Conservation laws

The fundamental conservation laws have been already introduced for mechanical systems in Section 2.1.3 and are now analyzed in the context of the proposed all entity contact formulation.

Therefore, all following explanations are referred to the semi-discrete setting, meaning that the problem is discrete in space but continuous in time.

4.8.3.1. Balance of linear momentum

First, as elaborated e.g. in Puso and Laursen [218] the requirement for linear momentum con-servation can be expressed as balance of all forces acting on the slave and master side. In the context of the newly developed all entity contact formulation this requirement reads:

f⁽¹⁾_c −f⁽²⁾_c =D^Tλ+f⁽¹⁾_× −(M^Tλ+f⁽²⁾_× ). (4.95)

4.8. All entity contact – combined formulation As explained in the previous sections, the slave and master side forces can be split into penalty force vectors f⁽ⁱ⁾_× and Lagrange multiplier force vectorsf⁽ⁱ⁾_λ . The overall balance of linear mo-mentum at the contact interface can be investigated by first considering the balance of penalty forces and then the balance of forces due to the Lagrange multipliers. Thus, the balance of penalty forces reads Since the displacement shape functions fulfill the fundamental requirement of partition of unity, i.e.^Pn_k=1⁽¹⁾N_k⁽¹⁾ = 1and^Pn_l=1⁽²⁾N_l⁽²⁾ = 1, the balance of linear momentum can be written as

1(f_pen−f_pen) =0. (4.97)

Here, it can be clearly seen that conservation of linear momentum isalwaysguaranteed for the penalty forces resulting from contact of crossing edges.

Conservation of linear momentum for the Lagrange multiplier force vectors can be stated as f⁽¹⁾_λ −f⁽²⁾_λ =

which is identical to the investigations in Popp [210] and Puso and Laursen [218]. In contrast to these publications, the mortar matricesDandMare created in a different way and contain now information from point contact, line contact and surface contact. In addition, the global Lagrange multiplier vector λ contains now information from all three contact scenarios. The expression in (4.98) can be reformulated for considering each Lagrange multiplier individually:

n⁽¹⁾ This means, that the sum of all contributions from slave and master side matrices associated with one Lagrange multiplier has to vanish. For the line contact terms, this reads in detail:

n⁽¹⁾

where it is not distinguished between the entries forDpp andDp⋆. When employing the aforemen-tioned partition of unity property for the displacement shape functions, the conservation of linear momentum finally reads

which is alwaysfulfilled when the integration of the slave and master side mortar matrix is per-formed over the same discrete domain. This was already stated in Puso and Laursen [218] and also holds for the line contact algorithm. It is also valid for the surface contact as elaborated in Popp [210]. For the point contact of vertices, the same reasoning holds without performing a numerical integration but rather a simple term evaluation. For the sake of brevity, these investi-gations are not outlined here in detail since they are in complete analogy to the explanations for the line contact.

4.8.3.2. Balance of angular momentum

Enforcing an exact conservation of angular momentum is rather challenging in the context of computational contact mechanics. The basic requirement for conservation of angular momentum is given as which means that the sum of slave and master interface momentum should vanish. In (4.102), the vectorsf⁽¹⁾_c,kandf⁽²⁾_c,l represent the nodal forces at slave nodekand master nodel, respectively. As discussed in Popp [210], Puso and Laursen [218], Yang et al. [301], expression (4.102) is zero when at least one of the two following requirements is fulfilled:

• the discrete form of the displacement jump vector (gap vector)gvanishes,

• the force vectors and the discrete displacement jump vector are collinear.

Since the discrete nodal force vectorsf⁽¹⁾_c,kandf⁽²⁾_c,l can be split into contributions from the penalty regularization of the contact interaction of crossing edges and Lagrange multiplier contributions, the above mentioned requirements for conservation of angular momentum are investigated sep-arately for these two types of force vectors. First, the penalty force vectors are considered. The discrete form of the displacement jump vector (gap vector) reads for the contact of non-parallel edges: which points per definition in the normal directionnˆ⁽¹⁾_× , see Section 4.4.5. The resulting forces due to the non-penetration condition point in the same direction, as can be seen in (4.60). Thus, all normal force vectors and the discrete gap vector are collinear and thus conservation of angular momentum isguaranteedfor contact without frictional effects. Note, that the other requirement of a vanishing gap vector can never be achieved due to the penalty regularization, i.e.ǫ_n → ∞. For frictional contact, the force vectors and the gap may not be collinear and thus conservation of angular momentum is not always fulfilled. This is due to the rate form of the frictional contact problem.

When considering the nodal forces resulting from Lagrange multipliers, the conservation of angular momentum in (4.102) can be rewritten as

m⁽¹⁾−m⁽²⁾= Thus, for contact force vectors resulting from Lagrange multipliers, the requirement of collinear force and displacement jump vectors can be reformulated into the requirement of collinearity be-tween the Lagrange multiplier vectors and the displacement jump vector, see Puso and Laursen [218] and Yang et al. [301]. For the presented mortar contact formulation from Chapter 3 and the

4.8. All entity contact – combined formulation current chapter, this is neither guaranteed for contact without friction nor for frictional contact, see again Popp [210] and Yang et al. [301]. A possible remedy for contact without frictional ef-fects would be a reformulation of the contact approach without an a-priori split of the Lagrange multipliers into a normal and a tangential part from (3.22). Instead, the normal direction should be included in the integrals for the mortar matricesD and Min order to account for the vary-ing normal direction over the integration domain. But, when dovary-ing so, the diagonal form of the subblocks inDcannot be achieved anymore with dual shape functions, and the computationally efficient solution procedure in Section 4.8.2 cannot be performed anymore. Furthermore, varia-tion of the mortar integrals when deriving the discrete contact virtual work contribuvaria-tion would lead to conservation of angular momentum, see Hesch and Betsch [106]. These variations are commonly neglected, since they require second derivatives of the mortar matrices D and M, and thus the computational complexity would strongly increase, see Popp et al. [214], Puso and Laursen [218] and Puso and Laursen [219]. However, the numerical example in Section 4.9.4 demonstrates that the violation of angular momentum conservation is very small and from an engineering point of view negligible in practice.

For investigating the collinearity condition, the corresponding displacement jump vector has to be stated for the Lagrange multiplier nodej:

g_h,j = It is again a quantity with different interpretations depending on the actual Lagrange multiplier.

For contact of vertices, this quantity becomes a distance measure, but for surface contact it rep-resents a volumetric measure. However, vanishing of the jump vector cannot be guaranteed since only the normal part of it is forced to zero. Thus, the approaches for Lagrange multiplier contact slightly violate both required conditions, which was already investigated by several authors for surface contact, see for example Popp [210] and Yang et al. [301]. Therefore, conservation of angular momentum may not be guaranteed for the all entity contact formulation.