**313**

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**Finite element approximation of fluid structure interaction using Taylor-Hood and Scott-Vogelius elements**

Vacek, Karel; Sváček, P.

2024 - English
This paper addresses the problem of fluid flow interacting a vibrating solid cylinder described by one degree of freedom system and with fixed airfoil. The problem is described by the incompressible Navier-Stokes equations written in the arbitrary Eulerian-Lagrangian (ALE) formulation. The ALE mapping is constructed with the use of a pseudo-elastic approach. The flow problem is numerically approximated by the finite element method (FEM). For discretization of the fluid flow, the results obtained by both the Taylor-Hood (TH) element and the Scott-Vogelius (SV) finite element are compared. The TH element satisfies the Babuška-Brezzi inf-sup condition, which guarantees the stability of the scheme. In the case of the SV element the mesh, that is created as a barycentric refinement of regular triangulation, is used to satisfy the Babuška-Brezzi condition. The numerical results for two benchmark problems are shown.
Keywords:
*finite element method; arbitrary Lagrangian-Eulerian method; Scott-Vogelius element; Taylor-Hood element*
Available in digital repository of the ASCR
Finite element approximation of fluid structure interaction using Taylor-Hood and Scott-Vogelius elements

This paper addresses the problem of fluid flow interacting a vibrating solid cylinder described by one degree of freedom system and with fixed airfoil. The problem is described by the incompressible ...

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**Numerical study of the steady airflow in the human respiratory system during inhaling and exhaling**

Lancmanová, Anna; Bodnár, Tomáš

2024 - English
This paper presents some of the initial results of the numerical simulations of a steady turbulent flow in human upper airways during inhalation and exhalation. The mathematical model is based on the system of Reynolds-Averaged incompressible Navier-Stokes equations complemented by the SST k − ω turbulence model. The simulations were performed using finite-volume open source solver OpenFOAM on a realistic three-dimensional geometry. The main aim of this particular study is to verify the computational setup with special focus on appropriate choice and implementation of boundary conditions. The prescribed boundary conditions are chosen to mimic the physiological conditions during normal breathing cycle. This study aims to gain an insight into the airflow behavior during the inhalation and exhalation process by comparing the results of two distinct simulations corresponding to two different (opposite) flow rates . The obtained local flow rates and flow fields for both cases are presented and mutually compared. This initial work should serve as a foundation for future more complex simulations that will include the time-dependent and compressible effects.
Keywords:
*human airways; incompressible Navier-Stokes; OpenFOAM*
Available in digital repository of the ASCR
Numerical study of the steady airflow in the human respiratory system during inhaling and exhaling

This paper presents some of the initial results of the numerical simulations of a steady turbulent flow in human upper airways during inhalation and exhalation. The mathematical model is based on the ...

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**Numerical evaluation of mass-diffusive compressible fluids flows models**

Bodnár, Tomáš; Fraunié, P.

2024 - English
This contribution presents first numerical tests of some recently published alternative models for solution of viscous compressible and nearly incompressible models. All models are solved by high resolution compact finite difference scheme with strong stability preserving RungeKutta time stepping. The two simple but challenging computational test cases are presented, based on the double-periodic shear layer and the Kelvin-Helmholtz instability. The obtained time-dependent flow fields are showing pronounced shear and vorticity layers being resolved by the standard as well as by the new mass-diffusive modified models. The preliminary results show that the new models are viable alternative to the well established classical models.
Keywords:
*compressible Navier-Stokes; nearly incompressible flow; mass diffusion; compact finite-difference*
Available in digital repository of the ASCR
Numerical evaluation of mass-diffusive compressible fluids flows models

This contribution presents first numerical tests of some recently published alternative models for solution of viscous compressible and nearly incompressible models. All models are solved by high ...

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**Motion of fluids in the moving domain**

Nečasová, Šárka

2024 - English
It is a survay paper where the problem of the existence of weak solutions of compressible barotropic solutions in a moving bounded domain is studied.
Keywords:
*compressible fluid; moving domain; weak solutions*
Available in digital repository of the ASCR
Motion of fluids in the moving domain

It is a survay paper where the problem of the existence of weak solutions of compressible barotropic solutions in a moving bounded domain is studied.

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**On the development of a numerical model for the simulation of air flow in the human airways**

Lancmanová, Anna; Bodnár, Tomáš; Sequeira, A.

2023 - English
This contribution reports on an ongoing study focusing on reduced order models for incompressible viscous fluid flow in two dimensional channels. A finite difference solver was developed using a simple implementation of the immersed boundary method to represent the channel geometry. The solver was validated for unsteady flow by comparing the obtained two-dimensional numerical solutions with analytical profiles computed from the Womersley solution. Finally the 2D model was coupled to a simple 1D extension simulating the flow in axisymmetric elastic vessel (tube). Some of the coupling principles and implementation issues are discussed in detail.
Keywords:
*reduced order model; incompressible Navier-Stokes equations; finite difference approximation; coupling method*
Available in digital repository of the ASCR
On the development of a numerical model for the simulation of air flow in the human airways

This contribution reports on an ongoing study focusing on reduced order models for incompressible viscous fluid flow in two dimensional channels. A finite difference solver was developed using a ...

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**Programs and Algorithms of Numerical Mathematics 21 : Jablonec nad Nisou, June 19-24, 2022 : Proceedings of Seminar**

Chleboun, J.; Kůs, Pavel; Papež, Jan; Rozložník, Miroslav; Segeth, Karel; Šístek, Jakub

2023 - English
These proceedings contain peer-reviewed papers that are based on the invited lectures, short communications, and posters presented at the 21st seminar Programs and Algorithms of Numerical Mathematics (PANM) held in Merkur Hotel, Jablonec nad Nisou, Czech Republic, June 19-24, 2022.\nThe seminar was organized by the Institute of Mathematics of the Czech Academy of Sciences under the auspices of EU-MATHS-IN.CZ, Czech Network for Mathematics in Industry, and with the nancial support provided by the RSJ Foundation. It continued the previous seminars on mathematical software and numerical methods held (biennially, with only one exception) in Alšovice, Bratříkov, Janov nad Nisou, Kořenov, Lázně Libverda, Dolní Maxov, Prague, and Hejnice in the period 1983-2020. The objective of this series of seminars is to provide a forum for presenting and discussing advanced topics in numerical analysis, computer implementation of numerical algorithms, new approaches to mathematical modeling, and single- or multi-processor applications of computational methods.
Keywords:
*numerical mathematics*
Available in digital repository of the ASCR
Programs and Algorithms of Numerical Mathematics 21 : Jablonec nad Nisou, June 19-24, 2022 : Proceedings of Seminar

These proceedings contain peer-reviewed papers that are based on the invited lectures, short communications, and posters presented at the 21st seminar Programs and Algorithms of Numerical Mathematics ...

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**Spherical basis function approximation with particular trend functions**

Segeth, Karel

2023 - English
The paper is concerned with the measurement of scalar physical quantities at nodes on the $(d-1)$-dimensional unit sphere surface in the hbox{$d$-dimensional} Euclidean space and the spherical RBF interpolation of the data obtained. In particular, we consider $d=3$. We employ an inverse multiquadric as the radial basis function and the corresponding trend is a polynomial of degree 2 defined in Cartesian coordinates. We prove the existence of the interpolation formula of the type considered. The formula can be useful in the interpretation of many physical measurements. We show an example concerned with the measurement of anisotropy of magnetic susceptibility having extensive applications in geosciences and present numerical difficulties connected with the high condition number of the matrix of the system defining the interpolation.
Keywords:
*spherical interpolation; spherical radial basis function; inverse multiquadric*
Available in digital repository of the ASCR
Spherical basis function approximation with particular trend functions

The paper is concerned with the measurement of scalar physical quantities at nodes on the $(d-1)$-dimensional unit sphere surface in the hbox{$d$-dimensional} Euclidean space and the spherical RBF ...

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**Interpolation with restrictions -- role of the boundary conditions and individual restrictions**

Valášek, Jan; Sváček, P.

2023 - English
The contribution deals with the remeshing procedure between two computational finite element meshes. The remeshing represented by the interpolation of an approximate solution onto a new mesh is needed in many applications like e.g. in aeroacoustics, here we are particularly interested in the numerical flow simulation of a gradual channel collapse connected with a~severe deterioration of the computational mesh quality. Since the classical Lagrangian projection from one mesh to another is a dissipative method not respecting conservation laws, a conservative interpolation method introducing constraints is described. The constraints have form of Lagrange multipliers enforcing conservation of desired flow quantities, like e.g. total fluid mass, flow kinetic energy or flow potential energy. Then the interpolation problem turns into an error minimization problem, such that the resulting quantities of proposed interpolation satisfy these physical properties while staying as close as possible to the results of Lagrangian interpolation in the L2 norm. The proposed interpolation scheme does not impose any restrictions on mesh generation process and it has a relatively low computational cost. The implementation details are discussed and test cases are shown.
Keywords:
*interpolation; Lagrange multiplier; Lagrange projection*
Available in digital repository of the ASCR
Interpolation with restrictions -- role of the boundary conditions and individual restrictions

The contribution deals with the remeshing procedure between two computational finite element meshes. The remeshing represented by the interpolation of an approximate solution onto a new mesh is needed ...

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**Validation of numerical simulations of a simple immersed boundary solver for fluid flow in branching channels**

Keslerová, R.; Lancmanová, Anna; Bodnár, Tomáš

2023 - English
This work deals with the flow of incompressible viscous fluids in a two-dimensional branching channel. Using the immersed boundary method, a new finite difference solver was developed to interpret the channel geometry. The numerical results obtained by this new solver are compared with the numerical simulations of the older finite volume method code and with the results obtained with OpenFOAM. The aim of this work is to verify whether the immersed boundary method is suitable for fluid flow in channels with more complex geometries with difficult grid generation.
Keywords:
*immersed boundary method; finite volume method; OpenFOAM*
Available in digital repository of the ASCR
Validation of numerical simulations of a simple immersed boundary solver for fluid flow in branching channels

This work deals with the flow of incompressible viscous fluids in a two-dimensional branching channel. Using the immersed boundary method, a new finite difference solver was developed to interpret the ...

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**On the problem of singular limit**

Caggio, Matteo; Ducomet, B.; Nečasová, Šárka; Tang, T.

2023 - English
We consider the problem of singular limit of the compressible Euler system confined to a straight layer Ωδ = (0, δ)×R², δ > 0. In the regime of low Mach number limit and reduction of dimension the convergence to the strong solution of the 2D incompressible Euler system is shown.
Keywords:
*compressible Euler equations; dissipative measure-valued solutions; low Mach number; thin domain*
Available in digital repository of the ASCR
On the problem of singular limit

We consider the problem of singular limit of the compressible Euler system confined to a straight layer Ωδ = (0, δ)×R², δ > 0. In the regime of low Mach number limit and reduction of dimension the ...

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