number of master points and the conservative load transfer algorithm 26 is

# Number of master points and the conservative load

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number of master points, and the conservative load transfer algorithm [26] is applied to compute the nodal 4
force f S j i at each slave node. Substituting eq. (2) and eq. (3) into eq. (4) leads to - δW F = n M X i =1 n i X j =1 f S j i · δ u M i + δ θ M i × R ( θ M i ) d j i , = n M X i =1 n i X j =1 f S j i · δ u M i + R ( θ M i ) d j i × f S j i · δ θ M i , = n M X i =1 f M i · δ u M i + p M i · δ θ M i , (5) Finally, noting that eq. (5) is the virtual work δW S of the one-dimensional beam structure, the energy is conservative in the decoupled master-slave procedure. Moreover, the master-slave kinematics does not require the matched master point M i to be coincident with a beam node. In such cases the conservative load transferred algorithm in [26] needs to be applied. Although we focus on cable subsystems with circular cross-sections in the present work, this procedure can also be generalized for more complicated surfaces, such as bridges [32], turbine blades and flexible aircraft. 4. Embedded computational framework The aforementioned cable-subsystem modeling approaches are particularly suitable for utilization in an Eulerian framework equipped with an Embedded or Immersed boundary method (EBMs or IBMs) [33, 34, 35, 36, 20, 18, 19, 21, 22, 37], also known as the fictitious domain method [38] and the Cartesian method [39, 40]. These methods effectively handle FSI applications featuring large structural deformations and/or topologi- cal changes [41, 21, 42], and therefore are an attractive option for cable subsystems characterized by large length-to-diameter ratios which generally undergo large deformations. Arbitrary Eulerian-Lagrangian (ALE) methods [43, 44] incorporating mesh motion algorithms or remeshing techniques [32, 10, 8] have also been applied to explore cable subsystem dynamics. In the present work, we focus mainly on the Eulerian frame- work. However, the cable subsystem modeling approaches discussed above can be readily incorporated into the ALE framework for problems with small or moderate displacements. 4.1. Governing equation Let Ω F denote the fixed fluid domain in the Eulerian computational framework, the conservative form of the Navier-Stokes equations can be written as W ∂t + ∇ · F ( W ) = ∇ · G ( V , V ) , in Ω F (6) where V and W are the vectors of the primitive and conservative variables describing the fluid state, respec- tively. F ( W ) and G ( V , V ) are respectively the inviscid and viscous flux tensor functions. Specifically, V = ρ v p , W = ρ ρ v E , F ( W ) = ρ v ρ v v + p I ( E + p ) v , and G ( V , V ) = 0 τ τ · v - q , (7) where ρ , v , p , and E denote the density, velocity, static pressure, and total energy per unit volume of the fluid, respectively. The velocity and total energy per unit volume are given by: v = ( v 1 , v 2 , v 3 ) T and E = ρe + 1 2 ρ ( v 2 1 + v 2 2 + v 2 3 ) , (8) 5
where e denotes the specific (i.e., per unit of mass) internal energy. In the physical inviscid flux tensor F , I ∈ R 3 × 3 is the identity matrix. In the viscous flux tensor