Abstract
In order to reduce the cost of oceanographic exploration, a new underwater vehicle is designed to sail the required distance with the lowest energy consumed. Since the new underwater vehicle is a complicated multidisciplinary system, it is firstly decomposed into four smaller disciplines and then a multidisciplinary design optimization (MDO) problem is built based on these disciplines. The Multidisciplinary Feasible (MDF) architecture is adopted as the solution strategy to this optimization problem considering that it is easily implemented and a multidisciplinary feasible solution is always guaranteed throughout the optimization process. To solve this optimization problem efficiently, the coupled adjoint method is firstly introduced to improve the efficiency of gradient calculation and then a discipline-merging method is proposed to further enhance the computational efficiency. After this, the discipline-merging method is verified against the finite difference method in two aspects of solution accuracy and computational costs and the results show it is an effective and high efficient gradient calculation method. Finally, the multidisciplinary design optimization of the new underwater vehicle is performed efficiently under the MDF architecture with the discipline-merging method to calculate gradients.
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This research was supported by the National Natural Science Foundation of China under Grant No. 51375389.
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Appendices
Appendix A: Earth-fixed and Body-fixed reference frames
In order to analyze the motion of the new underwater vehicle, two reference frames are introduced in this paper: earth-fixed reference frame O 0 x 0 y 0 z 0 and body-fixed reference frame O x y z, which are shown in Fig. 19. The earth-fixed reference frame is used to represent the motion of the new underwater vehicle relative to the ground. The axes of O 0 x 0 y 0 z 0 are fixed with the ground. Generally, the origin O 0 is chosen at the launch point on the ground, the axis O 0 x 0 points to the launch direction, the axis O 0 y 0 is vertically upwards and the axis O 0 z 0 is determined by right-hand rule. The body-fixed reference frame are used to describe the velocity and acceleration of the new underwater vehicle. Its origin O is usually set at the center of buoyancy. The axis O x is the longitudinal axis directed from the after to the forward end of the body, the axis O y is the normal axis directed from top to bottom and the axis O z is the transverse axis directed to starboard.
The earth-fixed and body-fixed reference frames
In order to measure the center of buoyancy, we introduce the front tip reference frame whose axes are parallel to the body-fixed reference frame but with the origin at the front tip point X 0, which are shown in Fig. 20.
The front tip reference frame
Appendix B: Motion state parameters
The motion of the new underwater vehicle with respect to the earth-fixed reference frame can be represented by the motion of body-fixed reference frame relative to the earth-fixed reference frame.
The new underwater vehicle has six DOF which can be determined by the vectors (x 0,y 0,z 0) and (𝜃,ψ,ϕ), respectively. The vector (x 0,y 0,z 0) represents the coordinate point of the origin of body-fixed reference frame in earth-fixed reference frame and the vector (𝜃,ψ,ϕ) describes the angular orientation of the body-fixed reference frame relative to the earth-fixed reference frame. Specifically, ψ is defined as the angle between the projection of O x on the plane x 0 O 0 z 0 and the axis O 0 x 0, which is positive when the head of the new underwater vehicle is at the left of O 0 x 0. ϕ is defined as the angle between the axis O y and the vertical plane which is through O x and perpendicular to the plane x 0 O 0 z 0, which is positive when the axis O y is at the right of the vertical plane observed from the after to the forward of the new underwater vehicle.
The vector (v x ,v y ,v z ) denotes the linear velocity of the center of buoyancy of the new underwater vehicle projected to the body-fixed reference frame. The vector (w x ,w y ,w z ) denotes the angular velocity of body-fixed reference frame projected to the body-fixed reference frame.
α and β respectively represent the angle of attack and the angle of side slip of the new underwater vehicle as shown in Fig. 21. α is the angle between the projection of velocity (v x ,v y ,v z ) on the plane x O y and the axis O x, which is positive when the velocity is below the head of the new underwater vehicle. β is the angle between the velocity and the plane x O y, which is positive when the velocity is at the right of the head of the new underwater vehicle.
The angles of attack and side slip of the new underwater vehicle
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Zhang, D., Song, B., Wang, P. et al. Multidisciplinary optimization design of a new underwater vehicle with highly efficient gradient calculation. Struct Multidisc Optim 55, 1483–1502 (2017). https://doi.org/10.1007/s00158-016-1575-2
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DOI: https://doi.org/10.1007/s00158-016-1575-2