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An adaptive volume constraint algorithm for topology optimization with a displacement-limit
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Advances in Engineering Software 39 (2008) 973–994 www.elsevier.com/locate/advengsoft
An adaptive volume constraint algorithm for topology optimization with a displacement-limit Chyi-Yeu Lin *, Fang-Ming Hsu Department of Mechanical Engineering, National Taiwan University of Science and Technology, 43 Keelung Road, Section 4, Taipei, Taiwan 10672, ROC Received 7 September 2007; received in revised form 24 September 2007; accepted 30 January 2008 Available online 17 March 2008
Abstract This paper proposes a novel adaptive volume constraint algorithm (AVC) to replace the fixed volume constraint (FVC) in the traditional topology optimization method, such that a minimum-compliance optimal structure that simultaneously meets the additional displacement limit can be searched. Optimal minimum-compliance structures are subject to the limit of a predetermined amount of material in traditional calculation methods, and often fail to meet practical stress and displacement constraints, or become unnecessarily strong. Without displacement sensitivities, the AVC algorithm iteratively adjusts allowable material usage based on the difference between the actual displacement, and the allowable displacement limit. In this way, topology optimization can efficiently construct a minimum-compliance structure that meets the allowable displacement limit. The regular volume constraint algorithm (RVC), which slightly changes the volume constraint each time at the end of FVC, is also executed in order to demonstrate the advantages of the AVC algorithm. The effectiveness of the AVC algorithm is demonstrated by two illustrative 2-D design problems: a cantilever beam and a simple beam. Ó 2008 Elsevier Ltd. All rights reserved. Keywords: Topology optimization; Adaptive volume constraint; Material distribution method; Displacement constraint; Density method
1. Introduction Bends/e and Kikuchi [1] developed the homogenization method for topology optimization, which is a milestone for structural optimization. Mlejnek and Schirrmacher [10] proposed the density function approach to attain a minimum-compliance structure under a volume usage constraint. Yang and Chuang [15] suggested the use of the normalized material density of each element as the design variable that reduces the number of design variables in a topology optimization problem. Since analytical forms of the gradients of compliance and volume usage for structures have been derived, most density-based topology optimization problems use the compliance as the objective function, and the volume usage as the only constraint. Various volume usage constraints in the density methods lead to different structural designs that consequently result *
Corresponding author. Tel.: +886 2 273 6494. E-mail address: [email protected] (C.-Y. Lin).
0965-9978/$ - see front matter Ó 2008 Elsevier Ltd. All rights reserved. doi:10.1016/j.advengsoft.2008.01.008
in different stress and displacement values. The minimumcompliance structure thus obtained can either become excessively strong or can fail to meet practical stress and displacement constraints. Without a proper volume usage constraint, the topology optimization will often fail to provide the correct structural topology when maximum stress/displacement considerations are taken into account. Much effort has been placed on developing algorithms that can find proper volume usage for a structure and/or that can find a structure that meets specified displacement limits. Yang [14] treated the weight as the objective function and used the displacement, frequency and compliance as inequality constraints. Mathematical programming methods or optimal criteria algorithms [13] have been adopted to solve structural topology optimization problems. Unless the analytical forms of the sensitivities for stress and displacement functions of the structure can be derived, it will be computationally prohibitive to consider stress or displacement constraints in topology optimization problems involving a large number of design variables.
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A simple formula for evolutionary structural optimization (ESO) [12] was proposed to gradually eliminate under-utilized elements in the design domain so as to form the fully stressed structure. Chu et al. [2,3] used nibbling techniques to iteratively eliminate elements with the lowest sensitivity values so as to create a structure that meets the stiffness constraint. ESO-related studies maximize various performance indexes of a structure subject to different constraints by gradually eliminating elements with the lowest strain energy values. Liang et al. [6,7] and Liang and Steven [8] developed the performance-based optimization method that uses performance indices to obtain the optimal designs of continuum structures. ESO is a simple method for topology optimization problems, which gradually removes elements from the structure based on the sensitivities of the objective and constraints to achieve the optimum. It is noted that the use of the element removal ratio of 1–2% in ESO-related methods gives satisfactory solutions. Larger removal ratios provide relatively unstable results and smaller ratios often require a high computational cost. This paper proposes a simple AVC algorithm to replace the FVC algorithm in density-based methods, so that without major algorithmic changes the optimization can seek a minimum-compliance design that simultaneously satisfies the additional displacement limit. It does not derive the analytical expression for the sensitivity of the displacement function or compute sensitivity by numerical techniques. The proposed algorithm will be used in a density method to solve the minimum-compliance structure under a displacement limit.
the stiffness of the structure. The sequential linear programming method (SLP) [11] is used to solve for the optimization problem in this work. If minimizing the displacement of the structure is the design goal, it is reasonable to pursue a structure that has the maximum stiffness. However, if maximum displacement is a design condition that needs to be met, a maximum-stiffness structure attained under a volume usage constraint is not necessarily the best design. For insufficient material usage, the optimal maximum-stiffness structure is still too flexible to meet the displacement constraint. For excessive volume usages, the overbuilt optimal structure easily satisfies the displacement constraint, but material is wasted. It is difficult to predict proper material usage for a given topology optimization problem under a displacement constraint. Often, the density method has to be conducted several times with different volume usage constraint definitions, and the lightest optimal structure satisfying the additional displacement constraint is selected as the proper structural configuration. Otherwise, an engineer arbitrarily
2. Continuum topology optimization The most common topology optimization formulation is to minimize compliance with a fixed volume constraint. By using the normalized density of each finite element that forms a structural domain as the design variable, the density method [10,15] simplifies the topology optimization problem as Minimize Subject to :
f ðqÞ n X i¼1
Side constraints : Ei And ¼ q2i Eo
ð1Þ qi vi =
n X
vi 6 V con
ð2Þ
i¼1
0 6 qi 6 1; i ¼ 1; n
ð3Þ ð4Þ
where f(q) is the objective function, qi is the normalized density of the ith element, vi is the volume of the ith element, Vcon is the volume usage limit, which is a percentage of the whole volume of the structure, and n is the number of design variables. Ei is an intermediate Young’s modulus of an element and Eo is the original Young’s modulus of structure. The objective function is selected as the compliance of the structure, which is equivalent to maximizing
Fig. 1. The flowchart of the topology optimization with an adaptive volume constraint (AVC).
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selects a volume usage to perform the density method and then treats the resultant structural configuration as the solution. Therefore, the structure fails to meet the displacement constraint and the structural configuration needs to be manually strengthened by placing more material on the existent structure. 3. Adaptive volume constraint algorithm The major drawback of using the fixed volume constraint algorithm (FVC) in traditional topology optimization methods is that use of an improper volume usage constraint leads to an excessively strong structure or a structure that fails to meet stress/displacement requirements. Therefore, the adaptive volume constraint algorithm (AVC) aims to simultaneously search for the most proper volume usage and the minimum-compliance structure using the allowable amount of material. With the AVC algorithm, the difference between the displacement of the specified location in the structure and the displacement limit at the end of several iterations of the optimization is used to consider adjustments to the current volume usage ratio. The optimization continually searches for the proper volume usage constraint according to the displacement status of current structures while the optimization seeks the best material distribution on a design domain with the newest volume usage constraint. The systematic flowchart of the execution of the topology optimization with the adaptive volume constraint algorithm is shown in Fig. 1. The detailed stepwise procedure for the topology optimization process with the adaptive volume constraint is as follows:
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Step 1: Define the design domain, the loading conditions, the boundary conditions and dlimit, which is the maximum displacement limit at a specific location of the structure. Set the initial volume constraint, V initial con , and an initial design. The densities,qi, of the initial design of the entire design domain are the same as the value of the initial volume constraint in this paper. Determine the convergent value, e, and the parameter c. Step 2: Run an iteration of the SLP topology optimization process. After the topology optimization problem is linearized, the optimal design is obtained in the move limit. Then, the specific displacement, dsp, and the compliance of the current structure of this optimal design can be attained. The entire process mentioned above is defined as an iteration of SLP [15]. Step 3: If both of the following two criteria (Criteria A) are met, go on to Step 4; otherwise go back to Step 2. k k1 (A-1) 0:1 6 C CC 6 0, the relative difference of k1 the compliance of structures between last two iterations, is between 10% and 0. The 0.1 value is determined empirically based on a large number of experiments. When the change of the compliance on two iterations is below 10%, it will be interpreted that the optimization under the current volume constraint has reached a stable status. It
Fig. 3. Design domain and conditions for a short cantilever beam (A).
Table 1 Performance comparison of AVC and RVC in Case A1: vertical limit is 0.5 in. at (16, 0) V initial con %
Algorithms
V final act %
d final sp
Num
20
AVC RVC AVC RVC AVC RVC AVC RVC
70.41 65.00 67.80 65.00 67.38 65.00 66.62 65.00
0.4789 0.4996 0.4785 0.4967 0.4789 0.5008 0.4843 0.4970
43 123 29 76 26 28 31 43
AVC RVC
68.05 65.00
0.4801 0.4985
40 60 80
Fig. 2. The flowchart of the topology optimization with a regular volume constraint (RVC).
Average
32.25 67.50
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is a good timing to switch into a new volume constraint, otherwise the computational cost will be continually spent on the optimization with the volume constraint far apart from the final volume
constraint. Criterion A-1 is one of the conditions determining the time to change volume constraint. (A-2) jV kact V kcon j 6 1%, the difference between the actual volume usage, V kact , and the current volume
Table 2 Histories of the actual displacement and the volume constraint and the images of AVC and RVC in Case A1: vertical limit is 0.5 in. at (16, 0)
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Table 2 (continued)
constraint, V kcon , is smaller than 1%. The 1% provides a good thresholding value for interpreting
the new volume constraint becomes active. A smaller value can be used but the interpretation will not
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be stable if the number is below 0.0001. The Criterion A-2 is to confirm that the topology design is substantially affected by the updated volume constraint during the optimization process. When the actual volume is close to the volume constraint, the optimization will continue to Step 4. Step 4: If any of the following three criteria (Criteria B) are met, go to Step 6. Otherwise, go on to Step 5 to adjust the volume constraint. d d (B-1) spd limitlimit 6 e is the relative difference between the displacement of the specified location of the structure, and the displacement limit is not over the criterion value, e. (B-2) The volume constraint either exceeds 90% or is less than 10% for five consecutive checks. This criterion is used to limit the range of the volume constraint in order for the gray level image, obtained by AVC, RVC or FVC, to be applied successfully to the practical design. In this paper, the number of elements of the design domain is 640. If the volume constraint is more than 90%, the densities of the entire design domain will become almost 1. The image is not referable. If the volume is less than 10%, the properties of the binary image, obtained by the penalty and threshold technique, is hard to closely compare to those of the gray level image. As the mesh resolution of the design domain gets finer, volume constraint becomes wider. Therefore, the reference number of the elements of the mesh recommended in the literature [4,5], is 2000. The range of the volume constraint can be wider than 80% (from 10% to 90% in this paper). Without the consideration of manufacturability and practicality of design, Criterion B-2 can be neglected. (B-3) The volume constraint has been adjusted over 10 times before Criterion B-1 is met. Criterion B-3 is required to avoid cases in which Criterion B1 is too severe to converge. Despite the fact that AVC does not converge after a maximum number of changes of the volume constraint, the volume constraint always oscillates, and the volume constraint approaches the proper value for volume usage. Therefore, AVC is forced to stop changing the volume constraint. Step 5: c d sp d limit m k k1 k1 V con ¼ V con þ ð1Þ min ; 0:2V con ; e d limit 0; d sp > d limit m¼ ð5Þ 1; d sp < d limit Increase or decrease the volume constraint by expression (5) and go back to Step 2. In the early periods of AVC, the relative difference between dsp and dlimit is usually outsized so that the term 0:2V k1 con of expression (5) dominates the change of the volume constraint. Since the minimum and
maximum volume constraints in the AVC method are set to as 10% and 90%, the maximum change of the volume constraint will be reasonably limited between 2% and 18%. Smaller or larger volume change ratios at this stage usually lead to inferior results. In the late period of AVC, the displacement, dsp, of the structure gradually becomes close to the displacement limit, dlimit. The term, c d sp d limit of expression (5) will then dominate e d limit , the volume constraint change. The parameter c is suggested to be limited between 1% and 3%. The empirical values 1% and 3% produced the most stable solutions based on a large number of test problems. In the later stages of the optimization the current design is always close to the constraint boundary, c smaller than 1% will lead to slow or false convergence, while larger than 3% will extend the time required for convergence. Step 6: Keep the current volume constraint unchanged and further run an iteration of SLP optimization. Step 7: If Criteria B-1 and B-2 continue to be met, go to Step 8. Otherwise, the maximum number of iteration is released to 15 and goes back to Step 5. Step 8: If the absolute value of the relative difference of compliance between the last two iterations is less k k1 than 0.1% (Criterion C), C CC k1 6 0:001, terminate the optimization and obtain the final displacefinal ment, d final sp , and final volume usage, V act . Otherwise, go back to Step 6. Criterion C is the major convergent condition for traditional topology optimization (FVC).
4. Regular volume constraint algorithm In general, in the topology optimization problem with displacement or stress constraints, the FVC algorithm will be applied directly by a regularly, gradually and slightly changing volume constraint. The topology optimization with a regular volume constraint (RVC) is executed in this paper in order to demonstrate the advantages of the AVC algorithm. The RVC algorithm solves a structural optimiTable 3 Performance comparison of AVC and RVC in Case A2: vertical limit is 0.5 in. at (12, 0) V initial con %
Algorithms
V final act %
d final sp
Num
20
AVC RVC AVC RVC AVC RVC AVC RVC
46.69 45.00 46.22 45.00 45.63 45.00 42.56 45.00
0.4810 0.4802 0.4796 0.4522 0.4850 0.4486 0.4860 0.4511
39 92 33 31 25 74 43 90
AVC RVC
45.28 45.00
0.4829 0.4580
35.00 71.75
40 60 80 Average
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zation problem with a displacement limit by slightly changing the volume constraint each time at the end of FVC. The main drawback of the RVC is that the value of change of
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the volume constraint is hard to determine. The value is too high to obtain precise volume usage or an image that meets the displacement limit criterion. The value of change
Table 4 Histories of the actual displacement and the volume constraint and the images of AVC and RVC in Case A2: vertical limit is 0.5 in. at (12, 0)
(continued on next page)
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Table 4 (continued)
of the volume constraint of RVC is 5% in this paper. The performances of the RVC depend on individual cases.
In the RVC algorithm, the major convergent criterion is Criterion B-1. The second one is whether or not the volume
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constraint violates the monotone property. The systematic flowchart of the execution of the RVC algorithm is shown in Fig. 2. The detailed stepwise procedure of the RVC algorithm process is stated as follows:
AVC and RVC, are executed with different initial volume constraints, 20%, 40%, 60%, and 80%, respectively.
Step 1: Define the design domain, the loading conditions, the boundary conditions, and dlimit: the displacement limit at a specific location of the structure. Set the initial volume constraint, V initial con , and define an initial design. Step 2: Run iterations for the SLP topology optimization process and then compute the specific displacement, dsp, and the compliance of the current structure, C. Step 3: If the absolute value of the relative difference of compliance between the last two iterations is less k k1 than 0.1% (Criterion C), C CC k1 6 0:001, terminate the optimization. Otherwise, go back to Step 2. Step 1 to Step 3 are defined as traditional topology optimization shown as gray blocks in Fig. 4. Step 4: If the Criterion B-1 is met, go to Stop. Otherwise, go to Step 5. Step 5: The volume constraint has to be monotone. If the monotone property is violated, then go to stop. Otherwise, go back to Step 2. (The volume constraint has to keep decreasing or increasing gradually. The property is defined as monotone.)
In the first illustrative example, a two-dimensional linear elastic example is shown. It involves a short cantilever beam with a fixed support subject to a displacement constraint in the vertical direction, as shown in Fig. 3. The design domain is divided into 32 20 quadrilateral elements. The concentrated load is applied to the bottom of the free end. Assuming the three different cases, the vertical displacement limits at the three locations, (16, 0), (12, 0) and (8, 0), are all 0.5 in.
5.1. Two-dimensional cantilever beam (A)
5.1.1. Case A1: vertical limit is 0.5 in. at (16, 0) The location of the limited displacement is the same as the load location. The performances of AVC and RVC are shown in Table 1. The histories of displacement and volume constraint and the final topological images are also presented in Table 2. The condition 80%-AVC is very stable to attain proper topologies by gradually decreasing the volume constraint. The final displacements of the images for all conditions meet Criterion B-1 and the designs are all feasible, with exception of the condition of 60%-RVC. Therefore, the two algorithms are successful in this example. The efficiency of AVC (32.25 iterations) is much better than that of RVC (67.5 iterations).
5. Numerical examples In the following two illustrative examples, the AVC algorithm will first be used to minimize the compliance of the structure with a vertical or horizontal displacement constraint at different locations. Secondly, the RVC algorithm is used to seek a minimum-compliance structure subject to volume constraints in both illustrative problems. The four indexes are used to compare AVC to RVC. The number of iterations, N, presents the efficiency of optimization. The coefficient of variation for the number of iterations, CVN, is the stability of efficiency. Final volume usage,V final act , is the proper usage attained by AVC or RVC. The performance index, PI, is used to classify the quality of the topological structure. The fewer the indexes, the better the algorithm is. The indexes are shown as expression (6) and (7): sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi P 2 ðN N Þ CVN ¼ =n ð6Þ n1 V act d sp PI ¼ ð7Þ V act d limit For the following two examples, the common conditions of the problems are as follows: Young’s modulus, E = 207,000 psi, and Poisson’s ratio, v = 0.3, are assumed. A concentrated load is 3000 lb, and the uniform thickness of the structure is 1 in. The convergent criterion, e, is 0.05. The parameter for c is 2.5%. The two algorithms,
5.1.2. Case A2: vertical limit is 0.5 in. at (12, 0) The distance between the limit location and the load location is 4 in. The efficiency of AVC (35 iterations) is much better than that of RVC (71.75 iterations), as shown in Table 3. The AVC algorithm also can efficiently obtain proper topologies that are feasible and meet Criterion B1. The value of changing the volume constraint in RVC is too rough because the images of RVC do not meet Criterion B-1. The qualities of images obtained in 40%, 60%, 80%-RVC and 80%-AVC conditions are better than those of others. The performance indexes (PI) of these images are all close to 0.9. Performance indexes in other cases
Table 5 Performance comparison of AVC and RVC in Case A3: vertical limit is 0.5 in. at (8, 0) V initial con %
Algorithms
V final act %
d final sp
Num
20
AVC RVC AVC RVC AVC RVC AVC RVC
34.41 35.00 34.72 30.00 32.00 35.00 31.94 35.00
0.5168 0.4370 0.4971 0.4693 0.4817 0.4609 0.4926 0.5169
50 72 40 70 36 96 58 97
AVC RVC
33.27 33.75
0.4970 0.4710
46.00 83.75
40 60 80 Average
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are about 1.0–1.1. The net structures of the images, shown in Table 4, are average distributions over the design domains.
5.1.3. Case A3: vertical limit is 0.5 in. at (8, 0) The distance between limit location and load location is 8 in. The efficiency of AVC (46 iterations) is also much bet-
Table 6 Histories of the actual displacement and the volume constraint and the images of AVC and RVC in Case A2: vertical limit is 0.5 in. at (8, 0)
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ter than that of RVC (83.75 iterations), as shown in Table 5. Although the oscillation phenomena, shown in Table 6 of the volume constraint at later periods of the 20% and 40%-AVC are serious, proper volume usages are obtained and the images still meet Criterion B-1. The 60% and 80%-AVC are very stable and successful. Despite the fact that Criterion B-1 is met in 20%-AVC and 80-RVC, the designs of these two conditions are not feasible. The displacements of the cases are 0.5168 and 0.5169, respectively. The images can be improved be using a binary technique. In cantilever beam example A, the AVC can obtain the proper volume usages and images, which meet Criterion B1 for all conditions, with different initial volume constraints and different location limits. The relationship between displacements vs. volume usage can be verified in the example. The efficiency of AVC (average 37.75 iterations) is much more than that of RVC (average 74.33 iterations). With respect to the stability of efficiency, CVN, the AVC is affected slightly by the initial volume constraint. The volume usages of the two algorithms show little difference. The performances of the AVC and RVC are listed in Table 7. The initial volume constraint of AVC is recommended at 60% or 80% to avoid oscillation phenomena. A change in the volume constraint each time of 5% in RVC is too rough to meet Criterion B-1 in the most conditions. However, the efficiency of RVC will decrease, as the change in the volume constraint is lower. Therefore, the AVC is a robust, stable and efficient algorithm in this example. 5.2. Two-dimensional simple beam (B) The second illustrative example [9] also involves a twodimensional linear elastic simple beam subject to a concentrated load applied at the bottom center as shown in Fig. 4. The design domain is divided into 40 16 quadrilateral elements. Also, vertical or horizontal displacement limits at different locations (10, 0), (5, 0) and (20, 0) are assumed in three different design cases. 5.2.1. Case B1: vertical limit is 0.5 in. at (10, 0) In this case, the displacement limit location and direction are the same as the load location. All performances are presented in Tables 8 and 9. The AVC (40 iterations) is much more efficient than the RVC (110.25 iterations). The images, obtained by AVC, all meet Criterion B-1, and the volume usage in the late periods of AVC have no oscillations. Changes in volume constraints do not occur over 10 times for all conditions of the AVC. The AVC is very successful in efficiency and stability. In the 60%-RVC condition, Criterion B-1 is not met and the algorithm is stopped until the displacement is over the limit. The final change of the volume constraint is not necessary. However, the lower the volume constraint is, the less efficient the FVC. Therefore, in the 20%-RVC condition, the first change of the volume constraint happens after 35 iterations. In this way, the performance of the AVC algorithm is more stable than the RVC algorithm.
Table 7 Performance comparison of AVC and RVC in cantilever beam of example A Limit locations
Algorithms
V final act %
CVN%
Num
A1 (16, 0)
AVC RVC AVC RVC AVC RVC
68.05 65.00 45.28 45.00 33.27 33.75
23.12 62.35 22.38 39.49 21.59 17.61
32.25 67.50 35.00 71.75 46.00 83.75
AVC RVC
48.87 47.92
22.36 39.82
37.75 74.33
A2 (12, 0) A3 (8, 0) Average
Fig. 4. Design domain and conditions for a simple beam (B).
Table 8 Performance comparison of AVC and RVC in case B1: vertical limit is 0.5 in. at (10, 0) V initial con %
Algorithms
V final act %
d final sp
Num
20
AVC RVC AVC RVC AVC RVC AVC RVC
33.68 35.00 30.03 30.00 29.02 30.00 29.51 30.00
0.4858 0.4229 0.4863 0.4940 0.4856 0.4713 0.4768 0.4776
30 78 39 83 47 144 44 136
AVC RVC
30.56 31.25
0.4836 0.4665
40.00 110.25
40 60 80 Average
5.2.2. Case B2: vertical limit is 0.5 in. at (5, 0) The location of limited displacement differs from that of the loaded. The distance of the two locations is 5 in. Proper volume usage is located from 15% to 25%. Oscillations in the late periods of the AVC are very serious because the volume usage becomes as low as 20% and the final designs are located in the infeasible domain. However, all of the images meet Criterion B-1. The efficiency of the AVC (average 70.25 iterations) is much better than that of RVC (average 143.5 iterations). In the condition of 60%AVC, despite 15 changes in volume, the final image still meets Criterion B-1. The AVC is determined to stop changing the volume constraint by Criterion B-1 and not B-3. In the 40%-RVC condition, the number of iterations is 209, but the quality of the image (PI = 0.3888) is outstanding. In other conditions, the performance indexes are formed from 0.9 to 1.1. In the future, it will be interesting
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to explore how good quality images are obtained in the multi-extreme value topology optimization problem. The performances are shown in Tables 10 and 11.
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5.2.3. Case B3: horizontal limit is 0.2 in. at (20, 0) In this case, the direction and location of the limit are both different from those of the load. The efficiency of
Table 9 Histories of the actual displacement and the volume constraint and the images of AVC and RVC in case B1: vertical limit is 0.5 in. at (10, 0)
(continued on next page)
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C.-Y. Lin, F.-M. Hsu / Advances in Engineering Software 39 (2008) 973–994 Table 10 Performance comparison of AVC and RVC in case B2: vertical limit is 0.5 in. at (5, 0) V initial con %
Algorithms
V final act %
d final sp
Num
20
AVC RVC AVC RVC AVC RVC AVC RVC
22.03 20.00 18.28 15.00 22.66 20.00 20.05 25.00
0.5158 0.5108 0.5016 0.2641 0.5190 0.5197 0.5157 0.3691
86 35 77 209 63 159 55 171
AVC RVC
20.76 20.00
0.5130 0.4159
70.25 143.50
40 60 80 Average
AVC (average 30 iterations) is better than that of RVC (average 53.25 iterations). The performances are shown in Tables 12 and 13. The sensitivity of displacement with respect to volume usage becomes small if the volume usage is greater than 40% in this case. The volume usages obtained by the AVC are distributed between 44.25% and 59.36%. The volume usages of the RVC are from 30% to 55%. The precise volume usage can be obtained by the severe convergent criterion in AVC. The AVC algorithm is executed again by
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the new severe criteria, e = 0.01, c = 2%. The performances are shown in Tables 14 and 15. The precision of the volume usage is improved. The average number of the iteration increases to 34.25 iterations. In the 60% and 80%-AVC conditions, the volume constraint is stopped by Criterion B-1. Although the oscillations are serious, the AVC still gradually converges to the proper value and the final image satisfies Criterion B-1. In example B for the cases, B1, B2 and B3, the performances of AVC and RVC are shown in Table 16. The efficiency of the AVC (average 46.75 iterations) is much better than that of the RVC (average 102.33 iterations). The AVC (CVN = 22.17%) is more stable than the RVC (CVN = 32.26%). The average volume usage of the AVC is almost same as that of the RVC. The images obtained by the AVC all meet Criterion B-1 but the RVC does not. For these two examples in this paper, the initial volume constraints range from 20% to 80%. Based on the above experiments, it is noted that the most suitable initial volume constraint is 80%. With such a large initial volume constraint, an initial design will easily evolve into a feasible design, which can then be successively reformed into the final optimal design in a stable and efficient manner. In the above experiments, the 80%-AVC case always obtains
Table 11 Histories of the actual displacement and the volume constraint and the images of AVC and RVC in case B2: vertical limit is 0.5 in. at (5, 0)
(continued on next page)
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Table 11 (continued)
a final feasible design with the smallest volume usage, with only one exception in the B1 case. Therefore, the initial vol-
ume constraint 80% is suggested when solving a new problem.
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989
Table 11 (continued)
5.2.4. Case B4: discussions in horizontal limits In this case, the purpose is to demonstrate the capability of the AVC in a displacement limit direction that is different from the load direction. This analysis of the topology optimization problem with different horizontal displacement limits, 0.2, 0.3, 0.4 and 0.5 in., demonstrates that the AVC algorithm still works successfully under these conditions. First, the FVC is applied to analyze this example with different volume constraints, 20%, 40%, 60% and 80%. Fig. 5 shows the relationship between displacement and volume usage. Secondly, the AVC is used to solve this problem and the initial volume constraint is chosen to be 80%. The convergent criterion, e, is 0.05. The parameter for c is 2.5%. The results are listed in Tables 17 and 18. The images are obtained successfully and meet Criterion B-1. The AVC is capable of dealing with the topology optimization problem with a displacement limit that differs from the load in direction. For these two examples, the AVC stops changing the volume constraint by Criterion B-1 but not B-2 for all cases. The number of changes to the volume constraint is not over 15 times (maximum in this paper). The AVC algorithm is capable of obtaining proper volume usage and gray level images, which satisfies Criterion B-1, despite oscillations that occur when the volume usage is less than 20%. The efficiency and stability of the AVC are outstanding.
With the consideration of a hole formed by the low densities of the elements in the design, the location of displacement limit is located in the inner portion of the hole of images obtained by both the AVC and the RVC. The elements of the hole still exist because the minimum density of elements (lower bound) is 0.001 in real programming of finite element analysis. Therefore, the displacement limit could be at any node of the design domain in the AVC algorithm. Both the AVC and RVC methods in this paper are performed on a 2D structure with a constant thickness. It is worthy noted that for 2D plane stress continuum structures, the element thickness has a significant effect on Table 12 Performance comparison of AVC and RVC in case B3: horizontal limit is 0.2 in. at (20, 0) V initial con %
Algorithms
V final act %
d final sp
Num
20%
AVC RVC AVC RVC AVC RVC AVC RVC
44.25 30.00 55.63 55.00 59.36 55.00 53.43 55.00
0.2085 0.2023 0.1964 0.2073 0.1904 0.1930 0.1910 0.1993
32 63 22 47 25 52 41 51
AVC RVC
53.17 48.75
0.1966 0.2004
30.00 53.25
40% 60% 80% Average
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Table 13 Histories of the actual displacement and the volume constraint and the images of AVC and RVC in case B3: horizontal limit is 0.2 in. at (20, 0)
the optimal design. The displacement constraints could be easily satisfied by changing the element thickness. The final designs generated by the AVC and RVC methods depend on the predetermined fixed value of the element
thickness. Because the element thickness is not considered as a design variable in the optimization problem, the optimal solution thus obtained is in a high possibility of a local optimum.
C.-Y. Lin, F.-M. Hsu / Advances in Engineering Software 39 (2008) 973–994 Table 13 (continued)
991
992
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Table 14 Performance of AVC in case B3: horizontal limit is 0.2 in. at (20, 0) with the severe criterion, e = 0.01, c = 2% V initial con %
V final act %
d final sp
Num
20 40 60 80
45.34 56.22 49.15 52.81
0.1983 0.1985 0.2002 0.1987
29 21 40 47
Average
50.88
0.1989
34.25
6. Concluding remarks A density-based topology optimization with a simple but robust AVC algorithm has proven its applicability and effectiveness in illustrative design problems. Although the AVC algorithm is exclusively illustrated in 2-D problems, this method can be easily applied to 3-D structural problems without any algorithmic changes. By modifying the traditional fixed volume constraint into an adjustable
Table 15 Histories of the actual displacement and the volume constraint and the images of AVC in case B3: horizontal limit is 0.2 in. at (20, 0) with the severe criterion, e = 0.01, c = 2%
C.-Y. Lin, F.-M. Hsu / Advances in Engineering Software 39 (2008) 973–994 Table 16 Performance comparison of AVC and RVC in simple beam of example B Limit locations
Algorithms
V final act %
CVN%
Num
B1 (10, 0)
AVC RVC AVC RVC AVC RVC
30.56 31.25 20.76 20.00 53.17 48.75
18.60 31.35 19.77 52.55 28.15 12.86
40.00 110.25 70.25 143.50 30.00 53.25
AVC RVC
34.83 33.33
22.17 32.26
46.75 102.33
B2 (5, 0) B3 (20, 0) Average
Fig. 5. The relationship between the displacements vs. volume usages.
993
Table 17 Performance of AVC in case B4: horizontal limits are 0.2, 0.3, 0.4 and 0.5 in. respectively dlimit (in.)
V final act %
d final sp
Num
0.5 0.4 0.3 0.2
18.79 22.94 26.00 53.43
0.5207 0.4023 0.2990 0.1993
56 54 45 51
volume constraint, the topology optimization becomes capable of seeking minimum-compliance structures that simultaneously meet additional displacement constraints. The AVC algorithm significantly extends the capabilities of traditional topology optimization methods to problems involving displacement constraints without the need for compensating for the sensitivities of displacement functions. The AVC algorithm, executed with different initial volume constraints, consistently obtains minimum-compliance structural configurations, which all meet Criterion B-1 in the two design problems with displacement constraints of varied location and direction. The number of iterations required to perform density-based topology optimization with the AVC algorithm is much less than that needed to execute RVC in all cases. The AVC method on the average
Table 18 Histories of the actual displacement and the volume constraint and the images of AVC in case B4: horizontal limits are 0.2, 0.3, 0.4 and 0.5 in. respectively
(continued on next page)
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Table 18 (continued)
requires only 50% of the computational time to locate the final solutions compared with the RVC method while RVC methods often provide slightly lighter solutions. The AVC algorithm has already been applied to topology optimization problems that seek minimum-compliance structures subject to maximum stress constraints. Furthermore, the AVC algorithm has been applied to problems considering both stress and displacement constraints, with other complicated constraining conditions currently under investigation. Acknowledgements This research was supported by the National Science Council of the Republic of China under Grant Number NSC92-2212-E-011-039. The authors would like to thank Dr. Ren-Jye Yang of the Ford Motor Company for his helpful review and suggestions for this paper. References [1] Bendsøe MP, Kikuchi N. Generating optimal topologies in structural design using a homogenization method. Comput Method Appl Mech Eng 1988;71:197–224. [2] Chu DN, Xie YM, Hira A, Steven GP. Evolutionary structural optimization for problems with stiffness constraints. Finite Elem Anal Des 1996;21:239–51. [3] Chu DN, Xie YM, Hira A, Steven GP. On various aspects of evolutionary structural optimization for problems with stiffness constraints. Finite Elem Anal Des 1997;24:197–212.
[4] Haber RB, Jog CS, Bendsøe MP. A new approach to variabletopology shape design using a constraint on perimeter. Struct Optim 1996;11:1–12. [5] Hsu MH, Hsu YL. Generalization of two- and three-dimensional structural topology optimization. Eng Optim 2005;37(1):83–102. [6] Liang QQ, Xie YM, Steven GP. Optimal topology selection of continuum structures with displacement constraints. Comput Struct 2000;77:635–44. [7] Liang QQ, Xie YM, Steven GP. A performance index for topology and shape optimization of plate bending problems with displacement constraints. Struct Multidiscip. Optim 2001;21:393–9. [8] Liang QQ, Steven GP. A performance-based optimization method for topology design of continuum structures with mean compliance constraints. Comput Methods Appl Mech Eng 2002;191: 1471–89. [9] Lin CY, Chao LS. Automated image interpretation for integrated topology and shape optimization. Struct Multidiscip. Optim 2000;20(2):125–37. [10] Mlejnek HP, Schirrmacher R. An engineer’s approach to optimal material distribution and shape finding. Comput Method Appl Mech Eng 1993;106:1–26. [11] Vanderplaats GN. Numerical optimization techniques for engineering design. McGraw-Hill; 1984. [12] Xie YM, Steven GP. A simple evolutionary procedure for structural optimization. Comput Struct 1993;49(5):885–96. [13] Yang D, Sui Y, Liu Z, Sun H. Topology optimization design of continuum structures under stress and displacement constraints. Appl Math Mech – English Ed 2000;21:19–26. [14] Yang RJ. Multidiscipline topology optimization. Comput Struct 1997;63(6):1205–12. [15] Yang RJ, Chuang CH. Optimal topology design using linear programming. Comput Struct 1994;52:265–75.
Advances in Engineering Software 39 (2008) 973–994 www.elsevier.com/locate/advengsoft
An adaptive volume constraint algorithm for topology optimization with a displacement-limit Chyi-Yeu Lin *, Fang-Ming Hsu Department of Mechanical Engineering, National Taiwan University of Science and Technology, 43 Keelung Road, Section 4, Taipei, Taiwan 10672, ROC Received 7 September 2007; received in revised form 24 September 2007; accepted 30 January 2008 Available online 17 March 2008
Abstract This paper proposes a novel adaptive volume constraint algorithm (AVC) to replace the fixed volume constraint (FVC) in the traditional topology optimization method, such that a minimum-compliance optimal structure that simultaneously meets the additional displacement limit can be searched. Optimal minimum-compliance structures are subject to the limit of a predetermined amount of material in traditional calculation methods, and often fail to meet practical stress and displacement constraints, or become unnecessarily strong. Without displacement sensitivities, the AVC algorithm iteratively adjusts allowable material usage based on the difference between the actual displacement, and the allowable displacement limit. In this way, topology optimization can efficiently construct a minimum-compliance structure that meets the allowable displacement limit. The regular volume constraint algorithm (RVC), which slightly changes the volume constraint each time at the end of FVC, is also executed in order to demonstrate the advantages of the AVC algorithm. The effectiveness of the AVC algorithm is demonstrated by two illustrative 2-D design problems: a cantilever beam and a simple beam. Ó 2008 Elsevier Ltd. All rights reserved. Keywords: Topology optimization; Adaptive volume constraint; Material distribution method; Displacement constraint; Density method
1. Introduction Bends/e and Kikuchi [1] developed the homogenization method for topology optimization, which is a milestone for structural optimization. Mlejnek and Schirrmacher [10] proposed the density function approach to attain a minimum-compliance structure under a volume usage constraint. Yang and Chuang [15] suggested the use of the normalized material density of each element as the design variable that reduces the number of design variables in a topology optimization problem. Since analytical forms of the gradients of compliance and volume usage for structures have been derived, most density-based topology optimization problems use the compliance as the objective function, and the volume usage as the only constraint. Various volume usage constraints in the density methods lead to different structural designs that consequently result *
Corresponding author. Tel.: +886 2 273 6494. E-mail address: [email protected] (C.-Y. Lin).
0965-9978/$ - see front matter Ó 2008 Elsevier Ltd. All rights reserved. doi:10.1016/j.advengsoft.2008.01.008
in different stress and displacement values. The minimumcompliance structure thus obtained can either become excessively strong or can fail to meet practical stress and displacement constraints. Without a proper volume usage constraint, the topology optimization will often fail to provide the correct structural topology when maximum stress/displacement considerations are taken into account. Much effort has been placed on developing algorithms that can find proper volume usage for a structure and/or that can find a structure that meets specified displacement limits. Yang [14] treated the weight as the objective function and used the displacement, frequency and compliance as inequality constraints. Mathematical programming methods or optimal criteria algorithms [13] have been adopted to solve structural topology optimization problems. Unless the analytical forms of the sensitivities for stress and displacement functions of the structure can be derived, it will be computationally prohibitive to consider stress or displacement constraints in topology optimization problems involving a large number of design variables.
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A simple formula for evolutionary structural optimization (ESO) [12] was proposed to gradually eliminate under-utilized elements in the design domain so as to form the fully stressed structure. Chu et al. [2,3] used nibbling techniques to iteratively eliminate elements with the lowest sensitivity values so as to create a structure that meets the stiffness constraint. ESO-related studies maximize various performance indexes of a structure subject to different constraints by gradually eliminating elements with the lowest strain energy values. Liang et al. [6,7] and Liang and Steven [8] developed the performance-based optimization method that uses performance indices to obtain the optimal designs of continuum structures. ESO is a simple method for topology optimization problems, which gradually removes elements from the structure based on the sensitivities of the objective and constraints to achieve the optimum. It is noted that the use of the element removal ratio of 1–2% in ESO-related methods gives satisfactory solutions. Larger removal ratios provide relatively unstable results and smaller ratios often require a high computational cost. This paper proposes a simple AVC algorithm to replace the FVC algorithm in density-based methods, so that without major algorithmic changes the optimization can seek a minimum-compliance design that simultaneously satisfies the additional displacement limit. It does not derive the analytical expression for the sensitivity of the displacement function or compute sensitivity by numerical techniques. The proposed algorithm will be used in a density method to solve the minimum-compliance structure under a displacement limit.
the stiffness of the structure. The sequential linear programming method (SLP) [11] is used to solve for the optimization problem in this work. If minimizing the displacement of the structure is the design goal, it is reasonable to pursue a structure that has the maximum stiffness. However, if maximum displacement is a design condition that needs to be met, a maximum-stiffness structure attained under a volume usage constraint is not necessarily the best design. For insufficient material usage, the optimal maximum-stiffness structure is still too flexible to meet the displacement constraint. For excessive volume usages, the overbuilt optimal structure easily satisfies the displacement constraint, but material is wasted. It is difficult to predict proper material usage for a given topology optimization problem under a displacement constraint. Often, the density method has to be conducted several times with different volume usage constraint definitions, and the lightest optimal structure satisfying the additional displacement constraint is selected as the proper structural configuration. Otherwise, an engineer arbitrarily
2. Continuum topology optimization The most common topology optimization formulation is to minimize compliance with a fixed volume constraint. By using the normalized density of each finite element that forms a structural domain as the design variable, the density method [10,15] simplifies the topology optimization problem as Minimize Subject to :
f ðqÞ n X i¼1
Side constraints : Ei And ¼ q2i Eo
ð1Þ qi vi =
n X
vi 6 V con
ð2Þ
i¼1
0 6 qi 6 1; i ¼ 1; n
ð3Þ ð4Þ
where f(q) is the objective function, qi is the normalized density of the ith element, vi is the volume of the ith element, Vcon is the volume usage limit, which is a percentage of the whole volume of the structure, and n is the number of design variables. Ei is an intermediate Young’s modulus of an element and Eo is the original Young’s modulus of structure. The objective function is selected as the compliance of the structure, which is equivalent to maximizing
Fig. 1. The flowchart of the topology optimization with an adaptive volume constraint (AVC).
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selects a volume usage to perform the density method and then treats the resultant structural configuration as the solution. Therefore, the structure fails to meet the displacement constraint and the structural configuration needs to be manually strengthened by placing more material on the existent structure. 3. Adaptive volume constraint algorithm The major drawback of using the fixed volume constraint algorithm (FVC) in traditional topology optimization methods is that use of an improper volume usage constraint leads to an excessively strong structure or a structure that fails to meet stress/displacement requirements. Therefore, the adaptive volume constraint algorithm (AVC) aims to simultaneously search for the most proper volume usage and the minimum-compliance structure using the allowable amount of material. With the AVC algorithm, the difference between the displacement of the specified location in the structure and the displacement limit at the end of several iterations of the optimization is used to consider adjustments to the current volume usage ratio. The optimization continually searches for the proper volume usage constraint according to the displacement status of current structures while the optimization seeks the best material distribution on a design domain with the newest volume usage constraint. The systematic flowchart of the execution of the topology optimization with the adaptive volume constraint algorithm is shown in Fig. 1. The detailed stepwise procedure for the topology optimization process with the adaptive volume constraint is as follows:
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Step 1: Define the design domain, the loading conditions, the boundary conditions and dlimit, which is the maximum displacement limit at a specific location of the structure. Set the initial volume constraint, V initial con , and an initial design. The densities,qi, of the initial design of the entire design domain are the same as the value of the initial volume constraint in this paper. Determine the convergent value, e, and the parameter c. Step 2: Run an iteration of the SLP topology optimization process. After the topology optimization problem is linearized, the optimal design is obtained in the move limit. Then, the specific displacement, dsp, and the compliance of the current structure of this optimal design can be attained. The entire process mentioned above is defined as an iteration of SLP [15]. Step 3: If both of the following two criteria (Criteria A) are met, go on to Step 4; otherwise go back to Step 2. k k1 (A-1) 0:1 6 C CC 6 0, the relative difference of k1 the compliance of structures between last two iterations, is between 10% and 0. The 0.1 value is determined empirically based on a large number of experiments. When the change of the compliance on two iterations is below 10%, it will be interpreted that the optimization under the current volume constraint has reached a stable status. It
Fig. 3. Design domain and conditions for a short cantilever beam (A).
Table 1 Performance comparison of AVC and RVC in Case A1: vertical limit is 0.5 in. at (16, 0) V initial con %
Algorithms
V final act %
d final sp
Num
20
AVC RVC AVC RVC AVC RVC AVC RVC
70.41 65.00 67.80 65.00 67.38 65.00 66.62 65.00
0.4789 0.4996 0.4785 0.4967 0.4789 0.5008 0.4843 0.4970
43 123 29 76 26 28 31 43
AVC RVC
68.05 65.00
0.4801 0.4985
40 60 80
Fig. 2. The flowchart of the topology optimization with a regular volume constraint (RVC).
Average
32.25 67.50
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is a good timing to switch into a new volume constraint, otherwise the computational cost will be continually spent on the optimization with the volume constraint far apart from the final volume
constraint. Criterion A-1 is one of the conditions determining the time to change volume constraint. (A-2) jV kact V kcon j 6 1%, the difference between the actual volume usage, V kact , and the current volume
Table 2 Histories of the actual displacement and the volume constraint and the images of AVC and RVC in Case A1: vertical limit is 0.5 in. at (16, 0)
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Table 2 (continued)
constraint, V kcon , is smaller than 1%. The 1% provides a good thresholding value for interpreting
the new volume constraint becomes active. A smaller value can be used but the interpretation will not
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be stable if the number is below 0.0001. The Criterion A-2 is to confirm that the topology design is substantially affected by the updated volume constraint during the optimization process. When the actual volume is close to the volume constraint, the optimization will continue to Step 4. Step 4: If any of the following three criteria (Criteria B) are met, go to Step 6. Otherwise, go on to Step 5 to adjust the volume constraint. d d (B-1) spd limitlimit 6 e is the relative difference between the displacement of the specified location of the structure, and the displacement limit is not over the criterion value, e. (B-2) The volume constraint either exceeds 90% or is less than 10% for five consecutive checks. This criterion is used to limit the range of the volume constraint in order for the gray level image, obtained by AVC, RVC or FVC, to be applied successfully to the practical design. In this paper, the number of elements of the design domain is 640. If the volume constraint is more than 90%, the densities of the entire design domain will become almost 1. The image is not referable. If the volume is less than 10%, the properties of the binary image, obtained by the penalty and threshold technique, is hard to closely compare to those of the gray level image. As the mesh resolution of the design domain gets finer, volume constraint becomes wider. Therefore, the reference number of the elements of the mesh recommended in the literature [4,5], is 2000. The range of the volume constraint can be wider than 80% (from 10% to 90% in this paper). Without the consideration of manufacturability and practicality of design, Criterion B-2 can be neglected. (B-3) The volume constraint has been adjusted over 10 times before Criterion B-1 is met. Criterion B-3 is required to avoid cases in which Criterion B1 is too severe to converge. Despite the fact that AVC does not converge after a maximum number of changes of the volume constraint, the volume constraint always oscillates, and the volume constraint approaches the proper value for volume usage. Therefore, AVC is forced to stop changing the volume constraint. Step 5: c d sp d limit m k k1 k1 V con ¼ V con þ ð1Þ min ; 0:2V con ; e d limit 0; d sp > d limit m¼ ð5Þ 1; d sp < d limit Increase or decrease the volume constraint by expression (5) and go back to Step 2. In the early periods of AVC, the relative difference between dsp and dlimit is usually outsized so that the term 0:2V k1 con of expression (5) dominates the change of the volume constraint. Since the minimum and
maximum volume constraints in the AVC method are set to as 10% and 90%, the maximum change of the volume constraint will be reasonably limited between 2% and 18%. Smaller or larger volume change ratios at this stage usually lead to inferior results. In the late period of AVC, the displacement, dsp, of the structure gradually becomes close to the displacement limit, dlimit. The term, c d sp d limit of expression (5) will then dominate e d limit , the volume constraint change. The parameter c is suggested to be limited between 1% and 3%. The empirical values 1% and 3% produced the most stable solutions based on a large number of test problems. In the later stages of the optimization the current design is always close to the constraint boundary, c smaller than 1% will lead to slow or false convergence, while larger than 3% will extend the time required for convergence. Step 6: Keep the current volume constraint unchanged and further run an iteration of SLP optimization. Step 7: If Criteria B-1 and B-2 continue to be met, go to Step 8. Otherwise, the maximum number of iteration is released to 15 and goes back to Step 5. Step 8: If the absolute value of the relative difference of compliance between the last two iterations is less k k1 than 0.1% (Criterion C), C CC k1 6 0:001, terminate the optimization and obtain the final displacefinal ment, d final sp , and final volume usage, V act . Otherwise, go back to Step 6. Criterion C is the major convergent condition for traditional topology optimization (FVC).
4. Regular volume constraint algorithm In general, in the topology optimization problem with displacement or stress constraints, the FVC algorithm will be applied directly by a regularly, gradually and slightly changing volume constraint. The topology optimization with a regular volume constraint (RVC) is executed in this paper in order to demonstrate the advantages of the AVC algorithm. The RVC algorithm solves a structural optimiTable 3 Performance comparison of AVC and RVC in Case A2: vertical limit is 0.5 in. at (12, 0) V initial con %
Algorithms
V final act %
d final sp
Num
20
AVC RVC AVC RVC AVC RVC AVC RVC
46.69 45.00 46.22 45.00 45.63 45.00 42.56 45.00
0.4810 0.4802 0.4796 0.4522 0.4850 0.4486 0.4860 0.4511
39 92 33 31 25 74 43 90
AVC RVC
45.28 45.00
0.4829 0.4580
35.00 71.75
40 60 80 Average
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zation problem with a displacement limit by slightly changing the volume constraint each time at the end of FVC. The main drawback of the RVC is that the value of change of
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the volume constraint is hard to determine. The value is too high to obtain precise volume usage or an image that meets the displacement limit criterion. The value of change
Table 4 Histories of the actual displacement and the volume constraint and the images of AVC and RVC in Case A2: vertical limit is 0.5 in. at (12, 0)
(continued on next page)
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Table 4 (continued)
of the volume constraint of RVC is 5% in this paper. The performances of the RVC depend on individual cases.
In the RVC algorithm, the major convergent criterion is Criterion B-1. The second one is whether or not the volume
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constraint violates the monotone property. The systematic flowchart of the execution of the RVC algorithm is shown in Fig. 2. The detailed stepwise procedure of the RVC algorithm process is stated as follows:
AVC and RVC, are executed with different initial volume constraints, 20%, 40%, 60%, and 80%, respectively.
Step 1: Define the design domain, the loading conditions, the boundary conditions, and dlimit: the displacement limit at a specific location of the structure. Set the initial volume constraint, V initial con , and define an initial design. Step 2: Run iterations for the SLP topology optimization process and then compute the specific displacement, dsp, and the compliance of the current structure, C. Step 3: If the absolute value of the relative difference of compliance between the last two iterations is less k k1 than 0.1% (Criterion C), C CC k1 6 0:001, terminate the optimization. Otherwise, go back to Step 2. Step 1 to Step 3 are defined as traditional topology optimization shown as gray blocks in Fig. 4. Step 4: If the Criterion B-1 is met, go to Stop. Otherwise, go to Step 5. Step 5: The volume constraint has to be monotone. If the monotone property is violated, then go to stop. Otherwise, go back to Step 2. (The volume constraint has to keep decreasing or increasing gradually. The property is defined as monotone.)
In the first illustrative example, a two-dimensional linear elastic example is shown. It involves a short cantilever beam with a fixed support subject to a displacement constraint in the vertical direction, as shown in Fig. 3. The design domain is divided into 32 20 quadrilateral elements. The concentrated load is applied to the bottom of the free end. Assuming the three different cases, the vertical displacement limits at the three locations, (16, 0), (12, 0) and (8, 0), are all 0.5 in.
5.1. Two-dimensional cantilever beam (A)
5.1.1. Case A1: vertical limit is 0.5 in. at (16, 0) The location of the limited displacement is the same as the load location. The performances of AVC and RVC are shown in Table 1. The histories of displacement and volume constraint and the final topological images are also presented in Table 2. The condition 80%-AVC is very stable to attain proper topologies by gradually decreasing the volume constraint. The final displacements of the images for all conditions meet Criterion B-1 and the designs are all feasible, with exception of the condition of 60%-RVC. Therefore, the two algorithms are successful in this example. The efficiency of AVC (32.25 iterations) is much better than that of RVC (67.5 iterations).
5. Numerical examples In the following two illustrative examples, the AVC algorithm will first be used to minimize the compliance of the structure with a vertical or horizontal displacement constraint at different locations. Secondly, the RVC algorithm is used to seek a minimum-compliance structure subject to volume constraints in both illustrative problems. The four indexes are used to compare AVC to RVC. The number of iterations, N, presents the efficiency of optimization. The coefficient of variation for the number of iterations, CVN, is the stability of efficiency. Final volume usage,V final act , is the proper usage attained by AVC or RVC. The performance index, PI, is used to classify the quality of the topological structure. The fewer the indexes, the better the algorithm is. The indexes are shown as expression (6) and (7): sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi P 2 ðN N Þ CVN ¼ =n ð6Þ n1 V act d sp PI ¼ ð7Þ V act d limit For the following two examples, the common conditions of the problems are as follows: Young’s modulus, E = 207,000 psi, and Poisson’s ratio, v = 0.3, are assumed. A concentrated load is 3000 lb, and the uniform thickness of the structure is 1 in. The convergent criterion, e, is 0.05. The parameter for c is 2.5%. The two algorithms,
5.1.2. Case A2: vertical limit is 0.5 in. at (12, 0) The distance between the limit location and the load location is 4 in. The efficiency of AVC (35 iterations) is much better than that of RVC (71.75 iterations), as shown in Table 3. The AVC algorithm also can efficiently obtain proper topologies that are feasible and meet Criterion B1. The value of changing the volume constraint in RVC is too rough because the images of RVC do not meet Criterion B-1. The qualities of images obtained in 40%, 60%, 80%-RVC and 80%-AVC conditions are better than those of others. The performance indexes (PI) of these images are all close to 0.9. Performance indexes in other cases
Table 5 Performance comparison of AVC and RVC in Case A3: vertical limit is 0.5 in. at (8, 0) V initial con %
Algorithms
V final act %
d final sp
Num
20
AVC RVC AVC RVC AVC RVC AVC RVC
34.41 35.00 34.72 30.00 32.00 35.00 31.94 35.00
0.5168 0.4370 0.4971 0.4693 0.4817 0.4609 0.4926 0.5169
50 72 40 70 36 96 58 97
AVC RVC
33.27 33.75
0.4970 0.4710
46.00 83.75
40 60 80 Average
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are about 1.0–1.1. The net structures of the images, shown in Table 4, are average distributions over the design domains.
5.1.3. Case A3: vertical limit is 0.5 in. at (8, 0) The distance between limit location and load location is 8 in. The efficiency of AVC (46 iterations) is also much bet-
Table 6 Histories of the actual displacement and the volume constraint and the images of AVC and RVC in Case A2: vertical limit is 0.5 in. at (8, 0)
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ter than that of RVC (83.75 iterations), as shown in Table 5. Although the oscillation phenomena, shown in Table 6 of the volume constraint at later periods of the 20% and 40%-AVC are serious, proper volume usages are obtained and the images still meet Criterion B-1. The 60% and 80%-AVC are very stable and successful. Despite the fact that Criterion B-1 is met in 20%-AVC and 80-RVC, the designs of these two conditions are not feasible. The displacements of the cases are 0.5168 and 0.5169, respectively. The images can be improved be using a binary technique. In cantilever beam example A, the AVC can obtain the proper volume usages and images, which meet Criterion B1 for all conditions, with different initial volume constraints and different location limits. The relationship between displacements vs. volume usage can be verified in the example. The efficiency of AVC (average 37.75 iterations) is much more than that of RVC (average 74.33 iterations). With respect to the stability of efficiency, CVN, the AVC is affected slightly by the initial volume constraint. The volume usages of the two algorithms show little difference. The performances of the AVC and RVC are listed in Table 7. The initial volume constraint of AVC is recommended at 60% or 80% to avoid oscillation phenomena. A change in the volume constraint each time of 5% in RVC is too rough to meet Criterion B-1 in the most conditions. However, the efficiency of RVC will decrease, as the change in the volume constraint is lower. Therefore, the AVC is a robust, stable and efficient algorithm in this example. 5.2. Two-dimensional simple beam (B) The second illustrative example [9] also involves a twodimensional linear elastic simple beam subject to a concentrated load applied at the bottom center as shown in Fig. 4. The design domain is divided into 40 16 quadrilateral elements. Also, vertical or horizontal displacement limits at different locations (10, 0), (5, 0) and (20, 0) are assumed in three different design cases. 5.2.1. Case B1: vertical limit is 0.5 in. at (10, 0) In this case, the displacement limit location and direction are the same as the load location. All performances are presented in Tables 8 and 9. The AVC (40 iterations) is much more efficient than the RVC (110.25 iterations). The images, obtained by AVC, all meet Criterion B-1, and the volume usage in the late periods of AVC have no oscillations. Changes in volume constraints do not occur over 10 times for all conditions of the AVC. The AVC is very successful in efficiency and stability. In the 60%-RVC condition, Criterion B-1 is not met and the algorithm is stopped until the displacement is over the limit. The final change of the volume constraint is not necessary. However, the lower the volume constraint is, the less efficient the FVC. Therefore, in the 20%-RVC condition, the first change of the volume constraint happens after 35 iterations. In this way, the performance of the AVC algorithm is more stable than the RVC algorithm.
Table 7 Performance comparison of AVC and RVC in cantilever beam of example A Limit locations
Algorithms
V final act %
CVN%
Num
A1 (16, 0)
AVC RVC AVC RVC AVC RVC
68.05 65.00 45.28 45.00 33.27 33.75
23.12 62.35 22.38 39.49 21.59 17.61
32.25 67.50 35.00 71.75 46.00 83.75
AVC RVC
48.87 47.92
22.36 39.82
37.75 74.33
A2 (12, 0) A3 (8, 0) Average
Fig. 4. Design domain and conditions for a simple beam (B).
Table 8 Performance comparison of AVC and RVC in case B1: vertical limit is 0.5 in. at (10, 0) V initial con %
Algorithms
V final act %
d final sp
Num
20
AVC RVC AVC RVC AVC RVC AVC RVC
33.68 35.00 30.03 30.00 29.02 30.00 29.51 30.00
0.4858 0.4229 0.4863 0.4940 0.4856 0.4713 0.4768 0.4776
30 78 39 83 47 144 44 136
AVC RVC
30.56 31.25
0.4836 0.4665
40.00 110.25
40 60 80 Average
5.2.2. Case B2: vertical limit is 0.5 in. at (5, 0) The location of limited displacement differs from that of the loaded. The distance of the two locations is 5 in. Proper volume usage is located from 15% to 25%. Oscillations in the late periods of the AVC are very serious because the volume usage becomes as low as 20% and the final designs are located in the infeasible domain. However, all of the images meet Criterion B-1. The efficiency of the AVC (average 70.25 iterations) is much better than that of RVC (average 143.5 iterations). In the condition of 60%AVC, despite 15 changes in volume, the final image still meets Criterion B-1. The AVC is determined to stop changing the volume constraint by Criterion B-1 and not B-3. In the 40%-RVC condition, the number of iterations is 209, but the quality of the image (PI = 0.3888) is outstanding. In other conditions, the performance indexes are formed from 0.9 to 1.1. In the future, it will be interesting
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to explore how good quality images are obtained in the multi-extreme value topology optimization problem. The performances are shown in Tables 10 and 11.
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5.2.3. Case B3: horizontal limit is 0.2 in. at (20, 0) In this case, the direction and location of the limit are both different from those of the load. The efficiency of
Table 9 Histories of the actual displacement and the volume constraint and the images of AVC and RVC in case B1: vertical limit is 0.5 in. at (10, 0)
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C.-Y. Lin, F.-M. Hsu / Advances in Engineering Software 39 (2008) 973–994 Table 10 Performance comparison of AVC and RVC in case B2: vertical limit is 0.5 in. at (5, 0) V initial con %
Algorithms
V final act %
d final sp
Num
20
AVC RVC AVC RVC AVC RVC AVC RVC
22.03 20.00 18.28 15.00 22.66 20.00 20.05 25.00
0.5158 0.5108 0.5016 0.2641 0.5190 0.5197 0.5157 0.3691
86 35 77 209 63 159 55 171
AVC RVC
20.76 20.00
0.5130 0.4159
70.25 143.50
40 60 80 Average
AVC (average 30 iterations) is better than that of RVC (average 53.25 iterations). The performances are shown in Tables 12 and 13. The sensitivity of displacement with respect to volume usage becomes small if the volume usage is greater than 40% in this case. The volume usages obtained by the AVC are distributed between 44.25% and 59.36%. The volume usages of the RVC are from 30% to 55%. The precise volume usage can be obtained by the severe convergent criterion in AVC. The AVC algorithm is executed again by
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the new severe criteria, e = 0.01, c = 2%. The performances are shown in Tables 14 and 15. The precision of the volume usage is improved. The average number of the iteration increases to 34.25 iterations. In the 60% and 80%-AVC conditions, the volume constraint is stopped by Criterion B-1. Although the oscillations are serious, the AVC still gradually converges to the proper value and the final image satisfies Criterion B-1. In example B for the cases, B1, B2 and B3, the performances of AVC and RVC are shown in Table 16. The efficiency of the AVC (average 46.75 iterations) is much better than that of the RVC (average 102.33 iterations). The AVC (CVN = 22.17%) is more stable than the RVC (CVN = 32.26%). The average volume usage of the AVC is almost same as that of the RVC. The images obtained by the AVC all meet Criterion B-1 but the RVC does not. For these two examples in this paper, the initial volume constraints range from 20% to 80%. Based on the above experiments, it is noted that the most suitable initial volume constraint is 80%. With such a large initial volume constraint, an initial design will easily evolve into a feasible design, which can then be successively reformed into the final optimal design in a stable and efficient manner. In the above experiments, the 80%-AVC case always obtains
Table 11 Histories of the actual displacement and the volume constraint and the images of AVC and RVC in case B2: vertical limit is 0.5 in. at (5, 0)
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Table 11 (continued)
a final feasible design with the smallest volume usage, with only one exception in the B1 case. Therefore, the initial vol-
ume constraint 80% is suggested when solving a new problem.
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Table 11 (continued)
5.2.4. Case B4: discussions in horizontal limits In this case, the purpose is to demonstrate the capability of the AVC in a displacement limit direction that is different from the load direction. This analysis of the topology optimization problem with different horizontal displacement limits, 0.2, 0.3, 0.4 and 0.5 in., demonstrates that the AVC algorithm still works successfully under these conditions. First, the FVC is applied to analyze this example with different volume constraints, 20%, 40%, 60% and 80%. Fig. 5 shows the relationship between displacement and volume usage. Secondly, the AVC is used to solve this problem and the initial volume constraint is chosen to be 80%. The convergent criterion, e, is 0.05. The parameter for c is 2.5%. The results are listed in Tables 17 and 18. The images are obtained successfully and meet Criterion B-1. The AVC is capable of dealing with the topology optimization problem with a displacement limit that differs from the load in direction. For these two examples, the AVC stops changing the volume constraint by Criterion B-1 but not B-2 for all cases. The number of changes to the volume constraint is not over 15 times (maximum in this paper). The AVC algorithm is capable of obtaining proper volume usage and gray level images, which satisfies Criterion B-1, despite oscillations that occur when the volume usage is less than 20%. The efficiency and stability of the AVC are outstanding.
With the consideration of a hole formed by the low densities of the elements in the design, the location of displacement limit is located in the inner portion of the hole of images obtained by both the AVC and the RVC. The elements of the hole still exist because the minimum density of elements (lower bound) is 0.001 in real programming of finite element analysis. Therefore, the displacement limit could be at any node of the design domain in the AVC algorithm. Both the AVC and RVC methods in this paper are performed on a 2D structure with a constant thickness. It is worthy noted that for 2D plane stress continuum structures, the element thickness has a significant effect on Table 12 Performance comparison of AVC and RVC in case B3: horizontal limit is 0.2 in. at (20, 0) V initial con %
Algorithms
V final act %
d final sp
Num
20%
AVC RVC AVC RVC AVC RVC AVC RVC
44.25 30.00 55.63 55.00 59.36 55.00 53.43 55.00
0.2085 0.2023 0.1964 0.2073 0.1904 0.1930 0.1910 0.1993
32 63 22 47 25 52 41 51
AVC RVC
53.17 48.75
0.1966 0.2004
30.00 53.25
40% 60% 80% Average
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Table 13 Histories of the actual displacement and the volume constraint and the images of AVC and RVC in case B3: horizontal limit is 0.2 in. at (20, 0)
the optimal design. The displacement constraints could be easily satisfied by changing the element thickness. The final designs generated by the AVC and RVC methods depend on the predetermined fixed value of the element
thickness. Because the element thickness is not considered as a design variable in the optimization problem, the optimal solution thus obtained is in a high possibility of a local optimum.
C.-Y. Lin, F.-M. Hsu / Advances in Engineering Software 39 (2008) 973–994 Table 13 (continued)
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Table 14 Performance of AVC in case B3: horizontal limit is 0.2 in. at (20, 0) with the severe criterion, e = 0.01, c = 2% V initial con %
V final act %
d final sp
Num
20 40 60 80
45.34 56.22 49.15 52.81
0.1983 0.1985 0.2002 0.1987
29 21 40 47
Average
50.88
0.1989
34.25
6. Concluding remarks A density-based topology optimization with a simple but robust AVC algorithm has proven its applicability and effectiveness in illustrative design problems. Although the AVC algorithm is exclusively illustrated in 2-D problems, this method can be easily applied to 3-D structural problems without any algorithmic changes. By modifying the traditional fixed volume constraint into an adjustable
Table 15 Histories of the actual displacement and the volume constraint and the images of AVC in case B3: horizontal limit is 0.2 in. at (20, 0) with the severe criterion, e = 0.01, c = 2%
C.-Y. Lin, F.-M. Hsu / Advances in Engineering Software 39 (2008) 973–994 Table 16 Performance comparison of AVC and RVC in simple beam of example B Limit locations
Algorithms
V final act %
CVN%
Num
B1 (10, 0)
AVC RVC AVC RVC AVC RVC
30.56 31.25 20.76 20.00 53.17 48.75
18.60 31.35 19.77 52.55 28.15 12.86
40.00 110.25 70.25 143.50 30.00 53.25
AVC RVC
34.83 33.33
22.17 32.26
46.75 102.33
B2 (5, 0) B3 (20, 0) Average
Fig. 5. The relationship between the displacements vs. volume usages.
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Table 17 Performance of AVC in case B4: horizontal limits are 0.2, 0.3, 0.4 and 0.5 in. respectively dlimit (in.)
V final act %
d final sp
Num
0.5 0.4 0.3 0.2
18.79 22.94 26.00 53.43
0.5207 0.4023 0.2990 0.1993
56 54 45 51
volume constraint, the topology optimization becomes capable of seeking minimum-compliance structures that simultaneously meet additional displacement constraints. The AVC algorithm significantly extends the capabilities of traditional topology optimization methods to problems involving displacement constraints without the need for compensating for the sensitivities of displacement functions. The AVC algorithm, executed with different initial volume constraints, consistently obtains minimum-compliance structural configurations, which all meet Criterion B-1 in the two design problems with displacement constraints of varied location and direction. The number of iterations required to perform density-based topology optimization with the AVC algorithm is much less than that needed to execute RVC in all cases. The AVC method on the average
Table 18 Histories of the actual displacement and the volume constraint and the images of AVC in case B4: horizontal limits are 0.2, 0.3, 0.4 and 0.5 in. respectively
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Table 18 (continued)
requires only 50% of the computational time to locate the final solutions compared with the RVC method while RVC methods often provide slightly lighter solutions. The AVC algorithm has already been applied to topology optimization problems that seek minimum-compliance structures subject to maximum stress constraints. Furthermore, the AVC algorithm has been applied to problems considering both stress and displacement constraints, with other complicated constraining conditions currently under investigation. Acknowledgements This research was supported by the National Science Council of the Republic of China under Grant Number NSC92-2212-E-011-039. The authors would like to thank Dr. Ren-Jye Yang of the Ford Motor Company for his helpful review and suggestions for this paper. References [1] Bendsøe MP, Kikuchi N. Generating optimal topologies in structural design using a homogenization method. Comput Method Appl Mech Eng 1988;71:197–224. [2] Chu DN, Xie YM, Hira A, Steven GP. Evolutionary structural optimization for problems with stiffness constraints. Finite Elem Anal Des 1996;21:239–51. [3] Chu DN, Xie YM, Hira A, Steven GP. On various aspects of evolutionary structural optimization for problems with stiffness constraints. Finite Elem Anal Des 1997;24:197–212.
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