NUMERICAL SIMULATION OF LID-DRIVEN CAVITY FLOW BY ISOGEOMETRIC ANALYSIS

Authors

  • Bohumír Bastl University of West Bohemia, Faculty of Applied Sciences, Department of Mathematics and European Centres of Excellence New Technologies for the Information Society, Univerzitní 8, 301 00, Plzeň, Czech Republic
  • Marek Brandner University of West Bohemia, Faculty of Applied Sciences, Department of Mathematics and European Centres of Excellence New Technologies for the Information Society, Univerzitní 8, 301 00, Plzeň, Czech Republic
  • Jiří Egermaier University of West Bohemia, Faculty of Applied Sciences, Department of Mathematics and European Centres of Excellence New Technologies for the Information Society, Univerzitní 8, 301 00, Plzeň, Czech Republic
  • Hana Horníková University of West Bohemia, Faculty of Applied Sciences, Department of Mathematics and European Centres of Excellence New Technologies for the Information Society, Univerzitní 8, 301 00, Plzeň, Czech Republic
  • Kristýna Michálková University of West Bohemia, Faculty of Applied Sciences, Department of Mathematics and European Centres of Excellence New Technologies for the Information Society, Univerzitní 8, 301 00, Plzeň, Czech Republic
  • Eva Turnerová University of West Bohemia, Faculty of Applied Sciences, Department of Mathematics and European Centres of Excellence New Technologies for the Information Society, Univerzitní 8, 301 00, Plzeň, Czech Republic

DOI:

https://doi.org/10.14311/AP.2021.61.0033

Keywords:

Isogeometric analysis, fluid flow simulation, B-splines, Navier-Stokes equations

Abstract

In this paper, we present numerical results obtained by an in-house incompressible fluid flow solver based on isogeometric analysis (IgA) for the standard benchmark problem for incompressible fluid flow simulation – lid-driven cavity flow. The steady Navier-Stokes equations are solved in their velocity-pressure formulation and we consider only inf-sup stable pairs of B-spline discretization spaces. The main aim of the paper is to compare the results from our IgA-based flow solver with the results obtained by a standard package based on finite element method with respect to degrees of freedom and stability of the solution. Further, the effectiveness of the recently introduced rIgA method for the steady Navier-Stokes equations is studied.
The authors dedicate the paper to Professor K. Kozel on the occasion of his 80th birthday.

References

H. Versteeg, W. Malalasekera. An Introduction to Computational Fluid Dynamics - Second Edition. Prentice Hall, 2007.

Y. Saad. Iterative Methods for Sparse Linear Systems. SIAM, Philadelphia, 2003. doi:10.1137/1.9780898718003.

H. C. Elman, D. Silvester, A. J. Wathen. Finite Elements and Fast Iterative Solvers With Applications in Incompressible Fluid Dynamics. Oxford University Press, 2014. doi:10.1093/acprof:oso/9780199678792.001.0001.

T. Hughes, J. Cottrell, Y. Bazilevs. Isogeometric analysis: CAD, finite elements, NURBS, exact geometry and mesh refinement. Computer Methods in Applied Mechanics and Engineering 194(39–41):4135–4195, 2005. doi:10.1016/j.cma.2004.10.008.

J. Cottrell, T. Hughes, Y. Bazilevs. Isogeometric analysis: Toward integration of CAD and FEA. John Wiley & Sons, Ltd 2009.

L. Piegl, W. Tiller. The NURBS book. Springer Verlag, 1997.

A. Falini, C. Giannelli, T. Kanduc, et al. An adaptive IgA-BEM with hierarchical B-splines based on quasi-interpolation quadrature schemes. International Journal for Numerical Methods in Engineering 117(10):1038–1058, 2018. doi:10.1002/nme.5990.

G. Farin, D. Hansford. Discrete Coons patches. Computer Aided Geometric Design 16:691–700, 1999. doi:10.1016/s0167-8396(99)00031-x.

J. Gravesen, A. Evgrafov, D. Nguyen, P. Nørtoft. Planar parametrization in isogeometric analysis. In International Conference on Mathematical Methods for Curves and Surfaces, pp. 189–212. Springer, 2012. doi:10.1007/978-3-642-54382-1_11.

X. Nian, F. Chen. Planar domain parameterization for isogeometric analysis based on Teichmüller mapping. Computer Methods in Applied Mechanics and Engineering 311:41–55, 2016. doi:10.1016/j.cma.2016.07.035.

A. Mantzaflaris, B. Jüttler, B. Khoromskij, U. Langer. Low rank tensor methods in Galerkin-based isogeometric analysis. Computer Methods in Applied Mechanics and Engineering 316:1062–1085, 2017. doi:10.1016/j.cma.2016.11.013.

M. Aigner, C. Heinrich, B. Jüttler, et al. Swept volume parameterization for isogeometric analysis. In IMA International Conference on Mathematics of Surfaces, pp. 19–44. Springer, 2009. doi:10.1007/978-3-642-03596-8_2.

P. Andel, B. Bastl, K. Slabá. Parameterizations of generalized NURBS volumes of revolution. Engineering Mechanics 19:293–306, 2012.

T. Hughes, A. Reali, G. Sangalli. Duality and unified analysis of discrete approximations in structural dynamics and wave propagation: Comparison of p-method finite elements with k-method NURBS. Computer Methods in Applied Mechanics and Engineering 197:4104–4124, 2008. doi:10.1016/j.cma.2008.04.006.

A. Tagliabue, L. Dedè, A. Quarteroni. Isogeometric Analysis and error estimates for high order partial differential equations in fluid dynamics. Computers & Fluids 102:277–303, 2014. doi:10.1016/j.compfluid.2014.07.002.

D. Benson, Y. Bazilevs, M. Hsu, T. Hughes. Isogeometric shell analysis: The Reissner-Mindlin shell. Computer Methods in Applied Mechanics and Engineering 199(5):276 – 289, 2010. Computational Geometry and Analysis, doi:10.1016/j.cma.2009.05.011.

F. Auricchio, L. B. da Veiga, A. Buffa, et al. A fully “locking-free” isogeometric approach for plane linear elasticity problems: A stream function formulation. Computer Methods in Applied Mechanics and Engineering 197(1):160 – 172, 2007. doi:10.1016/j.cma.2007.07.005.

A. Buffa, G. Sangalli, R. Vázquez. Isogeometric analysis in electromagnetics: B-splines approximation. Computer Methods in Applied Mechanics and Engineering 199(17):1143 – 1152, 2010. doi:10.1016/j.cma.2009.12.002.

H. Gómez, V. M. Calo, Y. Bazilevs, T. J. Hughes. Isogeometric analysis of the Cahn-Hilliard phase-field model. Computer Methods in Applied Mechanics and Engineering 197(49):4333 – 4352, 2008. doi:10.1016/j.cma.2008.05.003.

X. Qian. Full analytical sensitivities in NURBS based isogeometric shape optimization. Computer Methods in Applied Mechanics and Engineering 199(29):2059 – 2071, 2010. doi:10.1016/j.cma.2010.03.005.

N. D. Manh, A. Evgrafov, A. R. Gersborg, J. Gravesen. Isogeometric shape optimization of vibrating membranes. Computer Methods in Applied Mechanics and Engineering 200(13):1343 – 1353, 2011. doi:10.1016/j.cma.2010.12.015.

W. A. Wall, M. A. Frenzel, C. Cyron. Isogeometric structural shape optimization. Computer Methods in Applied Mechanics and Engineering 197(33):2976 – 2988, 2008. doi:10.1016/j.cma.2008.01.025.

Y.-D. Seo, H.-J. Kim, S.-K. Youn. Shape optimization and its extension to topological design based on isogeometric analysis. International Journal of Solids and Structures 47(11):1618 – 1640, 2010. doi:10.1016/j.ijsolstr.2010.03.004.

N. Liu, A. E. Jeffers. Isogeometric analysis of laminated composite and functionally graded sandwich plates based on a layerwise displacement theory. Composite Structures 176:143 – 153, 2017. doi:10.1016/j.compstruct.2017.05.037.

N. Liu, A. E. Jeffers. A geometrically exact isogeometric Kirchhoff plate: Feature-preserving automatic meshing and C1 rational triangular Bézier spline discretizations. International Journal for Numerical Methods in Engineering 115(3):395–409, 2018. https://onlinelibrary.wiley.com/doi/pdf/ 10.1002/nme.5809 doi:10.1002/nme.5809.

N. Liu, X. Ren, J. Lua. An isogeometric continuum shell element for modeling the nonlinear response of functionally graded material structures. Composite Structures 237:111893, 2020. doi:10.1016/j.compstruct.2020.111893.

N. Liu, A. E. Jeffers. Adaptive isogeometric analysis in structural frames using a layer-based discretization to model spread of plasticity. Computers & Structures 196:1 – 11, 2018. doi:10.1016/j.compstruc.2017.10.016.

N. Liu, P. A. Beata, A. E. Jeffers. A mixed isogeometric analysis and control volume approach for heat transfer analysis of nonuniformly heated plates. Numerical Heat Transfer, Part B: Fundamentals 75(6):347–362, 2019. doi:10.1080/10407790.2019.1627801.

N. Liu, A. E. Jeffers. Feature-preserving rational Bézier triangles for isogeometric analysis of higher-order gradient damage models. Computer Methods in Applied Mechanics and Engineering 357:112585, 2019. doi:10.1016/j.cma.2019.112585.

B. Bastl, M. Brandner, J. Egermaier, et al. IgA-based solver for turbulence modelling on multipatch geometries. Advances in Engineering Software 113:7–18, 2017. doi:10.1016/j.advengsoft.2017.06.012.

B. Bastl, M. Brandner, J. Egermaier, et al. Isogeometric analysis for turbulent flow. Mathematics and Computers in Simulation 145:3–17, 2018. doi:10.1016/j.matcom.2016.05.010.

Y. Bazilevs, V. Calo, J. Cottrell, et al. Variational multiscale residual-based turbulence modeling for large eddy simulation of incompressible flows. Computer methods in applied mechanics and engineering 197(1-4):173–201, 2007. doi:10.1016/j.cma.2007.07.016.

S. Hosseini, M. Möller, S. Turek. Isogeometric Analysis of the Navier-Stokes equations with Taylor-Hood B-spline elements. Appl Math Comput 267:264 – 281, 2015. doi:10.1016/j.amc.2015.03.104.

A. Falini, J. Špeh, B. Jüttler. Planar domain parameterization with THB-splines. Computer Aided Geometric Design 35–36:95–108, 2015. doi:10.1016/j.cagd.2015.03.014.

A. Mantzaflaris, B. Jüttler, B. N. Khoromskij, U. Langer. Matrix generation in isogeometric analysis by low rank tensor approximation. In International Conference on Curves and Surfaces, pp. 321–340. 2014. doi:10.1007/978-3-319-22804-4_24.

M. Barton, V. M. Calo. Gauss-Galerkin quadrature rules for quadratic and cubic spline spaces and their application to isogeometric analysis. Computer-Aided Design 82:57–67, 2017. doi:10.1016/j.cad.2016.07.003.

R. R. Hiemstra, G. Sangalli, M. Tani, et al. Fast formation and assembly of finite element matrices with application to isogeometric linear elasticity. Computer Methods in Applied Mechanics and Engineering 355:234–260, 2019. doi:10.1016/j.cma.2019.06.020.

N. Collier, D. Pardo, L. Dalcin, et al. The cost of continuity: A study of the performance of isogeometric finite elements using direct solvers. Comput Methods Appl Mech Engrg 213-216:353–361, 2012. doi:10.1016/j.cma.2011.11.002.

T. A. AbdelMigid, K. M. Saqr, M. A. Kotb, A. A. Aboelfarag. Revisiting the lid-driven cavity flow problem: Review and new steady state benchmarking results using gpu accelerated code. Alexandria Engineering Journal 56:123–135, 2017. doi:10.1016/j.aej.2016.09.013.

J. A. Evans, T. J. R. Hughes. Isogeometric divergence-conforming B-splines for the steady Navier–Stokes equations. Math Models Methods Appl Sci 23(8):1421–1478, 2013. doi:10.21236/ada560496.

V. John, L. Schumacher. A study of isogeometric analysis for scalar convection–diffusion equations. Applied Mathematics Letters 27:43–48, 2014. doi:10.1016/j.aml.2013.08.004.

D. Garcia, D. Pardo, L. Dalcin, et al. The value of continuity: Refined isogeometric analysis and fast direct solvers. Computer Methods in Applied Mechanics and Engineering 316:586–605, 2017. doi:10.1016/j.cma.2016.08.017.

D. Garcia, D. Pardo, V. M. Calo. Refined isogeometric analysis for fluid mechanics and electromagnetics. Computer Methods in Applied Mechanics and Engineering 356:598–628, 2019. doi:10.1016/j.cma.2019.06.011.

F. Brezzi, M. Fortin. Mixed and Hybrid Finite Element Methods. Springer-Verlag, 1991.

A. Bressan, G. Sangalli. Isogeometric discretizations of the Stokes problem: stability analysis by the macroelement technique. IMA Journal of Numerical Analysis 33(2):629–651, 2012. doi:10.1093/imanum/drr056.

P. N. Nielsen, A. R. Gersborg, J. Gravesen, N. L. Pedersen. Discretizations in isogeometric analysis of Navier–Stokes flow. Computer Methods in Applied Mechanics and Engineering 200(45):3242–3253, 2011. doi:10.1016/j.cma.2011.06.007.

V. John, P. Knobloch. On spurious oscillations at layers diminishing (SOLD) methods for convectiondiffusion equations: Part I - A review. Computer Methods in Applied Mechanics and Engineering 196(17):2197 – 2215, 2007. doi:10.1016/j.cma.2006.11.013.

A. N. Brooks, T. J. Hughes. Streamline upwind/Petrov-Galerkin formulations for convection dominated flows with particular emphasis on the incompressible Navier-Stokes equations. Computer methods in applied mechanics and engineering 32(1- 3):199–259, 1982. doi:10.1016/0045-7825(82)90071-8.

K. Takizawa, T. E. Tezduyar, Y. Otoguro. Stabilization and discontinuity-capturing parameters for space–time flow computations with finite element and isogeometric discretizations. Computational Mechanics 62(5):1169–1186, 2018. doi:10.1007/s00466-018-1557-x.

Y. Chai, J. Ouyang. Appropriate stabilized Galerkin approaches for solving two-dimensional coupled Burgers’ equations at high Reynolds numbers. Computers & Mathematics with Applications 79(5):1287–1301, 2019. doi:10.1016/j.camwa.2019.08.036.

V. John, E. Schmeyer. Finite element methods for time-dependent convection–diffusion–reaction equations with small diffusion. Computer methods in applied mechanics and engineering 198(3-4):475–494, 2008. doi:10.1016/j.cma.2008.08.016.

U. Ghia, K. Ghia, C. Shin. High-Re solutions for incompressible flow using the Navier-Stokes equations and a multigrid method. Journal of Computational Physics 48(3):387–411, 1982. doi:10.1016/0021-9991(82)90058-4.

O. Botella, R. Peyret. Benchmark spectral results on the lid-driven cavity flow. Computers & Fluids 27(4):421– 433, 1998. doi:10.1016/s0045-7930(98)00002-4.

E. Erturk, T. C. Corke, C. Gökçöl. Numerical solutions of 2-d steady incompressible driven cavity flow at high reynolds numbers. International Journal for Numerical Methods in Fluids 48(7):747–774, 2005. doi:10.1002/fld.953.

Y. Papadopoulos. A driven cavity exploration. <https://www.acenumerics.com/the-cavitysessions. html> [Online; accessed 30-Jan-2020].

C.-H. Bruneau, M. Saad. The 2D lid-driven cavity problem revisited. Computers & Fluids 35:326–348, 2006. doi:10.1016/j.compfluid.2004.12.004.

E. Erturk. Discussions on driven cavity flow. International Journal for Numerical Methods in Fluids 60(3):275–294, 2009. doi:10.1002/fld.1887.

Z. Cai, Y. Wang. An error estimate for two-dimensional stokes driven cavity flow. Mathematics of computation 78(266):771–787, 2009.

P. Matuszyk, M. Paszynski. Fully automatic hp adaptive finite element method for the Stokes problem in two dimensions. Computer Methods in Applied Mechanics and Engineering 197(51-52):4549–4558, 2008. doi:10.1016/j.cma.2008.05.027.

M. S. Alnaes, J. Blechta, J. Hake, et al. The FEniCS Project Version 1.5. Archive of Numerical Software 3:9–23, 2015.

B. Jüttler, U. Langer, A. Mantzaflaris, et al. Geometry + simulation modules: Implementing isogeometric analysis. In PAMM – Proceedings of Applied Mathematics and Mechanics, vol. 14, pp. 961–962. 2014. doi:10.1002/pamm.201410461.

I. S. Duff, J. K. Reid. The multifrontal solution of indefinite sparse symmetric linear. ACM Trans Math Softw 9(3):302–325, 1983. doi:10.1145/356044.356047.

Downloads

Published

2021-02-10

Issue

Section

Refereed Articles