doi:10.1016/j.cma.2007.09.004 
Copyright © 2007 Elsevier B.V. All rights reserved.
A large deformation mortar formulation of self contact with finite sliding
Bin Yanga and Tod A. Laursen
, a, 
aComputational Mechanics Laboratory, Department of Civil and Environmental Engineering, Duke University, Durham, NC 27708-0287, United States
Received 20 January 2007;
revised 4 June 2007;
accepted 7 September 2007.
Available online 18 September 2007.
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Abstract
This paper presents a new numerical method, in which self contact phenomena associated with a body undergoing large deformations and sliding can be described. In particular, the approach relies on a particular extension of the mortar approach appropriate for this class of problems. A bounding volume hierarchy (organized as a binary tree) is built for the self contact surface, based on the geometry and the mesh connectivity of the surface. A curvature criterion, using a new algorithm to detect subsurface adjacency, is used to accelerate the self contact searching procedure. To ensure that the mortar traction fields are properly defined on contiguous surface patches, a novel facet sorting algorithm is also proposed, based on the mesh connectivity of the contact element pairs found by the self contact searching algorithm. Several two- and three-dimensional numerical examples show the new self contact mortar formulation to be very efficient, and also demonstrate that it can be combined with multi-body contact algorithms to simulate a very general class of contact problems.
Keywords: Mortar methods; Self contact; Contact searching; Bounding volume hierarchy; Finite elements
Fig. 1. Notation for a large deformation self contact problem.
Fig. 2. Mortar segments as intersections of two opposing element surfaces, shown in (a) two dimensions; and (b) three dimensions.
Fig. 3. Different bounding volumes: (a) axis-aligned bounding box (AABB); (b) oriented bounding box (OBB); (c) k faced discrete orientation polytope (k-DOP).
Fig. 4. Proximity tests using different bounding volumes: (a) with AABB; (b) with OBB; (c) with k-DOP.
Fig. 5. An improved way to define the bounding volume.
Fig. 6. A bounding volume tree for a two-dimensional surface.
Fig. 7. Dual graph for a surface mesh. Each dual node represents an element or subsurface in the finite element mesh.
Fig. 8. Contraction of a dual edge to construct a new (parent) subsurface: (a) initial dual graph, and (b) new dual graph. The larger dot in (b) denotes the newly generated dual node, and the new dual node corresponds to a newly constructed subsurface indicated by the shaded area.
Fig. 9. A hierarchical (tree) structure, proceeding from the bottom level (a) to the top level (e).
Fig. 10. Curvature criterion for self contact. There is no self contact in this case.
Fig. 11. The curvature criterion (depicted in two-dimensional examples) applied for a single subsurface and two adjacent subsurfaces.
Fig. 12. Finding the qualified vectors from the bottom to the top of the hierarchy: (a) a full set of sample vectors; (b) a surface and qualified sample vectors for each element; (c) qualified sample vectors for all subsurfaces in the hierarchy from bottom to top.
Fig. 13. Discontinuous definition of slave and master elements (a double arrow denotes a contact element pair). Elements must be relabelled so that each side of each contact patch is contiguously defined.
Fig. 14. The contact element pairs after the facet sorting.
Fig. 15. The initial configuration of the 2D impact problem.
Fig. 16. Deformed configurations of the 2D impact problem at different time steps.
Fig. 17. The initial configuration of the cylinder post-buckling problem.
Fig. 18. Deformed configurations of the cylinder post-buckling problem at different time steps.
Fig. 19. The reaction force versus displacement curve.
Fig. 20. The initial configuration and mesh of the tire rolling problem.
Fig. 21. Deformed configurations of the tire rolling problem at different load steps; parts (c) and (d) only show one quarter of the tire.
Table 1.
The recursive algorithm to get qualified sample vectors for all subsurfaces (i.e. T-nodes)
Table 2.
Self contact searching algorithm
Table 3.
Initial procedure to get adjacent subsurfaces
Table 4.
Energy norms of convergence sequences at different load steps, and with different load increments
Table 5.
Energy norms of convergence sequences at different load steps of the tire rolling problem
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