Numerical evaluation of line contact

gp,j=

Z

γp⁽¹⁾,h

Φjgn,hdL j = 1, ..., n^(1)p . (4.70) Additionally, the weighted relative tangential velocity (v^p,τ,rel)j for line contact follows from discretizing the weak frictional sliding constraint in (4.33), viz.

(˜vp,τ,rel)j = (I₃−n_j⊗n_j)·

Again, frame indifference is achieved by formulating(v^p_,τ,rel)_j in terms of time derivatives of the mortar matrices.

4.6.2. Numerical evaluation of line contact

In contrast to the point contact formulation in Section 4.5, a numerical integration procedure has to be carried out to evaluate the mortar matrices in (4.67) and (4.69) and the kinematic

quantities (4.70) and (4.71). Since the mortar matrixMp, the weighted gap˜g^p_,j and the weighted relative tangential velocity (˜vp,τ,rel)j all require an integration over the slave side line contact boundaryγp⁽¹⁾,h with integrands containing quantities from both sides, an exact evaluation cannot be achieved by standard Gauss quadrature rules simply being applied on each slave line element.

This is due to the generally non-matching meshes that result from arbitrary line contact situ-ations in the finite deformation regime. To overcome this problem, a so-called segment-based integration scheme is employed, which is based on the idea of preventing all possible disconti-nuities in the integrands by creating smooth integrable segments. This idea was firstly outlined for classical segment-to-segment contact formulations in Simo et al. [253] and in Zavarise and Wriggers [304] and then applied in the context of mortar formulations in McDevitt and Laursen [168] and in Puso and Laursen [217]. Here, the basic principle is adopted for the line contact in-tegration. In order to create line segments that contain only C¹-continuous integrands in (4.69), (4.70) and (4.71), the nodes of a considered slave line element and a master element are pro-jected onto an auxiliary plane. Then, a line clipping algorithm is applied to determine the part of the line element that is located within the master element or the master element edges. The whole procedure is visualized in Figure 4.12.

Figure 4.12: Main steps of the segment-based integration scheme for the line contact algorithm:

Create an averaged normal vector at the middle of the slave line element (top left), project the averaged normal vector into the normal plane of the line element and construct an auxiliary plane (top right), project slave and master nodes onto the auxiliary plane (bottom left) and perform line clipping to identify line segments in which the numerical integration is performed (bottom right).

4.6. Line contact

Additionally, the evaluation process is given in the following algorithm:

Algorithm4.3. Segment-based integration for line contact

1. Create an averaged normal vectorn⁽¹⁾₀ based on the nodal normal vectorsn⁽¹⁾_k andn⁽¹⁾_k+1. 2. Project the averaged normal vectorn⁽¹⁾₀ into the normal plane of the considered line

ele-ment to create the normal vector˜n⁽¹⁾₀ . In detail:˜n⁽¹⁾₀ = (I3−τ⁽¹⁾₀ ⊗τ⁽¹⁾₀ )n⁽¹⁾₀ .

3. Construct an auxiliary plane for numerical integration based on the slave element cen-terx⁽¹⁾₀ and the corresponding normal vectorn˜⁽¹⁾₀ .

4. Project alln⁽²⁾_e master element nodesx⁽²⁾_l , l= 1, ..., n⁽²⁾_e alongn˜⁽¹⁾₀ onto the auxiliary plane to create the auxiliary master nodes˜x⁽²⁾_l .

5. Project alln⁽¹⁾_e slave line element nodesx⁽¹⁾_k , k = 1, ..., n⁽¹⁾_e along˜n⁽¹⁾₀ onto the auxiliary plane to create the auxiliary slave nodesx˜⁽¹⁾_k . This step is not required for first-order el-ements and can be considered as possible demand for extensions towards second-order elements.

6. Perform line clipping in the auxiliary plane in order to find the overlapping line seg-ment of projected slave and master nodes. Adequate line clipping algorithms can be found in Hughes et al. [119].

7. Define suitable integration points on the created line segment and find their counterparts on the slave and master element by an inverse mapping.

8. Perform numerical integration of the mortar matrices (4.67), (4.69), the weighted gap (4.70) and the weighted relative tangential velocity (4.71).

Figure 4.13: Special case of parallel edge-to-edge contact for segment based integration scheme:

Already constructed auxiliary plane (left), projected master and slave edge nodes onto auxiliary plane (middle) and line-to-line clipping (right).

In the presented algorithm, the edge-to-surface contact scenario is employed to explain the segmentation procedure. However, the proposed integration scheme is also valid for parallel

edges being in contact. This is visualized in Figure 4.13. Here, again an auxiliary plane is built in complete analogy to Figure 4.12. Then, a master line element is projected onto the slave side auxiliary plane, see middle part of Figure 4.13. Afterwards, a line-to-line clipping algo-rithm is performed, which shares a lot of similarities with segment-based integration schemes for 2D bodies with mortar contact, see Popp et al. [211] and Yang et al. [301]. Consequently, the integration segment end points can be directly identified as projected slave or master nodes.

For the edge-to-surface segmentation scheme, the integration segment end points could also be identified as crossing of projected element edges. More information concerning the node projec-tion and consistent linearizaprojec-tion of the geometrical procedures can be found in the Appendix B.

The robustness and accuracy of the segment-based integration scheme for edge-to-surface and edge-to-edge contact scenarios is demonstrated at several numerical examples in Section 4.9.

The algorithm explained above performs robustly and guarantees for highest accuracy in all tested numerical examples. However, an alternative segment-based integration scheme is given in the following, which requires less algorithmic steps and can thus be be implemented more efficiently. However, this increase in efficiency is dearly bought by the prize of less robustness compared to the first algorithm. For the sake of completeness, the alternative procedure is illus-trated in Figure 4.14 and the corresponding algorithm reads:

Algorithm4.4. Alternative segment-based integration for line contact

1. Construct an auxiliary plane for numerical integration based on the master element cen-terx⁽²⁾₀ and the corresponding element normal vectorn⁽²⁾₀ .

2. Project alln⁽²⁾_e master element nodesx⁽²⁾_l , l= 1, ..., n⁽²⁾_e alongn⁽²⁾₀ onto the auxiliary plane to create the auxiliary master nodes˜x⁽²⁾_l .

3. Project alln⁽¹⁾_e slave line element nodesx⁽¹⁾_k , k = 1, ..., n⁽¹⁾_e along their nodal normal n⁽¹⁾_k onto the auxiliary plane to create the auxiliary slave nodes˜x⁽¹⁾_k .

4. Perform line clipping in the auxiliary plane in order to find the overlapping line segments of projected slave and master nodes. Adequate line clipping algorithms can be found in Hughes et al. [119].

5. Define suitable integration points on the created line segment and find their counterparts on the slave and master element by an inverse mapping.

6. Perform numerical integration of the mortar matrices (4.67), (4.69), the weighted gap (4.70) and the weighted relative tangential velocity (4.71).

Here, the integration is performed on line segments being defined on the master side auxiliary plane. The calculation of the projection normal is much easier and consequently less terms to be linearized occur. Again, the alternative integration procedure performs less robust in the numer-ical examples but also guarantees an exact integral evaluation. Both algorithms are implemented in the employed in-house code BACI (cf. Wall et al. [282]), but it is recommended to use the first algorithm, see Figure 4.12.