41 DEIM for the Efficient Computation of Contact Interface Stresses 437 6 5 4 1 2 3 Γe,+ Γe,− jc,h ew ev eu Iez Iey Iex Γe jc,h ev eu η ξ 1 3 2 1 2 3 (0,0) (1,0) (0,1) jc,h a b c Fig. 41.2 Triangular shaped zero thickness element (a), arbitrary shaped triangle with local coordinate system.u; v/ (b) and uniform triangle with natural coordinates . ; / (c) For a discretization h of the domain with the nodes xh and the related nodal displacements uh Eq. (41.2) can be rewritten as sum over all elements e h h and element surfaces e h h. The virtual work of a single element not adjacent to the contact interface jc is givenby ıWe DZ e h Ru ıud CX i;j Z e h sijı ij d Z e t;h Sn ıud Z e h k ıud D0: (41.3) For each element the nodal displacement vector ue h uh can be defined. By using the Matrix of element shape functions Ne the displacement fieldue DNe ue h and the virtual displacement fieldıue DNeıue h can be formulated for each element. Evaluation of the integrals, sum over all elements defined by Eq. (41.3) and implementation of the boundary conditions, see [1], but without consideration of the contact stresses at jc and Cjc leads to the discretized equations of motion MRuh CKuh Dfext : (41.4) For an element adjacent to the contact surface jc the discretization h comprises a matching mesh jc;h and Cjc;h of the two sides jc and Cjc. To handle the contact within the joint interface jc the theory of zero thickness elements is a straight forward technique as only small relative movement of the contact partner is expected. Several formulations can be found in literature, e.g. [4] and [5]. For this contribution the stress equivalent nodal force vector of an isoparametric, triangular shaped zero thickness element, depicted in Fig. 41.2a, is formulated. One surface triangle connects to the discretization jc;h and the other one to the discretization C jc;h. For the nodal displacements of the element ue;C h D ue 1 T ue 2 T ue 3 T T and ue; h D ue 4 T ue 5 T ue 6 T T (41.5) a local coordinate system.u; v; w/ where theew vector is oriented in interface e jc;h normal direction as depicted in Fig. 41.2a is used. With these nodal displacements and the Matrix Ne D N1 N2 N3 with Ni Ddiag.Ni ;Ni ;Ni / (41.6) holding the shape functions N1 D 1 , N2 D and N3 D where the natural coordinates of a uniform triangle D .u; v/ and D .u; v/ are depending on the coordinatesuandvof the arbitrary shaped element, the displacement fields of both surface triangles ue;C DNeu e;C h and ue; DNeu e; h (41.7) can be described. Both surface triangles are interpolated using the same shape function definition. With consideration of Newton’s Third Law, the principle of actio and reactio te jc Dt e;C jc D t e; jc (41.8)
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