A mortar formulation for 3D large deformation contact using NURBS-based isogeometric analysis and the augmented Lagrangian method

Abstract

NURBS-based isogeometric analysis is applied to 3D frictionless large deformation contact problems. The contact constraints are treated with a mortar-based approach combined with a simplified integration method avoiding segmentation of the contact surfaces, and the discretization of the continuum is performed with arbitrary order NURBS and Lagrange polynomial elements. The contact constraints are satisfied exactly with the augmented Lagrangian formulation proposed by Alart and Curnier, whereby a Newton-like solution scheme is applied to solve the saddle point problem simultaneously for displacements and Lagrange multipliers. The numerical examples show that the proposed contact formulation in conjunction with the NURBS discretization delivers accurate and robust predictions. In both small and large deformation cases, the quality of the contact pressures is shown to improve significantly over that achieved with Lagrange discretizations. In large deformation and large sliding examples, the NURBS discretization provides an improved smoothness of the traction history curves. In both cases, increasing the order of the discretization is either beneficial or not influential when using isogeometric analysis, whereas it affects results negatively for Lagrange discretizations.

Isogeometric dual mortar methods for computational contact mechanics

“…Mortar methods are based on a variationally consistent discretization of the weak form (12) and (13), hence a suitable discrete approximation of the dual space M (11) is necessary. Given some still to be defined basis functions Φ as a basis of M h and discrete vector-valued Lagrange multipliers λ j at each control point on the potential contact surface, the Lagrange multiplier field on the slave side is approximated by…”

“…For contact algorithms based on Lagrangian finite elements such methods can be found in e.g. [36,[41][42][43][44] and within IGA in [12,13]. Using these methods based on a standard Lagrange multiplier interpolation, a system of increased size containing both displacement and Lagrange multiplier degrees of freedom has to be solved.…”

“…As a consequence, in the past five years various discretization techniques have been developed for IGA or transferred from finite element based contact mechanics to IGA, such as node-to-segment [4], Gauss-point-to-segment [5][6][7][8][9][10] and mortar methods [5,6,[11][12][13][14]. We refer to the recent review in [15] for a comprehensive discussion of such methods, comparisons to their finite element counterparts and further references.…”

“…While Gauss-point-to-segment approaches are, due to their lack of inf-sup stability (see e.g. [5,7] for numerical investigations), usually combined with a penalty approach, see [7][8][9][10], node-to-segment and mortar formulations can be combined with penalty methods [5,6], Uzawa-type algorithms [11], Lagrange multiplier methods [13,14] or augmented Lagrange methods [12]. In contrast to penalty methods, the other mentioned methods fulfill the contact constraints in a discrete sense exactly.…”

Hierarchical NURBS and a higher-order phase-field approach to fracture for finite-deformation contact problems

“…This work analyses a thermodynamically consistent framework including the energy transfer between the mechanical and the thermal field due to friction and the variationally consistent description of the contact interface on the basis of Mortar methods, see de Lorenzis et al [26,27] and Temizer [28,29] for further details on Mortar contact formulations for isogeometrical analysis. Hierarchical refinements for frictionless contact have been investigated in Temizer and Hesch [30].…”