Adjoint-based shape optimization of a ship hull using a Conditional Variational Autoencoder (CVAE) assisted propulsion surrogate model
Authors: Moloud Arian Maram, Georgios Bletsos, Thanh Tung Nguyen, Ahmed Hassan, Michael Palm, Thomas Rung
Organizations: J.M. Voith SE & Co. KG, Alexanderstrasse 2, Heidenheim, 89522, Baden-Württemberg, Germany · Institute for Fluid Dynamics and Ship Theory, Hamburg University of Technology, Am Schwarzenberg-Campus 4, Hamburg, 21073, Germany
Adjoint-based shape optimization of ship hulls is a powerful tool for addressing high-dimensional design problems in naval architecture, particularly in minimizing the ship resistance. However, its application to vessels that employ complex propulsion systems introduces significant challenges. They arise from the need for transient simulations extending over long periods of time with small time steps and from the reverse temporal propagation of the primal and adjoint solutions. These challenges place considerable demands on the required storage and computing power, which significantly hamper the use of adjoint methods in the industry. To address this issue, we propose a machine learning-assisted optimization framework that employs a Conditional Variational Autoencoder-based surrogate model of the propulsion system. The surrogate model replicates the time-averaged flow field induced by a Voith Schneider Propeller and replaces the geometrically and time-resolved propeller with a data-driven approximation. Primal flow verification examples demonstrate that the surrogate model achieves significant computational savings while maintaining the necessary accuracy of the resolved propeller. Optimization studies demonstrate that neglecting the propulsion system can result in hull designs whose performance is inferior to that of the initial shape when subsequently validated using a numerically resolved propulsor. In contrast, the proposed method produces shapes that actually achieve more than an 8% reduction in resistance.
Figures & tables
Figure 1 : Geometry of the investigated Service Operation Vessel (SOV) equipped with two propeller units (center), a single Voith Schneider Propeller (VSP) (left) and mesh components for a VSP unit with refinement near the propeller (right).
Figure 2 : Employed basic autoencoder scheme: The input X is compressed by the encoder network into a lower dimensional latent representation Z, and then reconstructed as X^ by the decoder network.
Figure 3 : The CVAE architecture used in this study. The 3D velocity field is provided as input to the encoder and concatenated with 11 conditional labels. The latent representation Z is reduced to two dimensions. The decoder reconstructs the 3D velocity field through three separate branches corresponding to the velocity components. Decoder residual blocks perform progressive spatial upsampling to recover the full resolution of the 3D field.
Figure 4 : Architecture of the model components. (top) Architecture of the self-attention mechanism, where the input is mapped to query, key, and value, and attention weights are computed using a softmax operation. The output is obtained by weighted summation. (bottom) basic residual block composed of convolution layers, batch normalization (BN), and ReLU activation, with a skip connection added to the output.
Figure 5 : Computational domain and boundary conditions. The ship is located at distances of 1, 2, 1.5, and 1.5 LOA from the inlet, outlet, lateral side walls and bottom boundary, respectively.
Figure 6 : Schematic representation of the Voith Schneider Propeller (VSP) kinematics, illustrating the azimuthal blade position ϕ and the cyclic pitch angle α . Each blade rotates around the propeller axis while oscillating about its own axis, resulting in a time-varying angle of attack.
Figure 7 : Computational mesh: the ship hull embedded in the unstructured polyhedral near-field mesh, full domain side view (top), with a close-up of the midship region (middle). Bottom row: close-up of the rotating VSP mesh cylinder containing the five blades (left), stern and VSP installation region (center), and bow region (right). The static unstructured polyhedral mesh ( ∼0.85 million cells) employs a base cell size of 1.0 m, a hull surface cell size of 0.2 m, and far-field cells up to 20 m, with a prism-layer first-cell height of y1=5×10−3 m. The rotating structured hexahedral mesh ( ∼1.34 million cells) consists of one outer cylinder enclosing five individual blades, with a prism-layer first-cell height of y1=1.5×10−4 m.
Figure 8 : Illustration of the employed geometric parameters, including key aftship (SAY1, and SAY2) and headbox features (HBLF, HBLB, HBAS,and HBH), cf. Table 1 .
Parameter
Unit
Value Range
Count of Unique Values
Ship Speed (in x -direction)
m/s
6.73–8.1
3.0
Stern Deadrise
deg
0.0
1.0
Stern Inclination
deg
0.40–1.40
10.0
Stern Angle Y1 (SAY1)
deg
1.88–6.55
10.0
Stern Angle Y2 (SAY2)
deg
5.29–17.95
10.0
HB_Length
-
1.0–2.0
6.0
Table 1 : Range of parameters values considered by the data set.
Figure 9 : Configuration of the meta-grid around the VSP. The meta-grid radius and height are set to 1.39 times the propeller diameter and height, respectively.
Figure 10 : 2D sketch of the meta-grid spacing and angular discretization. The angular coordinate is defined such that ϕ=0∘ corresponds to the forward (bow) direction and ϕ=180∘ to the aft (stern) direction. The propeller rotates counterclockwise.
abs. L2
rel. L2
abs. L∞
vx
1.05×10−1
1.31%
vy
5.46×10−2
3.74%
vz
3.20×10−2
2.92%
Velocity Divergence
6.69×10−2
6.43%
1.16
Velocity Magnitude
1.24%
Table 2 : Quantitative error metrics computed over the full test dataset.
Quantity
Mean
Median
95th Percentile
vx
7.63×10−2
5.49×10−2
8.99×10−2
vy
3.73×10−2
3.37×10−2
4.25×10−2
vz
2.12×10−2
1.89×10−2
2.56×10−2
∣v∣
7.45×10−2
5.37×10−2
8.75×10−2
Divergence
4.53×10−2
4.29×10−2
5.49×10−2
Table 3 : Distribution of per-sample spatial mean absolute errors across the test set. For each test sample, the mean absolute error (MAE) was computed over the 3D field, and the distribution of these errors across all test samples is summarized using the mean, median, and 95th percentile.
Parameter
Unit
Value
Ship Speed ( x -direction)
m/s
7.42
Stern Deadrise
deg
0.0
Stern Inclination
deg
1.067
SAY1
deg
4.99
SAY2
deg
13.87
HB_Length
-
1.2
Table 4 : Conditioning parameters used for the selected test sample.
Figure 11 : Comparison of x -velocity contours obtained from the time-averaged CFD (left) and the surrogate model (center) normalized with the ship speed in addition to the corresponding normalized error magnitude contours (right) for the test sample in the meta-grid.
Figure 12 : Comparison of velocity magnitude contours obtained from the time-averaged CFD (left) and the surrogate model (center) normalized with the ship speed in addition to the corresponding normalized error magnitude contours (right) for the test sample in the meta-grid.
Figure 13 : Comparison of velocity divergence contours obtained by 2nd-order Finite Differences from the time-averaged CFD (left) and the surrogate model results (center) with corresponding error magnitude contours (right) for a test sample in the meta-grid.
Parameter
Unit
Value
Ship Speed
m/s
6.73
Stern Deadrise
deg
0.0
Stern Inclination
deg
0.4
SAY1
deg
1.88
SAY2
deg
5.29
HB_Length
-
1.0
Table 5 : Conditioning parameters used for the test case.
Figure 14 : Spatial discretization of ship hull for the RP and SM simulations.
Figure 15 : Comparative side view of the three designs used for the parametric α^ study. Spatial discretization for SM simulations.
Figure 16 : Verification study for the propulsion model integration into CFD. Left : Error in percentage of the SM-predicted metrics for different values of α^ from the RP-predicted metrics.
Figure 17 : Velocity magnitude contours at 0.1m (left) and 0.9m (right) below the mounting point of the VSP. The dashed line denotes the symmetry plane. The top part is colored by the interpolated velocity magnitude coming from the surrogate model and the bottom part by the computed velocity magnitude as a result of the implicit forcing approach.
Figure 18 : Initial shape and discretized computational domain. Areas of the hull that are free for design are displayed in grey while those that are fixed to their initial position are displayed with red.
Figure 19 : Displacement field computed on the hull for the first optimization iteration ( k=0 ). Complete hull is used for visualization purposes. Dashed line indicates the symmetry plane.
Figure 20 : Optimization history. Vertical dashed lines highlight the optimization iteration in which the Armijo condition was for the first time not fulfilled.
Figure 21 : Framelines at three characteristic positions of the ship along its length. Continuous (green) line corresponds to the initial shape, dashed (blue) line to the optimized shape of the NP case, densely dashed (black) line to the optimized shape of the SP case Symmetry line at y=0 .
Parameter
Change - NP (%)
Change - SP (%)
Ship Speed ( x -direction)
-
-
Stern Deadrise
-
-
Stern Inclination
-
-
SAY1
-8.60
59.05
SAY2
-8.08
-8.07
HB_Length
-
-
Table 6 : Change of conditioning parameters from initial to optimized shape. Initial parameters correspond to those presented in Table 5 .
School of Ocean and Civil Engineering, Shanghai Jiao Tong University, Shanghai 200240, China · 2Eastern Institute of Technology, Ningbo, Ningbo 315200, Zhejiang, China · 3TenFong Technology Co., Ltd., Shenzhen 518000, Guangdong, China +1