Numerical Implementation of Two Nonconforming Finite Element Methods for Unilateral Contact

By Patrick Hild. 2000

125 citations recorded by the source corpus.

We consider the finite element approximation of the unilateral contact problem between elastic bodies. We are interested in a practical problem which often occurs in finite element computations concerning two independently discretized bodies in unilateral contact. It follows that the nodes of both bodies located on the contact surface do not fit together. We present two different approaches in order to define unilateral contact on nonmatching meshes. The first is an extension of the mortar finite element method to variational inequalities that defines the contact in a global way. On the contrary, the second one expresses local node-on-segment contact conditions. In both cases, the theoretical approximation properties are given. Then, we implement and compare the two methods.

Subfields: Civil and Structural Engineering, Computational Theory and Mathematics, Mechanics of Materials

View Paper: https://www.semanticscholar.org/paper/f2576e367591f042d78bea42abe1b96616ed1fb1

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In Civil and Structural Engineering

In Computational Theory and Mathematics

In Mechanics of Materials

Citations in the corpus

Quadratic finite element methods for unilateral contact problems (2002, background): “4) with linear finite elements (and linear non-interpenetration conditions corresponding to the theoretical results in [15, 7]).”
Isogeometric contact: a review (2014, background): “Early applications of mortar finite element methods for contact mechanics can be found in Belgacem et al. (1998), Belgacem (2000), Hild (2000), McDewitt and Laursen (2000), and Wohlmuth and Krause (2003) among others.”
Three-dimensional mortar-based frictional contact treatment in isogeometric analysis with NURBS (2012, background): “an exact evaluation of the surface integrals contributing to the weak formulation, combined with a discrete satisfaction of the continuous contact constraints through projected quantities [54,22,49,25,13,50,48,21,26,59].”
Two dimensional mortar contact methods for large deformation frictional sliding (2005, methodology): “Hild [8] compared the mortar element method and node-to-surface strategy for frictionless contact problems, and numerically demonstrated optimality of the mortar formulation (and lack of optimality for its node-to-surface counterpart).”
Efficient simulation of multi‐body contact problems on complex geometries: A flexible decomposition approach using constrained minimization (2009, background): “(12) This is in fact the coupling originally suggested in [7], apart from the inclusion of the contact mapping Φ.”
Frictionless 2D Contact formulations for finite deformations based on the mortar method (2005, methodology): “Several methods exist for the discrete detection of the active contact set: Hild (2000) describes a discrete Kuhn-Tucker type where the local contact element conditions are similar to the classical node-to-segment element conditions.”
A stabilized Lagrange multiplier method for the enriched finite-element approximation of contact problems of cracked elastic bodies (2012, methodology): “Concerning a priori error estimates for the contact problem of linearly elastic bodies approximated by a standard affine finite-element method, a rate of convergence of order h can be obtained for most methods (see [8,23,32] for instance).”