of Fcolt directly in the laboratory system is proposed here. Using equations (3.6), (4.16) and (4.17), the relative surface velocity gcp
pq at the new time level t + Δt is given by
gcppq= A· F + C , (C 1)
where the matrix A reads
A = Δt Ipmp I p(n21−1)−mpR2p(n22+n23) −n1n2(Ip+mpR2) −n1n3(Ip+mpR2) −n1n2(Ip+mpR2) Ip(n22−1)−mpR2p(n21+n23) −n2n3(Ip+mpR2) −n1n3(Ip+mpR2) −n2n3(Ip+mpR2) Ip(n23−1)−mpR2p(n21+n22) (C 2) and the vector C is
C = ⎛ ⎝u 1 p− u1q+ n3Rp(ω2p+ ωq2)− n2Rp(ω3p+ ωq3) up2− u2q+ n1Rp(ω3p+ ωq3)− n3Rp(ω1p+ ωq1) u3 p− u3q+ n2Rp(ω1p+ ωq1)− n1Rp(ω2p+ ωq2) ⎞ ⎠ . (C 3)
Requiring that the relative surface velocity is zero amounts to imposing
The contact force F in (C 4) also may have components in the direction normal to the surface. To suppress the normal components, the contact force would have to fulfil the restriction
F· n= 0! . (C 5)
With this condition, the desired contact force in tangential direction is given by
Ft= F . (C 6)
Equations (C 4), (C 5) and (C 6) constitute an over-determined linear system with four equations and three unknowns
⎛ ⎜ ⎜ ⎝ a11 a12 a13 a12 a22 a23 a13 a23 a33 n1 n2 n3 ⎞ ⎟ ⎟ ⎠ · ⎛ ⎝F 1 t F2 t Ft3 ⎞ ⎠ = − ⎛ ⎜ ⎜ ⎝ C1 C2 C3 0 ⎞ ⎟ ⎟ ⎠ (C 7) abbreviated as A∗· Ft= C∗ . (C 8)
Here, the linear least squares solution FRM St with
C∗− A∗· FRM S t
2= min (C 9)
is computed and used further on. The numerical effort for the solution of (C 9) is ex- tremely small compared to the cost of the fluid solver. The method (C 9) imposes FRM St to be very close to tangential if n A with . being a suitable norm. Since mp∝ R3p and Ip∝ R5
p equation (C 2) shows that A∝ R−3p . Therefore, in the current implemen- tation n is replaced by n∗ = n / (Ipmp), so that n∝ R−8p . Hence, for Rp 1, n is several orders of magnitudes larger than A so that the force FRM St being computed is the same as Fcolt according to (4.22) or (4.23) to 4 digits in the tests conducted.
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