the some reactor benchmarks. The results calculated with the GMsFEM are compared with the reference
transport calculation results.
