Skip to main content
SpringerLink
Log in

Multi-level hp-adaptivity: high-order mesh adaptivity without the difficulties of constraining hanging nodes

  • Original Paper
  • Published:
Computational Mechanics Aims and scope Submit manuscript

Abstract

The implementation of \(hp\)-adaptivity is challenging as hanging nodes, edges, and faces have to be constrained to ensure compatibility of the shape functions. For this reason, most \(hp\)-code frameworks restrict themselves to \(1\)-irregular meshes to ease the implementational effort. This work alleviates these difficulties by introducing a new formulation for high-order mesh adaptivity that provides full local \(hp\)-refinement capabilities at a comparably small implementational effort. Its main idea is the extension of the \(hp\)-\(d\)-method such that it allows for high-order overlay meshes yielding a hierarchical, multi-level \(hp\)-formulation of the Finite Element Method. This concept enables intuitive refinement and coarsening procedures, while linear independence and compatibility of the shape functions are guaranteed by construction. The proposed method is demonstrated to achieve exponential rates of convergence—both in terms of degrees of freedom and in run-time—for problems with non-smooth solutions. Furthermore, the scheme is used alongside the Finite Cell Method to simulate the heat flow around moving objects on a non-conforming background mesh and is combined with an energy-based refinement indicator for automatic \(hp\)-adaptivity.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9
Fig. 10
Fig. 11
Fig. 12
Fig. 13
Fig. 14
Fig. 15
Fig. 16
Fig. 17
Fig. 18
Fig. 19
Fig. 20
Fig. 21

Similar content being viewed by others

References

  1. Demkowicz L (2007) Computing with hp-adaptive finite elements, Vol. 1: one and two dimensional elliptic and maxwell problems., Applied mathematics and nonlinear science seriesChapman & Hall/CRC, Boca Raton

    Google Scholar 

  2. Solin P (2004) Higher-order finite element methods., Studies in advanced mathematicsChapman & Hall/CRC, Boca Raton

    MATH  Google Scholar 

  3. Demkowicz L, Oden JT, Rachowicz W, Hardy O (1989) Toward a universal h-p adaptive finite element strategy, part 1. Constrained approximation and data structure. Comput Methods Appl Mech Eng 77(1–2):79–112

    Article  MATH  MathSciNet  Google Scholar 

  4. Rachowicz W, Oden JT, Demkowicz L (1989) Toward a universal h-p adaptive finite element strategy part 3. Design of h-p meshes. Comput Methods Appl Mech Eng 77(1–2):181–212

    Article  MATH  MathSciNet  Google Scholar 

  5. Demkowicz L, Gerdes K, Schwab C, Bajer A, Walsh T (1998) HP90: a general and flexible Fortran 90 hp-FE code. Comput Vis Sci 1(3):145–163

    Article  MATH  Google Scholar 

  6. Demkowicz L, Bajer A, Rachowicz W, Gerdes K (1999) 3D hp-adaptive finite element package Fortran 90 implementation (3Dhp90), TICAM Report 99–29. The University of Texas at Austin, Texas Institute for Computational and Applied Mathematics

  7. Rachowicz W, Demkowicz L (2000) An hp-adaptive finite element method for electromagnetics: part 1: data structure and constrained approximation. Comput Methods Appl Mech Eng 187(1–2): 307–335

    Article  MATH  Google Scholar 

  8. Rachowicz W, Demkowicz L (2002) An hp-adaptive finite element method for electromagnetics—part II: a 3D implementation. Int J Numer Methods Eng 53(1):147–180

    Article  MATH  MathSciNet  Google Scholar 

  9. Paszyński M, Demkowicz L (2006) Parallel, fully automatic hp-adaptive 3D finite element package. Eng Comput 22(3–4):255–276

    Article  Google Scholar 

  10. Solin P, Cerveny J (2006) Automatic hp-adaptivity with arbitrary-level hanging nodes, Tech Rep Research Report No. 2006–07, The University of Texas at El Paso, Department of Mathematical Sciences

  11. Solin P, Cerveny J, Dolezel I (2008) Arbitrary-level hanging nodes and automatic adaptivity in the hp-FEM. Math Comput Simul 77(1):117–132

    Article  MATH  MathSciNet  Google Scholar 

  12. Kus P (2011) Automatic hp-adaptivity on meshes with arbitrary-level hanging nodes in 3D. Phd thesis, Charles University, Institute of Mathematics, Prague

  13. Schröder A (2011) Subdivisions and multi-level hanging nodes. In: Hesthaven JS, Ronquist EM (eds) Spectral and high order methods for partial differential equations. no. 76 in lecture notes in computational science and engineering. Springer, Heidelberg, pp 317–325

    Chapter  Google Scholar 

  14. Rank E (1992) Adaptive remeshing and h-p domain decomposition. Comput Methods Appl Mech Eng 101(1–3):299–313

    Article  MATH  Google Scholar 

  15. Schillinger D, Rank E (2011) An unfitted hp-adaptive finite element method based on hierarchical B-splines for interface problems of complex geometry. Comput Methods Appl Mech Eng 200(47–48):3358–3380

    Article  MATH  MathSciNet  Google Scholar 

  16. Schillinger D (2012) The p- and B-spline versions of the geometrically nonlinear finite cell method and hierarchical refinement strategies for adaptive isogeometric and embedded domain analysis. Doctoral thesis, Technische Universität München, Chair for Computation in Engineering

  17. Schillinger D, Düster A, Rank E (2012) The hp-d-adaptive finite cell method for geometrically nonlinear problems of solid mechanics. Int J Numer Methods Eng 89(9):1171–1202

    Article  MATH  Google Scholar 

  18. Schillinger D, Dedè L, Scott MA, Evans JA, Borden MJ, Rank E, Hughes TJ (2012) An isogeometric design-through-analysis methodology based on adaptive hierarchical refinement of NURBS, immersed boundary methods, and T-spline CAD surfaces. Comput Methods Appl Mech Eng 249–252:116–150

    Article  Google Scholar 

  19. Joulaian M, Düster A (2013) Local enrichment of the finite cell method for problems with material interfaces. Comput Mech 52(4):741–762

    Article  MATH  Google Scholar 

  20. Zienkiewicz O, Taylor R, Zhu J (2005) The finite element method: its basis and fundamentals, 6th edn. Butterworth-heinemann, Oxford

    Google Scholar 

  21. Franke DC, Düster A, Nübel V, Rank E (2010) A comparison of the h-, p-, hp-, and rp-version of the FEM for the solution of the 2D Hertzian contact problem. Computat Mech 45(5):513–522

    Article  MATH  Google Scholar 

  22. Hughes TJR (2000) The finite element method: linear static and dynamic finite element analysis. Dover Publications, Mineola

    Google Scholar 

  23. Bathe KJ (2007) Finite element procedures. Prentice Hall, Englewood Cliffs

    Google Scholar 

  24. Strang G (1973) An analysis of the finite element method. Prentice-Hall, Englewood Cliffs

    MATH  Google Scholar 

  25. Felippa CA (2013) Introduction to Finite Element Methods

  26. Ainsworth M, Senior B (1997) Aspects of an adaptive hp-finite element method: adaptive strategy, conforming approximation and efficient solvers. Comput Methods Appl Mech Eng 150(1–4): 65–87

    Article  MATH  MathSciNet  Google Scholar 

  27. Rivara M (1984) Mesh refinement processes based on the generalized bisection of simplices. SIAM J Numer Anal 21(3):604–613

    Article  MATH  MathSciNet  Google Scholar 

  28. Mitchell WF (1989) A comparison of adaptive refinement techniques for elliptic problems. ACM Trans Math Softw 15(4): 326–347

    Article  MATH  Google Scholar 

  29. Babuška I, Aziz A (1976) On the angle condition in the finite element method. SIAM J Numer Anal 13(2):214–226

    Article  MATH  MathSciNet  Google Scholar 

  30. Solin P, Dubcova L, Dolezel I (2010) Adaptive hp-FEM with arbitrary-level hanging nodes for Maxwell’s equations. Adv Appl Math Mech 2(4):518–532

    MathSciNet  Google Scholar 

  31. Schneiders R (2000) Algorithms for quadrilateral and hexahedral mesh generation. Proceedings of the VKI Lecture Series on Computational Fluid Dynamic, VKI-LS, vol. 4

  32. Niekamp R, Stein E (2002) An object-oriented approach for parallel two- and three-dimensional adaptive finite element computations. Comput Struct 80(3–4):317–328

    Article  Google Scholar 

  33. Fries T-P, Byfut A, Alizada A, Cheng KW, Schröder A (2011) Hanging nodes and XFEM. Int J Numer Methods Eng 86(4–5): 404–430

    Article  MATH  Google Scholar 

  34. Cheng K-W, Fries T-P (2012) XFEM with hanging nodes for two-phase incompressible flow. Comput Methods Appl Mech Eng 245–246:290–312

    Article  MathSciNet  Google Scholar 

  35. Szymczak A, Paszyńska A, Paszyński M, Pardo D (2013) Preventing deadlock during anisotropic 2D mesh adaptation in hp-adaptive FEM. J Comput Sci 4(3):170–179

    Article  Google Scholar 

  36. Mote CD (1971) Global-local finite element. Int J Numer Methods Eng 3(4):565–574

    Article  MATH  MathSciNet  Google Scholar 

  37. Noor AK (1986) Global-local methodologies and their application to nonlinear analysis. Finite Elem Anal Design 2(4):333–346

    Article  Google Scholar 

  38. Zienkiewicz OC, Craig A (1986) Adaptive refinement, error estimates, multigrid solution and hierarchic finite element method concepts. In: Babuska I, Zienkiewicz OC, Gago J, de Oliviera ER (eds) Accuracy estimates and adaptive refinements in finite element calculations. Wiley, New York, pp 25–55

    Google Scholar 

  39. Belytschko T, Fish J, Engelmann BE (1988) A finite element with embedded localization zones. Comput Methods Appl Mech Eng 70(1):59–89

    Article  MATH  Google Scholar 

  40. Fish J, Belytschko T (1988) Elements with embedded localization zones for large deformation problems. Comput Struct 30(1–2):247–256

    Article  MATH  Google Scholar 

  41. Fish J, Belytschko T (1990) A finite element with a unidirectionally enriched strain field for localization analysis. Comput Methods Appl Mech Eng 78(2):181–200

    Article  MATH  MathSciNet  Google Scholar 

  42. Belytschko T, Fish J, Bayliss A (1990) The spectral overlay on finite elements for problems with high gradients. Comput Methods Appl Mech Eng 81(1):71–89

    Article  MATH  Google Scholar 

  43. Rank E, Krause R (1997) A multiscale finite-element method. Comput Struct 64(1):139–144

    Article  MATH  Google Scholar 

  44. Fish J (1992) The s-version of the finite element method. Comput Struct 43(3):539–547

    Article  MATH  Google Scholar 

  45. Kim YH, Levit I, Stanley G (1991) A finite element adaptive mesh refinement technique that avoids multipoint constraints and transition zones. In: Parsons I, Nour-Omid B (eds) Iterative equation solvers for structural problems, vol CED-4. ASME, New York, pp 27–35

    Google Scholar 

  46. Fish J (1992) Hierarchical modelling of discontinuous fields. Commun Appl Numer Methods 8(7):443–453

    Article  MATH  Google Scholar 

  47. Fish J, Markolefas S (1993) Adaptive s-method for linear elastostatics. Comput Methods Appl Mech Eng 104(3):363–396

    Article  MATH  Google Scholar 

  48. Fish J, Markolefas S, Guttal R, Nayak P (1994) On adaptive multilevel superposition of finite element meshes for linear elastostatics. Appl Numer Math 14(1–3):135–164

    Article  MATH  MathSciNet  Google Scholar 

  49. Moore PK, Flaherty JE (1992) Adaptive local overlapping grid methods for parabolic systems in two space dimensions. J Comput Phys 98(1):54–63

    Article  MATH  MathSciNet  Google Scholar 

  50. Babuška I, Melenk JM (1997) The partition of Unity Method. Int J Numer Methods Eng 40(4):727–758

    Article  MATH  Google Scholar 

  51. Strouboulis T, Babuška I, Copps K (2000) The design and analysis of the generalized finite element method. Comput Methods Appl Mech Eng 181(1–3):43–69

    Article  MATH  Google Scholar 

  52. Fries T-P, Belytschko T (2010) The extended/generalized finite element method: an overview of the method and its applications. Int J Numer Methods Eng 84(3):253–304

    MATH  MathSciNet  Google Scholar 

  53. Krause R, Rank E (2003) Multiscale computations with a combination of the h- and p-versions of the finite-element method. Comput Methods Appl Mech Eng 192(35–36):3959–3983

    Article  MATH  Google Scholar 

  54. Düster A, Niggl A, Rank E (2007) Applying the hp–d version of the FEM to locally enhance dimensionally reduced models. Comput Methods Appl Mech Eng 196(37–40):3524–3533

    Article  MATH  Google Scholar 

  55. Schillinger D, Evans JA, Reali A, Scott MA, Hughes TJR (2013) Isogeometric collocation: cost comparison with Galerkin methods and extension to adaptive hierarchical NURBS discretizations. Comput Methods Appl Mech Eng 267:170–232

    Article  MATH  MathSciNet  Google Scholar 

  56. Szabó BA, Babuska I (1991) Finite element analysis. Wiley & Sons, New York

  57. Rank E, Zienkiewicz OC (1987) A simple error estimator in the finite element method. Commun Appl Numer Methods 3(3):243–249

    Article  MATH  Google Scholar 

  58. Szabó BA, Düster A, Rank E (2004) The p-version of the finite element method. In: Stein E (ed) Encyclopedia of computational mechanics. John Wiley & Sons Ltd, Chichester

    Google Scholar 

  59. Shewchuk JR (1994) An introduction to the conjugate gradient method without the agonizing pain. Tech Rep

  60. Parvizian J, Düster A, Rank E (2007) Finite cell method. Comput Mech 41(1):121–133

    Article  MATH  MathSciNet  Google Scholar 

  61. Düster A, Parvizian J, Yang Z, Rank E (2008) The finite cell method for three-dimensional problems of solid mechanics. Comput Methods Appl Mech Eng 197(45–48):3768–3782

    Article  MATH  Google Scholar 

  62. Zander N, Bog T, Elhaddad M, Espinoza R, Hu H, Joly A, Wu C, Zerbe P, Düster A, Kollmannsberger S, Parvizian J, Ruess M, Schillinger D, Rank E (2014) FCMLab: a finite cell research toolbox for MATLAB. Adv Eng Softw 74:49–63

    Article  Google Scholar 

  63. Parvizian J, Düster A, Rank E (2011) Topology optimization using the finite cell method. Optim Eng 13(1):57–78

    Article  Google Scholar 

  64. Schillinger D, Ruess M, Zander N, Bazilevs Y, Düster A, Rank E (2012) Small and large deformation analysis with the p- and B-spline versions of the finite cell method. Comput Mech 50(4):445–478

    Article  MATH  MathSciNet  Google Scholar 

  65. Yang Z, Ruess M, Kollmannsberger S, Düster A, Rank E (2012) An efficient integration technique for the voxel-based finite cell method. Int J Numer Methods Eng 91(5):457–471

    Article  Google Scholar 

  66. Yang Z, Kollmannsberger S, Düster A, Ruess M, Garcia EG, Burgkart R, Rank E (2012) Non-standard bone simulation: interactive numerical analysis by computational steering. Comput Vis Sci 14(5):207–216

    Article  Google Scholar 

  67. Ruess M, Tal D, Trabelsi N, Yosibash Z, Rank E (2012) The finite cell method for bone simulations: verification and validation. Biomech Model Mechanobiol 11(3–4):425–437

    Article  Google Scholar 

  68. Abedian A, Parvizian J, Düster A, Rank E (2013) The finite cell method for the J2 flow theory of plasticity. Finite Elem Anal Design 69:37–47

    Article  Google Scholar 

  69. Abedian A, Parvizian J, Düster A, Rank E (2014) Finite cell method compared to h-version finite element method for elasto-plastic problems. Appl Math Mech 35(10):1239–1248

    Article  MathSciNet  Google Scholar 

  70. Duczek S, Joulaian M, Düster A, Gabbert U (2013) Simulation of Lamb waves using the spectral cell method. pp. 86951U–86951U-11

  71. Cai Q (2013) Finite Cell Method for Transport Problems in Porous Media. Doctoral thesis, Technische Universität München, Munich

  72. Rank E, Kollmannsberger S, Sorger C, Düster A (2011) Shell finite cell method: a high order fictitious domain approach for thin-walled structures. Comput Methods Appl Mech Eng 200(45–46):3200–3209

    Article  MATH  Google Scholar 

  73. Rank E, Ruess M, Kollmannsberger S, Schillinger D, Düster A (2012) Geometric modeling, isogeometric analysis and the finite cell method. Comput Methods Appl Mech Eng 249–252:104–115

    Article  Google Scholar 

  74. Ruess M, Bazilevs Y, Schillinger D, Zander N, Rank E (2012) Weakly enforced boundary conditions for the NURBS-based Finite Cell Method. In: European Congress on Computational Methods in Applied Sciences and Engineering (ECCOMAS). Austria, Vienna

  75. Ruess M, Schillinger D, Bazilevs Y, Varduhn V, Rank E (2013) Weakly enforced essential boundary conditions for NURBS-embedded and trimmed NURBS geometries on the basis of the finite cell method. Int J Numer Methods Eng 95(10):811–846

    Article  MathSciNet  Google Scholar 

  76. Ruess M, Schillinger D, Özcan AI, Rank E (2014) Weak coupling for isogeometric analysis of non-matching and trimmed multi-patch geometries. Comput Methods Appl Mech Eng 269:46–71

    Article  MATH  Google Scholar 

  77. Zander N, Kollmannsberger S, Ruess M, Yosibash Z, Rank E (2012) The finite cell method for linear thermoelasticity. Comput Math Appl 64(11):3527–3541

    Article  MATH  MathSciNet  Google Scholar 

  78. Dauge M, Düster A, Rank E (2014) Theoretical and numerical investigation of the finite cell method. Tech Rep hal-00850602, version 2, Université de Rennes

  79. Schillinger D, Ruess M (2014) 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, pp. 1–65

  80. Sadd MH (2009) Elasticity theory, applications and numerics. Elsevier Butterworth-Heinemann, Burlington

    Google Scholar 

  81. Mitchell WF (2010) The hp-multigrid method applied to hp-adaptive refinement of triangular grids. Numer Linear Algebr Appl 17(2–3):211–228

    MATH  Google Scholar 

  82. Hu N, Guo X-Z, Katz IN (1997) Multi-p preconditioners. SIAM J Sci Comput 18(6):1676–1697

    Article  MATH  MathSciNet  Google Scholar 

  83. Wilson EL (1974) The static condensation algorithm. Int J Numer Methods Eng 8(1):198–203

    Article  Google Scholar 

  84. Szabó BA (1986) Estimation and control of error based on p convergence. In: Babuska I, Zienkiewicz OC, Gago J, de Oliviera ER (eds) Accuracy estimates and adaptive refinements in finite element calculations. Wiley, New York, pp 25–55

    Google Scholar 

  85. Niekamp R, Stein E (2001) The hierarchically graded multilevel finite element method. Comput Mech 27(4):302–304

    Article  MathSciNet  Google Scholar 

Download references

Acknowledgments

The first and the last author gratefully acknowledge the financial support of the German Research Foundation (DFG) under Grants RA 624/19-2 and RA 624/22-1.

Author information

Authors and Affiliations

  1. Chair for Computation in Engineering, Technische Universität München, Arcisstr. 21, 80333, München, Germany

    Nils Zander, Tino Bog, Stefan Kollmannsberger & Ernst Rank

  2. Department of Civil Engineering, University of Minnesota, 500 Pillsbury Drive S.E., Minneapolis, MN, 55455, USA

    Dominik Schillinger

Authors
  1. Nils Zander
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Tino Bog
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Stefan Kollmannsberger
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Dominik Schillinger
    View author publications

    You can also search for this author in PubMed Google Scholar

  5. Ernst Rank
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Nils Zander.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Zander, N., Bog, T., Kollmannsberger, S. et al. Multi-level hp-adaptivity: high-order mesh adaptivity without the difficulties of constraining hanging nodes. Comput Mech 55, 499–517 (2015). https://doi.org/10.1007/s00466-014-1118-x

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00466-014-1118-x

Keywords

Search

Navigation