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10Multiscale and high-performance computing for multiphysical problems 31
in a continuous medium) in the case of a gas-dust medium. The interaction of dust and gas was taken into account
using the IDIC method [1, 2] within the SPH method used to solve gas-dynamic equations.
An important feature of the work is the use of the open computational package OpenFPM, which makes it
easy to carry out parallel computations. The main advantage of this package is the ready-made (implemented
by the authors of the package) and intuitive, automatically parallelizable vector data structures, the use of
which is identical both in the case of calculations on a personal computer and in the case of using supercomputer
resources. The paper analyzes the efficiency of parallelization of numerical solutions of the considered
problems.
The research was supported by the Russian Science Foundation grant number 19-71-10026.
References
1. Stoyanovskaya et.al., Two-fluid dusty gas in smoothed particle hydrodynamics: Fast and implicit algorithm for stiff
linear drag // Astronomy and Computing, 2018, V. 25, P. 25-37.
2. Stoyanovskaya et.al., Fast method to simulate dynamics of two-phase medium with intense interaction between
phases by smoothed particle hydrodynamics: Gas-dust mixture with polydisperse particles, linear drag, one-dimensional
tests // J. of Computational Physics, 2021, V. 430.
Numerical solution of a 2d inverse gravimetric problem
D. Kh. Ivanov
Yakutsk branch of Regional Scientific and Educational Center �Far Eastern Center of Mathematical Research.
Email: ivanov.djulus@yandex.ru
DOI 10.24412/cl-35065-2021-1-02-80
A 2d inverse gravimetric problem is considered. The aim is to recover the shape of an homogeneous
ore body from the observation of the vertical gravity at the ground surface. To overcome the ill-posedness of
such problem we restrict the unknown body to a star shaped domain. The numerical solution of the direct
problem is based on solving an auxiliary boundary value problem in a bounded computational domain coupled
with the surface integral. According to that, for solution of the inverse problem we present an iterative algorithm
based on conjugate gradient method and specific regularization term. To demonstrate efficiencies of the
proposed method we investigate a model problem of a simplified body with the analytical solution and noised
input data.
Edge generalized multiscale finite element method for scattering problem in perforated domain
U. S. Kalachikova1, E. T. Chung2, M. V. Vasilyeva3, Y. R. Efendiev4
1
M. K. Ammosov North-Eastern Federal University, Yakutsk
2
The Chinese University of Hong Kong (CUHK), Hong Kong SAR
3
COIFPM, University of Wyoming, Laramie, WY 82071, USA
4
Texas A&M University, College Station, Texas, USA
Email: lanasemna@mail.ru
DOI 10.24412/cl-35065-2021-1-02-81
In this work we consider scattering problem in perforated domain. The mathematical model is described
by Helmholtz problem related to wave propagation with absorbing boundary condition. For the solution of the
problem using classic finite element method, we construct unstructured fine grid that resolve perforation on
the grid level. Such classic approximations lead to the large system of equations. To reduce size of the discrete
system, we construct a novel multiscale approximation on coarse grid. We use the Edge Generalized Mul
32 Mini-symposium
tiscale Finite Element Method, where we construct a multiscale space using solution of the local spectral problems
on the snapshot space related to the coarse grid edges. We present numerical results for the Helmholtz
problem in perforated domain with Dirichlet boundary condition on perforations. Proposed method are studied
for a different wave numbers and numbers of the edge multiscale basis functions.
Embedded discrete fracture model on structured grids
D. Y. Nikiforov
M. K. Ammosov North-Eastern Federal University, Yakutsk
Email: dju92@mail.com
DOI 10.24412/cl-35065-2021-1-02-82
An approximation of the embedded discrete fracture model EDFM by the finite element method is considered.
The paper proposes to use exponential functions instead of the Dirac delta function [1]. With this approach,
instead of a separate computational mesh for fractures, a mesh for a porous medium can be used. The
results of numerical experiments demonstrate the efficiency of the proposed approach.
This work was supported by the Ministry of science and higher education of the Russian Federation, supplementary
agreement N075-02-2020-1542/1, April 29, 2020.
References
1. Nikiforov D. Y., Stepanov S. P. Numerical simulation of the embedded discrete fractures by the finite element
method //J. of Physics: Conference Series. . IOP Publishing, 2019. Vol. 1158. No. 3. P. 032038.
Multiscale finite element technique for mathematical modelling of multi-physics processes
in the near-wellbore region
E. P. Shurina1,2, N. B. Itkina1,3, D. A. Arhipov1,2, D. V. Dobrolubova1,2, A. Yu. Kutishcheva1,2, S. I. Markov1,2,
N. V. Shtabel1,2, E. I. Shtanko1,2
1
The Trofimuk Institute of Petroleum Geology and Geophysics SB RAS
2
Novosibirsk State Technical University
3
Institute of Computational Technologies SBRAS
Email: shurina@online.sinor.ru
DOI 10.24412/cl-35065-2021-1-00-56
In borehole physic, the results of the direct mathematical modelling of multi-physical phenomena are
used to control drilling and well operation. Electromagnetic and acoustic measurements are the most accessible
indirect methods for determining the thermal, transport and mechanical properties of rock samples in the
near-wellbore zone. Mathematical modelling is one of the technologies used for solving multi-physical problems.
A multi-physical problem is formulated as a system of partial differential equations with special interface
conditions coupling mathematical models of physical processes. The near-wellbore region is characterized by a
multi-scale geometric structure, high contrast and anisotropy of physical parameters. The discretization method
should take into account the specifics of the problem and preserve the global regularity of mathematical
models at a discrete level. The paper presents modified variational formulations of multiscale non-conforming
finite element methods for mathematical modelling of electromagnetic and acoustic fields in fluid-saturated
media at various temperatures and mechanical loads. To solve the discretized mathematical models, special
multilevel solvers are developed. The results of three-dimensional mathematical modelling using model rock
samples from the near-wellbore zone are presented.
