Isogeometric analysis of 3D large deformation frictional contact problems and comparison with Lagrange and Bézier interpolations

NURBS-based isogeometric analysis is adopted to model 3D large deformation frictional contact problems. The proposed contact formulation is based on a mortar approach, extended to NURBS- based interpolations, and combined with a simple integration procedure which does not involve segmentation of the contact surfaces. The augmented Lagrangian method is chosen to obtain an exact satisfaction of the contact constraints. As proposed by previous researchers, a Newton-like solution scheme is applied to solve the saddle-point problem simultaneously for the displacements and the Lagrange multipliers. The performance of NURBS-based isogeometric analysis is compared to that of standard C0- continuous Lagrange finite element interpolations. A further comparison is carried out with a Bézier discretization based on C0-continuous Berstein polynomials, which are non-negative and possess the variation diminishing and convex hull properties like the NURBS basis. This comparison allows to identify the individual roles played by the higher continuity and by the other favorable properties of the NURBS basis functions. The presented examples deal with both small- and large-deformation cases. The quality of the solution is examined in terms of contact stress distributions in the small-deformation examples, and in terms of global load vs. displacement behavior for the large-deformation, large-sliding examples. In both cases, the results obtained with the isogeometric analysis and with Lagrange and Bézier discretizations are compared for varying resolution and order of the contact surfaces. Based on results obtained in this investigation, it can be concluded that the proposed contact mortar formulation using NURBS-based isogeometric analysis displays a significantly superior performance with respect to the same formulation using standard Lagrange polynomials. Bézier discretizations perform almost equally well as the NURBS ones for small deformation and small sliding examples, whereas NURBS are evidently superior in large deformation and large sliding cases. The contact pressure distributions stemming from the NURBS parameterizations are always non-negative, are practically insensitive to changes in the interpolation order, and improve monotonically as the mesh resolution increases. The respective distributions obtained from Lagrange parameterizations are highly sensitive to the interpolation order, display significant spurious oscillations and may attain large non-physical negative values. In 3D large frictionless sliding problems the time histories of the tractions obtained from the NURBS discretizations are remarkably smooth and improve in quality with increasing order of the parameterization. Conversely, the curves obtained from Lagrange parameterizations display irregular oscillations whose magnitude increases with the interpolation order and which may even prevent convergence.