Computational Mechanics

, Volume 60, Issue 1, pp 101–116 | Cite as

A new formulation for air-blast fluid–structure interaction using an immersed approach: part II—coupling of IGA and meshfree discretizations

  • Y. Bazilevs
  • G. Moutsanidis
  • J. Bueno
  • K. Kamran
  • D. Kamensky
  • M. C. Hillman
  • H. Gomez
  • J. S. Chen
Original Paper


In this two-part paper we begin the development of a new class of methods for modeling fluid–structure interaction (FSI) phenomena for air blast. We aim to develop accurate, robust, and practical computational methodology, which is capable of modeling the dynamics of air blast coupled with the structure response, where the latter involves large, inelastic deformations and disintegration into fragments. An immersed approach is adopted, which leads to an a-priori monolithic FSI formulation with intrinsic contact detection between solid objects, and without formal restrictions on the solid motions. In Part I of this paper, the core air-blast FSI methodology suitable for a variety of discretizations is presented and tested using standard finite elements. Part II of this paper focuses on a particular instantiation of the proposed framework, which couples isogeometric analysis (IGA) based on non-uniform rational B-splines and a reproducing-kernel particle method (RKPM), which is a meshfree technique. The combination of IGA and RKPM is felt to be particularly attractive for the problem class of interest due to the higher-order accuracy and smoothness of both discretizations, and relative simplicity of RKPM in handling fragmentation scenarios. A collection of mostly 2D numerical examples is presented in each of the parts to illustrate the good performance of the proposed air-blast FSI framework.


Air blast FSI Immersed methods FEM IGA RKPM 



YB was partially supported by the ARO W911NF-14-1-0296 award. DK was supported by AFOSR Award FA9550-12-1-0046. This support is gratefully acknowledged. The IGA-RKPM coupling was carried out in PetIGA, a software framework that implements NURBS-based IGA [42].


  1. 1.
    Hughes TJR, Cottrell JA, Bazilevs Y (2005) Isogeometric analysis: CAD, finite elements, NURBS, exact geometry, and mesh refinement. Comput Methods Appl Mech Eng 194:4135–4195MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Cottrell JA, Hughes TJR, Bazilevs Y (2009) Isogeometric analysis. Toward integration of CAD and FEA. Wiley, New YorkCrossRefGoogle Scholar
  3. 3.
    Farin GE (1999) NURBS: from projective geometry to practical use. AK Peters Ltd, NatickzbMATHGoogle Scholar
  4. 4.
    Piegl L, Tiller W (2012) The NURBS book. Springer, BerlinzbMATHGoogle Scholar
  5. 5.
    Rogers DF (2000) An introduction to NURBS: with historical perspective. Elsevier, AmsterdamGoogle Scholar
  6. 6.
    Chen JS, Belytschko T (2015) Meshless and meshfree methods. In: Encyclopedia of applied and computational mathematics. Springer, Berlin, pp 886–894Google Scholar
  7. 7.
    Liu WK, Jun S, Zhang YF (1995) Reproducing kernel particle methods. Int J Numer Methods Fluids 20(8–9):1081–1106MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Liu WK, Jun S, Li S, Adee J, Belytschko T (1995) Reproducing kernel particle methods for structural dynamics. Int J Numer Methods Eng 38(10):1655–1679MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Chen JS, Pan C, Wu C-T, Liu WK (1996) Reproducing kernel particle methods for large deformation analysis of non-linear structures. Comput Methods Appl Mech Eng 139(1):195–227MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    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 197:173–201MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Cottrell JA, Reali A, Bazilevs Y, Hughes TJR (2006) Isogeometric analysis of structural vibrations. Comput Methods Appl Mech Eng 195(41):5257–5296MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Lipton S, Evans JA, Bazilevs Y, Elguedj T, Hughes TJR (2010) Robustness of isogeometric structural discretizations under severe mesh distortion. Comput Methods Appl Mech Eng 199:357–373CrossRefzbMATHGoogle Scholar
  13. 13.
    Bardenhagen SG, Kober EM (2004) The generalized interpolation material point method. Comput Model Eng Sci 5(6):477–495Google Scholar
  14. 14.
    Bazilevs Y, Beirao da Veiga L, Cottrell JA, Hughes TJR, Sangalli G (2006) Isogeometric analysis: approximation, stability and error estimates for h-refined meshes. Math Models Methods Appl Sci 16(07):1031–1090MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Cottrell JA, Hughes TJR, Bazilevs Y (2009) Isogeometric analysis: toward integration of CAD and FEA. Wiley, New YorkCrossRefGoogle Scholar
  16. 16.
    Bazilevs Y, Calo VM, Zhang Y, Hughes TJR (2006) Isogeometric fluid–structure interaction analysis with applications to arterial blood flow. Comput Mech 38(4–5):310–322MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Bazilevs Y, Calo VM, Hughes TJR, Zhang Y (2008) Isogeometric fluid–structure interaction: theory, algorithms, and computations. Comput Mech 43(1):3–37MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Benson DJ, Bazilevs Y, Hsu M-C, Hughes TJR (2010) Isogeometric shell analysis: the Reissner–Mindlin shell. Comput Methods Appl Mech Eng 199(5):276–289MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Cox G (1972) The numerical evaluation of b-splines. IMA J Appl Math 10(2):134–149MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    De Boor C (1972) On calculating with b-splines. J Approx Theory 6(1):50–62MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Beirao da Veiga L, Cho D, Sangalli G (2012) Anisotropic NURBS approximation in isogeometric analysis. Comput Methods Appl Mech Eng 209–212:1–11MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Hughes TJR, Reali A, Sangalli G (2010) Efficient quadrature for NURBS-based isogeometric analysis. Comput Methods Appl Mech Eng 199:301–313MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Auricchio F, Calabrò F, Hughes TJR, Reali A, Sangalli G (2012) A simple algorithm for obtaining nearly optimal quadrature rules for NURBS-based isogeometric analysis. Comput Methods Appl Mech Eng 249–252:15–27MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    Nitsche J (1971) Über ein Variationsprinzip zur Lösung von Dirichlet-Problemen bei Verwendung von Teilräumen, die keinen Randbedingungen unterworfen sind. Abhandlungen aus dem Mathematischen Seminar der Universität Hamburg 36:9–15MathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    Chen JS, Liu WK, Hillman M, Chi SW, Lian Y, Bessa MA (2016) Reproducing kernel approximation and discretization. In: Stein E, de Borst R, Hughes TJR (eds) Encyclopedia of computational mechanics, 2nd edn. Wiley, New York (in press)Google Scholar
  26. 26.
    Liu WK, Li S, Belytschko T (1997) Moving least-square reproducing kernel methods (I): methodology and convergence. Comput Methods Appl Mech Eng 143(1–2):113–154MathSciNetCrossRefzbMATHGoogle Scholar
  27. 27.
    Chen J-S, Wu C-T, Yoon S, You Y (2001) A stabilized conforming nodal integration for Galerkin mesh-free methods. Int J Numer Methods Eng 50:435–466CrossRefzbMATHGoogle Scholar
  28. 28.
    Chen JS, Hillman M, Rüter M (2013) An arbitrary order variationally consistent integration for galerkin meshfree methods. Int J Numer Methods Eng 95(5):387–418MathSciNetCrossRefzbMATHGoogle Scholar
  29. 29.
    Guan PC, Chi S-W, Chen JS, Slawson TR, Roth MJ (2011) Semi-lagrangian reproducing kernel particle method for fragment-impact problems. Int J Impact Eng 38(12):1033–1047CrossRefGoogle Scholar
  30. 30.
    Hillman M, Chen JS, Chi SW (2014) Stabilized and variationally consistent nodal integration for meshfree modeling of impact problems. Comput Part Mech 1:245–256CrossRefGoogle Scholar
  31. 31.
    Hillman M, Chen JS, Bazilevs Y (2015) Variationally consistent domain integration for isogeometric analysis. Comput Methods Appl Mech Eng 284:521–540MathSciNetCrossRefGoogle Scholar
  32. 32.
    Hillman M, Chen JS (2016) An accelerated, convergent, and stable nodal integration in galerkin meshfree methods for linear and nonlinear mechanics. Int J Numer Methods Eng 107:603–630MathSciNetCrossRefzbMATHGoogle Scholar
  33. 33.
    Chen JS, Zhang X, Belytschko T (2004) An implicit gradient model by a reproducing kernel strain regularization in strain localization problems. Comput Methods Appl Mech Eng 193(27–29):2827–2844CrossRefzbMATHGoogle Scholar
  34. 34.
    Massing A, Larson MG, Logg A, Rognes ME (2014) A stabilized Nitsche fictitious domain method for the Stokes problem. J Sci Comput 61(3):604–628MathSciNetCrossRefzbMATHGoogle Scholar
  35. 35.
    Burman E (2010) Ghost penalty. Comptes Rendus Mathematique 348:1217–1220MathSciNetCrossRefzbMATHGoogle Scholar
  36. 36.
    Dettmer WG, Kadapa C, Peric D (2016) A stabilised immersed boundary method on hierarchical B-spline grids. Comput Methods Appl Mech Eng 311:415–437MathSciNetCrossRefGoogle Scholar
  37. 37.
    Belytschko T, Liu WK, Moran B (2000) Nonlinear finite elements for continua and structures. Wiley, New YorkzbMATHGoogle Scholar
  38. 38.
    Sod GA (1978) A survey of several finite difference methods for systems of nonlinear hyperbolic conservation laws. J Comput Phys 27:1–31MathSciNetCrossRefzbMATHGoogle Scholar
  39. 39.
    Sedov LI (1993) Similarity and dimensional methods in mechanics, 10th edn. CRC Press, Boca RatonGoogle Scholar
  40. 40.
    Giordano J, Jourdan G, Burtschell Y, Medale M, Zeitoun DE, Houas L (2005) Shock wave impacts on deforming panel, an application of fluid–structure interaction. Shock Waves 14(1–2):103–110Google Scholar
  41. 41.
    Deiterding R, Wood S (2013) Parallel adaptive fluid–structure interaction simulation of explosions impacting on building structures. Comput Fluids 88:719–729CrossRefGoogle Scholar
  42. 42.
    Collier N, Dalcin L, Calo VM (2013) PetIGA: high-performance isogeometric analysis. arXiv:1305.4452

Copyright information

© Springer-Verlag Berlin Heidelberg 2017

Authors and Affiliations

  • Y. Bazilevs
    • 1
  • G. Moutsanidis
    • 1
  • J. Bueno
    • 3
  • K. Kamran
    • 1
  • D. Kamensky
    • 1
  • M. C. Hillman
    • 2
  • H. Gomez
    • 4
  • J. S. Chen
    • 1
  1. 1.Department of Structural EngineeringUniversity of California, San DiegoLa JollaUSA
  2. 2.Department of Civil and Environmental EngineeringPennsylvania State UniversityState CollegeUSA
  3. 3.Departamento de Metodos Matematicos e de RepresentacionUniversidade da CoruñaA CoruñaSpain
  4. 4.School of Mechanical EngineeringPurdue UniversityWest LafayetteUSA

Personalised recommendations