Page 514 - 8th European Congress of Mathematics ∙ 20-26 June 2021 ∙ Portorož, Slovenia ∙ Book of Abstracts
P. 514
MULTISCALE MODELING AND METHODS: APPLICATION IN ENGINEERING,
BIOLOGY AND MEDICINE (MS-80)
ADI scheme for partially dimension reduced heat conduction models
Vytenis Šumskas, vytenis.sumskas@mif.vu.lt
Vilnius University, Lithuania
Coauthors: Raimondas Cˇ iegis, Grigory Panasenko, Konstantinas Pileckas
In this talk, an alternating direction implicit (ADI) type finite volume numerical scheme is pro-
posed to solve a non-classical non-stationary heat conduction problem set in a 3D tube with
radial symmetry. The original 3D model is reduced to a hybrid dimension model in a large
part of the domain. Special junction conditions are defined between 3D and 1D parts. The finite
volume method is applied to approximate spatial differential operators and ADI splitting is used
for time integration. The ADI scheme is unconditionally stable and under a mix of Dirichlet and
Neumann boundary conditions the approximation error is of second order in space and time. An
efficient factorization algorithm is presented to solve the obtained systems of equations. Results
of computational experiments confirm the theoretical error analysis. Visual representations and
computational times are compared for various sizes of reduced dimension zones, thus contribut-
ing to a conclusion that hybrid mathematical models can be used to simulate heat models for a
quite broad set of domains and coefficients.
This research is partially funded from European Social Fund (project No 09.3.3-LMT-K-
712-01-0012) under grant agreement with the Research Council of Lithuania (LMTLT).
FSI and reduced models for 3D hemodynamic simulations in
time-dependent domains
Yuri Vassilevski, yuri.vassilevski@gmail.com
Marchuk Institute of Numerical Mathematics RAS and Sechenov University,
Russian Federation
In the talk we briefly present our approach to fluid-structure interaction (FSI) simulations [6,
4] and consider three biomedical problems for flow in time-dependent domains which can be
solved by simpler formulations than the FSI formulation: blood flow in the human ventricles
[5, 3, 1], blood flow in the aortic bifurcation [2], coaptation characteristics of the aortic valve
[7]. The numerical schemes are summarized in the book [8].
This is the joint work with M.Olshanskii, A.Lozovskiy, A.Danilov, T.Dobroserdova,
G.Panassenko, V.Salamatova, A.Lyogkii.
References
[1] Lozovskiy A., Olshanskii M., Salamatova V., Vassilevski Yu. An unconditionally sta-
ble semi-implicit FSI finite element method. Comput.Methods Appl.Mech.Engrg., 297
(2015), 437-454.
[2] Lozovskiy A., Olshanskii M., Vassilevski Yu. Analysis and assessment of a monolithic
FSI finite element method. Comp. & Fluids, 179 (2019), 277-288.
[3] Danilov A., Lozovskiy A., Olshanskii M., Vassilevski Yu. A finite element method for the
Navier-Stokes equations in moving domain with application to hemodynamics of the left
ventricle. Russian J. Numer. Anal. Math. Modelling, 32:4 (2017), 225-236.
512
BIOLOGY AND MEDICINE (MS-80)
ADI scheme for partially dimension reduced heat conduction models
Vytenis Šumskas, vytenis.sumskas@mif.vu.lt
Vilnius University, Lithuania
Coauthors: Raimondas Cˇ iegis, Grigory Panasenko, Konstantinas Pileckas
In this talk, an alternating direction implicit (ADI) type finite volume numerical scheme is pro-
posed to solve a non-classical non-stationary heat conduction problem set in a 3D tube with
radial symmetry. The original 3D model is reduced to a hybrid dimension model in a large
part of the domain. Special junction conditions are defined between 3D and 1D parts. The finite
volume method is applied to approximate spatial differential operators and ADI splitting is used
for time integration. The ADI scheme is unconditionally stable and under a mix of Dirichlet and
Neumann boundary conditions the approximation error is of second order in space and time. An
efficient factorization algorithm is presented to solve the obtained systems of equations. Results
of computational experiments confirm the theoretical error analysis. Visual representations and
computational times are compared for various sizes of reduced dimension zones, thus contribut-
ing to a conclusion that hybrid mathematical models can be used to simulate heat models for a
quite broad set of domains and coefficients.
This research is partially funded from European Social Fund (project No 09.3.3-LMT-K-
712-01-0012) under grant agreement with the Research Council of Lithuania (LMTLT).
FSI and reduced models for 3D hemodynamic simulations in
time-dependent domains
Yuri Vassilevski, yuri.vassilevski@gmail.com
Marchuk Institute of Numerical Mathematics RAS and Sechenov University,
Russian Federation
In the talk we briefly present our approach to fluid-structure interaction (FSI) simulations [6,
4] and consider three biomedical problems for flow in time-dependent domains which can be
solved by simpler formulations than the FSI formulation: blood flow in the human ventricles
[5, 3, 1], blood flow in the aortic bifurcation [2], coaptation characteristics of the aortic valve
[7]. The numerical schemes are summarized in the book [8].
This is the joint work with M.Olshanskii, A.Lozovskiy, A.Danilov, T.Dobroserdova,
G.Panassenko, V.Salamatova, A.Lyogkii.
References
[1] Lozovskiy A., Olshanskii M., Salamatova V., Vassilevski Yu. An unconditionally sta-
ble semi-implicit FSI finite element method. Comput.Methods Appl.Mech.Engrg., 297
(2015), 437-454.
[2] Lozovskiy A., Olshanskii M., Vassilevski Yu. Analysis and assessment of a monolithic
FSI finite element method. Comp. & Fluids, 179 (2019), 277-288.
[3] Danilov A., Lozovskiy A., Olshanskii M., Vassilevski Yu. A finite element method for the
Navier-Stokes equations in moving domain with application to hemodynamics of the left
ventricle. Russian J. Numer. Anal. Math. Modelling, 32:4 (2017), 225-236.
512
