Cell-Based Smoothed Finite-Element Framework for Strongly Coupled Non-Newtonian Fluid–Structure Interaction
Publication: Journal of Engineering Mechanics
Volume 147, Issue 10
Abstract
This work presents a cell-based smoothed finite-element method (CS-FEM) for simulating strongly coupled non-Newtonian fluid–structure interactions. The governing equations of a Carreau-Yasuda fluid and an elastic solid are discretized by the CS-FEM, which softens all gradient-related terms. The stress equilibrium along the interface is also derived from the CS-FEM notion. After discussing a two-level mesh-updating strategy, the strong coupling between the two physical media is realized via the block Gauss-Seidel iterative procedure. Numerical examples are presented to demonstrate the performance of the proposed method.
Get full access to this article
View all available purchase options and get full access to this article.
Data Availability Statement
All data and models generated or used during the study are available from the corresponding author by request.
Acknowledgments
Support from the Natural Science Foundation of Shanghai (Grant No. 19ZR1437200) is gratefully acknowledged.
References
Alishahi, M., M. M. Alishahi, and H. Emdad. 2011. “Numerical simulation of blood flow in a flexible stenosed abdominal real aorta.” Sci. Iran. 18 (6): 1297–1305. https://doi.org/10.1016/j.scient.2011.11.021.
Anand, V., J. David, and I. C. Christov. 2019. “Non-Newtonian fluid–structure interactions: Static response of a microchannel due to internal flow of a power-law fluid.” J. Non-Newtonian Fluid Mech. 264 (Feb): 62–72. https://doi.org/10.1016/j.jnnfm.2018.12.008.
Apostolatos, A., G. De Nayer, K.-U. Bletzinger, M. Breuer, and R. Wüchner. 2019. “Systematic evaluation of the interface description for fluid–structure interaction simulations using the isogeometric mortar-based mapping.” J. Fluids Struct. 86 (Apr): 368–399. https://doi.org/10.1016/j.jfluidstructs.2019.02.012.
Bathe, K. J., E. Ramm, and E. L. Wilson. 1975. “Finite element formulations for large deformation dynamic analysis.” Int. J. Numer. Methods Eng. 9 (2): 353–386. https://doi.org/10.1002/nme.1620090207.
Bazilevs, Y., V. M. Calo, T. J. Hughes, and Y. Zhang. 2008. “Isogeometric fluid–structure interaction: theory, algorithms, and computations.” Comput. Mech. 43 (1): 3–37. https://doi.org/10.1007/s00466-008-0315-x.
Bazilevs, Y., K. Takizawa, and T. E. Tezduyar. 2013. Computational fluid–structure interaction: Methods and applications. West Sussex, UK: Wiley.
Bird, R. B. 1976. “Useful non-Newtonian models.” Annu. Rev. Fluid Mech. 8 (1): 13–34. https://doi.org/10.1146/annurev.fl.08.010176.000305.
Carreau, P. J. 1972. “Rheological equations from molecular network theories.” Trans. Soc. Rheol. 16 (1): 99–127. https://doi.org/10.1122/1.549276.
Cervera, M., R. Codina, and M. Galindo. 1996. “On the computational efficiency and implementation of block-iterative algorithms for nonlinear coupled problems.” Eng. Comput. 13 (6): 4–30. https://doi.org/10.1108/02644409610128382.
Chen, J. S., C. T. Wu, S. Yoon, and Y. You. 2001. “A stabilized conforming nodal integration for Galerkin mesh-free methods.” Int. J. Numer. Methods Eng. 50 (2): 435–466. https://doi.org/10.1002/1097-0207(20010120)50:2%3C435::AID-NME32%3E3.0.CO;2-A.
Chitra, K., T. Sundararajan, S. Vengadesan, and P. Nithiarasu. 2011. “Non-Newtonian blood flow study in a model cavopulmonary vascular system.” Int. J. Numer. Methods Fluids 66 (3): 269–283. https://doi.org/10.1002/fld.2256.
Cho, Y. I., and K. R. Kensey. 1991. “Effects of the non-Newtonian viscosity of blood on flows in a diseased arterial vessel. Part 1: Steady flows.” Biorheology 28 (3–4): 241–262. https://doi.org/10.3233/BIR-1991-283-415.
Chung, J., and G. M. Hulbert. 1993. “A time integration algorithm for structural dynamics with improved numerical dissipation: the generalized- method.” J. Appl. Mech. 60 (2): 371–375. https://doi.org/10.1115/1.2900803.
Crochet, M. J., and K. Walters. 1983. “Numerical methods in non-Newtonian fluid mechanics.” Annu. Rev. Fluid Mech. 15 (1): 241–260. https://doi.org/10.1146/annurev.fl.15.010183.001325.
Cui, X. Y., G. R. Liu, G. Y. Li, X. Zhao, T. T. Nguyen, and G. Y. Sun. 2008. “A smoothed finite element method (SFEM) for linear and geometrically nonlinear analysis of plates and shells.” Comput. Model. Eng. Sci. 28 (2): 109–126. https://doi.org/10.3970/cmes.2008.028.109.
Dai, K. Y., and G. R. Liu. 2007. “Free and forced vibration analysis using the smoothed finite element method (SFEM).” J. Sound Vib. 301 (3): 803–820. https://doi.org/10.1016/j.jsv.2006.10.035.
Degroote, J., R. Haelterman, S. Annerel, P. Bruggeman, and J. Vierendeels. 2010. “Performance of partitioned procedures in fluid–structure interaction.” Comput. Struct. 88 (7–8): 446–457. https://doi.org/10.1016/j.compstruc.2009.12.006.
Dettmer, W., and D. Perić. 2006. “A computational framework for fluid–structure interaction: Finite element formulation and applications.” Comput. Methods Appl. Mech. Eng. 195 (41–43): 5754–5779. https://doi.org/10.1016/j.cma.2005.10.019.
Farhat, C., and M. Lesoinne. 2000. “Two efficient staggered algorithms for the serial and parallel solution of three-dimensional nonlinear transient aeroelastic problems.” Comput. Methods Appl. Mech. Eng. 182 (3): 499–515. https://doi.org/10.1016/S0045-7825(99)00206-6.
Fernández, M. Á., and M. Moubachir. 2005. “A Newton method using exact Jacobians for solving fluid–structure coupling.” Comput. Struct. 83 (2–3): 127–142. https://doi.org/10.1016/j.compstruc.2004.04.021.
Ghalambaz, M., S. A. M. Mehryan, R. K. Feeoj, A. Hajjar, I. Hashim, and R. B. Mahani. 2020. “Free convective heat transfer of a non-Newtonian fluid in a cavity containing a thin flexible heater plate: An Eulerian–Lagrangian approach.” J. Therm. Anal. Calorim. 2020 (Nov): 1–16. https://doi.org/10.1007/s10973-020-10292-y.
Habchi, C., S. Russeil, D. Bougeard, J. L. Harion, T. Lemenand, A. Ghanem, D. D. Valle, and H. Peerhossaini. 2013. “Partitioned solver for strongly coupled fluid–structure interaction.” Comput. Fluids 71 (Jan): 306–319. https://doi.org/10.1016/j.compfluid.2012.11.004.
Han, Z., D. Zhou, and J. Tu. 2014. “Wake-induced vibrations of a circular cylinder behind a stationary square cylinder using a semi-implicit characteristic-based split scheme.” J. Eng. Mech. 140 (8): 04014059. https://doi.org/10.1061/(ASCE)EM.1943-7889.0000727.
He, T. 2018. “Towards straightforward use of cell-based smoothed finite element method in fluid–structure interaction.” Ocean Eng. 157 (Jun): 350–363. https://doi.org/10.1016/j.oceaneng.2018.03.054.
He, T. 2019a. “A cell-based smoothed CBS finite element formulation for computing the Oldroyd-B fluid flow.” J. Non-Newtonian Fluid Mech. 272 (Oct): 104162. https://doi.org/10.1016/j.jnnfm.2019.104162.
He, T. 2019b. “The cell-based smoothed finite element method for viscoelastic fluid flows using fractional-step schemes.” Comput. Struct. 222 (Oct): 133–147. https://doi.org/10.1016/j.compstruc.2019.07.007.
He, T. 2019c. “Insight into the cell-based smoothed finite element method for convection-dominated flows.” Comput. Struct. 212 (Feb): 215–224. https://doi.org/10.1016/j.compstruc.2018.10.021.
He, T. 2020a. “Cell-based smoothed finite element method for simulating vortex-induced vibration of multiple bluff bodies.” J. Fluids Struct. 98 (Oct): 103140. https://doi.org/10.1016/j.jfluidstructs.2020.103140.
He, T. 2020b. “An efficient selective cell-based smoothed finite element approach to fluid–structure interaction.” Phys. Fluids 32 (6): 067102. https://doi.org/10.1063/5.0010562.
He, T. 2020c. “Extending the cell-based smoothed finite element method into strongly coupled fluid–thermal–structure interaction.” Int. J. Numer. Methods Fluids 93 (4): 1–23. https://doi.org/10.1002/fld.4928.
He, T. 2020d. “A strongly-coupled cell-based smoothed finite element solver for unsteady viscoelastic fluid–structure interaction.” Comput. Struct. 235 (Jul): 106264. https://doi.org/10.1016/j.compstruc.2020.106264.
He, T. 2020e. “A truly mesh-distortion-enabled implementation of cell-based smoothed finite element method for incompressible fluid flows with fixed and moving boundaries.” Int. J. Numer. Methods Eng. 121 (14): 3227–3248. https://doi.org/10.1002/nme.6355.
He, T., T. Wang, and H. Zhang. 2018a. “The use of artificial compressibility to improve partitioned semi-implicit FSI coupling within the classical Chorin–Témam projection framework.” Comput. Fluids 166 (Apr): 64–77. https://doi.org/10.1016/j.compfluid.2018.01.022.
He, T., J. Yang, and C. Baniotopoulos. 2018b. “Improving the CBS-based partitioned semi-implicit coupling algorithm for fluid–structure interaction.” Int. J. Numer. Methods Fluids 87 (9): 463–486. https://doi.org/10.1002/fld.4501.
He, T., H. Zhang, and K. Zhang. 2018c. “A smoothed finite element approach for computational fluid dynamics: applications to incompressible flows and fluid–structure interaction.” Comput. Mech. 62 (5): 1037–1057. https://doi.org/10.1007/s00466-018-1549-x.
He, T., and K. Zhang. 2017. “An overview of the combined interface boundary condition method for fluid–structure interaction.” Arch. Comput. Methods Eng. 24 (4): 891–934. https://doi.org/10.1007/s11831-016-9193-0.
He, T., D. Zhou, and Y. Bao. 2012. “Combined interface boundary condition method for fluid–rigid body interaction.” Comput. Methods Appl. Mech. Eng. 223 (Jun): 81–102. https://doi.org/10.1016/j.cma.2012.02.007.
He, Z. C., G. R. Liu, Z. H. Zhong, G. Y. Zhang, and A. G. Cheng. 2011. “A coupled ES-FEM/BEM method for fluid–structure interaction problems.” Eng. Anal. Boundary Elem. 35 (1): 140–147. https://doi.org/10.1016/j.enganabound.2010.05.003.
Hirt, C. W., A. A. Amsden, and J. L. Cook. 1974. “An arbitrary Lagrangian–Eulerian computing method for all flow speeds.” J. Comput. Phys. 14 (3): 227–253. https://doi.org/10.1016/0021-9991(74)90051-5.
Hundertmark-Zaušková, A., and M. Lukáčová-Medvidová. 2010. “Numerical study of shear-dependent non-Newtonian fluids in compliant vessels.” Comput. Math. Appl. 60 (3): 572–590. https://doi.org/10.1016/j.camwa.2010.05.004.
Janela, J., A. Moura, and A. Sequeira. 2010a. “A 3D non-Newtonian fluid–structure interaction model for blood flow in arteries.” J. Comput. Appl. Math. 234 (9): 2783–2791. https://doi.org/10.1016/j.cam.2010.01.032.
Janela, J., A. Moura, and A. Sequeira. 2010b. “Absorbing boundary conditions for a 3D non-Newtonian fluid–structure interaction model for blood flow in arteries.” Int. J. Eng. Sci. 48 (11): 1332–1349. https://doi.org/10.1016/j.ijengsci.2010.08.004.
Jensen, O. E., and M. Heil. 2003. “High-frequency self-excited oscillations in a collapsible-channel flow.” J. Fluid Mech. 481 (9): 235–268. https://doi.org/10.1017/S002211200300394X.
Jiang, C., Z. Q. Zhang, X. Han, G. R. Liu, and T. Lin. 2018. “A cell-based smoothed finite element method with semi-implicit CBS procedures for incompressible laminar viscous flows.” Int. J. Numer. Methods Fluids 86 (1): 20–45. https://doi.org/10.1002/fld.4406.
Kassiotis, C., A. Ibrahimbegovic, R. Niekamp, and H. G. Matthies. 2011. “Nonlinear fluid–structure interaction problem. Part I: Implicit partitioned algorithm, nonlinear stability proof and validation examples.” Comput. Mech. 47 (3): 305–323. https://doi.org/10.1007/s00466-010-0545-6.
Kim, J., and S. Im. 2017. “Polygonal type variable-node elements by means of the smoothed finite element method for analysis of two-dimensional fluid–solid interaction problems in viscous incompressible flows.” Comput. Struct. 182 (Apr): 475–490. https://doi.org/10.1016/j.compstruc.2017.01.006.
Kim, N., and J. N. Reddy. 2016. “A spectral/hp least-squares finite element analysis of the Carreau–Yasuda fluids.” Int. J. Numer. Methods Fluids 82 (9): 541–566. https://doi.org/10.1002/fld.4230.
Kuhl, D., and M. A. Crisfield. 1999. “Energy-conserving and decaying algorithms in non-linear structural dynamics.” Int. J. Numer. Methods Eng. 45 (5): 569–599. https://doi.org/10.1002/(SICI)1097-0207(19990620)45:5%3C569::AID-NME595%3E3.0.CO;2-A.
Küttler, U., and W. A. Wall. 2008. “Fixed-point fluid–structure interaction solvers with dynamic relaxation.” Comput. Mech. 43 (1): 61–72. https://doi.org/10.1007/s00466-008-0255-5.
Lee, H., and S. Xu. 2016a. “Finite element error estimation for quasi-Newtonian fluid–structure interaction problems.” Appl. Math. Comput. 274 (Feb): 93–105. https://doi.org/10.1016/j.amc.2015.10.071.
Lee, H., and S. Xu. 2016b. “Fully discrete error estimation for a quasi-Newtonian fluid–structure interaction problem.” Comput. Math. Appl. 71 (11): 2373–2388. https://doi.org/10.1016/j.camwa.2015.12.024.
Lefrançois, E. 2008. “A simple mesh deformation technique for fluid–structure interaction based on a submesh approach.” Int. J. Numer. Methods Eng. 75 (9): 1085–1101. https://doi.org/10.1002/nme.2284.
Le Tallec, P., and J. Mouro. 2001. “Fluid structure interaction with large structural displacements.” Comput. Methods Appl. Mech. Eng. 190 (24): 3039–3067. https://doi.org/10.1016/S0045-7825(00)00381-9.
Lin, T., and G. R. Liu. 2017. “A development of a GSM-CFD solver for non-Newtonian flows.” Comput. Fluids 142 (Jan): 57–71. https://doi.org/10.1016/j.compfluid.2016.09.009.
Liu, G. R., K. Y. Dai, and T. T. Nguyen. 2007. “A smoothed finite element method for mechanics problems.” Comput. Mech. 39 (6): 859–877. https://doi.org/10.1007/s00466-006-0075-4.
Liu, G. R., and T. T. Nguyen. 2010. Smoothed finite element methods. Boca Raton, FL: CRC Press.
Lukáčová-Medvidová, M., and A. Zaušková. 2008. “Numerical modelling of shear-thinning non-Newtonian flows in compliant vessels.” Int. J. Numer. Methods Fluids 56 (8): 1409–1415. https://doi.org/10.1002/fld.1676.
Luo, X. Y., and T. J. Pedley. 1998. “The effects of wall inertia on flow in a two-dimensional collapsible channel.” J. Fluid Mech. 363 (May): 253–280. https://doi.org/10.1017/S0022112098001062.
Markou, G. A., Z. S. Mouroutis, D. C. Charmpis, and M. Papadrakakis. 2007. “The ortho-semi-torsional (OST) spring analogy method for 3D mesh moving boundary problems.” Comput. Methods Appl. Mech. Eng. 196 (4): 747–765. https://doi.org/10.1016/j.cma.2006.04.009.
Matthies, H. G., and J. Steindorf. 2002. “Partitioned but strongly coupled iteration schemes for nonlinear fluid–structure interaction.” Comput. Struct. 80 (27–30): 1991–1999. https://doi.org/10.1016/S0045-7949(02)00259-6.
Mok, D. P., and W. A. Wall. 2001. “Partitioned analysis schemes for the transient interaction of incompressible flows and nonlinear flexible structures.” In Trends in computational structural mechanics, edited by W. A. Wall, K.-U. Bletzinger, and K. Schweizerhof, 689–698. Barcelona, Spain: International Center for Numerical Methods in Engineering.
Muehlhausen, M.-P., U. Janoske, and H. Oertel. 2015. “Implicit partitioned cardiovascular fluid–structure interaction of the heart cycle using non-Newtonian fluid properties and orthotropic material behavior.” Cardiovasc. Eng. Technol. 6 (1): 8–18. https://doi.org/10.1007/s13239-014-0205-7.
Newmark, N. M. 1959. “A method of computation for structural dynamics.” J. Eng. Mech. 85 (3): 67–94. https://doi.org/10.1061/JMCEA3.0000098.
Nithiarasu, P., R. Codina, and O. C. Zienkiewicz. 2006. “The characteristic-based split (CBS) scheme—A unified approach to fluid dynamics.” Int. J. Numer. Methods Eng. 66 (10): 1514–1546. https://doi.org/10.1002/nme.1698.
Piperno, S. 1997. “Explicit/implicit fluid/structure staggered procedures with a structural predictor and fluid subcycling for 2D inviscid aeroelastic simulations.” Int. J. Numer. Methods Fluids 25 (10): 1207–1226. https://doi.org/10.1002/(SICI)1097-0363(19971130)25:10%3C1207::AID-FLD616%3E3.0.CO;2-R.
Qiao, Y., Y. Zeng, Y. Ding, J. Fan, K. Luo, and T. Zhu. 2019. “Numerical simulation of two-phase non-Newtonian blood flow with fluid–structure interaction in aortic dissection.” Comput. Methods Biomech. Biomed. Eng. 22 (6): 620–630. https://doi.org/10.1080/10255842.2019.1577398.
Shankar, P. N., and M. D. Deshpande. 2000. “Fluid mechanics in the driven cavity.” Annu. Rev. Fluid Mech. 32 (1): 93–136. https://doi.org/10.1146/annurev.fluid.32.1.93.
Thekkethil, N., and A. Sharma. 2019. “Level set function–based immersed interface method and benchmark solutions for fluid flexible-structure interaction.” Int. J. Numer. Methods Fluids 91 (3): 134–157. https://doi.org/10.1002/fld.4746.
Tian, F.-B. 2016. “Deformation of a capsule in a power-law shear flow.” Comput. Math. Methods Med. 2016 (Oct): 1–9. https://doi.org/10.1155/2016/7981386.
Wang, S., B. C. Khoo, G. R. Liu, and G. X. Xu. 2013. “An arbitrary Lagrangian–Eulerian gradient smoothing method (GSM/ALE) for interaction of fluid and a moving rigid body.” Comput. Fluids 71 (Jan): 327–347. https://doi.org/10.1016/j.compfluid.2012.10.028.
Wang, S., B. C. Khoo, G. R. Liu, G. X. Xu, and L. Chen. 2014. “Coupling GSM/ALE with ES-FEM-T3 for fluid–deformable structure interactions.” J. Comput. Phys. 276 (Nov): 315–340. https://doi.org/10.1016/j.jcp.2014.07.016.
Yamada, T., and S. Yoshimura. 2008. “Line search partitioned approach for fluid–structure interaction analysis of flapping wing.” Comput. Model. Eng. Sci. 24 (1): 51–60. https://doi.org/10.3970/cmes.2008.024.051.
Yao, J., G. R. Liu, D. A. Narmoneva, R. B. Hinton, and Z.-Q. Zhang. 2012. “Immersed smoothed finite element method for fluid–structure interaction simulation of aortic valves.” Comput. Mech. 50 (6): 789–804. https://doi.org/10.1007/s00466-012-0781-z.
Yaseen, D. T., and M. A. Ismael. 2020. “Analysis of power law fluid–structure interaction in an open trapezoidal cavity.” Int. J. Mech. Sci. 174 (May): 105481. https://doi.org/10.1016/j.ijmecsci.2020.105481.
Yasuda, K. Y., R. C. Armstrong, and R. E. Cohen. 1981. “Shear flow properties of concentrated solutions of linear and star branched polystyrenes.” Rheol. Acta 20 (2): 163–178. https://doi.org/10.1007/BF01513059.
Yoo, J. W., B. Moran, and J. S. Chen. 2004. “Stabilized conforming nodal integration in the natural-element method.” Int. J. Numer. Methods Eng. 60 (5): 861–890. https://doi.org/10.1002/nme.972.
Zeng, D., and C. R. Ethier. 2005. “A semi-torsional spring analogy model for updating unstructured meshes in 3D moving domains.” Finite Elem. Anal. Des. 41 (11): 1118–1139. https://doi.org/10.1016/j.finel.2005.01.003.
Zeng, W., and G. R. Liu. 2018. “Smoothed finite element methods (S-FEM): An overview and recent developments.” Arch. Comput. Methods Eng. 25 (2): 397–435. https://doi.org/10.1007/s11831-016-9202-3.
Zhang, Z.-Q., G. R. Liu, and B. C. Khoo. 2012. “Immersed smoothed finite element method for two dimensional fluid–structure interaction problems.” Int. J. Numer. Methods Eng. 90 (10): 1292–1320. https://doi.org/10.1002/nme.4299.
Zhang, Z.-Q., G. R. Liu, and B. C. Khoo. 2013. “A three dimensional immersed smoothed finite element method (3D IS-FEM) for fluid–structure interaction problems.” Comput. Mech. 51 (2): 129–150. https://doi.org/10.1007/s00466-012-0710-1.
Zhang, Z.-Q., J. Yao, and G. R. Liu. 2011. “An immersed smoothed finite element method for fluid–structure interaction problems.” Int. J. Comput. Methods 8 (4): 747–757. https://doi.org/10.1142/S0219876211002794.
Zhu, L. 2019. “An IB method for non-Newtonian-fluid flexible-structure interactions in three-dimensions.” Comput. Model. Eng. Sci. 119 (1): 125–143. https://doi.org/10.32604/cmes.2019.04828.
Zhu, L., X. Yu, N. Liu, Y. Cheng, and X. Lu. 2017. “A deformable plate interacting with a non-Newtonian fluid in three dimensions.” Phys. Fluids 29 (8): 083101. https://doi.org/10.1063/1.4996040.
Zienkiewicz, O. C., P. Nithiarasu, R. Codina, M. Vazquez, and P. Ortiz. 1999. “The characteristic-based-split procedure: An efficient and accurate algorithm for fluid problems.” Int. J. Numer. Methods Fluids 31 (1): 359–392. https://doi.org/10.1002/(SICI)1097-0363(19990915)31:1%3C359::AID-FLD984%3E3.0.CO;2-7.
Zorrilla, R., R. Rossi, R. Wüchner, and E. Oñate. 2020. “An embedded finite element framework for the resolution of strongly coupled fluid–structure interaction problems. Application to volumetric and membrane-like structures.” Comput. Methods Appl. Mech. Eng. 368 (Aug): 113179. https://doi.org/10.1016/j.cma.2020.113179.
Information & Authors
Information
Published In
Journal of Engineering Mechanics
Volume 147 • Issue 10 • October 2021
Copyright
© 2021 American Society of Civil Engineers.
History
Received: Dec 22, 2020
Accepted: Apr 5, 2021
Published online: Jul 23, 2021
Published in print: Oct 1, 2021
Discussion open until: Dec 23, 2021
Authors
Metrics & Citations
Metrics
Citations
Download citation
If you have the appropriate software installed, you can download article citation data to the citation manager of your choice. Simply select your manager software from the list below and click Download.
Cited by
- Tao He, Xu-Yan Zhang, Wen-Juan Yao, An Edge-Based Smoothed Finite-Element Method for Vortex-Induced Vibration in Generalized Newtonian Fluids, Journal of Engineering Mechanics, 10.1061/(ASCE)EM.1943-7889.0002164, 148, 11, (2022).
- Tao He, A stabilized cell‐based smoothed finite element method against severe mesh distortion in non‐Newtonian fluid–structure interaction, International Journal for Numerical Methods in Engineering, 10.1002/nme.6930, 123, 9, (2162-2184), (2022).
- Tao He, On the edge‐based smoothed finite element approximation of viscoelastic fluid flows, International Journal for Numerical Methods in Fluids, 10.1002/fld.5060, 94, 5, (423-442), (2022).