Liang, Q. Q., Xie, Y. M. and Steven, G. P., “A performance index for topology and shape optimization of plate bending problems with displacement constraints”, Structural and Multidisciplinary Optimization, 2001, 21(5), 393-399.
A performance index for topology and shape optimization of plate bending problems with displacement constraints
Q.Q. Liang and Y.M. Xie School of the Built Environment, Victoria University of Technology, PO Box 14428, Melbourne City MC, VIC 8001, Australia. E-mail:
[email protected] (Q. Q. Liang) G.P. Steven Department of Aeronautical Engineering, The University of Sydney, NSW 2006, Australia
Abstract This paper presents a performance index proposed using the scaling design approach for topology and shape optimization of plate bending problems with displacement constraints. The proposed performance index is used in the Evolutionary Structural Optimization (ESO) method for plates in bending to identify the optimum from the optimization process. This performance index can also be employed to compare the structural performance of topologies and shapes produced by different optimization methods. Several examples are provided to illustrate the validity and effectiveness of the proposed performance index for topology and shape optimization of bending plates. The results show that the shape optimization technique is more efficient in generating optimum design for plates in bending in terms of structural performance, computational cost and manufacture.
1 Introduction
The topology and shape optimization of continuum structures has attracted considerable attention in recent years. The topology optimization of continuum structures allows for the creation of cavities in the interior of the design whilst the shape optimization can only modify the design from the boundaries. Haftka and Grandhi (1986) have presented a survey on structural shape optimization. In
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Liang, Q. Q., Xie, Y. M. and Steven, G. P., “A performance index for topology and shape optimization of plate bending problems with displacement constraints”, Structural and Multidisciplinary Optimization, 2001, 21(5), 393-399.
the Homogenization method (Bendse and Kikuchi 1988; Tenek and Hagiwara 1993), the topology optimization of a continuum structure is treated as a composite material redistribution problem. The Solid Isotropic Microstructure with Penalty (SIMP) method (Zhou and Rozvany 1991; Rozvany et al. 1992) is efficient in producing solid-empty type topologies. The Evolutionary Structural Optimization (ESO) method presented by Xie and Steven (1993, 1997) offers a simple approach to topology and shape optimization by using material removal criteria. The ESO method has been further developed by Liang et al. (1999a, 1999b) as an efficient and reliable design tool for automatically generating optimal strut-and-tie models in reinforced and prestressed concrete structures.
Structural optimization is an effective tool for improving the structural performance of the design, but the performance of the design obtained is often limited by the methods used. Unfortunately, little work has been undertaken to assess the structural performance of topologies and shapes generated by different optimization methods. Recently, Burgess (1998) has extended the method outlined by Ashby (1992) to derive form factors for measuring the efficiency of the structural layouts of trusses and frames. However, it is difficult to extend this approach to the topology and shape optimization of continuum structures. In order to identify the optimum from the evolutionary path using the ESO method, performance indicators have been attempted by Querin (1997) and Zhao et al. (1998). However, these performance indicators, which do not consider any type of constraint, are not applicable to plates in bending.
This paper extends the methodology proposed by Liang et al. (1998a, 1998b) to developing a performance index for topology and shape optimization of plates in bending. The formulation of the performance index is given in section 2 and the ESO method for bending plates with displacement
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Liang, Q. Q., Xie, Y. M. and Steven, G. P., “A performance index for topology and shape optimization of plate bending problems with displacement constraints”, Structural and Multidisciplinary Optimization, 2001, 21(5), 393-399.
constraints is briefly outlined in section 3. In section 4, several examples are provided to demonstrate the validity and effectiveness of the proposed performance index for topology and shape optimization of plates in bending.
2 Formulation of performance index
The topology and shape optimization of a bending plate is often to find the optimal material layout that minimizes the weight of the plate under applied loads while the deflections are within acceptable limits. The optimum design criteria for plates in bending can be stated as follows:
minimize W
w t n
e 1
e
(1)
e
subject to u j u *j 0
j=1,…, m
(2)
where W is the total weight of the structure, we is the actual weight of the eth element, te is the thickness of the eth element, u j is the absolute value of the jth constrained displacement, u *j is the prescribed value of u j and m is the total number of constraints.
In order to obtain the optimum, the design can be scaled after each iteration in the optimization process so that the most critical constraint always reaches the prescribed limit (Kirch 1982; Liang et
al. 1998a, 1998b). For plates in bending, the stiffness matrix of the plate is the cubic root of the thickness of the plate. By uniformly scaling the thickness of the initial design, the scaled weight of the initial design domain can be expressed by
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Liang, Q. Q., Xie, Y. M. and Steven, G. P., “A performance index for topology and shape optimization of plate bending problems with displacement constraints”, Structural and Multidisciplinary Optimization, 2001, 21(5), 393-399.
u0 j W * u j s 0
1/ 3
W0
(3)
where W0 is the actual weight of the initial design domain and u 0 j is the absolute value of the jth constrained displacement that is the most critical in the initial design under the applied loads. In a same manner, the scaled weight of the current design at the ith iteration can be written as
u ij Wi * u j s
1/ 3
Wi
(4)
where Wi is the actual weight of the current design at the ith iteration and u ij is the absolute value of the jth constrained displacement that is the most critical in the design at the ith iteration under applied loads. The structural performance of the resulting material layout in a bending plate at the
ith iteration can be assessed by the performance index, which is defined by
* W0s u 0 j u j W0 u 0 j PI s Wi uij u *j 1 / 3Wi uij 1/ 3
1/ 3
W0 Wi
(5)
It can be seen from (5) that the performance index indicates the structural performance in terms of material efficiency in resisting deflections. The structural performance of a bending plate will be gradually improved when inefficient material is removed bit by bit from the design. The optimum design criteria can be achieved by maximizing the performance index in an optimization process.
3 Evolutionary topology and shape optimization for plates in bending
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Liang, Q. Q., Xie, Y. M. and Steven, G. P., “A performance index for topology and shape optimization of plate bending problems with displacement constraints”, Structural and Multidisciplinary Optimization, 2001, 21(5), 393-399.
3.1 Material removal criterion
The material removal criterion can be established by undertaking the sensitivity analysis, which is to study the effect of material removal on the changes in constrained displacements. In the finite element analysis, the equilibrium equation for a static structure is expressed by
Ku P
(6)
in which K is the stiffness matrix of the structure, u is the nodal displacement vector and P is the nodal load vector. If the eth element is removed from a structure, (6) can be written as
(K K )(u u) P
(7)
where K is the change in the stiffness matrix and u is the change of displacement vector. The change of the displacement vector due to element removal can be obtained by subtracting (6) from (7) and neglecting the higher-order term as
u K 1 Ku K 1k e u
(8)
in which k e is the stiffness matrix of the eth element. To find the change of the jth constrained displacement due to element removal, a virtual unit load is applied to the direction of the jth constrained displacement. By multiplying (8) with the virtual unit load vector F Tj , the change of the constrained displacement can be obtained as
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Liang, Q. Q., Xie, Y. M. and Steven, G. P., “A performance index for topology and shape optimization of plate bending problems with displacement constraints”, Structural and Multidisciplinary Optimization, 2001, 21(5), 393-399.
u j F j K 1k e u u Tej k e u e T
(9)
where u Tej is the nodal displacement vector of the eth element under the virtual unit load and u e is the displacement vector of the eth element under the real loads. The sensitivity number for the eth element for a displacement constraint as adopted by Chu et al. (1996) is defined as
e u Tej k e u e
(10)
The sensitivity number of each element in a structure under multiple displacement constraints is obtained by using the weighted average approach as
m
e j u Tej k e u e
(11)
j 1
where the weighting parameter j is defined as u j / u *j and m is the total number of constraints. The sensitivity number of each element indicates the effect of its removal from the design on the change in the constrained displacements. Therefore, the material removal criterion can be stated that elements with the lowest sensitivity numbers should be removed from the structure at each iteration to obtain the stiffest structure. For plates in bending under a symmetrical geometry, loading and boundary condition about the two in-plane axises, a symmetry criterion has been added to the extended ESO algorithm to maintain the symmetry of the resulting topology and shape.
3.2 Incorporating PI into the ESO procedure
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Liang, Q. Q., Xie, Y. M. and Steven, G. P., “A performance index for topology and shape optimization of plate bending problems with displacement constraints”, Structural and Multidisciplinary Optimization, 2001, 21(5), 393-399.
The proposed performance index is incorporated into the following evolutionary material removal procedure:
Step 1: Model the plate using a fine mesh of finite elements; Step 2: Analyze the plate for the real loads and virtual unit loads; Step 3: Calculate the performance index (PI); Step 4: Calculate the sensitivity number for each element; Step 5: Remove a small number of elements with the lowest e ; Step 6: Repeat Steps 2 to 5 until the PI < 1 or keeps constant in later iterations.
The Element Removal Ratio (ERR) is defined as the ratio of the number of elements to be removed to the total number of elements in the initial design domain. The displacement limits are usually set to large values so that the optimum is included in the optimization process. The optimal topology or shape that corresponds to the maximum performance index can be identified from the performance index history.
4 Numerical examples
4.1 Clamped plate under concentrated loading
The design domain of a clamped square plate under a concentrated load of 500 N applied to the centre of the plate is shown in Fig. 1. A displacement constraint is imposed on the loaded point in the vertical direction. The design domain is divided into a 50 × 50 mesh using four-node plate elements. The Young’s modulus E = 200 GPa, Poisson’s ratio v = 0.3 and the thickness of the plate
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Liang, Q. Q., Xie, Y. M. and Steven, G. P., “A performance index for topology and shape optimization of plate bending problems with displacement constraints”, Structural and Multidisciplinary Optimization, 2001, 21(5), 393-399.
t = 5 mm are assumed. The Element Removal Ratio ERR = 1% is adopted.
The performance index histories for the topology and shape optimization of the clamped plate are presented in Fig. 2. It can be seen that performance indices are gradually increased and almost identical up to iteration 59 while elements are deleted from the design. The shape optimization generates the optimum with a maximum performance index of 2.13 whilst it is only 2.09 by topology optimization. The topology and shape optimization histories for the plate are shown in Figs. 3 and 4, respectively. It is seen that cavities in the interior of the plate are created by the topology optimization whilst no holes in the interior of the plate are generated by the shape optimization. Based on the consideration of manufacture and structural performance, the shape optimization technique should be used in optimizing plates in bending. Table 1 gives a comparison of material volumes required for the initial design and shapes at different iterations shown in Fig. 3 for various displacement limits. It is seen from the table that the structural performance of the optimal shape does not depend on the magnitude of the displacement limits.
4.2 Simply supported plate under area loading
Figure 5 shows the design domain of a simply supported plate under a local area pressure of 0.1 MPa normal to the surface of the plate. A displacement constraint is imposed on the centre of the plate. The mesh and material properties used are the same as the first example. The loading region is frozen where on element is removed. ERR=1% is also used. It is seen from the performance index histories shown in Fig. 6 that the performance indices are almost identical up to iteration 32 for the plate using both optimization techniques. Using the shape optimization for this plate results in a higher maximum performance index of 1.53 whilst it is only 1.34 by the topology optimization
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Liang, Q. Q., Xie, Y. M. and Steven, G. P., “A performance index for topology and shape optimization of plate bending problems with displacement constraints”, Structural and Multidisciplinary Optimization, 2001, 21(5), 393-399.
method.
The topology and shape optimization histories are presented in Figs. 7 and 8, respectively. It is seen that hinge lines are formed between the four corners and the central region in the optimal topology. However, no hinge lines are observed in the optimal shape because elements are only removed from the boundaries of the plate in shape optimization. Moreover, it is observed that checkerboard patterns appear in the resulting topologies whilst no checkerboard pattern is present in the shapes obtained. Topologies with checkerboard patterns cause difficulty in manufacturing process. Although checkerboard patterns can be eliminated by introducing an intuitive smoothing scheme into the ESO, the computational cost will be penalized considerably. From the manufacturing, computational cost and structural performance points of view, it is suggested that the shape optimization technique should be used to optimize plates in bending. The effects of boundary conditions on the optimal topologies and shapes of bending plates can be observed from the first example and this example presented.
4.3 Clamped plate under strip loading
The design domain of a clamped square plate under the strip pressures of 0.1 MPa is illustrated in Fig. 9. A displacement constraint imposed on the centre of the plate is considered. The mesh and material properties are the same as used in the first example. The elements in the two loaded strips are frozen. The ERR=1% is employed in this problem.
Figure 10 shows the performance index histories for the topology and shape optimization of the clamped plate. It can be observed that the performance index curve obtained using the shape
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Liang, Q. Q., Xie, Y. M. and Steven, G. P., “A performance index for topology and shape optimization of plate bending problems with displacement constraints”, Structural and Multidisciplinary Optimization, 2001, 21(5), 393-399.
optimization method is smoother than the one generated using the topology optimization scheme. This may be the effect of holes that are created in the interior of the plate in topology optimization process. However, it is shown that these two optimization methods provide the optimal design with the same maximum performance index of 5.44, which is constant in later iterations. This is because the loading strips are frozen so that no element can be removed from the strips after eliminating the rest from the design. The topology and shape optimization histories for the plate are presented in Figs. 11 and 12, respectively. Both techniques generate the same optimum, which suggests that the most efficient design can be achieved by using beams to support the loading strips.
4.4 Plate with multiple displacement constraints
This example illustrates the application of the proposed performance index to bending plates with multiple displacement constraints. Fig. 13 shows the design domain of a simply supported plate under multiple displacement constraints of the same limit imposed on points A, B and C, where three point loads of 10 kN are placed at these points respectively. The design domain is divided into a 60 × 30 mesh using four-node plate elements. Four elements around each loaded point are frozen. The Young’s modulus E = 28.6 GPa, Poisson’s ratio v = 0.2 and the thickness of the plate t = 100 mm are assumed. ERR = 1% is used in the shape optimization process. The structural performance of resulting shape at each iteration is monitored by the constrained displacements imposed on points A and B respectively, as shown in Fig. 14. The maximum performance indices calculated using the constrained displacements at points A and B are 2.78 and 2.09, respectively. It is obvious that the optimal shape occurs at iteration 65 at point B, which is shown in Fig. 15.
4.5 Structural performance of topology and shape
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Liang, Q. Q., Xie, Y. M. and Steven, G. P., “A performance index for topology and shape optimization of plate bending problems with displacement constraints”, Structural and Multidisciplinary Optimization, 2001, 21(5), 393-399.
The structural performance of topologies and shapes generated by different optimization methods is compared by using the proposed performance index. The topology optimization of a simply supported square plate (200 × 200) under a point load of 0.04 N applied to the centre of the plate is undertaken by using the performance index and the ESO method. The design domain is divided into 6400 three-node elements. A displacement constraint is imposed on the loaded point. The Young’s modulus E = 174.72 GPa, Poisson’s ratio v= 0.3 and the thickness of the plate t = 0.1 mm are assumed, as used by Atrek (1989). The Element Removal Ratio EER = 1% is employed.
The maximum performance index obtained by the present study is 1.64, which corresponds to the optimal topology shown in Fig. 16(a). The topology given by Chu et al. (1996) using the ESO technique for this problem as illustrated in Fig. 16(b) has a performance index of 1.39. Fig. 16 (c) illustrates the regenerated result of the optimized plate undertaken by Atrek (1989). Based on the data of the initial design and final topology provided by Atrek, the performance index is determined as 1.61 by (5). By comparisons, the structural performance of the topology obtained by present study is higher than that given by Atrek (1989) and Chu et al. (1996). The mesh and element type have a considerable effect on the structural performance of the optimal topology by the ESO method. This example also shows that the efficiency of various optimization algorithms can be evaluated by comparing the results generated by them via the proposed performance index.
5 Concluding remarks
This paper has demonstrated that the proposed performance index can be incorporated in any structural optimization method such as the ESO approach to monitor the optimization process, from which the optimal topology and shape of bending plates can be easily identified. Moreover, using
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Liang, Q. Q., Xie, Y. M. and Steven, G. P., “A performance index for topology and shape optimization of plate bending problems with displacement constraints”, Structural and Multidisciplinary Optimization, 2001, 21(5), 393-399.
the performance index can objectively assess the structural performance of topologies and shapes as well as the efficiency of various optimization methods. Furthermore, it is shown that the shape optimization is more efficient than the topology optimization for optimizing plates in bending from the structural performance, computational cost and manufacturing points of view.
Acknowledgments
This paper forms part of a program of research into the evolutionary structural optimization as an efficient and reliable design tool being undertaken at Victoria University of Technology and The University of Sydney, Australia. This program is funded by the Australian Research Council under the Large Grants Scheme. The first author is supported by an Australian Postgraduate Award and a Faculty of Engineering and Science Scholarship.
References
Ashby, M.F. 1992: Materials Selection in Mechanical Design. Pergamon Press, Oxford.
Atrek, E. 1989: Shape: a program for shape optimization of continuum structures. In: Brebbia, C.A.; Hernandez, S. (eds) Computer Aided Optimization Design of Structures: Applications, pp. 135-144. Southampton: Computational Mechanics Publications.
Bendse, M.P.; Kikuchi, N. 1988: Generating optimal topologies in structural design using a homogenization method. Comp. Meth. Appl. Mech. Engrg, 71, 197-224.
Burgess, S.C. 1998a: The ranking of efficiency of structural layouts using form factors. Part 1:
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Liang, Q. Q., Xie, Y. M. and Steven, G. P., “A performance index for topology and shape optimization of plate bending problems with displacement constraints”, Structural and Multidisciplinary Optimization, 2001, 21(5), 393-399.
design for stiffness. Proc. Instn Mech. Engrs, Part C, J. Mech. Engrg Sci. 212(C2), 117-128.
Chu, D.N.; Xie, Y.M.; Hira, A.; Steven, G.P. 1996: Evolutionary structural optimization for problems with stiffness constraints. Finite Elements in Analysis and Design. 21, 239-251.
Haftka, R.T.; Grandhi, R.V. 1986: Structural shape optimization-A survey. Comp. Meth. Appl.
Mech. Engrg, 57(1), 91-106.
Kirsch, U. 1982: Optimal design based on approximate scaling. J. Struct. Engrg, 108(ST4), 888909.
Liang, Q.Q.; Xie, Y.M.; Steven, G.P. 1998a: Optimal selection of topologies for the minimumweight design of continuum structures with stress constraints. Proc. Instn Mech. Engrs, Part C, J.
Mech. Engrg Sci. (accepted).
Liang, Q.Q.; Xie, Y.M.; Steven, G.P. 1998b: Optimal topology selection of continuum structures with displacement constraints. Comput. & Struct. (submitted).
Liang, Q.Q.; Xie, Y.M.; Steven, G.P. 1999a: Topology optimization of strut-and-tie models in reinforced concrete structures using an evolutionary procedure. ACI Structural Journal (submitted).
Liang, Q.Q.; Xie, Y.M.; Steven, G.P. 1999b: Optimization of strut-and-tie models in prestressed concrete beams. Proc. Instn Civ. Engrs Structs & Bldgs (submitted).
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Liang, Q. Q., Xie, Y. M. and Steven, G. P., “A performance index for topology and shape optimization of plate bending problems with displacement constraints”, Structural and Multidisciplinary Optimization, 2001, 21(5), 393-399.
Querin, O.M. 1997: Evolutionary Structural Optimization: Stress Based Formulation and
Implementation. PhD Thesis, Department of Aeronautical Engineering, The University of Sydney.
Rozvany, G.I.N.; Zhou, M.; Birker, T. 1992: Generalized shape optimization without homogenization. Struct. Optim. 4, 250-252.
Tenek, L.H.; Hagiwara, I. 1993: Static and vibrational shape and topology optimization using homogenization and mathematical programming. Compt. Meth. Appl. Mech. Engrg, 109, 143-154.
Xie, Y.M.; Steven, G.P. 1993: A simple evolutionary procedure for structural optimization.
Comput. & Struct. 49(5), 885-896.
Xie, Y.M.; Steven, G.P. 1997: Evolutionary Structural Optimization. Springer-Verlag, Berlin.
Zhou, M; Rozvany, G.I.N. 1991: The COC algorithm, Part II: topological, geometrical and generalized shape optimization. Comp. Meth. Appl. Engrg, 89, 309-336.
Zhao, C.B.; Hornby, P.; Steven, G.P.; Xie, Y.M. 1998: A generalized evolutionary method for numerical topology optimization of structures under static loading conditions. Struct. Optim. 15, 251-260.
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Liang, Q. Q., Xie, Y. M. and Steven, G. P., “A performance index for topology and shape optimization of plate bending problems with displacement constraints”, Structural and Multidisciplinary Optimization, 2001, 21(5), 393-399.
Figures and Tables
Table 1 Material volumes required for the design at different iteration for various displacement limits (mm) 0.5 0.75 1.0
V0s
5
3
(10 mm ) 5.85 5.11 4.65
V20s
5
3
V40s
(10 mm ) 4.70 4.11 3.73
5
s Vopt
3
(10 mm ) 3.69 3.22 2.93
(10 mm3) 2.75 2.4 2.18
PI max
5
2.13 2.13 2.13
Fig. 1. Design domain of the clamped plate under concentrated loading
1.000311 1.010002 2.5 1.021426 1.033095 1.043234 2 1.055173 1.066879 1.5 1.077525 1.090136 1 1.102729 1.113772 0.5 1.124683 1.137622 0 1.149341 1.162950 1.174858 Performance Index PI
u *j
1.000311 1.011607 1.023091 Topology 1.034696 Shape 1.046526 1.05619 1.068531 1.079059 1.091253 1.10227 1.112977 1.125904 1.137431 1.150661 10 20 30 40 1.164504 Iteration 1.176983
50
60
70
Fig. 2. Performance index history of the clamped plate under concentrated loading
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Liang, Q. Q., Xie, Y. M. and Steven, G. P., “A performance index for topology and shape optimization of plate bending problems with displacement constraints”, Structural and Multidisciplinary Optimization, 2001, 21(5), 393-399.
(a) Topology at iteration 10
(b) Topology at iteration 20
(c) Topology at iteration 40
(d) Optimal topology
Fig. 3. Topology optimization history of the clamped plate under concentrated loading
(a) Shape at iteration 10
(b) Shape at iteration 20
(c) Shape at iteration 40
(d) Optimal shape
Fig. 4. Shape optimization history of the clamped plate under concentrated loading
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Liang, Q. Q., Xie, Y. M. and Steven, G. P., “A performance index for topology and shape optimization of plate bending problems with displacement constraints”, Structural and Multidisciplinary Optimization, 2001, 21(5), 393-399.
Fig. 5. Design domain of the simply supported plate under area loading
Performance Index PI
0.999798 1.009366 1.018783 2 1.028322 1.038075 1.5 1.047625 1.056743 1 1.066855 1.076478 1.086279 0.5 1.094574 1.104655 0 1.115179 1.12528 1.135592 1.138792
0.999798 1.009241 1.018882 1.028549Topology 1.037679Shape 1.047387 1.057048 1.066479 1.075361 1.084777 1.095417 1.103417 1.112527 01.123491 10 20 30 1.133573 Iteration 1.142479
40
50
Fig. 6. Performance index history of the simply supported plate under area loading
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Liang, Q. Q., Xie, Y. M. and Steven, G. P., “A performance index for topology and shape optimization of plate bending problems with displacement constraints”, Structural and Multidisciplinary Optimization, 2001, 21(5), 393-399.
(a) Topology at iteration 10
(b) Topology at iteration 20
(c) Topology at iteration 25
(d) Optimal topology
Fig. 7. Topology optimization history of the simply supported plate under area loading
(a) Shape at iteration 10
(b) Shape at iteration 20
(c) Shape at iteration 25
(c) Optimal shape
Fig. 8. Shape optimization history of the simply supported plate under area loading
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Liang, Q. Q., Xie, Y. M. and Steven, G. P., “A performance index for topology and shape optimization of plate bending problems with displacement constraints”, Structural and Multidisciplinary Optimization, 2001, 21(5), 393-399.
Fig. 9. Design domain of the clamped plate under strip loading
Performance Index PI
0.999282 1.008959 1.020328 6 1.030231 5 1.041995 1.051796 4 1.065499 3 1.075707 1.086242 2 1.098515 1.111223 1 1.121856 0 1.13494 1.1464440 1.157276 1.169683
0.999282 1.010554 1.021989 1.031791Topology 1.045211Shape 1.054563 1.06679 1.077131 1.088993 1.099832 1.112176 1.122733 1.136109 1.146828 20 40 60 1.158122 Iteration 1.172521
80
100
Fig. 10. Performance index history of the clamped plate under strip loading
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Liang, Q. Q., Xie, Y. M. and Steven, G. P., “A performance index for topology and shape optimization of plate bending problems with displacement constraints”, Structural and Multidisciplinary Optimization, 2001, 21(5), 393-399.
(a) Topology at iteration 20
(b) Topology at iteration 40
(c) Topology at iteration 80
(d) Optimal topology
Fig. 11. Topology optimization history of the clamped plate under strip loading
(a) Shape at iteration 20
(b) Shape at iteration 40
(c) Shape at iteration 80
(d) Optimal shape
Fig. 12. Shape optimization history of the clamped plate under strip loading
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Liang, Q. Q., Xie, Y. M. and Steven, G. P., “A performance index for topology and shape optimization of plate bending problems with displacement constraints”, Structural and Multidisciplinary Optimization, 2001, 21(5), 393-399.
Fig. 13. Design domain of the plate under multiple displacement constraints
Performance Index PI
0.999951 1.010838 3 1.02174 1.032615 1.043393 1.053002 2 1.065266 1.076778 1.088675 1 1.099999 1.111168 1.12302 1.135666 0 1.1489 0 1.161859 1.171957
0.999862 3.91E-01 1.010497 3.91E-01 1.021511 3.92E-01 1.032337 3.93E-01 Point A & C 1.042999 Point B 3.95E-01 1.052691 3.96E-01 1.065581 3.98E-01 1.076284 3.99E-01 1.087977 4.01E-01 1.099807 4.03E-01 1.111844 4.06E-01 1.124088 4.08E-01 1.135149 4.10E-01 1.146421 20 4.12E-01 40 1.159154 4.14E-01 Iteration 1.170619 4.20E-01
5.16E-01 5.17E-01 5.18E-01 5.19E-01 5.21E-01 5.23E-01 5.25E-01 5.28E-01 5.30E-01 5.32E-01 5.35E-01 5.37E-01 5.42E-01 5.47E-01 60 5.50E-01 5.56E-01
20 40 60 80 98 120 140 160 180 200 220 240 80260 280 300
Fig. 14. Performance index history of the plate under multiple displacement constraints
Fig. 15. Optimal shape of the plate under multiple displacement constraints
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Liang, Q. Q., Xie, Y. M. and Steven, G. P., “A performance index for topology and shape optimization of plate bending problems with displacement constraints”, Structural and Multidisciplinary Optimization, 2001, 21(5), 393-399.
(a)
(b)
(c)
Fig. 16. Comparison of the structural performance of topologies: (a) topology by present study, PI=1.64 (b) topology by Chu et al. (1996), PI=1.39 (c) topology by Atrek (1989), PI=1.61
22
