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16) 22 3 Non-Smooth Mechanics Either the relative acceleration γ˙i is zero and the friction element remains sticking with |λi | < ai , or the friction element begins to slide, where the relative acceleration γ˙i points in the opposite direction of the sliding force λi . e. Stribeck friction. 16) is combined with an external force, which depends on the relative velocities γi . This external force has no set-valued character and its value is zero for γi = 0, see Sect. 2 or [41]. e. a unilateral contact with friction.

This criterion can be evaluated for a global choice of ri = r ∀i, see also [32]. Let μ G and μ TJOR be vectors which contain the eigenvalues of G and TJOR , respectively. Note that μ G 0 because G is at least positive semidefinite. 54) which allows for writing the Lipschitz constant as a function of μ G , L = ρ (TJOR ) = max(|1 − r max(μ G )|, |1 − r min(μ G )|). 55) The Lipschitz constant is minimal for 1 − r max(μ G ) = 1 − r min(μ G ) ⇒ r= 2 . 56) Note that if the matrix G is only positive semidefinite, then at least one eigenvalue is equal to zero.

30) j=1 which we call the JORprox scheme. Note that this JORprox scheme corresponds to the Jacobi-relaxation (JOR) scheme [79] λhν +1 = λhν − ωh m ( ∑ Ghk λkν + ch ), Ghh k=1 h = 1 . . 2 Augmented Lagrangian Approach m ∑ k=1 k=h 37 Ghk <1 Ghh ∀h = 1 . . m. 32) The scalar ωh is the relaxation parameter which must be chosen as 0 < ωh < 2. 31) in order to solve the linear system Gλ + c = 0 and adds a projection to ensure that the new update of the force λ νi +1 is within the set Ci of admissible forces.

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