Научная статья на тему 'Generalized Multiscale Finite Element Method for the poroelasticity problem in multicontinuum media'

Generalized Multiscale Finite Element Method for the poroelasticity problem in multicontinuum media Текст научной статьи по специальности «Математика»

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Текст научной работы на тему «Generalized Multiscale Finite Element Method for the poroelasticity problem in multicontinuum media»

the 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: cepe2a@inbox.ru

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: aa.tyrylgin@mail.ru

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: haska87@gmail.com

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.

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