The second application refers to a Double Ended Ferry (DEF), as depicted in Figs. 5.8 and 5.9. The hull is optimized w.r.t. total drag for a scale 1:8.75 model. Since a double ended ferry moves both forward (V1∗ < 0) and backward (V1∗ > 0), the performance op-timization must mimic this symmetry and preserve the longitudinal symmetry along the central x∗2−x∗3 plane (mid ship section) of the initial hull, cf. Fig. 5.9 (a). Floatation is
Figure 5.8: Double Ended Ferry (DEF): Perspective 3D representation of the initial geometry.
not considered, and geometrical constraints aim to preserve the initial main dimensions, to maintain the displacement, and to guarantee the symmetry of the hull w.r.t. an identical forward and backward cruise performance. Simply speaking, the latter requires the refor-mulation of the force objective from Sec. 3.2, in which the forces of the forward (J∗+) and backward (J∗−) cruise need to be evaluated separately to subsequently form an equally weighted sum, viz. J∗+,− = 0.5J∗++ 0.5J∗−. However, in case of a perfectly symmetric initial geometry and domain, the determination of the sensitivity for one cruise direction, e.g. J∗+, using an unmirrored geometry is sufficient, if the sensitivity is subsequently mirrored around the mid ship plane. The mirrored sensitivity should precisely correspond to the reversed flow sensitivity, e.g. J∗−, and thus to the opposite travel direction. Both sensitivity distributions are combined to guide the gradient-based optimization. A per-fectly mirrored grid with an identical control volume distribution on both sides of the mid ship symmetry plane is therefore much appreciated. In this situation, an interpolation of mirrored sensitivities could be bypassed by a simple injection, and the optimized shape naturally remains symmetric throughout the optimization.
The DEF consists of a hull, bulkwark and deck as conceptually sketched in Fig. 5.9, where the coordinate system is placed at the bottom of the hull in the mid ship section and the free surface is initialized in thex∗1−x∗2 plane atx∗3/L∗D= 5/146 based on the DEF lengthL∗D. The study is performed for a turbulent flow at ReL =V1∗L∗D/ν∗b = 6.26·106 and Fn = V1∗/√
G∗L∗D = 0.25, based on the gravitational acceleration G∗, the inflow velocity V1∗ and the kinematic viscosity of the water ν∗b. A dimensionless wave length of λ = λ∗/L∗D = 2πFn2 = 4.4 is expected. The three-dimensional domain has a length, height and width of 10L∗D, 4L∗D as well as 4L∗D. The geometry is symmetric around the x∗1−x∗3 andx∗2−x∗3 plane. Therefore only half of the lateralx∗2-extend is considered and the numerical grid is mirrored around the mid shipx∗2−x∗3plane by construction. The inlet and
x∗1 x∗3 L∗D 10L∗D
4L∗D
2L∗D 4.5L∗D
5/146L∗D gi∗
(a) (b)
Figure 5.9: Double Ended Ferry (DEF) case (ReL = 6.26 · 106 and Fn = 0.25): (a) Schematic drawing of the initial configuration and (b) unstructured numeri-cal grid around the stern and the free surface.
outlet boundaries are located 4.5 geometry-lengths away from the origin. The symmetric unstructured numerical grid is displayed in Fig. 5.9 (b) and consists of approximately 5·106 control volumes. The fully turbulent simulations employ a wall-function-basedk∗−ω∗SST model and convective terms for momentum and turbulence are approximated using the QUICK scheme. The HRIC scheme is used for the compressive concentration transport.
The simulation applies a main flow against thex∗1-axis. Velocity and concentration values are prescribed at the inlet and slip walls are used along the top, bottom as well as remote lateral planes. A hydrostatic pressure boundary is employed along the outlet located at the minimum x∗1-position. A symmetry condition is imposed along the central lateral plane.
The dimensionless wall normal distance of the first grid layer reads x+2 = y+ ≈ 50 and the free surface refinement employs approximately δx∗1/λ∗ = δx∗3/λ∗ = 1/50 cells in the longitudinal as well as lateral and δx∗2/λ∗ = 1/500 cells in the normal direction. The tangential resolution of the free surface region is refined within a Kelvin-wedge to capture the wave field generated by the DEF geometry. To ensure a symmetric mesh update, the utilized numerical mesh and thus the Kelvin-wedge is mirrored, cf. Fig. 5.10. According to Alg. (2) the integration in pseudo-time applies a time step size based on Cotar = 0.35 embedded in five sub-cycles.
Adjoint simulations employ a pure DDS approximation of the convective adjoint con-centration transport and use an apparent viscosity based on the primal estimation from (2.53) together with an adjoint time step size ∆t∗adjthat is significantly reduced by a factor
∆t∗adj/∆t∗pri = 1/100 compared to the primal time step size ∆t∗pri.
During a first series of computations, the symmetry condition is not considered for the formulation of the objective functional and only the forward cruise (J∗+) force on the hull is minimized for three different step sizes. The cost functional convergence is shown in Fig.
5.11 (a). All optimizations minimize the force by about (J∗+−J∗+,ini)/J∗+,ini ≈ 2.75%.
The normalized initial deformation field is shown in Fig. 5.12 (top). Due to the asymmetric shape update, the forces of the forward (J∗+) and rearward (J∗−) cruise diverge for the optimized shape. To validate this, the inlet and outlet boundary conditions are reversed
Figure 5.10: Double Ended Ferry (DEF) case (ReL = 6.26·106 and Fn = 0.25): Symmetric numerical grid in the still water plane.
on both the initial as well as the optimized grid, so that the DEF now effectively travels in the reverse direction that offers access to the J∗− performance. The resulting normalized drag force is shown in Fig. 5.11 (b) for a certain time period, where the solid [dashed]
lines correspond to the total resistance from forward (J∗+) [backward (J∗−)] travel. In line with the expectation, the initial grid provides an identical resistance for both cases (i.e. J∗+,ini =J∗−,ini). Nonetheless, the drag convergence as well as the final value of the optimized shape differ significantly (i.e. J∗+,opt ̸=J∗−,opt) and the optimized resistance of the backward travel even exceeds that of the initial forward cruise, i.e. J∗−,opt > J∗−,ini (=J∗+,ini).
0 10 20
−3
−2
−1 0
nopt [-]
(J∗+ −J∗+,ini )/J∗+,ini ·102 [%]
α∗di= 1L∗O/103 α∗di= 2L∗O/103 α∗di= 4L∗O/103
(a)
10 25 50 75 90
0.9 0.95 1 1.05 1.1
nTS/103 [-]
|J∗(·) /J∗(·),ini |[-]
J∗+,ini J∗−,ini J∗+,opt J∗−,opt
(b)
Figure 5.11: Double Ended Ferry (DEF) case (ReL = 6.26·106and Fn = 0.25) with unsym-metric shape update: (a) Relative objective convergences and (b) comparison of initial and optimized shape based on forward and backward cruise.
Subsequently, three additional optimizations are performed, which mirror the
sensitiv-g∗i+ni
||g∗i+ni||
g∗i+,−ni
||g∗i+,−ni||
1.0
0.5
0.0
Figure 5.12: Double Ended Ferry (DEF) case (ReL= 6.26·106 and Fn = 0.25): Normalized initial deformation field along the hull for (top) the straight forward drag minimization and (bottom) its symmetrized analogue.
ity derivative around the mid ship after each adjoint simulation in order to minimize J∗+,− = 0.5J∗++ 0.5J∗−. An exemplary normalized deformation field is shown in Fig.
5.12 (bottom) and indicates the symmetrized character in contrast to the former, non-symmetric approach. For all three step sizes used, the cost functional convergence is plotted in Fig. 5.13 (a). Again, all optimizations converge to nearly the same gain of (J∗+,−−J∗+,−,ini)/J∗+,−,ini ≈3.7%, which is, interestingly, below that of the non-symmetric optimization. To emulate the reverse travel, the optimized shape based on the smallest step size is simulated with reversed inlet and outlet boundary conditions. The resulting total resistance force convergence over an exclusive time period is shown in Fig. 5.13 (b), where the solid [dashed] lines correspond to the total resistance from forward (J∗+) [backward (J∗−)] cruise. As for the initial shape, the symmetrically optimized shape has an identical cost functional convergence for both forward and backward travel, i.e.
J∗+,opt = J∗−,opt < J∗−,ini = J∗+,ini and thus J∗+,−,opt < J∗+,ini = J∗−,ini. To summarize:
The mirroring of the shape sensitivity derivative after an adjoint simulation of the DEF forward cruise allows a symmetry preservation of the shape optimization for the purpose of an equivalent backward traveling performance. For the optimizations based on the smallest step size, slices of the optimized shapes with and without symmetrization are depicted in Fig. 5.14.
0 10 20 30
−4
−3
−2
−1 0
nopt [-]
(J∗+,− −J∗+,−,ini )/J∗+,−,ini ·102 [%]
α∗di= 1L∗O/103 α∗di= 2L∗O/103 α∗di= 4L∗O/103
(a)
10 25 50 75 90
0.9 0.95 1 1.05
nTS/103 [-]
|J∗(·) /J∗(·),ini |[-]
J∗+,ini J∗−,ini J∗+,opt J∗−,opt
(b)
Figure 5.13: Double Ended Ferry (DEF) case (ReL = 6.26·106 and Fn = 0.25) with sym-metric shape update: (a) Relative objective convergence and (b) comparison of initial and optimized shape based on forward and backward cruise.
0 0.025 0.05 0.075 0.1 0.125 0
0.025 0.05 0.075 0.1
x∗2/L∗D [-]
x∗ 3/L∗D [-]
initial
−0.4 −0.3 −0.2 −0.1 0 0.1 0.2 0.3 0.4 0
0.025 0.05 0.075 0.1
x∗1/L∗D [-]
x∗ 3/L∗D [-]
unsymmetric
−0.4 −0.3 −0.2 −0.1 0 0.1 0.2 0.3 0.4 0
0.025 0.05 0.075 0.1 0.125
x∗1/L∗D [-]
x∗ 2/L∗D [-]
symmetric
Figure 5.14: Double Ended Ferry (DEF) case (ReL = 6.26·106 and Fn = 0.25): (Top) Frames, (center) waterlines and (bottom) buttocks for the initial (black) and symmetric (blue, dashed) as well as unsymmetric optimized (orange, dotted) geometry.