Advertisement

Isogeometric Analysis of Solids in Boundary Representation

  • Sven KlinkelEmail author
  • Margarita Chasapi
Chapter
Part of the CISM International Centre for Mechanical Sciences book series (CISM, volume 597)

Abstract

In this chapter, we present boundary-oriented numerical methods to analyze three-dimensional solid structures. For the analysis, the original geometry of the solid is employed according to the isogeometric paradigm. For the parametrization of the domain, the idea of the scaled boundary finite element method is adopted. Hence, the boundary of the solid is sufficient to describe the entire domain. The presented approaches employ analytical and numerical solution methods such as the Galerkin and collocation methods. To illustrate the applicability in the analysis procedure, three formulations are elaborated and demonstrated by means of numerical examples. The advantages compared to standard numerical methods are discussed thoroughly.

References

  1. Abaqus. (2001). 6.7. User’s manual. Dassault Systemes.Google Scholar
  2. Apostolatos, A., Schmidt, R., Wüchner, R., & Bletzinger, K. U. (2014). A Nitsche-type formulation and comparison of the most common domain decomposition methods in isogeometric analysis. International Journal for Numerical Methods in Engineering, 97(7), 473–504.MathSciNetCrossRefGoogle Scholar
  3. Auricchio, F., da Veiga, L. B., Hughes, T. J. R., Reali, A., & Sangalli, G. (2010). Isogeometric collocation methods. Mathematical Models and Methods in Applied Sciences, 20(11), 2075–2107.MathSciNetCrossRefGoogle Scholar
  4. Auricchio, F., da Veiga, L. B., Hughes, T. J. R., Reali, A., & Sangalli, G. (2012). Isogeometric collocation for elastostatics and explicit dynamics. Computer Methods in Applied Mechanics and Engineering, 249, 2–14.MathSciNetCrossRefGoogle Scholar
  5. Bazilevs, Y., Long, C. C., Akkerman, I., Benson, D. J., & Shashkov, M. J. (2014). Isogeometric analysis of lagrangian hydrodynamics: Axisymmetric formulation in the rz-cylindrical coordinates. Journal of Computational Physics, 262, 244–261.MathSciNetCrossRefGoogle Scholar
  6. Behnke, R., Mundil, M., Birk, C., & Kaliske, M. (2014). A physically and geometrically nonlinear scaled-boundary-based finite element formulation for fracture in elastomers. International Journal for Numerical Methods in Engineering, 99, 966–999.MathSciNetCrossRefGoogle Scholar
  7. Breitenberger, M., Apostolatos, A., Philipp, B., Wüchner, R., & Bletzinger, K. U. (2015). Analysis in computer aided design: Nonlinear isogeometric B-Rep analysis of shell structures. Computer Methods in Applied Mechanics and Engineering, 284, 401–457.MathSciNetCrossRefGoogle Scholar
  8. Chasapi, M., & Klinkel, S. (2018). A scaled boundary isogeometric formulation for the elasto-plastic analysis of solids in boundary representation. Computer Methods in Applied Mechanics and Engineering, 333, 475–496.MathSciNetCrossRefGoogle Scholar
  9. Chen, L., Dornisch, W., & Klinkel, S. (2015). Hybrid collocation-Galerkin approach for the analysis of surface represented 3D-solids employing SB-FEM. Computer Methods in Applied Mechanics and Engineering, 295, 268–289.MathSciNetCrossRefGoogle Scholar
  10. Chen, L., Simeon, B., & Klinkel, S. (2016). A NURBS based Galerkin approach for the analysis of solids in boundary representation. Computer Methods in Applied Mechanics and Engineering, 305, 777–805.MathSciNetCrossRefGoogle Scholar
  11. Cottrell, J. A., Hughes, T. J. R., & Bazilevs, Y. (2009). Isogeometric analysis: Toward integration of CAD and FEA. John Wiley & Sons.Google Scholar
  12. De Lorenzis, L., Evans, J. A., Hughes, T. J. R., & Reali, A. (2015). Isogeometric collocation: Neumann boundary conditions and contact. Computer Methods in Applied Mechanics and Engineering, 282, 21–54.MathSciNetCrossRefGoogle Scholar
  13. Dornisch, W., Klinkel, S., & Simeon, B. (2013). Isogeometric Reissner-Mindlin shell analysis with exactly calculated director vectors. Computer Methods in Applied Mechanics and Engineering, 253, 491–504.MathSciNetCrossRefGoogle Scholar
  14. Dornisch, W., Vitucci, G., & Klinkel, S. (2015). The weak substitution method - an application of the mortar method for patch coupling in NURBS-based isogeometric analysis. International Journal for Numerical Methods in Engineering, 103, 205–234.MathSciNetCrossRefGoogle Scholar
  15. Düster, A., Parvizian, J., Yang, Z., & Rank, E. (2008). The finite cell method for three-dimensional problems of solid mechanics. Computer Methods in Applied Mechanics and Engineering, 197(45), 3768–3782.MathSciNetCrossRefGoogle Scholar
  16. Gomez, H., & De Lorenzis, L. (2016). The variational collocation method. Computer Methods in Applied Mechanics and Engineering, 309, 152–181.MathSciNetCrossRefGoogle Scholar
  17. Hughes, T. J. R. (2000). The finite element method: Linear static and dynamic finite element analysis. Courier Dover Publications.Google Scholar
  18. Hughes, T. J. R., Cottrell, J. A., & Bazilevs, Y. (2005). Isogeometric analysis: CAD, finite elements, NURBS, exact geometry and mesh refinement. Computer Methods in Applied Mechanics and Engineering, 194, 4135–4195.MathSciNetCrossRefGoogle Scholar
  19. Hughes, T. J. R., Reali, A., & Sangalli, G. (2010). Efficient quadrature for NURBS-based isogeometric analysis: Computational Geometry and Analysis. Computer Methods in Applied Mechanics and Engineering, 199, 301–313.MathSciNetCrossRefGoogle Scholar
  20. Kiendl, J., Auricchio, F., da Veiga, L. B., Lovadina, C., & Reali, A. (2015). Isogeometric collocation methods for the Reissner-Mindlin plate problem. Computer Methods in Applied Mechanics and Engineering, 284, 489–507.MathSciNetCrossRefGoogle Scholar
  21. Klinkel, S., Chen, L., & Dornisch, W. (2015). A NURBS based hybrid collocation-Galerkin method for the analysis of boundary represented solids. Computer Methods in Applied Mechanics and Engineering, 284, 689–711.MathSciNetCrossRefGoogle Scholar
  22. Lin, G., Zhang, Y., Hu, Z., & Zhong, H. (2014). Scaled boundary isogeometric analysis for 2D elastostatics. Science China Physics, Mechanics and Astronomy, 57(3), 286–300.CrossRefGoogle Scholar
  23. Lin, Z., & Liao, S. (2011). The scaled boundary FEM for nonlinear problems. Communications in Nonlinear Science and Numerical Simulation, 16(1), 63–75.MathSciNetCrossRefGoogle Scholar
  24. Natarajan, S., Wang, J. C., Song, C., & Birk, C. (2015). Isogeometric analysis enhanced by the scaled boundary finite element method. Computer Methods in Applied Mechanics and Engineering, 283, 733–762.MathSciNetCrossRefGoogle Scholar
  25. Ooi, E., Song, C., & Tin-Loi, F. (2014). A scaled boundary polygon formulation for elasto-plastic analyses. Computer Methods in Applied Mechanics and Engineering, 268, 905–937.MathSciNetCrossRefGoogle Scholar
  26. Piegl, L. & Tiller, W. (1997). The NURBS book. Monographs in visual communications. Springer.Google Scholar
  27. Rank, E., Ruess, M., Kollmannsberger, S., Schillinger, D., & Düster, A. (2012). Geometric modeling, isogeometric analysis and the finite cell method. Computer Methods in Applied Mechanics and Engineering, 249, 104–115.CrossRefGoogle Scholar
  28. Reali, A., & Gomez, H. (2015). An isogeometric collocation approach for Bernoulli-Euler beams and Kirchhoff plates. Computer Methods in Applied Mechanics and Engineering, 284, 623–636.MathSciNetCrossRefGoogle Scholar
  29. Ruess, M., Schillinger, D., Özcan, A. I., & Rank, E. (2014). Weak coupling for isogeometric analysis of non-matching and trimmed multi-patch geometries. Computer Methods in Applied Mechanics and Engineering, 269, 46–71.MathSciNetCrossRefGoogle Scholar
  30. Schillinger, D., Evans, J. A., Reali, A., Scott, M. A., & Hughes, T. J. R. (2013). Isogeometric collocation: Cost comparison with Galerkin methods and extension to adaptive hierarchical NURBS discretizations. Computer Methods in Applied Mechanics and Engineering, 267, 170–232.MathSciNetCrossRefGoogle Scholar
  31. Schmidt, R., Wüchner, R., & Bletzinger, K. U. (2012). Isogeometric analysis of trimmed NURBS geometries. Computer Methods in Applied Mechanics and Engineering, 241–244, 93–111.MathSciNetCrossRefGoogle Scholar
  32. Song, C. (2004). A matrix function solution for the scaled boundary finite-element equation in statics. Computer Methods in Applied Mechanics and Engineering, 193(23), 2325–2356.MathSciNetCrossRefGoogle Scholar
  33. Song, C., & Wolf, J. P. (1997). The scaled boundary finite-element method–alias consistent infinitesimal finite-element cell method–for elastodynamics. Computer Methods in Applied Mechanics and Engineering, 147, 329–355.MathSciNetCrossRefGoogle Scholar
  34. Song, C., & Wolf, J. P. (1998). The scaled boundary finite-element method: analytical solution in frequency domain. Computer Methods in Applied Mechanics and Engineering, 164(1–2), 249–264.MathSciNetCrossRefGoogle Scholar
  35. Stroud, I. (2006). Boundary representation modelling techniques. Springer.Google Scholar
  36. Temizer, I., Wriggers, P., & Hughes, T. J. R. (2012). Three-dimensional mortar-based frictional contact treatment in isogeometric analysis with NURBS. Computer Methods in Applied Mechanics and Engineering, 209, 115–128.MathSciNetCrossRefGoogle Scholar
  37. Timoshenko, S. (1951). Theory of elasticity. Engineering societies monographs: McGraw-Hill.zbMATHGoogle Scholar

Copyright information

© CISM International Centre for Mechanical Sciences 2020

Authors and Affiliations

  1. 1.RWTH Aachen UniversityAachenGermany

Personalised recommendations