Lagrangian step

In this section, the treatment of wear phenomena within the Lagrangian step is explained. Basi-cally, the implicit partitioned algorithm described in the previous section also works when wear phenomena are not considered within the Lagrangian step. Then, a standard contact problem as described in Chapter 3 is solved without any modification. However, the wear depth occurring during one load step is taken into account in order to accelerate the partitioned algorithm. Thus, in complete analogy to fretting wear modeling in Section 5.4.1, wear effects are considered as additional contribution to the gap function, which leads to a modification of the Hertz-Signorini-Moreau conditions:

g_n^w≥0, pn≤0, png_n^w= 0. (5.60) Again, g_n^w represents the modified gap function. The modified constraints in (5.60) together with the boundary value problem (5.52), (5.53) and (5.54) state the problem formulation for the Lagrangian step.

Remark 5.4. Note that the wear depth w⁽ⁱ⁾, which is utilized to modify the gap function ac-cording to (5.34) or (5.43), is generally not the absolute wear depth per time increment, but the increment of the solution procedure with respect to the last Lagrangian step of the parti-tioned algorithm. By introducing the notation(·)^p for the current iteration step of the partitioned algorithm, the wear depth reads

w⁽ⁱ⁾ =w^(i),p−w^(i),p−1. (5.61)

For the first iterationp = 1, the initial wear guess is set to zero, i.e.w^(i),0 = 0. However, for the sake of brevity, the iteration counter is skipped for the partitioned algorithm in the following and relation(5.61)is implicitly used.

In the following, the implicit treatment of the gap modification for the Lagrangian step is ex-plained in detail for the primary variable approach and briefly introduced for the internal state variable approach. However, the primary variable approach is the preferred spatial discretization approach because the wear depth, which is a transfer variable in the implicit partitioned algo-rithm, is directly represented by an additional variable. This is very beneficial in order to iterate between the Lagrangian step and the shape evolution step. The internal state variable approach is only applicable for an explicit partitioned algorithm without iterating between the steps.

5.5.3.1. Primary variable approach

Starting point for the primary variable approach is the discrete residual for the slave and master side wear:

r⁽ⁱ⁾_w =E⁽ⁱ⁾w⁽ⁱ⁾−T⁽ⁱ⁾λ=0. (5.62)

For the application of iterative nonlinear solvers based on a Newton-Raphson scheme, the wear residual (5.21) has to be linearized. Thus, in complete analogy to the explanation for standard structural problems in Section 2.2.4, a truncated Taylor series expansion is carried out

Linr_w(dⁱ,λⁱ,wⁱ) =r_w(dⁱ,λⁱ,wⁱ) + ∂rw Here, the index i denotes the previously calculated Newton step. This results in the linearized wear residual

−r_w(dⁱ,λⁱ,wⁱ) =Sⁱ∆dⁱ⁺¹ +Fⁱ∆λⁱ⁺¹+Eⁱ∆wⁱ⁺¹. (5.64) Details on the directional derivatives for the linearized wear residual are omitted here, since they are strongly related to the linearizations well-known for surface mortar contact, see Popp [210] and Popp et al. [212]. The semi-discrete problem formulation in (3.39) remains unchanged, despite the occurring wear phenomena, but the complementarity functions in (3.58) and (3.60) change due to the modified gap. Thus, the linearized system of equations to be solved within each Newton-Raphson step, which is explained in Section 3.5.3, must be extended due to wear considerations. Concretely, linearizations of the complementarity functions with respect to wear and the linearized wear equation itself are introduced. In the following, for the sake of brevity, the algebraic representation of the finite wear problem formulation is given for slave-sided wear.

However, an extension towards both-sided wear is straightforward and the described solution procedures are also applicable to both-sided wear. Thus, an exemplary matrix representation of the system of equations, when only slave-sided wear phenomena are considered, looks as follows:

Here, the solution vector contains increments of discrete displacements ∆d, Lagrange multi-pliers ∆λ and wear quantities∆w. The displacements are split into inner nodes (·)_N, master nodes(·)_M and slave nodes(·)_S, whereas the Lagrange multipliers and wear quantities are only defined on the slave side anyway. The stiffness blocksKarise from linearization of the internal force vector, and the tilde symbols indicate additional linearizations due to the contact force vec-tor. The first three rows of the system can be identified as linearized algebraic form of the force equilibrium and row four represents the linearized contact constraints. In addition to the stan-dard frictional contact problem, the last row arises due to the additional wear consideration. As wear is taken into account as an additional gap within the Hertz-Signorini-Moreau conditions, coupling terms only arise in the row associated with the contact constraints and a direct coupling into the force equilibrium is not to be expected.

While this system of equations could be solved directly, this does not seem advisable, since it not only contains displacement degrees of freedom, but also Lagrange multipliers as well as unknowns due to wear. Therefore, it is of increased and even of varying size. Moreover, the introduced Lagrange multipliers cause a saddle point structure of the linearized system of

5.5. Formulation for finite wear – ALE formulation equations, which may cause difficulties for the performance of common iterative solvers and preconditioners. To get rid of these problems, the condensation procedure explained in Gitterle et al. [88], Popp et al. [211] and Popp et al. [212] and already employed in a similar form in Chapters 3 and 4 is used. Here, dual shape functions for the Lagrange multiplier interpolation are employed to generate a diagonal structure of the mortar matrix D, which then allows for a computationally cheap condensation of the Lagrange multipliers via

∆λⁱ⁺¹ =D^i,−T(−rⁱ_S −Kⁱ_SN∆dⁱ⁺¹_N −K˜ⁱ_SM∆dⁱ⁺¹_M −K˜ⁱ_SS∆dⁱ⁺¹_S ). (5.66) Here, the projection operatorsPare defined as

Pⁱ_d =M^i,TD^i,−T, Pⁱ_λ =Cⁱ_λD^i,−T, Pⁱ_w =FⁱD^i,−T. (5.68) This condensation concept can now be extended to the wear unknowns by using dual shape functions also for the discrete weighting of the wear residual, which then yields a diagonal E matrix. Consequently, an additional condensation step can be performed, which expresses the wear unknowns in terms of discrete displacements:

∆wⁱ⁺¹ =E^i,−1(−˜rⁱ_w+Pⁱ_wrⁱ_S+Pⁱ_wKⁱ_SN∆dⁱ⁺¹_N −(Sⁱ_M−Pⁱ_wK˜ⁱ_SM)∆dⁱ⁺¹_M

−(Sⁱ_S−Pⁱ_wK˜ⁱ_SS)∆dⁱ⁺¹_S ). (5.69)

The resulting condensed system of equations is not explicitly given here for the sake of brevity.

However, as the main result, it is only solved for displacement degrees of freedom and all contact and wear information is included in the modified system matrix. This matrix is of constant size and no saddle point structure occurs anymore. Discrete Lagrange multiplier and wear variables can be obtained by simple post-processing steps based on (5.66) and (5.69).

5.5.3.2. Internal state variable approach

The introduced internal state variable approach is also applicable to finite wear problems. This has already been demonstrated for 2D problem settings in Gitterle [87]. However, the primary variable approach is the preferred wear discretization scheme since the internal state variable is not directly controllable in the partitioned algorithm 5.2. This is due to the inherent feature of the internal state variable approach of no additional unknowns entering the system of equations for the Lagrangian step. Thus, the methodology for the internal state variable approach is only briefly outlined. When considering wear as internal state variable in an implicit solution scheme and as additional contribution to the gap function as explained in 5.2, only the linearizations of

the nonlinear complementarity functions are directly affected. The other parts of the system of equations remain unchanged (cf. (3.62)). After the Lagrangian step is completed, wear must be enforced as Dirichlet condition for the shape evolution step. In contrast to the primary variable approach, the internal state wear quantity represents a surface-weighted wear depth, i.e. a worn volume. Consequently, is has to be expressed in distance measure by dividing it by the support area of the attached node. This has already been realized in (5.41). Such a computation is the natural result of a weak enforcement of Dirichlet boundary condition, i.e. wear depth.

The internal state variable approach has been implemented and carefully validated by the author. When no iteration between Lagrangian and shape evolution step is required, no change in solution quality and robustness for both wear discretization schemes are noticeable. However, as already mentioned, the primary variable approach easily allows for a completely implicit partitioned iteration scheme and is thus the preferred discretization technique for finite wear simulations.