Abstract
This paper presents a new stabilized form of incompressible Navier–Stokes equations for weak enforcement of Dirichlet boundary conditions at immersed boundaries. The boundary terms are derived via the Variational Multiscale (VMS) method which involves solving the fine-scale variational problem locally within a narrow band along the boundary. The fine-scale model is then variationally embedded into the coarse-scale form that yields a stabilized method which is free of user defined parameters. The derived boundary terms weakly enforce the Dirichlet boundary conditions along the immersed boundaries that may not align with the inter-element edges in the mesh. A unique feature of this rigorous derivation is that the structure of the stabilization tensor which emerges is naturally endowed with the mathematical attributes of area-averaging and stress-averaging. The method is implemented using 4-node quadrilateral and 8-node hexahedral elements. A set of 2D and 3D benchmark problems is presented that investigate the mathematical attributes of the method. These test cases show that the proposed method is mathematically robust as well as computationally stable and accurate for modeling boundary layers around immersed objects in the fluid domain.
Similar content being viewed by others
References
Baiges J, Codina R, Henke F, Shahmiri S, Wall WA (2012) A symmetric method for weakly imposing Dirichlet boundary conditions in embedded finite element meshes. Int J Numer Methods Eng 90:636–658
Bazilevs Y, Michler C, Calo VM, Hughes TJR (2007) Weak Dirichlet boundary conditions for wall-bounded turbulent flows. Comput Methods Appl Mech Eng 196:4853–4862
Bazilevs Y, Calo VM, Cottrell JA, Hughes TJR, Reali A, Scovazzi G (2007) Variational multiscale residual-based turbulence modeling for large eddy simulation of incompressible flows. Comput Methods Appl Mech Eng 197:173–201
Becker R, Burman E, Hansbo P (2009) A Nitsche extended finite element method for incompressible elasticity with discontinuous modulus of elasticity. Comput Methods Appl Mech Eng 198:3352–3360
Burman E, Hansbo P (2012) Fictitious domain finite element methods using cut elements: II. A stabilized Nitsche method. Appl Numer Math 62:328–341
Burman E, Claus S, Hansbo P, Larson MG, Massing A (2015) CutFEM: Discretizing geometry and partial differential equations. Int J Numer Methods Eng 104:472–501
Calderer R, Masud A (2010) A multiscale stabilized ALE formulation for incompressible flows with moving boundaries. Comput Mech 46–1:185–197
Calderer R, Masud A (2013) Residual-based variational multiscale turbulence models for unstructured tetrahedral meshes. Comput Methods Appl Mech Eng 254:238–253
Calhoun D (2002) A Cartesian grid method for solving the two-dimensional streamfunction-vorticity equations in irregular regions. J Comput Phys 176:231–275
Chen P, Truster TJ, Masud A (2018) Interfacial stabilization at finite strains for weak and strong discontinuities in multi-constituent materials. Comput Methods Appl Mech Eng 328:717–751
Choi J-I, Oberoi RC, Edwards JR, Rosati JA (2007) An immersed boundary method for complex incompressible flows. J Comput Phys 224:757–784
Constantinescu GS, Squires KD (2003) LES and DES investigations of turbulent flow over a sphere at Re = 10,000. Flow Turbul Combust 70:267–298
Dauge M, Düster A, Rank E (2015) Theoretical and numerical investigation of finite cell method. J Sci Comput 65:1039–1064
de Prenter F, Verhoosel CV, van Brummelen EH (2019) Preconditioning immersed isogeometric finite element methods with application to flow problems. Comput Methods Appl Mech Eng 348:604–631
Embar A, Dolbow J, Harari I (2010) Imposing Dirichlet boundary conditions with Nitsche’s method and spline-based finite elements. Int J Numer Methods Eng 83:877–898
Hansbo P, Larson MG, Massing A (2017) A stabilized cut finite element method for the Darcy problem on surfaces. Comput Methods Appl Mech Eng 326:298–318
Jeong J, Hussain F (1995) On the identification of a vortex. J Fluid Mech 285:69–94
Johnson TA, Patel VC (1999) Flow past a sphere up to a Reynolds number of 300. J Fluid Mech 378:19–70
Kamensky D, Hsu M-C, Schillinger D, Evans JA, Aggarwal A, Bazilevs Y, Sacks MS, Hughes TJR (2015) An immersogeometric variational framework for fluid–structure interaction: application to bioprosthetic heart valves. Comput Methods Appl Mech Eng 284:1005–1053
Kamensky D, Hsu M-C, Yu Y, Evans JA, Sacks MS, Hughes TJR (2017) Immersogeometric cardiovascular fluid–structure interaction analysis with divergence-conforming B-splines. Comput Methods Appl Mech Eng 314:408–472
Kim J, Kim D, Choi H (2001) An immersed-boundary finite-volume method for simulations of flow in complex geometries. J Comput Phys 171:132–150
Kwack J, Masud A (2014) A stabilized mixed finite element method for shear-rate dependent non-Newtonian fluids: 3D benchmark problems and application to blood flow in bifurcating arteries. Comput Mech 53:751–776
Lee S (2000) A numerical study of the unsteady wake behind a sphere in a uniform flow at moderate Reynolds numbers. Comput Fluids 29:639–667
Lehrenfeld C, Reusken A (2017) Optimal preconditioners for Nitsche-XFEM discretizations of interface problems. Numer Math 135:313–332
Main A, Scovazzi G (2018) The shifted boundary method for embedded domain computations. Part I: Poisson and Stokes problems. J Comput Phys 372:972–995
Main A, Scovazzi G (2018) The shifted boundary method for embedded domain computations. Part II: linear advection–diffusion and incompressible Navier–Stokes equations. J Comput Phys 372:996–1026
Marella S, Krishnan S, Liu H, Udaykumar HS (2005) Sharp interface Cartesian grid method I: an easily implemented technique for 3D moving boundary computations. J Comput Phys 210:1–31
Massing A, Larson MG, Logg A, Rognes ME (2014) A stabilized Nitsche fictitious domain method for the Stokes problem. J Sci Comput 61:604–628
Masud A, Calderer R (2011) A variational multiscale method for incompressible turbulent flows: bubble functions and fine scale fields. Comput Methods Appl Mech Eng 200:2577–2593
Masud A, Calderer R (2013) Residual-based turbulence models for moving boundary flows: hierarchical application of variational multiscale method and three-level scale separation. Int J Numer Methods Fluids 73:284–305
Masud A, Kwack J (2011) A stabilized mixed finite element method for the incompressible shear-rate dependent non-Newtonian fluids: variational Multiscale framework and consistent linearization. Comput Methods Appl Mech Eng 200:577–596
Masud A, Truster TJ, Bergman LA (2012) A unified formulation for interface coupling and frictional contact modeling with embedded error estimation. Int J Numer Methods Eng 92–2:141–177
Mittal R (1999) A Fourier–Chebyshev spectral collocation method for simulating flow past spheres and spheroids. Int J Numer Methods Fluids 30:921–937
Parvizian J, Düster A, Rank E (2007) Finite cell method: h- and p-extension for embedded domain problems in solid mechanics. Comput Mech 41:121–133
Pinelli A, Naqavi IZ, Piomelli U, Favier J (2010) Immersed-boundary methods for general finite-difference and finite-volume Navier–Stokes solvers. J Comput Phys 229:9073–9091
Ploumhans P, Winckelmans GS, Salmon JK, Leonard A, Warren MS (2002) Vortex methods for direct numerical simulation of three-dimensional bluff body flows: application to the sphere at Re = 300, 500, and 1000. J Comput Phys 178:427–463
Pontaza JP, Reddy JN (2003) Spectral/hp least-squares finite element formulation for the Navier–Stokes equations. J Comput Phys 190:523–549
Prabhakar V, Pontaza JP, Reddy JN (2012) A collocation penalty least-squares finite element formulation for incompressible flows. Comput Methods Appl Mech Eng 197:449–463
Rajani BN, Kandasamy A, Majumdar S (2009) Numerical simulation of laminar flow past a circular cylinder. Appl Math Model 33:1228–1247
Schillinger D, Ruess M (2015) The finite cell method: a review in the context of higher-order structural analysis of CAD and image-based geometric models. Arch Comput Methods Eng 22:391–455
Schillinger D, Harari I, Hsu M-C, Kamensky D, Stoter SKF, Yu Y, Zhao Y (2016) The non-symmetric Nitsche method for the parameter-free imposition of weak boundary and coupling conditions in immersed finite elements. Comput Methods Appl Mech Eng 306:625–652
Schott B, Wall WA (2014) A new face-oriented stabilized XFEM approach for 2D and 3D incompressible Navier–Stokes equations. Comput Methods Appl Mech Eng 276:233–265
Song T, Main A, Scovazzi G, Ricchiuto M (2018) The shifted boundary method for hyperbolic systems: embedded domain computations of linear waves and shallow water flows. J Comput Phys 369:45–79
Tomboulides AG, Orszag SA (2000) Numerical investigation of transitional and weak turbulent flow past a sphere. J Fluid Mech 416:45–73
Truster TJ, Masud A (2014) Primal interface formulation for coupling multiple PDEs: a consistent derivation via the Variational Multiscale method. Comput Methods Appl Mech Eng 268:194–224
Truster TJ, Masud A (2016) Discontinuous Galerkin method for frictional interface dynamics. J Eng Mech 142–11:04016084
Truster TJ, Chen P, Masud A (2015) Finite strain primal interface formulation with consistently evolving stabilization. Int J Numer Methods Eng 102:278–315
Tumkur RKR, Calderer R, Masud A, Pearlstein AJ (2013) Computational study of vortex-induced vibration of a sprung rigid circular cylinder with a strongly nonlinear internal attachment. J Fluids Struct 40:214–232
Varduhn V, Hsu M-C, Ruess M, Schillinger D (2016) The tetrahedral finite cell method: higher-order immersogeometric analysis on adaptive non-boundary-fitted meshes. Int J Numer Methods Eng 107:1054–1079
Wang S, Zhang X (2011) An immersed boundary method based on discrete stream function formulation for two- and three-dimensional incompressible flows. J Comput Phys 230:3479–3499
Wang Z, Fan J, Cen K (2009) Immersed boundary method for the simulation of 2D viscous flow based on vorticity–velocity formulations. J Comput Phys 228:1502–1504
Xu F, Schillinger D, Kamensky D, Varduhn V, Wang C, Hsu M-C (2016) The tetrahedral finite cell method for fluids: immersogeometric analysis of turbulent flow around complex geometries. Comput Fluids 141:135–154
Zhu L, Goraya SA, Masud A (2019) Interface-capturing method for free-surface plunging and breaking waves. J Eng Mech 145(11):04019088
Acknowledgements
This research was partly supported by NIH Grant No. 1R01GN135921-01, and Carle-UIUC Foundation Grant No. SPA-088888. Computing resources were provided by NSF under Grant AC121/DMS 16-20231, for Advanced Computing on the Frontera Supercomputing Platform housed at the Texas Advanced Computing Center (TACC), at The University of Texas at Austin. This support is gratefully acknowledged.
Author information
Authors and Affiliations
Corresponding author
Ethics declarations
Conflict of interest
The authors declare that they have no conflict of interest.
Additional information
In honor of Professor J.N. Reddy for his 75th Birthday.
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Appendix: Interfacial edge basis functions
Appendix: Interfacial edge basis functions
In the derivation of the VMIB formulation (31) we have assumed that the interfacial fine scales vanish at the element edges, except for the case when the interface itself is coincident with the element edge. In the former case, interface traverses through the element and therefore a part of the element belongs to the fluid subdomain while the other part belongs to the fictitious subdomain. Consequently, the interface function is non-zero within the element and becomes zero at the element boundaries, and therefore retains the definition of a bubble function. However, when the interface \( \varGamma_{I}^{e} \) is coincident with element edge Γe, the interface function becomes an edge function. For the 1D case, the interface basis functions are shown in Fig. 30.
The design guideline is to modify the regular bubble function by moving its peak in accordance with the location of the interface. In our implementation for 2D quadrilateral and 3D hexahedral elements, the fine-scale basis function \( b_{I}^{e} \) is obtained from the underlying conventional polynomial bubble function be by shifting the position of its peak point from the center of the element to the middle point of the interface segment.The fine-scale basis function \( b_{I}^{e} \) is defined as
where
where \( \xi_{i} { (}i = 1,2,3 ) \) are the natural coordinates in the parent element and \( \bar{\xi }_{i} { (}i = 1,2,3 ) \) are the natural coordinates for the middle point of the interface segment \( \varGamma_{I}^{e} \). The 1D basis functions \( b_{I}^{e} (\xi ) = \psi (\xi ) \) are presented in Fig. 30.
Rights and permissions
About this article
Cite this article
Kang, S., Masud, A. A Variational Multiscale method with immersed boundary conditions for incompressible flows. Meccanica 56, 1397–1422 (2021). https://doi.org/10.1007/s11012-020-01227-w
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11012-020-01227-w