- Email: [email protected]
Design of bulkhead reinforcement of trimaran based on topological optimization
Ocean Engineering 191 (2019) 106498
Contents lists available at ScienceDirect
Ocean Engineering journal homepage: www.elsevier.com/locate/oceaneng
Design of bulkhead reinforcement of trimaran based on topological optimization Dejun Jia, Fanchun Li * School of Ship and Ocean Engineering, Dalian Maritime University, Dalian, 116026, China
A R T I C L E I N F O
A B S T R A C T
Keywords: Trimaran Bulkhead reinforcement Finite element method Topology optimization Lightweight design
A design method of bulkhead reinforcement of trimaran based on topological optimization is proposed, which can reduce the mass of bulkhead and reinforcement as much as possible while ensuring that bulkhead has suf ficient structural strength. Based on Rules for The Classification of Trimarans, seven structural strength checking conditions are designed. Stress distribution of trimaran bulkhead structure and reinforcement before and after optimization under different conditions are calculated by using finite element method and variable density to pology optimization method. The results show that: The maximum stress distribution area of bulkhead and reinforcement occurs near the connecting bridge and the large opening. When the volume constraints are changed, the stress distribution and maximum stress of the optimized bulkhead and reinforcement will change accordingly. There is no positive correlation between the maximum stress and the volume reduction ratio of reinforcement. The maximum bulkhead stress cannot be reduced by increasing the volume retention ratio of the reinforcement. The distribution of the optimized reinforcement is similar to that of the real bulkhead. On the premise that the bulkhead has sufficient structural strength, the lightweight design of trimaran structure can be realized based on the topological optimization method.
1. Introduction Some advantages like good stability and seakeeping, low resistance under high-speed navigation, etc. are within trimaran. Lightweight design of trimaran structure can effectively improve its comprehensive performance on the premise of ensuring structural safety. Some studies attempt to apply rules-based design method (Zhang and Wang, 2007; Zhen et al., 2012; Fuentes et al., 2015), design method based on existing ships (Fuentes et al., 2015), and auxiliary design method based on environmental load and finite element calculation (Ren et al., 2012) to optimize the structure of trimaran. These design methods have contributed to ensuring the safety of hull structures, but their contribution to lightweight design of ships is very limited. Some studies attempt to lighten the bridge structure of trimaran, aiming at ensuring structural strength and weight reduction, and obtain the cor responding optimal structure (Yang, 2008, 2010; Deng, 2008) under specific working conditions. With the development of intelligent optimization methods, re searchers have gradually applied intelligent optimization methods, such as particle swarm optimization (Ehlers, 2012) and genetic algorithm (Du
et al., 2018), to ship structure lightweight design. Intelligent optimiza tion method can improve material utilization (Ehlers, 2012), effectively reduce the quality of ship structure and obtain hull structure with target quality (Du et al., 2018), and ensure that the optimized structure has sufficient strength (Ehlers, 2012; Du et al., 2018). Compared with intelligent optimization method, topology optimi zation method has more prominent advantages in structural lightweight design, especially in structural lightweight design for continuum. Based on the principle of computational mechanics, the continuous structure topology optimization design method (Bendsoe and Kijuchi, 1998; Bendsoe, 1989; Xie and Steven, 1993; Allaire et al., 2002) and the discrete structure topology optimization design method (Cheng and Guo, 1997; Su et al., 2009) can realize the lightweight design of the structure on the premise of ensuring the safety of engineering structures. At present, these kinds of methods have been widely used in civil en gineering (Xu et al., 2018), aerospace science and technology (Munk and Verstraete, 2017; Oktay and Akay, 2011), large deformation flexible structure (Zhu et al., 2019) and biomechanics (Chen et al., 2018). In addition, through the latest research (Wang and Zhang, 2018), it can be seen that the topology optimization design method has been applied to
* Corresponding author. E-mail address: [email protected] (F. Li). https://doi.org/10.1016/j.oceaneng.2019.106498 Received 2 March 2019; Received in revised form 15 August 2019; Accepted 25 September 2019 Available online 1 October 2019 0029-8018/© 2019 Elsevier Ltd. All rights reserved.
D. Jia and F. Li
Ocean Engineering 191 (2019) 106498
Table 1 Main parameters of bulkhead. Parameters
Value
Yield strength Allowable stress Young’s modulus Poisson’s ratio Density Bulkhead thickness
235 MPa 199.75 MPa 2.1 � 105MPa 0.3 7850 kg/m3 10 mm
Fig. 3. Longitudinal torque.
of this paper are compared with those of the actual ship bulkhead reinforcement structure distribution, and the results of the trimaran transverse bulkhead topology optimization are evaluated to provide technical reference for the trimaran main structure topology optimiza tion design and the bulkhead reinforcement structure distribution area design.
Fig. 1. Geometric shape of bulkhead.
2. Topology optimization theory of continuous structures based on finite element method The optimization region is divided into several finite elements, and the relative density ρi of each element is taken as the design variable: � 1 i 2 Ωmat ρi ¼ (1) 0 i 2 Ω=Ωmat here, i is the number of the finite element; Ωmat is the retaining material area and Ω is the whole design area. According to ρi , we have: Z ρi dΩ � V0 ; 0 � ρi � 1 ; i2Ω (2)
Fig. 2. Horizontal moment.
Ω
the design of high precision instruments. The objective of this method is no longer the traditional structural mechanical response, but the per formance response parameters of the instrument. At present, the appli cation of topology optimization method in ship structure optimization design is relatively limited, and the optimization zone is small, but the contribution is obvious. For example, Zhang (Zhang and Yang, 2015) applied shape optimization method and topology optimization method to small-scale optimization design of ship bottom frame structure, and the weight of optimized structure was reduced by 15.82% compared with that of un-optimized structure. At present, the application of to pology optimization method to the optimization design of large-scale ship structures (such as bulkhead and bulkhead reinforcement struc tures of trimaran ships) is rare. In view of the current research situation, this paper carries out structural strength checking design on the basis of Rules for The Classification of Trimarans, which is the code of the Lloyd’s Classification Society of Britain. The preliminary design of the trans verse bulkhead of a specific trimaran is a prototype of structural opti mization, and the static response of the bulkhead structure is calculated by the finite element method. Subsequently, on the premise of guaran teeing that the structural strength conforms to the design specifications, the topology optimization of bulkhead structure is carried out with the minimum flexibility as the optimization objective. Finally, the opti mized bulkhead structure is taken as the reinforcement structure. The strength of the optimized bulkhead structure is analyzed, and the stress distributions under different working conditions and different weight reduction conditions are compared. In addition, the optimization results
here, V0 is the total volume of the design domain. The optimization problem expression with the objective of minimizing structural flexi bility under volume constraints is: 8 Find ρi ; i ¼ 1; 2; … m > > > > min C ¼ FT u < s:t: V=V0 f � 0 (3) > > Ku ¼ F > > : 0 < ρmin < ρi � 1 here, m is the total element number of units; the objective function C represents the total structural compliance; F is the overall load vector and u is the displacement vector; V is the optimized total volume and f is the volume ratio. K is the global stiffness matrix. In order to prevent the singularity of stiffness matrix, the lower limit of relative density of element ρmin is introduced, ρmin ¼ 0.001. After established the model, the sensitivity values of the constraint and the objective function are required. Applying solid isotropic mi crostructures penalty (SIMP) model can avoid solving the discrete value design problem (Zhang and Yang, 2015; Olsen and Vanderplaats, 1989). Bendsoe and Sigmund (2003); Zhou and Rozvany (1991) gave the conditions that the real SIMP difference needs to satisfy: � � 1 γ0 3 1 γ0 p � p* ¼ max 15 (4) ; 7 5γ0 2 1 2γ0 here, γ0 is Poisson’s Ratio; p* is penalty factor lower limit and p is 2
D. Jia and F. Li
Ocean Engineering 191 (2019) 106498
3. Structural strength checking of trimaran bulkhead 3.1. Bulkhead information and load case design In this section, a special trimaran bulkhead structure is taken as the optimization object to optimize its structure topology. The main pa rameters of bulkhead are shown in Table 1, and the geometric shape of bulkhead is shown in Fig. 1. Because the transverse bulkhead of trimaran has the symmetry of longitudinal-mid-section of the ship, the half bulkhead model shown in Fig. 1 will be selected in this section for finite element calculation of bulkhead, and the topological optimization of bulkhead will be carried out based on the calculation results. In the absence of hydrodynamic loads on wet surface of trimaran, the working conditions can be designed and checked in combination with Rules for The Classification of Trimarans. Combining with the bulkhead model in this paper, the load in this section is selected as follows: hor izontal bending moment (Fig. 2), longitudinal torque (Fig. 3), arch state in lateral separation moment (Fig. 4) and vertical state in lateral sepa ration moment (Fig. 5). According to the Rules for The Classification of Trimarans, the approximate formula for calculating horizontal bending moment is as follows:
Fig. 4. Arch state in lateral separation moment.
Fig. 5. Vertical state of lateral separation moment.
penalty factor. In the following optimization work, based on ANSYS Parameter Design Language and combined with the above theory, the structural topology optimization design of trimaran bulkhead will be carried out in this paper.
Mh ¼ Df Lf fserv L2R DðCb þ 0:7Þ
Fig. 6. Finite element analysis of bulkhead structure (stress unit: MPa). 3
(5)
D. Jia and F. Li
Ocean Engineering 191 (2019) 106498
Fig. 7. Optimized area and optimized result.
" Lf ¼
# � �2 LR 70 þ 5LR � 10 100
3
(6)
Msps ¼ 9:81fserv Wsh
� Δ
� � 2Δsh az ysh 2
Bmh 2
� (8)
here, Msph is the lateral separation bending moment (mid-arch); Wsh is the weight of a single hull, which can be obtained by multiplying the total weight of the trimaran by the percentage of the drainage of a single hull; az is the vertical acceleration; ysh is the distance from the center line of the hull to the center line of the main hull; Bmh is the main hull width; Δis the overall hull drainage and Δsh is the main hull drainage. The transverse bending moment at the joint of the connecting bridge and the sheet is calculated as follows:
here, Mh is the horizontal bending moment; Df is the longitudinal dis tribution factor, 0 is taken at the bow and stern, 1 is taken at the bow and stern, and other positions are obtained by interpolation; Lf is the captain change factor; fserv is the service area coefficient; LR is the length be tween the main vertical lines; D is the depth of the main hull; Cb is the square coefficient. The expression for calculating the lateral bending moment at the connection between the bridge and the main body is as follows: � � Bmh Msph ¼ 9:81fserv Wsh ð1 þ az Þ ysh (7) 2
Msph ¼ 9:81fserv Wsh ð1 þ az Þðysh
4
y0 Þ
(9)
D. Jia and F. Li
Ocean Engineering 191 (2019) 106498
Fig. 8. Maximum stress of composite structures.
5
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Ocean Engineering 191 (2019) 106498
Fig. 9. Stress distribution of bulkhead structures with different volume retention ratio.
6
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Ocean Engineering 191 (2019) 106498
Fig. 10. Stress distribution of reinforcement with different volume retention ratio.
7
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Ocean Engineering 191 (2019) 106498
bulkhead opening. In CASE5, the maximum stress of bulkhead structure is 270.04 MPa. The dangerous area is the bottom of connecting bridge and the vicinity of bulkhead opening. In this case, the bulkhead will be damaged. Therefore, it is necessary to add reinforcement structure to bulkhead. 4. Optimization design of bulkhead structure Although the stress value of bulkhead structure corresponding to CASE 5 is the largest, and the bulkhead structure will be destroyed only in this case, this section will carry out the topological optimization design of bulkhead structure under seven working conditions. The optimized bulkhead structure is taken as the original reinforced struc ture of the un-optimized bulkhead. The optimal design will minimize the volume of the reinforced structure (i.e. the optimized bulkhead) as much as possible on the premise of ensuring the structural safety of bulkhead and reinforced structure. In Fig. 7 (a), the blue area is the optimum area and the gray area is the preservation area. Under the constraints of 50% volume reduction in the optimized area, the optimized and smoothed trimaran bulkhead structure is shown in Fig. 7(b)–(h). Change the shape preservation of trimaran bulkhead optimization area. The volume retention ratio of the optimized area is set to 0%, 50%, 55%, 60%, 65%, respectively. 95%, 100%, and the bulkhead topology optimization. The optimized bulkhead is used as a reinforcement structure and “pasted” on the un-optimized bulkhead. The strength of the new structure is checked again. Fig. 8 shows the maximum stress of composite structures under different volume constraints. From the results of Fig. 8, it can be seen that the maximum stress of the optimized composite structure are different under different volume constraints. There is no positive correlation between the maximum stress and the volume retention ratio of composite structures. For example, in CASE 7, the maximum stress of bulkhead structure increases first and then decreases when the volume retention ratio of optimized area increases from 60% to 70%. When the retention ratio increases from 65% to 75%, the maximum stress of bulkhead structure will first decrease and then increase. In addition, Fig. 8 shows that the maximum stress of bulkhead reinforcement structure (i.e., the optimized bulkhead structure with “sticking” on the unoptimized bulkhead) is generally higher than that of bulkhead. In Section 3, the maximum stress of bulkhead in CASE 5 is far greater than the allowable stress of bulkhead. From Fig. 8 (e), it can be seen that the structural stress of bulkhead decreases obviously with the attach ment of reinforcement structure, and the maximum stress of bulkhead and the maximum stress of reinforcement structure will change with the change of retention ratio of optimized area volume. Fig. 9 shows the stress distribution of bulkhead structure with different volume confor mity under condition 5, and Fig. 10 shows the corresponding stress distribution of reinforced structure. Combining Figs. 8 (e), Figs. 9 and 10, we can see that: The maximum stress distribution area of the composite structure is concentrated under the connecting bridge and near the large opening. The maximum stresses of bulkhead structure and reinforced structure are lower than the allowable stresses when the reinforced structure is added. When the conformity ratio is reduced from 100% to 50%, the stress of the com posite structure does not increase significantly, which indicates that if the structure is optimized reasonably, the mass of the structure can be greatly reduced and the safety of the structure can be ensured. In addition, although the optimum region is far away from the large opening, the change of its structural form will also influence the stress distribution in the dangerous region.
Fig. 11. Construction site of Austal’s trimaran.
� Msps ¼ 9:81fserv Wsh
Δ
� 2Δsh az ðysh 2
y0 Þ
(10)
here, y0 is the transverse distance between the longitudinal section of the main hull and the connecting bridge. The expression for calculating the longitudinal torque is as follows: � � Vmhs Mlt ¼ 7:5Tf fserv ρ Vsh þ Vcd þ (11) ycs ahcave 2 here, Mlt is the longitudinal torque; Tf is the longitudinal torque distri bution coefficient; Vsh is the volume of the slice, Vcd is the volume of the connecting bridge, Vmhs is the volume of the main body as the length of the slice; ycs is the transverse section of the slice, the distance between the central line of the connecting bridge and the central line of the main hull; ahcave is the pitching acceleration. Combining with the optimized object in this paper, the checking conditions in this section are as follows: CASE1.
0.3 � Transverse Separation Moment (mid Arch)
CASE 2. 0.3 � Transverse Separated CASE3. Transverse Separated Bending Moment (Mid Arch) þ0.2 � Longitudinal Torque CASE4. Lateral Separation Moment (Mid-vertical) þ0.2 � Longitudi nal Torque CASE5. 0.3 � Horizontal Moment þ 0.4 � Lateral Separation Moment (Middle Arch) þ Longitudinal Torque CASE6. Horizontal Moment þ 0.4 � Transverse Separation Moment (Mid Arch) CASE7. 0.2 � Horizontal Moment þ 0.6 � Lateral Separation Moment (Mid Arch) 3.2. Finite element analysis of bulkhead structure of trimaran In this section, the hydrostatic bending moment and transverse tor sion are neglected in the calculation of bulkhead structural stress. The structural stress of bulkhead under seven working conditions are calculated by ANSYS. Higher-order triangular mesh and quadrilateral mesh with intermediate nodes are used to mesh bulkhead (as shown in Fig. 6 (a)). The results of stress distribution of bulkhead structure are shown in Fig. 6(b)–(h). The bulkhead structure will be loaded based on Figs. 2–5 and Equations (5)–(11). According to the calculation results of Fig. 6, the maximum stress area of bulkhead mainly occurs near the connecting bridge and the
5. Optimized contrast and rationality verification Fig. 11 is the construction site of Austal’s trimaran. The distribution of bulkhead structure and reinforcement can be seen in the figure. Ac cording to the structure in the drawing, the finite element model is 8
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Ocean Engineering 191 (2019) 106498
Fig. 12. Optimization results of bulkhead with 60% volume retention ratio.
established and the static analysis under seven working conditions are carried out. Based on the calculation results, the bulkhead is topology optimized, and the optimization results are shown in Fig. 12. By com parison, the results of optimized bulkheads in CASE1~CASE5 are highly similar to those reinforcement in Fig. 11, which shows that the light weight design of trimaran bulkheads based on topological optimization method is feasible.
out topological optimization design and stress analysis for a particular trimaran bulkhead structure. The optimized bulkhead can be used as a reinforcement structure to “paste” on the unoptimized bulkhead to obtain a new composite structure. Compared with the original structure, the stress of the new structure is significantly reduced, and in most cases, the stress of the reinforced structure is higher than that of the bulkhead. The optimized bulkhead structure has an effect on the stress distribution of the new structure. There is no positive correlation between the stress amplitude of the new structure and the conformation ratio of the optimized region, which indicates that the potential factors affecting the stress amplitude of the structure may include the location and shape of the material area
6. Discussion and conclusion Based on Rules for the Classification of Trimarans, this paper designs seven checking conditions for trimaran bulkhead structure, and carries 9
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Ocean Engineering 191 (2019) 106498
removed. The influence of these factors on structural strength needs to be verified by further research. It should be noted that, from the point of view of technology, the optimized bulkhead is not easy to be “pasted” on the bulkhead as a reinforced structure. However, the material distribution area of the optimized structure can be used as a reference for the design of bulkhead reinforcement structures, that is, the reinforcement structures can be arranged in these areas. In addition, the optimization method in this
paper can also be used as the optimization method of non-watertight bulkhead. By comparison, the results of the optimized bulkhead are highly similar to the distribution of the actual bulkhead reinforcement struc ture in some cases, which shows that the lightweight design of trimaran bulkhead and the design of the reinforced structure based on the topo logical optimization method are feasible.
Appendix Based on the results of the topological optimization in Fig. 12, the appendix establishes the reinforcement structure of the bulkhead according to the results of the material distribution of the topological optimization. The bulkhead surface has a reinforced structure perpendicular to the bulkhead, with a height of 0.2 m and a thickness of 10 mm. The bulkhead has a reinforced structure with a thickness of 8 mm and a width of 0.1 m. Fig. A1 shows the stress analysis results of the reinforced bulkhead.
10
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Ocean Engineering 191 (2019) 106498
Similar to the results in section 4, the distribution of reinforcement structure will affect the stress distribution and stress amplitude of bulkhead structure. The maximum stress is still distributed near the bulkhead connection bridge and the large opening.
References
Olsen, G.R., Vanderplaats, G.N., 1989. Method for nonlinear optimization with discrete design variables. AIAA J. 27 (11), 1584–1589. Ren, H., Zhen, C., Feng, G., et al., 2012. Experimental study on fatigue strength of trimaran connecting bridge. J. Huazhong Univ. Sci. Technol. 40 (8), 62–66. Su, R., Gui, L., Fan, Z., 2009. Topology and Sizing Optimization of Truss Structures Using Adaptive Genetic Algorithm with Node Matrix encoding[C]//Fifth International Conference on Natural (Tianjin, China). Wang, R., Zhang, X., 2018. Parameters optimization and experiment of a planar parallel 3-DOF nanopositioning system. IEEE Trans. Ind. Electron. 65 (8), 6487–6496. Xie, Y., Steven, G.P., 1993. A simple evolutionary procedure for structural optimization. Comput. Struct. 49 (5), 885–896. Xu, Z., Huang, Q., Zhao, Z., et al., 2018. Design of quiet structure based on topological optimization. J. Huazhong Univ. Sci. Technol. 38 (4), 86–87. Yang, Z., 2008. Research on Lightweight Design of High-Speed Trimaran Structure. Doctoral Dissertation. Wuhan University of Technology, Wuhan, China. Yang, D., 2010. Structural Design and Strength Evaluation of High-Speed Trimaran. Doctoral Dissertation, Harbin Engineering University, Harbin, China. Zhang, L., Wang, D., 2007. Optimum design of mid-ship profile of 3100 TEU container ship based on LSIGHT. Mar. Eng. 180 (5), 16–18. Zhang, H., Yang, D., 2015. Topology and shape optimization design of typical ship grillage. China Ship Res. 59 (6), 27–33, 59. Zhen, C., Ren, H., Zhang, C., et al., 2012. Direct calculation and analysis of structural strength of trimaran. J. Dalian Marit. Univ. 38 (3), 31–35. Zhou, M., Rozvany, G.I.N., 1991. The COC algorithm, Part II:topological, geometrical and generalized shape optimization. Comput. Methods Appl. Mech. Eng. 89 (1), 309–336. Zhu, B., Chen, Q., Li, H., Zhang, H., Zhang, X., 2019. Design of planar large-deflection compliant mechanisms with decoupled multi-input-output using topology optimization. J. Mech. Robot. 11 (3), 031015-031015-7.
Allaire, G., Jouve, F., Toader, A.M., 2002. A level-set method for shape optimization. Compt. Rendus Math. 334 (12), 1125–1130. Bendsoe, M.P., 1989. Optimal shape design as a material distribution problem. Struct. Multidiscip. Optim. 1 (4), 193–202. Bendsoe, M.P., Kijuchi, N., 1998. Generating optimal topologies in structural design using a homogenization method. Comput. Methods Appl. Mech. Eng. 71 (2), 197–224. Bendsoe, M.P., Sigmund, O., 2003. Topology optimization:Theory, Methods and Applications. Springer, New York. Chen, Q., Zhang, X., Zhu, B., 2018. Topology optimization of fusiform muscles with a maximum contraction. Int. J. Numer. Methods Biomed. Eng. 34 (8), e3096. Cheng, G., Guo, X., 1997. ε - relaxed approach in structural topology optimization. Struct. Optim. 13 (4), 258–266. Deng, L., 2008. Structural and Mechanical Characteristics of High-Speed Trimaran. Doctoral Dissertation, Wuhan University of Technology, Wuhan, China. Du, L., Pinto, G.O., Hefazi, H., et al., 2018. Trimaran structural weight optimization based on classification rules. J. Ship Prod. Des. 00 (0), 1–10. Ehlers, S., 2012. A particle swarm algorithm-based optimization for high-strength steel structures. J. Ship Prod. 28 (1), 1–9. Fuentes, D., Salas, M., Tampier, G., Troncoso, C., 2015. In: Structural Design and Optimisation of an Aluminium Trimaran. Analysis and Design of Marine Structures Guedes Soares & Shenoi. Taylor & Francis Group, London, ISBN 978-1-138-02789-3. Munk, D.J., Verstraete, D., 2017. Effect of fluid-thermal–structural interactions on the topology optimization of a hypersonic transport aircraft wing. J. Fluids Struct. 75 (11), 45–76. Oktay, E., Akay, H.U., 2011. Parallelized structural topology optimization and CFD coupling for design of aircraft wing structures. Comput. Fluids 49 (10), 141–145.
11
Contents lists available at ScienceDirect
Ocean Engineering journal homepage: www.elsevier.com/locate/oceaneng
Design of bulkhead reinforcement of trimaran based on topological optimization Dejun Jia, Fanchun Li * School of Ship and Ocean Engineering, Dalian Maritime University, Dalian, 116026, China
A R T I C L E I N F O
A B S T R A C T
Keywords: Trimaran Bulkhead reinforcement Finite element method Topology optimization Lightweight design
A design method of bulkhead reinforcement of trimaran based on topological optimization is proposed, which can reduce the mass of bulkhead and reinforcement as much as possible while ensuring that bulkhead has suf ficient structural strength. Based on Rules for The Classification of Trimarans, seven structural strength checking conditions are designed. Stress distribution of trimaran bulkhead structure and reinforcement before and after optimization under different conditions are calculated by using finite element method and variable density to pology optimization method. The results show that: The maximum stress distribution area of bulkhead and reinforcement occurs near the connecting bridge and the large opening. When the volume constraints are changed, the stress distribution and maximum stress of the optimized bulkhead and reinforcement will change accordingly. There is no positive correlation between the maximum stress and the volume reduction ratio of reinforcement. The maximum bulkhead stress cannot be reduced by increasing the volume retention ratio of the reinforcement. The distribution of the optimized reinforcement is similar to that of the real bulkhead. On the premise that the bulkhead has sufficient structural strength, the lightweight design of trimaran structure can be realized based on the topological optimization method.
1. Introduction Some advantages like good stability and seakeeping, low resistance under high-speed navigation, etc. are within trimaran. Lightweight design of trimaran structure can effectively improve its comprehensive performance on the premise of ensuring structural safety. Some studies attempt to apply rules-based design method (Zhang and Wang, 2007; Zhen et al., 2012; Fuentes et al., 2015), design method based on existing ships (Fuentes et al., 2015), and auxiliary design method based on environmental load and finite element calculation (Ren et al., 2012) to optimize the structure of trimaran. These design methods have contributed to ensuring the safety of hull structures, but their contribution to lightweight design of ships is very limited. Some studies attempt to lighten the bridge structure of trimaran, aiming at ensuring structural strength and weight reduction, and obtain the cor responding optimal structure (Yang, 2008, 2010; Deng, 2008) under specific working conditions. With the development of intelligent optimization methods, re searchers have gradually applied intelligent optimization methods, such as particle swarm optimization (Ehlers, 2012) and genetic algorithm (Du
et al., 2018), to ship structure lightweight design. Intelligent optimiza tion method can improve material utilization (Ehlers, 2012), effectively reduce the quality of ship structure and obtain hull structure with target quality (Du et al., 2018), and ensure that the optimized structure has sufficient strength (Ehlers, 2012; Du et al., 2018). Compared with intelligent optimization method, topology optimi zation method has more prominent advantages in structural lightweight design, especially in structural lightweight design for continuum. Based on the principle of computational mechanics, the continuous structure topology optimization design method (Bendsoe and Kijuchi, 1998; Bendsoe, 1989; Xie and Steven, 1993; Allaire et al., 2002) and the discrete structure topology optimization design method (Cheng and Guo, 1997; Su et al., 2009) can realize the lightweight design of the structure on the premise of ensuring the safety of engineering structures. At present, these kinds of methods have been widely used in civil en gineering (Xu et al., 2018), aerospace science and technology (Munk and Verstraete, 2017; Oktay and Akay, 2011), large deformation flexible structure (Zhu et al., 2019) and biomechanics (Chen et al., 2018). In addition, through the latest research (Wang and Zhang, 2018), it can be seen that the topology optimization design method has been applied to
* Corresponding author. E-mail address: [email protected] (F. Li). https://doi.org/10.1016/j.oceaneng.2019.106498 Received 2 March 2019; Received in revised form 15 August 2019; Accepted 25 September 2019 Available online 1 October 2019 0029-8018/© 2019 Elsevier Ltd. All rights reserved.
D. Jia and F. Li
Ocean Engineering 191 (2019) 106498
Table 1 Main parameters of bulkhead. Parameters
Value
Yield strength Allowable stress Young’s modulus Poisson’s ratio Density Bulkhead thickness
235 MPa 199.75 MPa 2.1 � 105MPa 0.3 7850 kg/m3 10 mm
Fig. 3. Longitudinal torque.
of this paper are compared with those of the actual ship bulkhead reinforcement structure distribution, and the results of the trimaran transverse bulkhead topology optimization are evaluated to provide technical reference for the trimaran main structure topology optimiza tion design and the bulkhead reinforcement structure distribution area design.
Fig. 1. Geometric shape of bulkhead.
2. Topology optimization theory of continuous structures based on finite element method The optimization region is divided into several finite elements, and the relative density ρi of each element is taken as the design variable: � 1 i 2 Ωmat ρi ¼ (1) 0 i 2 Ω=Ωmat here, i is the number of the finite element; Ωmat is the retaining material area and Ω is the whole design area. According to ρi , we have: Z ρi dΩ � V0 ; 0 � ρi � 1 ; i2Ω (2)
Fig. 2. Horizontal moment.
Ω
the design of high precision instruments. The objective of this method is no longer the traditional structural mechanical response, but the per formance response parameters of the instrument. At present, the appli cation of topology optimization method in ship structure optimization design is relatively limited, and the optimization zone is small, but the contribution is obvious. For example, Zhang (Zhang and Yang, 2015) applied shape optimization method and topology optimization method to small-scale optimization design of ship bottom frame structure, and the weight of optimized structure was reduced by 15.82% compared with that of un-optimized structure. At present, the application of to pology optimization method to the optimization design of large-scale ship structures (such as bulkhead and bulkhead reinforcement struc tures of trimaran ships) is rare. In view of the current research situation, this paper carries out structural strength checking design on the basis of Rules for The Classification of Trimarans, which is the code of the Lloyd’s Classification Society of Britain. The preliminary design of the trans verse bulkhead of a specific trimaran is a prototype of structural opti mization, and the static response of the bulkhead structure is calculated by the finite element method. Subsequently, on the premise of guaran teeing that the structural strength conforms to the design specifications, the topology optimization of bulkhead structure is carried out with the minimum flexibility as the optimization objective. Finally, the opti mized bulkhead structure is taken as the reinforcement structure. The strength of the optimized bulkhead structure is analyzed, and the stress distributions under different working conditions and different weight reduction conditions are compared. In addition, the optimization results
here, V0 is the total volume of the design domain. The optimization problem expression with the objective of minimizing structural flexi bility under volume constraints is: 8 Find ρi ; i ¼ 1; 2; … m > > > > min C ¼ FT u < s:t: V=V0 f � 0 (3) > > Ku ¼ F > > : 0 < ρmin < ρi � 1 here, m is the total element number of units; the objective function C represents the total structural compliance; F is the overall load vector and u is the displacement vector; V is the optimized total volume and f is the volume ratio. K is the global stiffness matrix. In order to prevent the singularity of stiffness matrix, the lower limit of relative density of element ρmin is introduced, ρmin ¼ 0.001. After established the model, the sensitivity values of the constraint and the objective function are required. Applying solid isotropic mi crostructures penalty (SIMP) model can avoid solving the discrete value design problem (Zhang and Yang, 2015; Olsen and Vanderplaats, 1989). Bendsoe and Sigmund (2003); Zhou and Rozvany (1991) gave the conditions that the real SIMP difference needs to satisfy: � � 1 γ0 3 1 γ0 p � p* ¼ max 15 (4) ; 7 5γ0 2 1 2γ0 here, γ0 is Poisson’s Ratio; p* is penalty factor lower limit and p is 2
D. Jia and F. Li
Ocean Engineering 191 (2019) 106498
3. Structural strength checking of trimaran bulkhead 3.1. Bulkhead information and load case design In this section, a special trimaran bulkhead structure is taken as the optimization object to optimize its structure topology. The main pa rameters of bulkhead are shown in Table 1, and the geometric shape of bulkhead is shown in Fig. 1. Because the transverse bulkhead of trimaran has the symmetry of longitudinal-mid-section of the ship, the half bulkhead model shown in Fig. 1 will be selected in this section for finite element calculation of bulkhead, and the topological optimization of bulkhead will be carried out based on the calculation results. In the absence of hydrodynamic loads on wet surface of trimaran, the working conditions can be designed and checked in combination with Rules for The Classification of Trimarans. Combining with the bulkhead model in this paper, the load in this section is selected as follows: hor izontal bending moment (Fig. 2), longitudinal torque (Fig. 3), arch state in lateral separation moment (Fig. 4) and vertical state in lateral sepa ration moment (Fig. 5). According to the Rules for The Classification of Trimarans, the approximate formula for calculating horizontal bending moment is as follows:
Fig. 4. Arch state in lateral separation moment.
Fig. 5. Vertical state of lateral separation moment.
penalty factor. In the following optimization work, based on ANSYS Parameter Design Language and combined with the above theory, the structural topology optimization design of trimaran bulkhead will be carried out in this paper.
Mh ¼ Df Lf fserv L2R DðCb þ 0:7Þ
Fig. 6. Finite element analysis of bulkhead structure (stress unit: MPa). 3
(5)
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Ocean Engineering 191 (2019) 106498
Fig. 7. Optimized area and optimized result.
" Lf ¼
# � �2 LR 70 þ 5LR � 10 100
3
(6)
Msps ¼ 9:81fserv Wsh
� Δ
� � 2Δsh az ysh 2
Bmh 2
� (8)
here, Msph is the lateral separation bending moment (mid-arch); Wsh is the weight of a single hull, which can be obtained by multiplying the total weight of the trimaran by the percentage of the drainage of a single hull; az is the vertical acceleration; ysh is the distance from the center line of the hull to the center line of the main hull; Bmh is the main hull width; Δis the overall hull drainage and Δsh is the main hull drainage. The transverse bending moment at the joint of the connecting bridge and the sheet is calculated as follows:
here, Mh is the horizontal bending moment; Df is the longitudinal dis tribution factor, 0 is taken at the bow and stern, 1 is taken at the bow and stern, and other positions are obtained by interpolation; Lf is the captain change factor; fserv is the service area coefficient; LR is the length be tween the main vertical lines; D is the depth of the main hull; Cb is the square coefficient. The expression for calculating the lateral bending moment at the connection between the bridge and the main body is as follows: � � Bmh Msph ¼ 9:81fserv Wsh ð1 þ az Þ ysh (7) 2
Msph ¼ 9:81fserv Wsh ð1 þ az Þðysh
4
y0 Þ
(9)
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Ocean Engineering 191 (2019) 106498
Fig. 8. Maximum stress of composite structures.
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Ocean Engineering 191 (2019) 106498
Fig. 9. Stress distribution of bulkhead structures with different volume retention ratio.
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Fig. 10. Stress distribution of reinforcement with different volume retention ratio.
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bulkhead opening. In CASE5, the maximum stress of bulkhead structure is 270.04 MPa. The dangerous area is the bottom of connecting bridge and the vicinity of bulkhead opening. In this case, the bulkhead will be damaged. Therefore, it is necessary to add reinforcement structure to bulkhead. 4. Optimization design of bulkhead structure Although the stress value of bulkhead structure corresponding to CASE 5 is the largest, and the bulkhead structure will be destroyed only in this case, this section will carry out the topological optimization design of bulkhead structure under seven working conditions. The optimized bulkhead structure is taken as the original reinforced struc ture of the un-optimized bulkhead. The optimal design will minimize the volume of the reinforced structure (i.e. the optimized bulkhead) as much as possible on the premise of ensuring the structural safety of bulkhead and reinforced structure. In Fig. 7 (a), the blue area is the optimum area and the gray area is the preservation area. Under the constraints of 50% volume reduction in the optimized area, the optimized and smoothed trimaran bulkhead structure is shown in Fig. 7(b)–(h). Change the shape preservation of trimaran bulkhead optimization area. The volume retention ratio of the optimized area is set to 0%, 50%, 55%, 60%, 65%, respectively. 95%, 100%, and the bulkhead topology optimization. The optimized bulkhead is used as a reinforcement structure and “pasted” on the un-optimized bulkhead. The strength of the new structure is checked again. Fig. 8 shows the maximum stress of composite structures under different volume constraints. From the results of Fig. 8, it can be seen that the maximum stress of the optimized composite structure are different under different volume constraints. There is no positive correlation between the maximum stress and the volume retention ratio of composite structures. For example, in CASE 7, the maximum stress of bulkhead structure increases first and then decreases when the volume retention ratio of optimized area increases from 60% to 70%. When the retention ratio increases from 65% to 75%, the maximum stress of bulkhead structure will first decrease and then increase. In addition, Fig. 8 shows that the maximum stress of bulkhead reinforcement structure (i.e., the optimized bulkhead structure with “sticking” on the unoptimized bulkhead) is generally higher than that of bulkhead. In Section 3, the maximum stress of bulkhead in CASE 5 is far greater than the allowable stress of bulkhead. From Fig. 8 (e), it can be seen that the structural stress of bulkhead decreases obviously with the attach ment of reinforcement structure, and the maximum stress of bulkhead and the maximum stress of reinforcement structure will change with the change of retention ratio of optimized area volume. Fig. 9 shows the stress distribution of bulkhead structure with different volume confor mity under condition 5, and Fig. 10 shows the corresponding stress distribution of reinforced structure. Combining Figs. 8 (e), Figs. 9 and 10, we can see that: The maximum stress distribution area of the composite structure is concentrated under the connecting bridge and near the large opening. The maximum stresses of bulkhead structure and reinforced structure are lower than the allowable stresses when the reinforced structure is added. When the conformity ratio is reduced from 100% to 50%, the stress of the com posite structure does not increase significantly, which indicates that if the structure is optimized reasonably, the mass of the structure can be greatly reduced and the safety of the structure can be ensured. In addition, although the optimum region is far away from the large opening, the change of its structural form will also influence the stress distribution in the dangerous region.
Fig. 11. Construction site of Austal’s trimaran.
� Msps ¼ 9:81fserv Wsh
Δ
� 2Δsh az ðysh 2
y0 Þ
(10)
here, y0 is the transverse distance between the longitudinal section of the main hull and the connecting bridge. The expression for calculating the longitudinal torque is as follows: � � Vmhs Mlt ¼ 7:5Tf fserv ρ Vsh þ Vcd þ (11) ycs ahcave 2 here, Mlt is the longitudinal torque; Tf is the longitudinal torque distri bution coefficient; Vsh is the volume of the slice, Vcd is the volume of the connecting bridge, Vmhs is the volume of the main body as the length of the slice; ycs is the transverse section of the slice, the distance between the central line of the connecting bridge and the central line of the main hull; ahcave is the pitching acceleration. Combining with the optimized object in this paper, the checking conditions in this section are as follows: CASE1.
0.3 � Transverse Separation Moment (mid Arch)
CASE 2. 0.3 � Transverse Separated CASE3. Transverse Separated Bending Moment (Mid Arch) þ0.2 � Longitudinal Torque CASE4. Lateral Separation Moment (Mid-vertical) þ0.2 � Longitudi nal Torque CASE5. 0.3 � Horizontal Moment þ 0.4 � Lateral Separation Moment (Middle Arch) þ Longitudinal Torque CASE6. Horizontal Moment þ 0.4 � Transverse Separation Moment (Mid Arch) CASE7. 0.2 � Horizontal Moment þ 0.6 � Lateral Separation Moment (Mid Arch) 3.2. Finite element analysis of bulkhead structure of trimaran In this section, the hydrostatic bending moment and transverse tor sion are neglected in the calculation of bulkhead structural stress. The structural stress of bulkhead under seven working conditions are calculated by ANSYS. Higher-order triangular mesh and quadrilateral mesh with intermediate nodes are used to mesh bulkhead (as shown in Fig. 6 (a)). The results of stress distribution of bulkhead structure are shown in Fig. 6(b)–(h). The bulkhead structure will be loaded based on Figs. 2–5 and Equations (5)–(11). According to the calculation results of Fig. 6, the maximum stress area of bulkhead mainly occurs near the connecting bridge and the
5. Optimized contrast and rationality verification Fig. 11 is the construction site of Austal’s trimaran. The distribution of bulkhead structure and reinforcement can be seen in the figure. Ac cording to the structure in the drawing, the finite element model is 8
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Fig. 12. Optimization results of bulkhead with 60% volume retention ratio.
established and the static analysis under seven working conditions are carried out. Based on the calculation results, the bulkhead is topology optimized, and the optimization results are shown in Fig. 12. By com parison, the results of optimized bulkheads in CASE1~CASE5 are highly similar to those reinforcement in Fig. 11, which shows that the light weight design of trimaran bulkheads based on topological optimization method is feasible.
out topological optimization design and stress analysis for a particular trimaran bulkhead structure. The optimized bulkhead can be used as a reinforcement structure to “paste” on the unoptimized bulkhead to obtain a new composite structure. Compared with the original structure, the stress of the new structure is significantly reduced, and in most cases, the stress of the reinforced structure is higher than that of the bulkhead. The optimized bulkhead structure has an effect on the stress distribution of the new structure. There is no positive correlation between the stress amplitude of the new structure and the conformation ratio of the optimized region, which indicates that the potential factors affecting the stress amplitude of the structure may include the location and shape of the material area
6. Discussion and conclusion Based on Rules for the Classification of Trimarans, this paper designs seven checking conditions for trimaran bulkhead structure, and carries 9
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removed. The influence of these factors on structural strength needs to be verified by further research. It should be noted that, from the point of view of technology, the optimized bulkhead is not easy to be “pasted” on the bulkhead as a reinforced structure. However, the material distribution area of the optimized structure can be used as a reference for the design of bulkhead reinforcement structures, that is, the reinforcement structures can be arranged in these areas. In addition, the optimization method in this
paper can also be used as the optimization method of non-watertight bulkhead. By comparison, the results of the optimized bulkhead are highly similar to the distribution of the actual bulkhead reinforcement struc ture in some cases, which shows that the lightweight design of trimaran bulkhead and the design of the reinforced structure based on the topo logical optimization method are feasible.
Appendix Based on the results of the topological optimization in Fig. 12, the appendix establishes the reinforcement structure of the bulkhead according to the results of the material distribution of the topological optimization. The bulkhead surface has a reinforced structure perpendicular to the bulkhead, with a height of 0.2 m and a thickness of 10 mm. The bulkhead has a reinforced structure with a thickness of 8 mm and a width of 0.1 m. Fig. A1 shows the stress analysis results of the reinforced bulkhead.
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Similar to the results in section 4, the distribution of reinforcement structure will affect the stress distribution and stress amplitude of bulkhead structure. The maximum stress is still distributed near the bulkhead connection bridge and the large opening.
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