# Isogeometric Analysis of Solids in Boundary Representation

• 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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
16. Gomez, H., & De Lorenzis, L. (2016). The variational collocation method. Computer Methods in Applied Mechanics and Engineering, 309, 152–181.
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.
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.
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.
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.
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.
23. Lin, Z., & Liao, S. (2011). The scaled boundary FEM for nonlinear problems. Communications in Nonlinear Science and Numerical Simulation, 16(1), 63–75.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
37. Timoshenko, S. (1951). Theory of elasticity. Engineering societies monographs: McGraw-Hill.