On contact modelling in isogeometric analysis
DOI:
https://doi.org/10.1080/17797179.2017.1354575Keywords:
Isogeometric analysis, NURBS, contact analysis, sheet metal formingAbstract
IsoGeometric Analysis (IGA) has proved to be a reliable numerical tool for the simulation of structural behaviour and fluid mechanics. The main reasons for this popularity are essentially due to: (i) the possibility of using higher order polynomials for the basis functions; (ii) the high convergence rates possible to achieve; (iii) the possibility to operate directly on CAD geometry without the need to resort to a mesh of elements. The major drawback of IGA is the non-interpolatory characteristic of the basis functions, which adds a difficulty in handling essential boundary conditions and makes it particularly challenging for contact analysis. In this work, the IGA is expanded to include frictionless contact procedures for sheet metal forming analyses. Non-Uniform Rational B-Splines (NURBS) are going to be used for the modelling of rigid tools as well as for the modelling of the deformable blank sheet. The contactmethods developed are based on a two-step contact search scheme, where during the first step a global search algorithm is used for the allocation of contact knots into potential contact faces and a second (local) contact search scheme where point inversion techniques are used for the calculation of the contact penetration gap. For completeness, elastoplastic procedures are also included for a proper description of the entire IGA of sheet metal forming processes.
Downloads
References
Bazilevs, Y., Calo, V. M., Cottrell, J. A., Evans, J. A., Hughes, T. J. R., Lipton, S., ... Sederberg, T.
W. (2010). Isogeometric analysis using T-Splines. Computer Methods in AppliedMechanics
and Engineering, 199, 229–263.
Benson, D. J., Bazilevs, Y., Hsu, M. C., & Hughes, T. J. R. (2010). Isogeometric shell analysis:
The Reissner–Mindlin shell. Computer Methods in Applied Mechanics and Engineering,
, 276–289.
Cardoso, R. P. R. (2002). Development of one point quadrature shell elements with anisotropic
material models for sheet metal forming analysis (PhD thesis), University of Aveiro, Portugal.
Cardoso, R. P. R., & Yoon, J.W. (2005). One point quadrature shell elements for sheet metal
forming analysis. Archives of Computational Methods in Engineering, 12, 3–66.
Cardoso, R. P. R., & Yoon, J. W. (2007). One point quadrature shell elements: A study on
convergence and patch tests. Computational Mechanics, 40, 871–883.
Cardoso, R. P. R., & Cesar de Sa, J. M. A. (2012). The enhanced assumed strainmethod for the
isogeometric analysis of nearly incompressible deformation of solids. International Journal
for Numerical Methods in Engineering, 92, 56–78.
Cardoso, R. P. R., & Cesar de Sa, J. M. A. (2014). Blending moving least squares techniques
withNURBSbasis functions for nonlinear isogeometric analysis. ComputationalMechanics,
, 1327–1340.
Cesar de Sa, J.M.A.,&Owen,D. R. J. (1986). The imposition of the incompressible constraints
in finite elements – A review of approaches with a new insight to the locking phenomenon.
In Numerical methods for non-linear problems. Pineridge Press.
Cesar de Sa, J. M. A., & Natal Jorge, R. M. (1999). New enhanced strain elements for
incompressible problems. International Journal for Numerical Methods in Engineering, 44,
–248.
Cottrell, J. A., Reali, A., Bazilevs, Y., & Hughes, T. J. R. (2006). Isogeometric analysis of
structural vibrations. Computer Methods in AppliedMechanics and Engineering, 195, 5257–
Cottrell, J. A., Hughes, T. J. R., & Reali, A. (2007). Studies of refinement and continuity in
isogeometric structural analysis. Computer Methods in AppliedMechanics and Engineering,
, 4160–4183.
Cottrell, J. A., Hughes, T. J. R., & Bazilevs, Y. (2009). Isogeometric analysis, toward integration
of CAD and FEA. Wiley.
Deseri, L., & Owen, D. R. (2002). Invertible structured deformations and the geometry of
multiple slip in single crystals. International Journal of Plasticity, 18, 833–849.
Deseri, L., & Owen, D. R. (2016). Submacroscopic disarrangements induce a unique, additive
and universal decomposition of continuum fluxes. Journal of Elasticity, 122, 223–230.
Echter, R., & Bischoff, M. (2010). Numerical efficiency, locking and unlocking of NURBS
finite elements. Computer Methods in Applied Mechanics and Engineering, 199, 374–382.
Hughes, T. J. R., Cohen, M., & Haroun, M. (1978). Reduced and selective integration
techniques in finite element analysis of plates. Nuclear Engineering Design, 46, 203–22.
Hughes, T. J. R. (2000). The finite element method: Linear static and dynamic finite element
analysis (2nd ed.). Englewood Cliffs, NJ: Prentice-Hall.
Hughes, T. J. R., Cottrel, 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.
Hughes, T. J. R., Reali, A., & Sangalli, G. (2008). Duality and unified analysis of discrete
approximations in structural dynamics and wave propagation: Comparison of p-method finite elements with k-method NURBS. Computer Methods in Applied Mechanics and
Engineering, 197, 4104–4124.
Hughes, T. J. R., Reali, A., & Sangalli, G. (2010). Efficient quadrature for NURBS-based
isogeometric analysis. Computer Methods in AppliedMechanics and Engineering, 199, 301–
Laursen, T. A., & Simo, J. C. (1993). Algorithmic symmetrization of Coulomb frictional
problems using augmented Lagrangians. Computer Methods in Applied Mechanics and
Engineering, 108, 133–146.
Laursen, T. A. (2002). Computational contact and impact mechanics: Fundamentals of
modeling interfacial phenomena in nonlinear finite element analysis. Berlin: Springer-
Verlag.
Oldenburg, M., & Nilsson, L. (1994). The position code algorithm for contact searching.
International Journal for Numerical Methods in Engineering, 37, 359–386.
Ortiz, M., & Simo, J. C. (1986). An analysis of a new class of integration algorithms for
elastoplastic relations. International Journal for Numerical Methods in Engineering, 23,
–366.
Piegl, L., & Tiller, W. (1996). The NURBS Book. Berlin: Springer-Verlag.
Simo, J. C., & Hughes, T. J. R. (1998). Computational inelasticity. Interdisciplinary applied
mathematics (Vol. 7). New York, NY: Springer.
Taylor, R. L. (2010). Isogeometric analysis of nearly incompressible solids. International
Journal for Numerical Methods in Engineering, 87, 273–288.
Yoon, J.W., Yang, D. Y., & Chung, K. (1999). Elasto-plastic finite element method based on
incremental deformation theory and continuum based shell elements for planar anisotropic
sheet materials. Computer Methods in Applied Mechanics and Engineering, 174, 23–56.
Wriggers, P. (2002). Computational contact mechanics. Berlin: Springer-Verlag.
