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Edge-based anisotropic mesh adaptation of unstructured meshes with applications to compressible flows

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Abstract

An adaptation procedure using refinement, coarsening and smoothing with anisotropic considerations for unstructured meshes and its coupling with a compressible flow solver is presented in this work. The developed refinement strategy combines a Riemannian centred split of the edges. The coarsening methodology uses an edge collapse procedure where the collapse point is evaluated using anisotropic quality considerations. An accurate and anisotropic vertex smoothing algorithm that operates along the direction of the edges is presented. Edge and face swapping are also used together with the refinement coarsening and smoothing strategies to enhance anisotropic quality. The developed approach makes use of only simple mesh information and is applicable to two and three-dimensional meshes. The performance of the methodology is analysed for two-dimensional triangular meshes and three-dimensional tetrahedral meshes on analytical test cases and to solve steady-state inviscid compressible flows. The transonic flow past an airfoil, the supersonic flow around a sphere and the flow around a complex real-life application are performed.

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References

  1. Frey PJ, Alauzet F (2005) Anisotropic mesh adaptation for CFD computations. Comput Methods Appl Mech Eng 194(48–49):5068–5082

    Article  MathSciNet  MATH  Google Scholar 

  2. Stein E, Rust W (1991) Mesh adaptations for linear 2d finite-element discretizations in structural mechanics, especially in thin shell analysis. J Comput Appl Math 36(1):107–129

    Article  MATH  Google Scholar 

  3. Klein RI (1999) Star formation with 3-D adaptive mesh refinement: the collapse and fragmentation of molecular clouds. J Comput Appl Math 109(1–2):123–152

    Article  MATH  Google Scholar 

  4. Frey PJ (2004) Generation and adaptation of computational surface meshes from discrete anatomical data. Int J Numer Methods Eng 60(6):1049–1074

    Article  MATH  Google Scholar 

  5. Castro-Diaz MJ, Hecht F, Mohammadi B, Pironneau O (1997) Anisotropic unstructured mesh adaptation for flow simulation. Int J Numer Methods Fluids 25(4):475–491

    Article  MATH  Google Scholar 

  6. Vallet MG (1992) Generation de maillages elements finis anisotropes et adaptatifs. Ph.D. Thesis, Universite Pierre et Marie Currie, Paris VI, France

  7. Loseille A, Alauzet F (2011) Continuous mesh framework part I: well-posed continuous interpolation error. SIAM J Numer Anal 49(1):38–60

    Article  MathSciNet  MATH  Google Scholar 

  8. Bottasso CL (2004) Anisotropic mesh adaption by metric-driven optimization. Int J Numer Methods Eng 60(3):597–639

    Article  MathSciNet  MATH  Google Scholar 

  9. Alauzet F (2014) A changing-topology moving mesh technique for large displacements. Eng Comput 30(2):175–200

    Article  Google Scholar 

  10. Alexandra C, Ducrot V (2009) Levelsets and anisotropic mesh adaptation. Discrete Contin Dyn Syst 23(1–2):165–183

    MathSciNet  MATH  Google Scholar 

  11. Loseille A, Löhner R (2009) On 3D anisotropic local remeshing for surface, volume, and boundary layers. In: Proceedings in 18th international meshing roundtable, Springer-Verlag, Salt-Lake City, UT, USA, pp 611–630

  12. Quan D-L, Toulorge T, Bricteux G, Remacle J-F, Marchandise E (2014) Anisotropic adaptive nearly body-fitted meshes for CFD. Eng Comput 30(4):517–533

    Article  Google Scholar 

  13. Peraire J, Peiro J, Morgan K (1992) Adaptive remeshing for three dimensional compressible flow computation. J Comput Phys 103(2):269–285

    Article  MATH  Google Scholar 

  14. Borouchaki H, George PL, Hecht F, Laug P, Saltel E (1997) Delaunay mesh generation governed by metric specificationspart I: algorithms. Finite Elements Anal Design 25(1–2):61–83

    Article  MathSciNet  MATH  Google Scholar 

  15. Löhner R, Baum JD (1990) Numerical simulation of shock interactions with complex geometry three-dimensional structures using a new adaptive h-refinement scheme on unstructured grids. In: Proceedings in 28th aerospace science meeting, AIAA-90-0700

  16. Liu A, Joe B (1996) Quality local refinements of tetrahedral meshed based on 8-subtetrahedron subdivision. Math Comput 65(215):1183–1200

    Article  MATH  Google Scholar 

  17. Rivara MC (1997) New longest-edge algorithms for the refinement and/or improvement of unstructured triangulations. Int J Numer Methods Eng 40(18):3313–3324

    Article  MathSciNet  MATH  Google Scholar 

  18. Bey J (1995) Tetrahedral grid refinement. Computing 55(4):355–378

    Article  MathSciNet  MATH  Google Scholar 

  19. Maubach JM (1995) Local bisection refinement for n-simplicial grids generated by reflection. SIAM J Sci Comput 16(1):210–227

    Article  MathSciNet  MATH  Google Scholar 

  20. De Cougny HL, Shephard MS (1999) Parallel refinement and coarsening of tetrahedral meshes. Int J Numer Methods Eng 46:1101–1125

    Article  MathSciNet  MATH  Google Scholar 

  21. Rassineux A, Breitkopf P, Villon P (2003) Simultaneous surface and tetrahedron mesh adaptation using mesh-free techniques. Int J Numer Methods Eng 57:371–389

    Article  MATH  Google Scholar 

  22. Habashi WG, Dompierre J, Bourgault Y, Ait-Ali-Yahia D, Fortin M, Vallet MG (2000) Anisotropic mesh adaptation: towards user-independent, mesh-independent and solver independent CFD. part i: general principles (no. 6). J Numer Methods Fluids 32:725–744

    Article  MathSciNet  MATH  Google Scholar 

  23. Pain CC, Umpleby AP, de Oliveira CRE (2001) Tetrahedral mesh optimization and adaptivity for steady-state and transient finite element calculations. Comput Methods Appl Mech Eng 190:3771–3796

    Article  MATH  Google Scholar 

  24. Waltz J (2004) Parallel adaptive refinement for unsteady flow calculations on 3D unstructured grids. Int J Numer Methods Fluids 46:37–57

    Article  MATH  Google Scholar 

  25. Rivara M-C (1989) Selective refinement/derefinement algorithms for sequences of nested triangulations. Int J Numer Methods Eng 28(12):2889–2906

    Article  MathSciNet  MATH  Google Scholar 

  26. Ollivier-Gooch C (2003) Coarsening unstructured meshes by edge contraction. Int J Numer Methods Eng 57(3):391–414

    Article  MATH  Google Scholar 

  27. Walter MAT, Abdu AAQ, da Silva LFF, Azevedo JL (2005) Evaluation of adaptive mesh refinement and coarsening for the computation of compressible flow on unstructured meshes. Int J Numer Methods Fluids 49(9):999–1014

    Article  MATH  Google Scholar 

  28. Jones MT, Plassmann PE (1997) Adaptive refinement of unstructured finite-element meshes. Finite Elements Anal Design 25(1–2):41–60

    Article  MathSciNet  MATH  Google Scholar 

  29. Li X, Shephard MS, Beall MW (2003) Accounting for curved domains in mesh adaptation. Int J Numer Methods Eng 58(2):247–276

    Article  MATH  Google Scholar 

  30. Freitag LA, Ollivier-Gooch C (1997) Tetrahedral mesh improvement using swapping and smoothing. Int J Numer Methods Eng 40(21):3979–4002

    Article  MathSciNet  MATH  Google Scholar 

  31. Tam A, Ait-Ali-Yahia D, Robichaud MP, Moore M, Kozel V, Habashi WG (2000) Anisotropic mesh adaptation for 3d flows on unstructured grids. Comput Methods Appl Mech Eng 189(4):1025–1230

    Article  MATH  Google Scholar 

  32. Sastry SP, Shontz SM (2012) Performance characterization of nonlinear optimization methods for mesh quality improvement. Eng Comput 28(3):269–286

    Article  Google Scholar 

  33. Jones W, Nielsen E, Park M (2006) Validation of 3d adjoint based error estimation and mesh adaptation for sonic boom prediction. In: Proceedings in 44th aerospace science meeting and exhibit, AIAA-2006-1150

  34. Alauzet F, Olivier G (2011) Extension of metric-based anisotropic mesh adaptation to time-dependent problems involving moving geometries. In: Proceedings in 49th aerospace science meeting and exhibit, AIAA-2011-0896

  35. Zienkiewicz OC, Codina R (1995) A general algorithm for compressible and incompressible flow-part I: the split, characteristic-based scheme. Int J Numer Methods Fluids 20(8–9):869–885

    Article  MATH  Google Scholar 

  36. Zienkiewicz OC, Taylor RL, Nithiarasu P (2005) The finite element method for fluid dynamics, 6th edn. Butterworth-Heinemann, Oxford

    MATH  Google Scholar 

  37. Vallet MG, Manole CM, Dompierre J, Dufour S, Guibault F (2007) Numerical comparison of some Hessian recovery techniques. Int J Numer Methods Eng 72(8):987–1007

    Article  MathSciNet  MATH  Google Scholar 

  38. Loseille A, Alauzet F (2009) Optimal 3D highly anisotropic mesh adaptation based on the continuous mesh framework. In: Proceedings in 18th international meshing roundtable, Springer-Verlag, Salt-Lake City, UT, USA, pp 575–594

  39. Ruprecht D, Muller H (1998) A Scheme for edge-based adaptive tetrahedron subdivision. Mathematical visualization. Springer, Berlin

    MATH  Google Scholar 

  40. Alauzet F, Mehrenberger M (2010) P1-conservative solution interpolation on unstructured triangular meshes. Int J Numer Methods Eng 84(13):1552–1588

    Article  MATH  Google Scholar 

  41. Farrell PE, Piggott MD, Pain CC, Gorman GJ, Wilson CR (2009) Conservative interpolation between unstructured meshes via supermesh construction. Comput Methods Appl Mech Eng 84(33–36):2632–2642

    Article  MathSciNet  MATH  Google Scholar 

  42. Nithiarasu P, Codina R, Zienkiewicz OC (2006) The characteristic-based split (CBS) scheme-a unified approach to fluid dynamics. Int J Numer Methods Eng 66(10):1514–1546

    Article  MathSciNet  MATH  Google Scholar 

  43. Nithiarasu P, Zienkiewicz OC, Satya-Sai BVK, Morgan K, Vzquez M (1998) Shock capturing viscosities for the general fluid mechanics algorithm. Int J Numer Methods Fluids 28(9):1325–1353

    Article  MATH  Google Scholar 

  44. Hetmaniuk U, Knupp P (2011) A mesh optimization algorithm to decrease the maximum interpolation error of linear triangular finite elements. Eng Comput 27(1):3–15

    Article  Google Scholar 

  45. Yoshihara H, Sacher P (1985) Test cases for Inviscid flow field methods. AGARD-AR-221, Paris

  46. Bailey A, Hiatt J (1971) Free-flight measurements of sphere drag at subsonic, transonic, supersonic, and hypersonic speeds for continuum, transition, and near free molecular flow conditions. Arnold Engng Development Ctr., Arnold Air Force Station, TN, Rep. AEDC-TR-70-291

  47. Charters A, Thomas R (1945) The aerodynamic performance of small spheres from subsonic to high supersonic velocities. J Aeronaut Sci 12(4):468–476

    Article  Google Scholar 

  48. Nazarov N, Hoffman J (2010) An adaptive finite element method for inviscid compressible flow. Int J Numer Methods Fluids 64(10–12):1102–1128

    Article  MathSciNet  MATH  Google Scholar 

Download references

Acknowledgements

The authors wish to thank to CAPES and CNPq (Brazilian Research Committees) for their financial support and they thank also to CESUP (UFRGS Supercomputing Center) for its important contributions.

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  1. PPGEC/UFRGS, 99 Osvaldo Aranha Ave. 3rd floor, Porto Alegre, RS, 90035-190, Brazil

    Renato V. Linn & Armando M. Awruch

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  1. Renato V. Linn
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Correspondence to Renato V. Linn.

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Linn, R.V., Awruch, A.M. Edge-based anisotropic mesh adaptation of unstructured meshes with applications to compressible flows. Engineering with Computers 33, 1007–1025 (2017). https://doi.org/10.1007/s00366-017-0513-2

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