Abstract
Isogeometric analysis was applied very successfully to many problem classes like linear elasticity, heat transfer and incompressible flow problems but its application to compressible flows is very rare. However, its ability to accurately represent complex geometries used in industrial applications makes IGA a suitable tool for the analysis of compressible flow problems that require the accurate resolution of boundary layers. The convection-diffusion solver presented in this chapter, is an indispensable step on the way to developing a compressible solver for complex viscous industrial flows. It is well known that the standard Galerkin finite element method and its isogeometric counterpart suffer from spurious oscillatory behaviour in the presence of shocks and steep solution gradients. As a remedy, the algebraic flux correction paradigm is generalized to B-Spline basis functions to suppress the creation of oscillations and occurrence of non-physical values in the solution. This work provides early results for scalar conservation laws and lays the foundation for extending this approach to the compressible Euler equations in the next chapter.
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References
Hughes, T.J.R., Cottrell, J.A., Bazilevs, Y.: Isogeometric analysis: CAD, finite elements, NURBS, exact geometry and mesh refinement. Comput. Methods Appl. Mech. Eng. 194(39–41), 4135–4195 (2005)
Cottrell, J.A., Hughes, T.J.R., Bazilevs, T.J.R.: Isogeometric Analysis: Toward Integration of CAD and FEA. Wiley, Hoboken (2009)
Trontin, P.: Isogeometric analysis of Euler compressible flow. Application to aerodynamics. Conference: 50th AIAA Aerospace Sciences Meeting Including the New Horizons Forum and Aerospace Exposition (2012)
Jaeschke, A.: Isogeometric analysis for compressible flows with application in turbomachinery. Master’s Thesis, Delft University of Technology (2015)
Brooks, A.N., Hughes, T.J.R.: Streamline upwind/Petrov-Galerkin formulations for convection dominated flows with particular emphasis on the incompressible Navier-Stokes equations. Comput. Methods Appl. Mech. Eng. 32(1–3), 199–259 (1982)
Kuzmin, D., Turek, S.: Flux correction tools for finite elements. J. Comput. Phys. 175(2), 525–558 (2002)
Kuzmin, D., Turek, S.: High-resolution FEM-TVD schemes based on a fully multidimensional flux limiter. J. Comput. Phys. 198(1), 131–158 (2004)
Kuzmin, D., Möller, M.: Flux-corrected transport, chapter algebraic flux correction I. In: Scalar Conservation Laws. Springer, Berlin (2005)
Kuzmin, D.: On the design of general-purpose flux limiters for finite element schemes. I. Scalar Convection. J. Comput. Phys. 219(2), 513–531 (2006)
Kuzmin, D.: Algebraic flux correction for finite element discretizations of coupled systems. In: Proceedings of the International Conference on Computational Methods for Coupled Problems in Science and Engineering II, CIMNE, Barcelona, pp. 653–656 (2007)
Kuzmin, D., Möller, M., Shadid, J.N., Shashkov, M.: Failsafe flux limiting and constrained data projections for equations of gas dynamics. J. Comput. phys. 229(23), 8766–8779, 11 (2010)
Kuzmin, D.: Flux-corrected transport, chapter Algebraic flux correction I. In: Scalar Conservation Laws. Springer, Berlin (2012)
Kuzmin, D., Möller, M., Gurris, M.: Flux-corrected transport, chapter Algebraic flux correction II. Compressible Flow Problems. Springer, Berlin (2012)
de Boor, C.: Subroutine package for calculating with B-splines. Technical Report LA-4728-MS. Los Alamos National Laboratory, New Mexico (1971)
Fletcher, C.A.J.: The group finite element formulation. Comput. Methods Appl. Mech. Eng. 37, 225–243 (1983)
Farago, I., Horvath, R., Korotov, S.: Discrete maximum principle for linear parabolic problems solved on hybrid meshes. Appl. Numer. Math. 53(2–4), 249–264, 05 (2004)
Kuzmin, D., Shashkov, M.J., Svyatski, D.: A constrained finite element method satisfying the discrete maximum principle for anisotropic diffusion problems on arbitrary meshes. J. Comput. Phys. 228, 3448–3463 (2009)
Godunov, S.K.: Finite difference method for numerical computation of discontinuous solutions of the equations of fluid dynamics. Mat. Sbornik 47, 271–306 (1959)
Walker, H.F., Ni, P.: Anderson acceleration for fixed-point iterations. SIAM J. Numer. Anal. 49(4),1715–1735, 08 (2011)
Möller, M.: Efficient solution techniques for implicit finite element schemes with flux limiters. Int. J. Numer. Methods Fluids 55(7), 611–635, 11 (2007)
Jüttler, B., Langer, U., Mantzaflaris, A., Moore, A., Zulehner, W.: Geometry + simulation modules: implementing isogeometric analysis. Proc. Appl. Math. Mech. 14(1), 961–962 (2014); Special Issue: 85th Annual Meeting of the Int. Assoc. of Appl. Math. and Mech. (GAMM), Erlangen (2014)
Giannelli, C., Jüttler, B., Speleers, H.: Thb-splines: the truncated basis for hierarchical splines. Comput. Aided Geom. Des. 29(7), 485–498, 10 (2012)
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This work has been supported by the European Unions Horizon 2020 research and innovation programme under grant agreement No. 678727.
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Jaeschke, A., Möller, M. (2020). High-Order Isogeometric Methods for Compressible Flows. In: van Brummelen, H., Corsini, A., Perotto, S., Rozza, G. (eds) Numerical Methods for Flows. Lecture Notes in Computational Science and Engineering, vol 132. Springer, Cham. https://doi.org/10.1007/978-3-030-30705-9_3
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DOI: https://doi.org/10.1007/978-3-030-30705-9_3
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