Abstract
In this paper, we develop a new immersed finite element method (IFEM) for two-phase incompressible Stokes flows. We allow the interface to cut the finite elements.
On the noninterface element, the standard Crouzeix–Raviart element and the
Funding source: National Research Foundation of Korea
Award Identifier / Grant number: 2020R1C1C1A01005396
Award Identifier / Grant number: 2021R1A2C1003340
Funding statement: The first author is supported by the National Research Foundation of Korea (NRF) grant funded by the Korea government (MSIT) (No. 2020R1C1C1A01005396). The second author is supported by NRF funded by MSIT (No. 2021R1A2C1003340).
References
[1] S. Adjerid, N. Chaabane and T. Lin, An immersed discontinuous finite element method for Stokes interface problems, Comput. Methods Appl. Mech. Engrg. 293 (2015), 170–190. 10.1016/j.cma.2015.04.006Suche in Google Scholar
[2] S. Adjerid, N. Chaabane, T. Lin and P. Yue, An immersed discontinuous finite element method for the Stokes problem with a moving interface, J. Comput. Appl. Math. 362 (2019), 540–559. 10.1016/j.cam.2018.07.033Suche in Google Scholar
[3] R. Becker, E. Burman and P. Hansbo, A Nitsche extended finite element method for incompressible elasticity with discontinuous modulus of elasticity, Comput. Methods Appl. Mech. Engrg. 198 (2009), no. 41–44, 3352–3360. 10.1016/j.cma.2009.06.017Suche in Google Scholar
[4] T. Belytschko and T. Black, Elastic crack growth in finite elements with minimal remeshing, Internat. J. Numer. Methods Engrg. 45 (1999), no. 5, 601–620. 10.1002/(SICI)1097-0207(19990620)45:5<601::AID-NME598>3.0.CO;2-SSuche in Google Scholar
[5] J. Chessa and T. Belytschko, An extended finite element method for two-phase fluids, Trans. ASME J. Appl. Mech. 70 (2003), no. 1, 10–17. 10.1115/1.1526599Suche in Google Scholar
[6] S.-H. Chou, D. Y. Kwak and K. T. Wee, Optimal convergence analysis of an immersed interface finite element method, Adv. Comput. Math. 33 (2010), no. 2, 149–168. 10.1007/s10444-009-9122-ySuche in Google Scholar
[7] M. Crouzeix and P.-A. Raviart, Conforming and nonconforming finite element methods for solving the stationary Stokes equations. I, Rev. Française Automat. Informat. Recherche Opérationnelle Sér. Rouge 7 (1973), no. R3, 33–75. 10.1051/m2an/197307R300331Suche in Google Scholar
[8] R. Ghandriz, K. Hart and J. Li, StressExtended finite element method (XFEM) modeling of fracture in additively manufactured polymers, Additive Manuf. 31 (2020), Article ID 100945. 10.1016/j.addma.2019.100945Suche in Google Scholar
[9] S. Groß and A. Reusken, An extended pressure finite element space for two-phase incompressible flows with surface tension, J. Comput. Phys. 224 (2007), no. 1, 40–58. 10.1016/j.jcp.2006.12.021Suche in Google Scholar
[10] S. Gross and A. Reusken, Finite element discretization error analysis of a surface tension force in two-phase incompressible flows, SIAM J. Numer. Anal. 45 (2007), no. 4, 1679–1700. 10.1137/060667530Suche in Google Scholar
[11] A. Hansbo and P. Hansbo, An unfitted finite element method, based on Nitsche’s method, for elliptic interface problems, Comput. Methods Appl. Mech. Engrg. 191 (2002), no. 47–48, 5537–5552. 10.1016/S0045-7825(02)00524-8Suche in Google Scholar
[12] P. Hansbo, M. G. Larson and S. Zahedi, A cut finite element method for a Stokes interface problem, Appl. Numer. Math. 85 (2014), 90–114. 10.1016/j.apnum.2014.06.009Suche in Google Scholar
[13] G. Jo and D. Y. Kwak, An IMPES scheme for a two-phase flow in heterogeneous porous media using a structured grid, Comput. Methods Appl. Mech. Engrg. 317 (2017), 684–701. 10.1016/j.cma.2017.01.005Suche in Google Scholar
[14] G. Jo and D. Y. Kwak, Recent development of immersed FEM for elliptic and elastic interface problems, J. Korean Soc. Ind. Appl. Math. 23 (2019), no. 2, 65–92. Suche in Google Scholar
[15] G. Jo and D. Y. Kwak, A reduced Crouzeix–Raviart immersed finite element method for elasticity problems with interfaces, Comput. Methods Appl. Math. 20 (2020), no. 3, 501–516. 10.1515/cmam-2019-0046Suche in Google Scholar
[16] D. Jones and X. Zhang, A class of nonconforming immersed finite element methods for Stokes interface problems, J. Comput. Appl. Math. 392 (2021), Paper No. 113493. 10.1016/j.cam.2021.113493Suche in Google Scholar
[17] R. Kafafy, T. Lin, Y. Lin and J. Wang, Three-dimensional immersed finite element methods for electric field simulation in composite materials, Internat. J. Numer. Methods Engrg. 64 (2005), no. 7, 940–972. 10.1002/nme.1401Suche in Google Scholar
[18]
D. Y. Kwak, S. Jin and D. Kyeong,
A stabilized
[19]
D. Y. Kwak, K. T. Wee and K. S. Chang,
An analysis of a broken
[20] I. Kwon, D. Y. Kwak and G. Jo, Discontinuous bubble immersed finite element method for Poisson–Boltzmann–Nernst–Planck model, J. Comput. Phys. 438 (2021), Paper No. 110370. 10.1016/j.jcp.2021.110370Suche in Google Scholar
[21] G. Legrain, N. Moës and E. Verron, Stress analysis around crack tips in finite strain problems using the eXtended finite element method, Internat. J. Numer. Methods Engrg. 63 (2005), no. 2, 290–314. 10.1002/nme.1291Suche in Google Scholar
[22] Z. Li, T. Lin, Y. Lin and R. C. Rogers, An immersed finite element space and its approximation capability, Numer. Methods Partial Differential Equations 20 (2004), no. 3, 338–367. 10.1002/num.10092Suche in Google Scholar
[23] Z. Li, T. Lin and X. Wu, New Cartesian grid methods for interface problems using the finite element formulation, Numer. Math. 96 (2003), no. 1, 61–98. 10.1007/s00211-003-0473-xSuche in Google Scholar
[24] T. Lin, Y. Lin, R. Rogers and M. L. Ryan, A rectangular immersed finite element space for interface problems, Scientific Computing and Applications (Kananaskis 2000), Adv. Comput. Theory Pract. 7, Nova Science, Huntington (2001), 107–114. Suche in Google Scholar
[25] T. Lin and J. Wang, The immersed finite element method for plasma particle simulation, 41 st AIAA Aerospace Sciences Meeting and Exhibit, AIAA, Reno (2003), https://arc.aiaa.org/doi/abs/10.2514/6.2003-842. Suche in Google Scholar
[26] N. Moës and T. Belytschko, Extended finite element method for cohesive crack growth, Eng. Fracture Mech. 69 (2002), no. 7, 813–833. 10.1016/S0013-7944(01)00128-XSuche in Google Scholar
[27] N. Moës, J. Dolbow and T. Belytschko, A finite element method for crack growth without remeshing, Internat. J. Numer. Methods Engrg. 46 (1999), no. 1, 131–150. 10.1002/(SICI)1097-0207(19990910)46:1<131::AID-NME726>3.0.CO;2-JSuche in Google Scholar
[28] A. Reusken, Analysis of an extended pressure finite element space for two-phase incompressible flows, Comput. Vis. Sci. 11 (2008), no. 4–6, 293–305. 10.1007/s00791-008-0099-8Suche in Google Scholar
[29] S.-N. Roth, P. Léger and A. Soulaïmani, Strongly coupled XFEM formulation for non-planar three-dimensional simulation of hydraulic fracturing with emphasis on concrete dams, Comput. Methods Appl. Mech. Engrg. 363 (2020), Article ID 112899. 10.1016/j.cma.2020.112899Suche in Google Scholar
[30] S. Vallaghé and T. Papadopoulo, A trilinear immersed finite element method for solving the electroencephalography forward problem, SIAM J. Sci. Comput. 32 (2010), no. 4, 2379–2394. 10.1137/09075038XSuche in Google Scholar
[31] N. Wang and J. Chen, A nonconforming Nitsche’s extended finite element method for Stokes interface problems, J. Sci. Comput. 81 (2019), no. 1, 342–374. 10.1007/s10915-019-01019-9Suche in Google Scholar
[32] L. T. Zhang and M. Gay, Immersed finite element method for fluid-structure interactions, J. Fluids Struct. 23 (2007), no. 6, 839–857. 10.1016/j.jfluidstructs.2007.01.001Suche in Google Scholar
© 2023 Walter de Gruyter GmbH, Berlin/Boston
Artikel in diesem Heft
- Frontmatter
- Approximate Deconvolution with Correction – A High Fidelity Model for Magnetohydrodynamic Flows at High Reynolds and Magnetic Reynolds Numbers
- Well-Posedness and Convergence Analysis of PML Method for Time-Dependent Acoustic Scattering Problems Over a Locally Rough Surface
- A New Immersed Finite Element Method for Two-Phase Stokes Problems Having Discontinuous Pressure
- Simultaneous Recovery of Two Time-Dependent Coefficients in a Multi-Term Time-Fractional Diffusion Equation
- 𝐻2-Conformal Approximation of Miura Surfaces
- A Second-Order Difference Scheme for Generalized Time-Fractional Diffusion Equation with Smooth Solutions
- Volume Integral Equations and Single-Trace Formulations for Acoustic Wave Scattering in an Inhomogeneous Medium
- An Adaptive Two-Grid Solver for DPG Formulation of Compressible Navier–Stokes Equations in 3D
- An Efficient Discretization Scheme for Solving Nonlinear Ill-Posed Problems
- Tensor-Product Space-Time Goal-Oriented Error Control and Adaptivity With Partition-of-Unity Dual-Weighted Residuals for Nonstationary Flow Problems
- Identification of an Inverse Source Problem in a Fractional Partial Differential Equation Based on Sinc-Galerkin Method and TSVD Regularization
- A Formulation for a Nonlinear Axisymmetric Magneto-Heat Coupling Problem with an Unknown Nonlocal Boundary Condition
- Landweber Iterative Method for an Inverse Source Problem of Time-Space Fractional Diffusion-Wave Equation
Artikel in diesem Heft
- Frontmatter
- Approximate Deconvolution with Correction – A High Fidelity Model for Magnetohydrodynamic Flows at High Reynolds and Magnetic Reynolds Numbers
- Well-Posedness and Convergence Analysis of PML Method for Time-Dependent Acoustic Scattering Problems Over a Locally Rough Surface
- A New Immersed Finite Element Method for Two-Phase Stokes Problems Having Discontinuous Pressure
- Simultaneous Recovery of Two Time-Dependent Coefficients in a Multi-Term Time-Fractional Diffusion Equation
- 𝐻2-Conformal Approximation of Miura Surfaces
- A Second-Order Difference Scheme for Generalized Time-Fractional Diffusion Equation with Smooth Solutions
- Volume Integral Equations and Single-Trace Formulations for Acoustic Wave Scattering in an Inhomogeneous Medium
- An Adaptive Two-Grid Solver for DPG Formulation of Compressible Navier–Stokes Equations in 3D
- An Efficient Discretization Scheme for Solving Nonlinear Ill-Posed Problems
- Tensor-Product Space-Time Goal-Oriented Error Control and Adaptivity With Partition-of-Unity Dual-Weighted Residuals for Nonstationary Flow Problems
- Identification of an Inverse Source Problem in a Fractional Partial Differential Equation Based on Sinc-Galerkin Method and TSVD Regularization
- A Formulation for a Nonlinear Axisymmetric Magneto-Heat Coupling Problem with an Unknown Nonlocal Boundary Condition
- Landweber Iterative Method for an Inverse Source Problem of Time-Space Fractional Diffusion-Wave Equation