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3D Stabilized FEM Solution of the MHD Equations in an External Medium and Around a Solid

Year 2023, , 547 - 560, 29.09.2023

Selçuk Han Aydın , Mahir Ceylan Erdoğan

https://doi.org/10.17776/csj.1293551

Abstract

In this study, we consider 3-D MagnetoHydroDynamic (MHD) flow problems with different configurations which are mathematically expressed by system of coupled partial differential equation with coupled boundary conditions. These equations are solved numerically using one of the most popular schemes named as the finite element method (FEM) with SUPG type stabilized version in order to obtain accurate and stable solutions especially for the high values of the problem parameters. Obtained numerical solutions are visualized in terms of figures by taking the 2-D slices of the 3-D data in order to emphasize the accuracy of the proposed formulation.

Keywords

3D-FEM Stabilization MHD flow Exterior medium Conducting solid

References

  • [1] Iwona A.W., Lucyna B., Lukasz D., Marek M., Stanislaw H.I., Modelling 3D dynamics of offshore lattice jib cranes by means of the rigid finite element method, Journal of Ocean Engineering and Marine Energy, 9 (2023) 495-513.
  • [2] Saimi A., Bensaid I., Fellah A., Effect of crack presence on the dynamic and buckling responses of bidirectional functionally graded beams based on quasi-3D beam model and differential quadrature finite element method, Archive of Applied Mechanics, 93 (2023) 3131–3151.
  • [3] Joshi K.K., Kar V.R., Elastoplastic Behaviour of Multidirectional Porous Functionally Graded Panels: A Nonlinear FEM Approach, Iran J. Sci. Technol. Trans. Mech. Eng., (2023).
  • [4] Shao Z., Li X.S., Xiang P., A new computational scheme for structural static stochastic analysis based on Karhunen–Loève expansion and modified perturbation stochastic finite element method, Computational Mechanics, 71 (2023) 917-933.
  • [5] Gatica G.N., Nunez N., Ruiz-Baier R., Mixed-Primal Methods for Natural Convection Driven Phase Change with Navier–Stokes–Brinkman Equations, Journal of Scientific Computing, 95 (2023) 79.
  • [6] Vantyghem G., Ooms T., Corte W.D., FEM modelling techniques for simulation of 3D concrete printing, (2020).
  • [7] Liu W.K., Li S., Park H.S., Eighty Years of the Finite Element Method: Birth, Evolution, and Future, Arch. Computat. Methods Eng., 29 (2022) 4431-4453.
  • [8] Xu H., Zou D., Kong X., Hu Z., Study on the effects of hydrodynamic pressure on the dynamic stresses in slabs of high CFRD based on the scaled boundary finite-element method, Soil Dynamics and Earthquake Engineering, 88 (2016) 223–236.
  • [9] Hell S., Becker W., The scaled boundary finite element method for the analysis of 3D crack interaction, Journal of Computational Science, 9(7) (2015) 76–81.
  • [10] Anjos, G.R., Borhani,N., Mangiavacchi, N., Thome J.R., A 3D moving mesh Finite Element Method for two-phase flows, Journal of Computational Physics, 270 (2014) 366–377.
  • [11] Schott B., Wall W.A., A new face-oriented stabilized XFEM approach for 2D and 3D incompressible Navier-Stokes equations, Comput. Methods Appl. Mech. Engrg., 276 (2014) 233–265.
  • [12] Gravenkamp H., Man H., Song C., Prager J., The computation of dispersion relations for three-dimensional elastic waveguides using the Scaled Boundary Finite Element Method, Journal of Sound and Vibration, 332 (2013) 3756–3771.
  • [13] Stephan E.P., Maischak M., Leydecker F., An hp-adaptive finite element/boundary element coupling method for electromagnetic problems, Comput. Mech., 39 (2007) 673–680.
  • [14] Geramy A., Sharafoddin F., Abfraction: 3D analysis by means of the finite element method, Dental Research, 34(7) (2003) 526–533.
  • [15] Rachowicz W., Demkowicz L., An hp-adaptive finite element method for electromagnetics: Part II. A 3D implementation, Int. J. Numer. Meth. Engng., 53 (2002) 147–180.
  • [16] Chakraborty S., Bhattacharyya B., An efficient 3D stochastic finite element method, International Journal of Solids and Structures, 39 (2002) 2465–2475.
  • [17] Hartmann J., Theory of the laminar flow of an electrically conductive liquid in a homogeneous magnetic field, K. Dan. Vidensk. Selsk. Mat. Fys. Medd., 15(6) (1937) 1-28.
  • [18] Shercliff J.A., Steady motion of conducting fluids in pipes under transverse magnetic fields, Math. Proc. Cambridge, 49 (1953) 136–144.
  • [19] Dragoş L., Magnetofluid Dynamics, Abacus Pres, 1975, 92-99.
  • [20] Brooks A.N., Hughes T.J.R., Streamline upwind/Petrov-Galerkin formulations for convection dominated flows with particular emphasis on the incompressible Navier-Stokes equations, Comput. Methods Appl. Mech. Engrg., 32 (1982) 199-259.
  • [21] Rao S., A Numerical Study on Unsteady MHD Williamson Nanofluid Flow past a Permeable Moving Cylinder in the presence of Thermal Radiation and Chemical Reaction, Biointerface Research in Applied Chemistry, 13(5) (2023) 436.
  • [22] Zhao Y., Global well-posedness for the compressible non-resistive MHD equations in a 3D infinite slab, Nonlinear Analysis, 227 (2023) 113162.
  • [23] Patel A., Bhattacharyay R., 3D Thermo-fluid MHD simulation in a complex flow geometry, Fusion Engineering and Design, 191 (2023) 113558.
  • [24] Wang Z., Liu H., Global well-posedness for the 3-D generalized MHD equations, Applied Mathematics Letters, 140 (2023) 108585.
  • [25] Tezer-Sezgin M., Aydın S.H., Stabilized FEM solution of MHD duct flow with conducting cracks in the insulation, Journal of Computational and Applied Mathematics, 4230 (2023) 114936.
  • [26] Aggul M., Eroglu F.G., Kaya S., Artificial compression method for MHD system in Elsässer variables, Applied Numerical Mathematics, 185 (2023) 72-87.
  • [27] Chen Y., Peng Y., Shi X., A new blowup criterion for a generalized Hall-MHD system concerning the deformation tensor, Applied Mathematics Letters, 140 (2023) 108567.
  • [28] Fu L., An Efficient Low-Dissipation High-Order TENO Scheme for MHD Flows, Journal of Scientific Computing, 90(1) (2022) 1-24.
  • [29] Faizan, M., Ali, F., Loganathan, K. Zaib, A., Reddy, C.A., Abdelsalam, S.I., Entropy analysis of sutterby nanofluid flow over a riga sheet with gyrotactic microorganisms and Cattaneo-Christov double diffusion, Mathematics, 10(17) (2022) 3157.
  • [30] Luo Y., Fan X., Kim C.N., MHD flows in a U-channel under the influence of the spatially different channel-wall electric conductivity and of the magnetic field orientation, Journal of Mechanical Science and Technology, 35 (2021) 4477-4487.
  • [31] Wang H., Chen L., Zhang N.M., Ni M.J., Numerical simulations of MHD flows around a 180-degree sharp bend under a strong transverse magnetic field, Nuclear Fusion, 61(12) (2011) 126069.
  • [32] Aydin S.H., Nesliturk A.I., Tezer-Sezgin M., Two-level finite element method with a stabilizing subgrid for the incompressible MHD equations, International Journal for Numerical Methods in Fluids, 62(2) (2010) 188–210.
  • [33] Nesliturk A.I., Tezer-Sezgin M., Finite element method solution of electrically driven magnetohydrodynamic flow, Journal of Computational and Applied Mathematics, 192 (2006) 339–352.
  • [34] Codina R., Silva N.H., Stabilized finite element approximation of the stationary magneto-hydrodynamics equations, Computational Mechanics, 38 (2006) 344–355.
  • [35] Lungu E., Pohoata A., Finite element-boundary element approach of MHD pipe flow, Proc. of Conf. on Fluid Mech. and Technical Appl., Bucharest, Romania, 2005, 79-88.
  • [36] Tezer-Sezgin M., Han Aydın S., BEM Solution of MHD Flow in a Pipe Coupled with Magnetic Induction of Exterior Region, Computing 95(1) (2013) 751–770.
  • [37] Han Aydın S., Tezer-Sezgin M., DRBEM Solution of MHD Pipe Flow in a Conducting Medium, Journal of Computational and Applied Mathematics, 259(B) (2014) 720–729.
  • [38] Han Aydın S., Selvitopi H., Stabilized FEM-BEM coupled solution of MHD pipe flow in an unbounded conducting medium, Engineering Analysis with Boundary Elements, 87(2) (2018) 122–132.
  • [39] Aydin, S.H., Stabilized solution of the 3-D MHD flow problem with FEM-BEM coupling approach, Engineering Analysis with Boundary Elements, 140 (2022) 519–530.
  • [40] Aydın S.H., Erdoğan M.C., Stabilization in 3-D FEM and solution of the MHD equations, Mathematical Methods in the Applied Sciences, (2023).

Year 2023, , 547 - 560, 29.09.2023

Selçuk Han Aydın , Mahir Ceylan Erdoğan

https://doi.org/10.17776/csj.1293551

Abstract

References

  • [1] Iwona A.W., Lucyna B., Lukasz D., Marek M., Stanislaw H.I., Modelling 3D dynamics of offshore lattice jib cranes by means of the rigid finite element method, Journal of Ocean Engineering and Marine Energy, 9 (2023) 495-513.
  • [2] Saimi A., Bensaid I., Fellah A., Effect of crack presence on the dynamic and buckling responses of bidirectional functionally graded beams based on quasi-3D beam model and differential quadrature finite element method, Archive of Applied Mechanics, 93 (2023) 3131–3151.
  • [3] Joshi K.K., Kar V.R., Elastoplastic Behaviour of Multidirectional Porous Functionally Graded Panels: A Nonlinear FEM Approach, Iran J. Sci. Technol. Trans. Mech. Eng., (2023).
  • [4] Shao Z., Li X.S., Xiang P., A new computational scheme for structural static stochastic analysis based on Karhunen–Loève expansion and modified perturbation stochastic finite element method, Computational Mechanics, 71 (2023) 917-933.
  • [5] Gatica G.N., Nunez N., Ruiz-Baier R., Mixed-Primal Methods for Natural Convection Driven Phase Change with Navier–Stokes–Brinkman Equations, Journal of Scientific Computing, 95 (2023) 79.
  • [6] Vantyghem G., Ooms T., Corte W.D., FEM modelling techniques for simulation of 3D concrete printing, (2020).
  • [7] Liu W.K., Li S., Park H.S., Eighty Years of the Finite Element Method: Birth, Evolution, and Future, Arch. Computat. Methods Eng., 29 (2022) 4431-4453.
  • [8] Xu H., Zou D., Kong X., Hu Z., Study on the effects of hydrodynamic pressure on the dynamic stresses in slabs of high CFRD based on the scaled boundary finite-element method, Soil Dynamics and Earthquake Engineering, 88 (2016) 223–236.
  • [9] Hell S., Becker W., The scaled boundary finite element method for the analysis of 3D crack interaction, Journal of Computational Science, 9(7) (2015) 76–81.
  • [10] Anjos, G.R., Borhani,N., Mangiavacchi, N., Thome J.R., A 3D moving mesh Finite Element Method for two-phase flows, Journal of Computational Physics, 270 (2014) 366–377.
  • [11] Schott B., Wall W.A., A new face-oriented stabilized XFEM approach for 2D and 3D incompressible Navier-Stokes equations, Comput. Methods Appl. Mech. Engrg., 276 (2014) 233–265.
  • [12] Gravenkamp H., Man H., Song C., Prager J., The computation of dispersion relations for three-dimensional elastic waveguides using the Scaled Boundary Finite Element Method, Journal of Sound and Vibration, 332 (2013) 3756–3771.
  • [13] Stephan E.P., Maischak M., Leydecker F., An hp-adaptive finite element/boundary element coupling method for electromagnetic problems, Comput. Mech., 39 (2007) 673–680.
  • [14] Geramy A., Sharafoddin F., Abfraction: 3D analysis by means of the finite element method, Dental Research, 34(7) (2003) 526–533.
  • [15] Rachowicz W., Demkowicz L., An hp-adaptive finite element method for electromagnetics: Part II. A 3D implementation, Int. J. Numer. Meth. Engng., 53 (2002) 147–180.
  • [16] Chakraborty S., Bhattacharyya B., An efficient 3D stochastic finite element method, International Journal of Solids and Structures, 39 (2002) 2465–2475.
  • [17] Hartmann J., Theory of the laminar flow of an electrically conductive liquid in a homogeneous magnetic field, K. Dan. Vidensk. Selsk. Mat. Fys. Medd., 15(6) (1937) 1-28.
  • [18] Shercliff J.A., Steady motion of conducting fluids in pipes under transverse magnetic fields, Math. Proc. Cambridge, 49 (1953) 136–144.
  • [19] Dragoş L., Magnetofluid Dynamics, Abacus Pres, 1975, 92-99.
  • [20] Brooks A.N., Hughes T.J.R., Streamline upwind/Petrov-Galerkin formulations for convection dominated flows with particular emphasis on the incompressible Navier-Stokes equations, Comput. Methods Appl. Mech. Engrg., 32 (1982) 199-259.
  • [21] Rao S., A Numerical Study on Unsteady MHD Williamson Nanofluid Flow past a Permeable Moving Cylinder in the presence of Thermal Radiation and Chemical Reaction, Biointerface Research in Applied Chemistry, 13(5) (2023) 436.
  • [22] Zhao Y., Global well-posedness for the compressible non-resistive MHD equations in a 3D infinite slab, Nonlinear Analysis, 227 (2023) 113162.
  • [23] Patel A., Bhattacharyay R., 3D Thermo-fluid MHD simulation in a complex flow geometry, Fusion Engineering and Design, 191 (2023) 113558.
  • [24] Wang Z., Liu H., Global well-posedness for the 3-D generalized MHD equations, Applied Mathematics Letters, 140 (2023) 108585.
  • [25] Tezer-Sezgin M., Aydın S.H., Stabilized FEM solution of MHD duct flow with conducting cracks in the insulation, Journal of Computational and Applied Mathematics, 4230 (2023) 114936.
  • [26] Aggul M., Eroglu F.G., Kaya S., Artificial compression method for MHD system in Elsässer variables, Applied Numerical Mathematics, 185 (2023) 72-87.
  • [27] Chen Y., Peng Y., Shi X., A new blowup criterion for a generalized Hall-MHD system concerning the deformation tensor, Applied Mathematics Letters, 140 (2023) 108567.
  • [28] Fu L., An Efficient Low-Dissipation High-Order TENO Scheme for MHD Flows, Journal of Scientific Computing, 90(1) (2022) 1-24.
  • [29] Faizan, M., Ali, F., Loganathan, K. Zaib, A., Reddy, C.A., Abdelsalam, S.I., Entropy analysis of sutterby nanofluid flow over a riga sheet with gyrotactic microorganisms and Cattaneo-Christov double diffusion, Mathematics, 10(17) (2022) 3157.
  • [30] Luo Y., Fan X., Kim C.N., MHD flows in a U-channel under the influence of the spatially different channel-wall electric conductivity and of the magnetic field orientation, Journal of Mechanical Science and Technology, 35 (2021) 4477-4487.
  • [31] Wang H., Chen L., Zhang N.M., Ni M.J., Numerical simulations of MHD flows around a 180-degree sharp bend under a strong transverse magnetic field, Nuclear Fusion, 61(12) (2011) 126069.
  • [32] Aydin S.H., Nesliturk A.I., Tezer-Sezgin M., Two-level finite element method with a stabilizing subgrid for the incompressible MHD equations, International Journal for Numerical Methods in Fluids, 62(2) (2010) 188–210.
  • [33] Nesliturk A.I., Tezer-Sezgin M., Finite element method solution of electrically driven magnetohydrodynamic flow, Journal of Computational and Applied Mathematics, 192 (2006) 339–352.
  • [34] Codina R., Silva N.H., Stabilized finite element approximation of the stationary magneto-hydrodynamics equations, Computational Mechanics, 38 (2006) 344–355.
  • [35] Lungu E., Pohoata A., Finite element-boundary element approach of MHD pipe flow, Proc. of Conf. on Fluid Mech. and Technical Appl., Bucharest, Romania, 2005, 79-88.
  • [36] Tezer-Sezgin M., Han Aydın S., BEM Solution of MHD Flow in a Pipe Coupled with Magnetic Induction of Exterior Region, Computing 95(1) (2013) 751–770.
  • [37] Han Aydın S., Tezer-Sezgin M., DRBEM Solution of MHD Pipe Flow in a Conducting Medium, Journal of Computational and Applied Mathematics, 259(B) (2014) 720–729.
  • [38] Han Aydın S., Selvitopi H., Stabilized FEM-BEM coupled solution of MHD pipe flow in an unbounded conducting medium, Engineering Analysis with Boundary Elements, 87(2) (2018) 122–132.
  • [39] Aydin, S.H., Stabilized solution of the 3-D MHD flow problem with FEM-BEM coupling approach, Engineering Analysis with Boundary Elements, 140 (2022) 519–530.
  • [40] Aydın S.H., Erdoğan M.C., Stabilization in 3-D FEM and solution of the MHD equations, Mathematical Methods in the Applied Sciences, (2023).
There are 40 citations in total.

Details

Primary Language English
Subjects Mathematical Sciences
Journal Section Natural Sciences
Authors

Selçuk Han Aydın 0000-0002-1419-9458

Mahir Ceylan Erdoğan 0000-0001-9775-5271

Publication Date September 29, 2023
Submission Date May 6, 2023
Acceptance Date August 30, 2023
Published in Issue Year 2023

Cite

APA Aydın, S. H., & Erdoğan, M. C. (2023). 3D Stabilized FEM Solution of the MHD Equations in an External Medium and Around a Solid. Cumhuriyet Science Journal, 44(3), 547-560. https://doi.org/10.17776/csj.1293551