Point contact

Firstly, the spatial discretization and numerical evaluation of point contact is considered. As stated in Section 4.3, there are two strategies to enforce the point contact scenario depending on the involved geometrical entities. Real point contact with vertices being involved is realized via the vector-valued Lagrange multiplierλ⋆, while the point contact that occurs due to crossing edges is treated by a penalty regularization.

4.5.1. Vertex contact

For the Lagrange multiplier constraint enforcement, the discrete counterpart to the vectorλ⋆ is required. It is based on the discrete Lagrange multiplier space M_⋆,h being an approximation ofM_⋆. The notation for the discrete point Lagrange multiplier reads

λ⋆,h =

n⋆

X

j=1

Λjλ_⋆,j . (4.51)

In (4.51), the shape functionsΛjof the point Lagrange multiplier interpolation reduce to impulse functions being 1 at the nodes of physical slave verticesn⁽¹⁾⋆ and zero at all other points:

Λj =







1 at x_j ,

0 else. (4.52)

Therefore, no interpolation functions are necessary between these points. The use of these La-grange multipliers at slave vertices can easily be interpreted as the well-known node-to-segment formulation for point contact, see for example Bathe and Chaudhary [14], Erhart et al. [67], Hal-lquist et al. [96], Laursen and Simo [154], Papadopoulos and Taylor [195], Simo et al. [253]

and Wriggers et al. [296]. Nevertheless, in the following, computational details on the numerical evaluation are briefly given. When inserting the introduced finite element discretizations (4.36) and (4.51) into the contact virtual work corresponding to the point contact contribution (4.31), the point contact matricesD_⋆ ∈R^{3n(1)⋆ ×3n(1)⋆} andM_⋆ ∈R^{3n(1)⋆ ×3n(2)} can be computed by merging the nodal blocks

D_⋆[j, k] =D⋆,jkI₃ = ΛjN_k⁽¹⁾ I₃ =I₃ , j = 1, ..., n⁽¹⁾_⋆ , k = 1, ..., n⁽¹⁾_⋆ , (4.53) M_⋆[j, l] =M_⋆,jlI₃ = Λ_j(N_l⁽²⁾◦χh)I₃ , j = 1, ..., n⁽¹⁾_⋆ , l = 1, ..., n⁽²⁾ . (4.54)

4.5. Point contact Herein,I₃ ∈ R^3×3 is the identity andχh : γ_c,h⁽¹⁾ →γ⁽²⁾_c,h represents a suitable discrete approxima-tion of the mappingχbetween the contact sides, see e.g. Dickopf and Krause [60] and Puso [216]

for more details. Discretization of the non-penetration constraint in (4.32) yields the discrete gap functiong_⋆,j at each nodej:

g_⋆,j =gn,h=n_j^hˆx⁽²⁾(x⁽¹⁾_j )−x⁽¹⁾_j ⁱ j = 1, ..., n⁽¹⁾_⋆ . (4.55) Here,ˆx⁽²⁾(x⁽¹⁾_j )is the discrete point on the master side that results from the projection of the slave node positionx_j, whilen_j is the discrete nodal normal at nodej. The discrete relative tangential velocity(v_⋆,τ,rel)j at nodej yields with the time derivative (·˙) being shifted from the nodal positions to the contact matrices to guarantee the satisfaction of the fundamental requirement of frame indifference, see Gitterle [87] for a detailed explanation. Finally, the algorithm to evaluate the point contact contributions for one pair of vertex node and possibly contacting master element reads:

Algorithm4.1. Vertex contact

1. Project the slave node x⁽¹⁾_j that corresponds to the vertex along its unit normal n⁽¹⁾_j onto the master element to obtain the projected position on the master elementˆx⁽²⁾(x⁽¹⁾_j ).

2. Evaluate the contact matrices (4.53) and (4.54), gap function (4.55) and relative tangential velocity (4.56) at these points.

4.5.2. Contact of non-parallel edges

The penalty regularization of the point contact scenario resulting from edge-to-edge contact is considered in the following. Since the non-parallel edge-to-edge scenario results in contact points that are generally not coincident with finite element nodes, stability requirements for the Lagrange multipliers at these points are hardly predictable. Additionally, a point Lagrange multiplier would be located in the interior of the support of the already defined edge Lagrange multipliers. Thus, from an engineering point of view, these situations could be described as be-ing over-constrained. Therefore, the exact (point-wise) enforcement of the contact constraints onγ_×,his relaxed via a penalty regularization.

Remark 4.1. Note that the scenario of non-parallel edges being in contact is the only inevitable situation in the overall algorithm, where a penalty regularization is needed. To the best of knowl-edge of the author, no suitable Lagrange multiplier space can be a priori constructed for such a scenario and consequently the penalty approach cannot be avoided at this point.

By inserting the spatial discretization (4.36) into (4.35), the discrete penalty force vector of crossing edges results in

f_× =f⁽¹⁾_× −f⁽²⁾_× , (4.57)

with the discrete slave force vector f⁽¹⁾_× and the discrete master force vector f⁽²⁾_× . These can be computed by merging the nodal vectors

f⁽¹⁾_× [k] =f_penN_k⁽¹⁾( ˆξ_×⁽¹⁾), k= 1, ..., n⁽¹⁾, (4.58) f⁽²⁾_× [l] =f_penN_l⁽²⁾( ˆξ_×⁽²⁾), l= 1, ..., n⁽²⁾. (4.59) Here, the expression in (4.35) reduces to a point-wise evaluation at the parameter space coor-dinates ξˆ_×⁽¹⁾ and ξˆ_×⁽²⁾. These points represent the parameter space counterparts to the points in physical spaceˆx⁽¹⁾_× andˆx⁽²⁾_× at which the closest distance between two line elements can be mea-sured, see Section 4.4.5. Generally, these points are not coincident with finite element nodes and thus they have to be computed via a closest-point-projection between two line elements.

In (4.58) and (4.59), the discrete penalty force vectorf_pen can be split into its normal partf_pen,n and its tangential part f_pen,τ. The normal force can be obtained by inserting the finite element discretization into (4.25): defined along the connecting line betweenˆx⁽¹⁾_× andˆx⁽²⁾_× , but pointing in outward direction of the slave body, see again Figure 4.9. For defining the discrete frictional penalty force, the discrete relative tangential velocity atˆx⁽¹⁾_× has to be defined as

v_×,τ,rel = (I₃−nˆ⁽¹⁾_× ⊗ˆn⁽¹⁾_× )· Again, the time derivative is shifted to the discrete interpolation, which guarantees frame in-difference, see Gitterle [87]. There, the time derivative stems from the changing geometrical projection, which can be directly expressed as change of the parameter space coordinate. The Lie derivative in (4.26) is defined as

Lf_pen,τ = (I₃−ˆn⁽¹⁾_× ⊗nˆ⁽¹⁾_× ) ˙f_pen,τ. (4.62) This expression contains only material time derivatives of the penalty force itself and no time derivatives of base vectors are present. Thus, the Lie derivative in (4.62) is frame indifferent. For the calculation of the Coulomb frictional forces at the edge-to-edge contact points, a trial state-return map strategy is employed, which is an algorithmic time stepping procedure, see Laursen [151] and Yang et al. [301]. Here, a trial state is computed by assuming a perfect stick state during the time increment∆t:

f^trial_pen,τ

n+1 =f^trial_pen,τ

n −ǫτv_×,τ,rel. (4.63)

4.6. Line contact