# American Institute of Mathematical Sciences

March  2021, 14(3): 819-833. doi: 10.3934/dcdss.2020349

## Computational study of immersed boundary - lattice Boltzmann method for fluid-structure interaction

 a. Department of Mathematics, Faculty of Nuclear Sciences and Physical Engineering, Czech Technical University in Prague, Trojanova 13,120 00 Prague, Czech Republic b. Department of Heat Engineering and Environment Protection, AGH University of Science and Technology, Al. Mickiewicza 30, 30-059 Krakow, Poland

* Corresponding author: eichlpa1@fjfi.cvut.cz

Received  January 2019 Revised  December 2019 Published  March 2021 Early access  May 2020

In this article, we deal with the numerical immersed boundary-lattice Boltzmann method for simulation of the fluid-structure interaction problems in 2D. We consider the interaction of incompressible, Newtonian fluid in an isothermal system with an elastic fiber, which represents an immersed body boundary. First, a short introduction to the lattice Boltzmann and immersed boundary method is presented and the combination of these two methods is briefly discussed. Then, the choice of the smooth approximation of the Dirac delta function and the discretization of the immersed body is discussed. One of the significant drawbacks of immersed boundary method is the penetrative flow through the immersed impermeable boundary. The effect of the immersed body boundary discretization is investigated using two benchmark problems, where an elastic fiber is deformed. The results indicate that the restrictions placed on the discretization in literature are not necessary.

Citation: Pavel Eichler, Radek Fučík, Robert Straka. Computational study of immersed boundary - lattice Boltzmann method for fluid-structure interaction. Discrete & Continuous Dynamical Systems - S, 2021, 14 (3) : 819-833. doi: 10.3934/dcdss.2020349
##### References:
 [1] F. Bashforth and J. C. Adams, An attempt to test the theories of capillary action: By comparing the theoretical and measured forms of drops of fluid, Cambridge University Press, 1883.   Google Scholar [2] S. R. Blair and Y. W. Kwon, Modeling of fluid–structure interaction using lattice Boltzmann and finite element methods, J. Pressure Vessel Technol., 137 (2015), 9pp. doi: 10.1115/1.4027866.  Google Scholar [3] S. Chikatamarla, S. Ansumali and I. Karlin, Entropic lattice Boltzmann models for hydrodynamics in three dimensions, Phys. Rev. Lett., 97 (2006), 4pp. doi: 10.1103/PhysRevLett.97.010201.  Google Scholar [4] B. S. H. Connell and D. K. P. Yue, Flapping dynamics of a flag in a uniform stream, J. Fluid Mech., 581 (2007), 33-67.  doi: 10.1017/S0022112007005307.  Google Scholar [5] D. d'Humières, Generalized lattice-Boltzmann equations, Prog. Astronautics Aeronautics, 159 (1994). doi: 10.2514/5.9781600866319.0450.0458.  Google Scholar [6] R. Fučík, P. Eichler, R. Straka, P. Pauš, J. Klinkovský and T. Oberhuber, On optimal node spacing for immersed boundary lattice Boltzmann method in 2D and 3D, Comput. Math. Appl., 77 (2019), 1144-1162.  doi: 10.1016/j.camwa.2018.10.045.  Google Scholar [7] M. Geier, A. Greiner and J. G. Korvink, Cascaded digital lattice Boltzmann automata for high Reynolds number flow, Phys. Rev. E, 73 (2006). doi: 10.1103/PhysRevE.73.066705.  Google Scholar [8] M. Geier, M. Schönherr, A. Pasquili and M. Krafczyk, The cumulant lattice Boltzmann equation in three dimensions: Theory and validation, Comput. Math. Appl., 70 (2015), 507-547.  doi: 10.1016/j.camwa.2015.05.001.  Google Scholar [9] Z. Guo and C. Shu, Lattice Boltzmann Method and its Applications in Engineering, Advances in Computational Fluid Dynamics, 3, World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, 2013. doi: 10.1142/8806.  Google Scholar [10] B. Kallemov, A. P. S. Bhalla, B. E. Griffith and A. Donev, An immersed boundary method for rigid bodies, Commun. Appl. Math. Comput. Sci., 11 (2016), 79-141.  doi: 10.2140/camcos.2016.11.79.  Google Scholar [11] I. Karlin, F. Bösch and S. Chikatamarla, Gibbs' principle for the lattice-kinetic theory of fluid dynamics, Phys. Rev. E, 90 (2014). doi: 10.1103/PhysRevE.90.031302.  Google Scholar [12] Y. Kim and C. S. Peskin, Penalty immersed boundary method for an elastic boundary with mass, Phys. Fluids, 19 (2007). doi: 10.1063/1.2734674.  Google Scholar [13] T. Krüger, H. Kusumaatmaja, A. Kuzmin, O. Shardt, G. Orest and E. Viggen, The Lattice Boltzmann Method: Principles and Practice, Graduate Texts in Physics, Springer, Cham, 2017. doi: 10.1007/978-3-319-44649-3.  Google Scholar [14] T. Krüger, F. Varnik and D. Raabe, Efficient and accurate simulations of deformable particles immersed in a fluid using a combined immersed boundary lattice Boltzmann finite element method, Comput. Math. Appl., 61 (2011), 3485-3505.  doi: 10.1016/j.camwa.2010.03.057.  Google Scholar [15] H. Lewy, K. Friedrichs and R. Courant, Über die partiellen Differenzengleichungen der mathematischen Physik, Math. Ann., 100 (1928), 32-74.  doi: 10.1007/BF01448839.  Google Scholar [16] C. S. Peskin, The immersed boundary method, Acta Numer., 11 (2002), 479-517.  doi: 10.1017/S0962492902000077.  Google Scholar [17] K. N. Premnath and S. Banerjee, Incorporating forcing terms in cascaded lattice Boltzmann approach by method of central moments, Phys. Rev. E, 80 (2009). doi: 10.1103/PhysRevE.80.036702.  Google Scholar [18] K. V. Sharman, R. Straka and F. W. Tavares, New Cascaded Thermal Lattice Boltzmann Method for simulations of advection-diffusion and convective heat transfer, Internat. J. Thermal Sciences, 118 (2017), 259-277.  doi: 10.1016/j.ijthermalsci.2017.04.020.  Google Scholar [19] M. C. Sukop and D. J. Thorne Jr., Lattice Boltzmann Modeling: An Introduction for Geoscientists and Engineers, Springer, Berlin, Heidelberg, 2006. doi: 10.1007/978-3-540-27982-2.  Google Scholar [20] K. Taira and T. Colonius, The immersed boundary method: A projection approach, J. Comput. Phys., 225 (2007), 2118-2137.  doi: 10.1016/j.jcp.2007.03.005.  Google Scholar [21] R. Temam, Navier-Stokes Equations: Theory and Numerical Analysis, Studies in Mathematics and its Applications, 2, North-Holland Publishing Co., Amsterdam-New York-Oxford, 1977.  Google Scholar [22] F.-B. Tian, H. Luo, L. Zhu, J. C. Liao and X.-Y. Lu, An efficient immersed boundary-lattice Boltzmann method for the hydrodynamic interaction of elastic filaments, J. Comput. Phys., 230 (2011), 7266-7283.  doi: 10.1016/j.jcp.2011.05.028.  Google Scholar [23] J. Tölke, Implementation of a Lattice Boltzmann kernel using the Compute Unified Device Architecture developed by nVIDIA, Comput. Visualization Science, 13 (2010), 29-39.  doi: 10.1007/s00791-008-0120-2.  Google Scholar [24] V. S. Vladimirov, Methods of the Theory of Generalized Functions, Analytical Methods and Special Functions, 6, Taylor & Francis, London, 2002.  Google Scholar [25] X. Yang, X. Zhang, Z. Li and G.-W. He, A smoothing technique for discrete delta functions with application to immersed boundary method in moving boundary simulations, J. Comput. Phys., 228 (2009), 7821-7836.  doi: 10.1016/j.jcp.2009.07.023.  Google Scholar [26] Z.-Q. Zhang, G. R. Liu and B. C. Khoo, Immersed smoothed finite element method for two dimensional fluid-structure interaction problems, Internat. J. Numer. Methods Engrg., 90 (2012), 1292-1320.  doi: 10.1002/nme.4299.  Google Scholar

show all references

##### References:
 [1] F. Bashforth and J. C. Adams, An attempt to test the theories of capillary action: By comparing the theoretical and measured forms of drops of fluid, Cambridge University Press, 1883.   Google Scholar [2] S. R. Blair and Y. W. Kwon, Modeling of fluid–structure interaction using lattice Boltzmann and finite element methods, J. Pressure Vessel Technol., 137 (2015), 9pp. doi: 10.1115/1.4027866.  Google Scholar [3] S. Chikatamarla, S. Ansumali and I. Karlin, Entropic lattice Boltzmann models for hydrodynamics in three dimensions, Phys. Rev. Lett., 97 (2006), 4pp. doi: 10.1103/PhysRevLett.97.010201.  Google Scholar [4] B. S. H. Connell and D. K. P. Yue, Flapping dynamics of a flag in a uniform stream, J. Fluid Mech., 581 (2007), 33-67.  doi: 10.1017/S0022112007005307.  Google Scholar [5] D. d'Humières, Generalized lattice-Boltzmann equations, Prog. Astronautics Aeronautics, 159 (1994). doi: 10.2514/5.9781600866319.0450.0458.  Google Scholar [6] R. Fučík, P. Eichler, R. Straka, P. Pauš, J. Klinkovský and T. Oberhuber, On optimal node spacing for immersed boundary lattice Boltzmann method in 2D and 3D, Comput. Math. Appl., 77 (2019), 1144-1162.  doi: 10.1016/j.camwa.2018.10.045.  Google Scholar [7] M. Geier, A. Greiner and J. G. Korvink, Cascaded digital lattice Boltzmann automata for high Reynolds number flow, Phys. Rev. E, 73 (2006). doi: 10.1103/PhysRevE.73.066705.  Google Scholar [8] M. Geier, M. Schönherr, A. Pasquili and M. Krafczyk, The cumulant lattice Boltzmann equation in three dimensions: Theory and validation, Comput. Math. Appl., 70 (2015), 507-547.  doi: 10.1016/j.camwa.2015.05.001.  Google Scholar [9] Z. Guo and C. Shu, Lattice Boltzmann Method and its Applications in Engineering, Advances in Computational Fluid Dynamics, 3, World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, 2013. doi: 10.1142/8806.  Google Scholar [10] B. Kallemov, A. P. S. Bhalla, B. E. Griffith and A. Donev, An immersed boundary method for rigid bodies, Commun. Appl. Math. Comput. Sci., 11 (2016), 79-141.  doi: 10.2140/camcos.2016.11.79.  Google Scholar [11] I. Karlin, F. Bösch and S. Chikatamarla, Gibbs' principle for the lattice-kinetic theory of fluid dynamics, Phys. Rev. E, 90 (2014). doi: 10.1103/PhysRevE.90.031302.  Google Scholar [12] Y. Kim and C. S. Peskin, Penalty immersed boundary method for an elastic boundary with mass, Phys. Fluids, 19 (2007). doi: 10.1063/1.2734674.  Google Scholar [13] T. Krüger, H. Kusumaatmaja, A. Kuzmin, O. Shardt, G. Orest and E. Viggen, The Lattice Boltzmann Method: Principles and Practice, Graduate Texts in Physics, Springer, Cham, 2017. doi: 10.1007/978-3-319-44649-3.  Google Scholar [14] T. Krüger, F. Varnik and D. Raabe, Efficient and accurate simulations of deformable particles immersed in a fluid using a combined immersed boundary lattice Boltzmann finite element method, Comput. Math. Appl., 61 (2011), 3485-3505.  doi: 10.1016/j.camwa.2010.03.057.  Google Scholar [15] H. Lewy, K. Friedrichs and R. Courant, Über die partiellen Differenzengleichungen der mathematischen Physik, Math. Ann., 100 (1928), 32-74.  doi: 10.1007/BF01448839.  Google Scholar [16] C. S. Peskin, The immersed boundary method, Acta Numer., 11 (2002), 479-517.  doi: 10.1017/S0962492902000077.  Google Scholar [17] K. N. Premnath and S. Banerjee, Incorporating forcing terms in cascaded lattice Boltzmann approach by method of central moments, Phys. Rev. E, 80 (2009). doi: 10.1103/PhysRevE.80.036702.  Google Scholar [18] K. V. Sharman, R. Straka and F. W. Tavares, New Cascaded Thermal Lattice Boltzmann Method for simulations of advection-diffusion and convective heat transfer, Internat. J. Thermal Sciences, 118 (2017), 259-277.  doi: 10.1016/j.ijthermalsci.2017.04.020.  Google Scholar [19] M. C. Sukop and D. J. Thorne Jr., Lattice Boltzmann Modeling: An Introduction for Geoscientists and Engineers, Springer, Berlin, Heidelberg, 2006. doi: 10.1007/978-3-540-27982-2.  Google Scholar [20] K. Taira and T. Colonius, The immersed boundary method: A projection approach, J. Comput. Phys., 225 (2007), 2118-2137.  doi: 10.1016/j.jcp.2007.03.005.  Google Scholar [21] R. Temam, Navier-Stokes Equations: Theory and Numerical Analysis, Studies in Mathematics and its Applications, 2, North-Holland Publishing Co., Amsterdam-New York-Oxford, 1977.  Google Scholar [22] F.-B. Tian, H. Luo, L. Zhu, J. C. Liao and X.-Y. Lu, An efficient immersed boundary-lattice Boltzmann method for the hydrodynamic interaction of elastic filaments, J. Comput. Phys., 230 (2011), 7266-7283.  doi: 10.1016/j.jcp.2011.05.028.  Google Scholar [23] J. Tölke, Implementation of a Lattice Boltzmann kernel using the Compute Unified Device Architecture developed by nVIDIA, Comput. Visualization Science, 13 (2010), 29-39.  doi: 10.1007/s00791-008-0120-2.  Google Scholar [24] V. S. Vladimirov, Methods of the Theory of Generalized Functions, Analytical Methods and Special Functions, 6, Taylor & Francis, London, 2002.  Google Scholar [25] X. Yang, X. Zhang, Z. Li and G.-W. He, A smoothing technique for discrete delta functions with application to immersed boundary method in moving boundary simulations, J. Comput. Phys., 228 (2009), 7821-7836.  doi: 10.1016/j.jcp.2009.07.023.  Google Scholar [26] Z.-Q. Zhang, G. R. Liu and B. C. Khoo, Immersed smoothed finite element method for two dimensional fluid-structure interaction problems, Internat. J. Numer. Methods Engrg., 90 (2012), 1292-1320.  doi: 10.1002/nme.4299.  Google Scholar
The setup of the computational domain in the case of the Kármán vortex flow problem. Two obstacles are immersed in the fluid. The first one is a rigid cylinder with diameter $d$. The last one is an elastic fiber with one fixed end (filled circle) and one free end (empty circle)
and 2b, the free end trajectory is shown for two methods: (A) Forward Euler method (red), (B) The third order Runge-Kutta method (blue), and the results are compared with those in [22] (black circles). In Fig. 2c, the superposition of the elastic fibers is shown">Figure 2.  The study of the elastic fiber deformation in the Kármán vortex street region. In Figs. 2a and 2b, the free end trajectory is shown for two methods: (A) Forward Euler method (red), (B) The third order Runge-Kutta method (blue), and the results are compared with those in [22] (black circles). In Fig. 2c, the superposition of the elastic fibers is shown
and 3b the free end trajectory is shown for two methods: (A) Forward Euler method (red), (B) The third order Runge-Kutta method (blue), and the results are compared with results in [22] (black triangles). In Fig. 3c, the superposition of the elastic fibers is shown">Figure 3.  The study of the elastic fiber deformation in the Kármán vortex street region with neglected inertial forces of the elastic fiber. In Figs. 3a and 3b the free end trajectory is shown for two methods: (A) Forward Euler method (red), (B) The third order Runge-Kutta method (blue), and the results are compared with results in [22] (black triangles). In Fig. 3c, the superposition of the elastic fibers is shown
The setup of the computational domain for the cavity flow problem
Evolution of $E_{rkl}$ w.r.t. ${\sigma}/{h}$ for different values of the stiffness coefficient $k_S$ and for different mesh spacings $h$: $h_1 = 1.6 \cdot 10^{-4}\;\mathrm{m}$, $h_2 = 7.84\cdot 10^{-5}\;\mathrm{m}$, and $h_3 = 5.24\cdot 10^{-5}\;\mathrm{m}$. The results computed at the final time $T = 10\;\mathrm{s}$ when the steady state is reached
Schematic visualization of the fluid penetration through the flexible fiber for three values of ${\sigma}/{h}$. In the figures, the velocity vectors (white arrows) and values of the fluid velocity magnitude (color scale) are shown. The results are at the final time $T = 10\;\mathrm{s}$ when the steady state is reached. All results are computed using $k_S = 100\;\mathrm{kg\, s^{-2}}$ and $h = 5.24\cdot 10^{-5}\; \mathrm{m}$
Schematic visualization of the fiber deformation at the final time $T = 10\, \mathrm{s}$ and for $h = 5.24 \cdot 10^{-5}$ m. Different colors of the fibers represent different values of ${\sigma}/{h}$
The setup of the computational domain in the case of elastic bump deformation
Investigation of fluid flow around the bump. In Figs. (A) and (B), the relation between the surface area $\mu\left(\Omega_u\right)$ and ${\sigma}/{h}$ is shown. Figs. (C) and (D) illustrate the evolution of $E_{rkl}$ with respect to ${\sigma}/{h}$. For $U_\infty = 0.5\, \mathrm{m\, s^{-1}}$, the values are time-averaged over the time interval of $(1, 9.5)$ s. For $U_\infty = 0.5\, \mathrm{m\, s^{-1}}$, the values are computed at the final time $T = 10\; \mathrm{s}$
Schematic visualization of the fluid flow around an impermeable flexible fiber in time $t = 9.5\, \mathrm{s}$. In the figures, the velocity vectors (white arrows) and the values of the fluid velocity magnitude (color scale) are shown
 [1] Qiang Du, Manlin Li. On the stochastic immersed boundary method with an implicit interface formulation. Discrete & Continuous Dynamical Systems - B, 2011, 15 (2) : 373-389. doi: 10.3934/dcdsb.2011.15.373 [2] Harvey A. R. Williams, Lisa J. Fauci, Donald P. Gaver III. Evaluation of interfacial fluid dynamical stresses using the immersed boundary method. Discrete & Continuous Dynamical Systems - B, 2009, 11 (2) : 519-540. doi: 10.3934/dcdsb.2009.11.519 [3] Robert H. Dillon, Jingxuan Zhuo. Using the immersed boundary method to model complex fluids-structure interaction in sperm motility. Discrete & Continuous Dynamical Systems - B, 2011, 15 (2) : 343-355. doi: 10.3934/dcdsb.2011.15.343 [4] Daniele Boffi, Lucia Gastaldi. Discrete models for fluid-structure interactions: The finite element Immersed Boundary Method. Discrete & Continuous Dynamical Systems - S, 2016, 9 (1) : 89-107. doi: 10.3934/dcdss.2016.9.89 [5] Champike Attanayake, So-Hsiang Chou. An immersed interface method for Pennes bioheat transfer equation. Discrete & Continuous Dynamical Systems - B, 2015, 20 (2) : 323-337. doi: 10.3934/dcdsb.2015.20.323 [6] Jian Hao, Zhilin Li, Sharon R. Lubkin. An augmented immersed interface method for moving structures with mass. Discrete & Continuous Dynamical Systems - B, 2012, 17 (4) : 1175-1184. doi: 10.3934/dcdsb.2012.17.1175 [7] So-Hsiang Chou. An immersed linear finite element method with interface flux capturing recovery. Discrete & Continuous Dynamical Systems - B, 2012, 17 (7) : 2343-2357. doi: 10.3934/dcdsb.2012.17.2343 [8] Sheng Xu. Derivation of principal jump conditions for the immersed interface method in two-fluid flow simulation. Conference Publications, 2009, 2009 (Special) : 838-845. doi: 10.3934/proc.2009.2009.838 [9] Derrick Jones, Xu Zhang. A conforming-nonconforming mixed immersed finite element method for unsteady Stokes equations with moving interfaces. Electronic Research Archive, , () : -. doi: 10.3934/era.2021032 [10] Xiao-Yu Zhang, Qing Fang. A sixth order numerical method for a class of nonlinear two-point boundary value problems. Numerical Algebra, Control & Optimization, 2012, 2 (1) : 31-43. doi: 10.3934/naco.2012.2.31 [11] Boshi Tian, Xiaoqi Yang, Kaiwen Meng. An interior-point $l_{\frac{1}{2}}$-penalty method for inequality constrained nonlinear optimization. Journal of Industrial & Management Optimization, 2016, 12 (3) : 949-973. doi: 10.3934/jimo.2016.12.949 [12] Yibing Lv, Tiesong Hu, Jianlin Jiang. Penalty method-based equilibrium point approach for solving the linear bilevel multiobjective programming problem. Discrete & Continuous Dynamical Systems - S, 2020, 13 (6) : 1743-1755. doi: 10.3934/dcdss.2020102 [13] Canghua Jiang, Zhiqiang Guo, Xin Li, Hai Wang, Ming Yu. An efficient adjoint computational method based on lifted IRK integrator and exact penalty function for optimal control problems involving continuous inequality constraints. Discrete & Continuous Dynamical Systems - S, 2020, 13 (6) : 1845-1865. doi: 10.3934/dcdss.2020109 [14] Matthias Eller. A remark on Littman's method of boundary controllability. Evolution Equations & Control Theory, 2013, 2 (4) : 621-630. doi: 10.3934/eect.2013.2.621 [15] Yosra Boukari, Houssem Haddar. The factorization method applied to cracks with impedance boundary conditions. Inverse Problems & Imaging, 2013, 7 (4) : 1123-1138. doi: 10.3934/ipi.2013.7.1123 [16] Zhonghua Qiao, Xuguang Yang. A multiple-relaxation-time lattice Boltzmann method with Beam-Warming scheme for a coupled chemotaxis-fluid model. Electronic Research Archive, 2020, 28 (3) : 1207-1225. doi: 10.3934/era.2020066 [17] Wei Zhu. A numerical study of a mean curvature denoising model using a novel augmented Lagrangian method. Inverse Problems & Imaging, 2017, 11 (6) : 975-996. doi: 10.3934/ipi.2017045 [18] Yanhong Yuan, Hongwei Zhang, Liwei Zhang. A penalty method for generalized Nash equilibrium problems. Journal of Industrial & Management Optimization, 2012, 8 (1) : 51-65. doi: 10.3934/jimo.2012.8.51 [19] Regina S. Burachik, C. Yalçın Kaya. An update rule and a convergence result for a penalty function method. Journal of Industrial & Management Optimization, 2007, 3 (2) : 381-398. doi: 10.3934/jimo.2007.3.381 [20] Zhongwen Chen, Songqiang Qiu, Yujie Jiao. A penalty-free method for equality constrained optimization. Journal of Industrial & Management Optimization, 2013, 9 (2) : 391-409. doi: 10.3934/jimo.2013.9.391

2020 Impact Factor: 2.425

## Metrics

• PDF downloads (207)
• HTML views (324)
• Cited by (1)

## Other articlesby authors

• on AIMS
• on Google Scholar

[Back to Top]