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Design optimisation of composite panel structures with stiffening ribs under multiple loading cases
Computers and Structures 78 (2000) 637±647
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Design optimisation of composite panel structures with stiening ribs under multiple loading cases Jing-Sheng Liu a, Len Hollaway b,* b
a Department of Engineering, Cambridge University, Cambridge CB2 1PZ, UK Department of Civil Engineering, Composite Structures Research Unit, University of Surrey, Guildford, Surrey GU2 5XH, UK
Received 31 July 1998; accepted 8 December 1999
Abstract An optimisation procedure for composite panel structures with stiening ribs is presented in this paper. The procedure employs standard FE structural analysis, structural system pro®le analysis, and multi-factor optimisation techniques to predict an optimum structural design. A composite antenna re¯ector structure in space environment is optimised. Laminate, sandwich con®guration and rib shape in the structure are optimised in this application. The optimisation improves antenna performance under multiple loading cases and minimises structural mass simultaneously. Ó 2000 Elsevier Science Ltd. All rights reserved. Keywords: Structural con®gurations; Optimisation; Re¯ector; Advanced polymer composites; Panel; Stiening rib
1. Introduction Sandwich systems, which are manufactured by cocuring or secondary bonding, are the most used structural solution for high accuracy lightweight panel structures. A honeycomb sandwich panel is a layered construction typically formed by bonding two thin face sheets to a thick core. This sandwich construction provides a very lightweight structural con®guration for many load conditions. The face sheets take the membrane and bending loads while the core resists the shear loads. On the other hand, carbon-®bre-reinforced composite materials provide an extremely small coecient of thermal expansion and one of the lightest weight material systems due to their excellent speci®c stiness. Therefore, a sandwich panel employing ®bre-reinforced composite material faces brings together the excellence of both the structural con®guration and materials system.
*
Corresponding author. Tel.: +44-1483-879280; fax: +441483-450984. E-mail address: [email protected] (L. Hollaway).
In this work, a composite space antenna structure with parabolic re¯ector has been optimised. The panels of the re¯ector are stiened with ribs. Both the panels and the ribs are constructed from sandwich con®guration utilising aluminium honeycomb cores; the face sheets of both the panels and the ribs are fabricated from laminates. The individual plies are ``composites'' of carbon ®bre ®laments encapsulated unidirectionally in a matrix material giving them orthogonal material properties. An optimum design for the structure and con®guration was sought. In the optimisation, the re¯ector surface accuracy (typically, the RMS value of the displacements from the ideal re¯ector surface), structural strength, stiness, natural frequencies, and structural mass were optimised simultaneously, while considering the thermal deformations due to the temperature changes in space environment and a launch case. The possibility of achieving a design that eciently optimises multiple performances, coupled with the dif®culty in selecting the values of a large set of design variables, makes structural optimisation an important tool for the design of laminated composite structures. The proposed method takes advantage of the concept of the goal programming (GP) method which minimises
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the deviations between the achievement of the goals and their aspiration levels. Linear goal programming (LGP) was initiated by Charnes and Cooper [1] and the LGP algorithms were further developed by Ijiri [2], Lee [3] and many others. The extension of GP to non-linear optimisation problems has been considered by Ignizio [4], Rao et al. [5], El-Sayed and Jang [6]. However, the method proposed here diers from general GP methods mainly in that · It treats complex engineering design problems, which may have multiple objectives and multiple loading cases, in a systematic way by employing a parameter pro®le analysis. · It introduces an assessment system which brings scores and indexes into the range of 0±10 for all the non-commensurable performances (i.e. have dierent units) and loading cases. · It makes no dierence for performance constraints and objective functions in the formulation. A multi-objective optimisation computer program, M O S T (multifactor optimisation of structures technique), has been developed to accommodate and implement the optimisation method. The M O S T system is a synergistic combination of ®nite element static/dynamic analysis, system assessment, and optimisation techniques, together with some pre-phase programs of design optimisation, input/output, model update, and interfacing programs. It utilises A B A Q U S and the theory of laminated plate to build a comprehensive analysis/design capability for structural composites. The optimisation system control program is written in U N I X shell scripts. It can control optimisation ¯ow and execute application programs in U N I X environment so that all the functions can be integrated in a system. It can also simulate the electromagnetic analysis of distorted antenna systems (a speci®cally useful feature in optimising antenna systems) [7]. The program can account for various environmental conditions and loading cases. These features make M O S T a powerful, cost eective and reliable tool to optimise complex structural systems.
2. The analyses of a composite re¯ector panel structure The antenna is an oset system. The re¯ector has a projected aperture size of 3:6 2:6 m2 and a highly accurate surface. The nominal surface is a section of a paraboloid having a focal length of 1.8 m and oset by 0.4 m from the paraboloid axis (Fig. 1). The re¯ector dish is fabricated from a graphite composite honeycomb sandwich panel structure stiened by a ribbed backing structure which is formed from a lattice of beams, also of honeycomb sandwich construction. Fig. 2 shows the backside of the re¯ector. All these ribs are assembled and are oset bonded towards the rear of the re¯ector.
Fig. 1. An oset re¯ector.
Fig. 2. The original re¯ector geometry.
The ribs can vary in height over the structure. In the original structure, all the ribs have a height of 0.1 m. The dish sandwich panel is a 0.01 m thick aluminium alloy honeycomb core covered with graphite ®bre reinforced epoxy (GFRE) face sheets. The sheets on both sides of the core are constructed with 0.0001 m thick GFRE layers in a [0/90/45/)45] lay-up. Usually, the front surface is coated with a metalised material to provide the required high radio frequency (RF) re¯ectivity and to minimise the temperature excursions and the resulting distortions of the panels. The rib sandwich panel is a 0.02 m thick aluminium alloy honeycomb core covered with face sheets in a [0/90/45/)45] lay-up with 0.0001 m GFRE layers. The overall mass of the re¯ector is 18.6 kg. All the thermal and dynamic loads are carried by both the surface sandwich shell and backing structure. In its deployed position, the re¯ector has a four-point interface with the spacecraft, consisting of two antenna deployment mechanisms and two release assemblies mounted on the spacecraft. The backing structure provides suitable attachment points for the hold-down and release mechanisms. In the stowed state, there are two more interface points of the re¯ector with the spacecraft.
J.-S. Liu, L. Hollaway / Computers and Structures 78 (2000) 637±647
These two points are located in the upper middle area of the re¯ector front surface. The structure is analysed to determine the distortions of the parabolic re¯ector surface, subject to solar heating in a synchronous orbit. The in¯uence of thermal strains on surface accuracy is complex and dependent to a great extent on the detailed design. As the temperatures change throughout the orbit, the structural elements expand or contract depending on their thermal expansion properties and the change in element temperature relative to its undeformed temperature (22°C for this analysis). The properties of surface shell elements and rib plate elements were taken from the physical structure of the sandwiches. The lay-ups of these elements are assumed to be constant across the whole re¯ector. A simpli®cation of the analysis was achieved by neglecting the anisotropy of the honeycomb cores. The following composite physical properties are used in the analysis: Face sheets: E1 289 GPa, E2 6:1 GPa, G12 G13 G23 4:21 GPa, ls 0:29, qs 1750 kg/m3 , a11 ÿ1:15 10ÿ6 =°C, a22 a33 36:2 10ÿ6 =°C. Honeycomb core: E1 200 MPa, E2 200 MPa, G12 G13 G23 140 MPa, lc 0:3, qc 32 kg/ m3 , a 22 10ÿ6 =°C. Three thermal patterns were investigated, these were uniform cooling, uniform heating and a transition temperature ®eld (Fig. 3) resulting from re¯ector shadow-
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Table 1 Some static analysis results of a space re¯ector structure
RMS (mm) rmax (MN/m2 ) dmax (mm)
Case 1 ()180°C)
Case 2 (+115°C)
Case 3 (thermal gradient)
Case 4 (launch case)
2.28 136.0 4.88
1.05 62.7 2.25
1.98 136.0 4.27
1.43 33.0 4.75
ing. Temperature ranges were estimated from previously published work [8,9], and brief transients were not considered in this study. Overall variations of ÿ180°C to 115°C were used to estimate structural distortions due to thermal loads. The re¯ector assembly has been analysed in the deployed and stowed con®gurations with thermal and launch loads. The launch loads are simulated by giving accelerations in the x=y=z directions [10]. Four different worst space-loading cases are selected to give maximum and minimum absolute temperatures, maximum thermal gradient, and launch accelerations. These cases are Case 1: Extremely low temperature (ÿ180°C). Case 2: Extremely high temperature (115°C). Case 3: A temperature gradient distribution from 0°C to ÿ180°C in the structure. Case 4: Stowed re¯ector in 8/11.2/30 Gs accelerations in the x=y=z directions in a launch case. The results of the analyses are presented as displacement and Von Mises stress for the selected loading conditions. The re¯ector surface RMS deviations, maximum stresses, and maximum displacements of the structure at the selected four worse loading cases are listed in Table 1, where the RMS values are the deviations of deformed re¯ector surface measured with normal deviation. Ordinarily, the stiness of a spacecraft structure is expressed in terms of its natural frequency. Normal mode analyses are performed entirely on the models. The lowest vibration frequencies of the structure in deployed and stowed con®gurations are 4.35 and 24.3 Hz, respectively.
3. Design assessment ± a systematic method
Fig. 3. Re¯ector isotherms for a sun angle.
A systematic method for evaluating engineering design is used. Using the concept of parameter pro®les [11,12], the procedure reviews, in a non-dimensional manner, the pro®le of the performance of structural system with respect to dierent loading cases. This method evaluates a structural design by considering many individual performance parameters at a spectrum of working/loading cases simultaneously in addition to considering mass and cost.
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3.1. Performance data matrix The basis of the analysis method is a matrix of data which describes system performance under dierent working/loading cases of the structure, (see the PDM given later). The matrix, called performance data matrix (PDM), is a schematic representation of a collection of data. The matrix lists every item of the loading cases considered and also every performance parameter relevant to the individual loading cases. The matrix is de®ned by the set of performance parameters Pi included in the analysis at loading cases Cj considered. Thus, the data point dij is the performance of the structure with respect to performance Pi at case Cj .
P1 P .. 2 . Pm
C1
C2
...
Cn
d11 ..d21 . dm1
d12 ..d22 . dm2
... ...
d1n ..d2n . dmn
...
Here, Pi the ith parameter describing the system performance, Cj the jth loading case of the structure, dij the data point.
C1
C2
...
Cn
D11 ..D21 . Dm1
D12 ..D22 . Dm2
... ... . . . ...
D1n ..D2n . Dmn
i 1; 2; . . . ; m;
n X 1 1 : Ui D ij j1
2
1
where dij is the actual value of the performance taken from the PDM shown earlier; lij and bij are the lower
3
When the ith parameter is very vulnerable, i.e., some data points Dij are close to 0, Ui and (PPI)i will be close to 0. Similarly, CPIj Vj m;
j 1; 2; . . . ; n;
4
where m 1 X 1 : Vj D ij i1
The calculation of the data point Dij for only one acceptable limit (e.g. lower limit) is, in principle, as follows: dij ÿ lij 10; bij ÿ lij
The information obtained from the parameter pro®le matrix makes it possible to evaluate the structural system. For each row and column, the mean and standard deviation (SD) for each parameter and loading case are calculated. The SD is a measure of the degree of the dispersion of the data around the mean. A well-designed system should have a low SD and a high mean which is close to 10. The existence of high SDs signi®es that the system will be likely to have signi®cant problematic areas. Therefore, a high SD for a row indicates a variable performance at dierent loading cases in the system for a particular parameter. A high SD for a column indicates that the system, at that loading case, will have signi®cant problematic performance. It is possible to analyse the system at a more advanced level. A parameter performance index (PPI) and a case performance index (CPI) can be de®ned:
where
The character of a structural system is assessed by a review of the pro®le of the performance parameters at dierent loading cases, and with respect to the proximity of actual performance to the acceptable limit and the best level value of the performance. An evaluation matrix called parameter pro®le matrix (PPM) is used in the assessment method. The data point Dij in the PPM (see the PPM given later) is a non-dimensional number in the range 0±10 which is determined by the closeness of the actual performance dij to the acceptable limit and the best level value of the performance.
Dij
3.3. Performance assessment by the analysis of the matrix
PPIi Ui n;
3.2. Parameter pro®le matrix
P1 P .. 2 . Pm
limit and best level value, respectively. Expression (1) is valid for lij < dij < bij ; for dij > bij , Dij 10, and for dij < lij , Dij 0. The data point Dij for the cases of acceptable upper limit and double acceptable limits can be calculated in a similar way.
5
When the structure system is vulnerable at a particular jth loading case, then Vj and (CPI)j will be close to 0. The mean values, SDs, PPIs and CPIs give an overall performance rating for each system performance and each loading case, respectively. The indices are calculated by summing up the inverse of the data points to avoid the eect of any particularly low scores being hidden by high scores in other respects which is possible when only the mean is calculated. For ease of analysis, the performance indices are brought into the range 0±10, no matter how many data points are used in each calculation. This enables dierent parameters and loading cases to be compared in order to gain an overall per-
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spective of the character of the system. The system may be reviewed by using this information as follows: 1. A comparison of PPIs will indicate whether the system performs better with respect to some performances than others. 2. A comparison of CPIs will show whether the system performs signi®cantly better at some loading cases than others. By comparing the indices CPIs, the weakest loading case can be identi®ed. Once this is given, it is possible to determine which performance parameter has the most in¯uence on that weak behaviour. If this is given for one item only, then it is sucient to search the particular column for the lowest performance data point. This will identify the parameter whose performance needs to be improved.
4. The optimisation problem The results in performance data matrix represent a variety of performance parameters of a complicated structural system at various loading conditions. Optimisation of such a structural system signi®es the improvement of these performance parameters under the loading cases considered and the side constraints of the design variables. The stated non-linear programming problem has a large number of objectives, variables and constraints. In real problems, these objectives are usually antagonist functions. In mathematic±analytic terms, optimising some of the functions may degrade others. For this reason and owing to the complexity of the structure and of the material, it may not be possible to ®nd a simple closed-form relationship that includes these functions and the design variables. In this work, a goal system is established by transforming every performance parameter into a set of goal functions in the range of 0±10 with respect to loading cases. For every performance parameter, the target has the same numerical value and is speci®ed as 10. The values of goal functions re¯ect closeness to the predetermined targets. Hence, the original optimisation problem is converted to the problem of minimising the deviations between all these goal functions and their pseudotargets. This is equivalent to minimising the distance between the performance Pi and its given best value Pi ; i 1; . . . ; m; i.e. min kPi ÿ Pi k;
i 1; . . . ; m:
6
From matrix pro®le analysis, it has been known that the PPI is a measure of the vulnerability of each performance parameter and the CPI is a measure of the vulnerability of each loading case. Therefore, the integration of PPI and CPI should be a measure of the
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vulnerability of the particular parameter/loading-case combination. High vulnerability results in low indices and high superiority results in high indices. A single multiplication therefore seems most appropriate: Sij PPIi CPIj ; i 1; 2; . . . ; m; j 1; 2; . . . ; n:
7
Each performance parameter and loading case may be weighted according to its importance, and the data points are calculated as Sij W pi PPIi W cj CPIj ; i 1; 2; . . . ; m; j 1; 2; . . . ; n;
8
where Wpi and Wcj are weighting factors in the range of 0±1 re¯ecting the preference for dierent parameters and dierent loading cases, respectively. This design synthesis concept provides a framework for formulating the quanti®able portion of a system design on which advanced optimisation techniques can be brought to bear. The optimisation objective function should be an overall measurement of design quality of a structural system. An overall performance index (OPI), which is a qualitative score, can be established for the system considering all the performances and all the loading cases. Mathematically, this is expressed as (for the weighted case) OPI
m X n 100 X Sij : m n i1 j1
9
The OPI is in the range of 0±100. This function comprises all the m parameters considered under dierent total n loading conditions. Optimisation techniques make it possible to force the performances to approach their best level values. The nearer the performances are to their acceptable limits, the more severe will be Ôthe punishmentsÕ. The optimisation problem stated above is complicated by the fact that the objective does not always have continuous ®rst and second derivatives for some engineering problems. Consequently, the problem has been tackled by means of numerical processes. However, the numerical calculation of the gradient and Hessian matrix may be costly if not impossible. Therefore, the optimisation problem is solved by maximising the objective function using the eective zero-order method which utilises conjugate search directions. An eective polynomial interpolation unidimensional search technique is also used in the algorithm. The procedure consists of the steps of analysis of a prior established starting design, sensitivity analysis, and the development of preferential values of the design variables. The analysis step may be mathematically ÔexactÕ, but non-linearities in the systemÕs response with respect to changes in the design variables make the development step approximate.
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J.-S. Liu, L. Hollaway / Computers and Structures 78 (2000) 637±647
These two steps are repeated iteratively to achieve the ®nal design. 5. The optimisation of a composite re¯ector structure Minimising structural distortions caused by temperature changes encountered during orbital ¯ight are the most critical aspect in designing structural and material systems for space antenna structures. Surface pro®le accuracy is of prime importance for performance of antenna re¯ectors for space communication satellites. Deviations of the re¯ector shape from the ideal shape could cause a change in position to the focal point, reduction in the peak antenna gain, and an increase in side-lobe level. The surface RMS error !1=2 n X 2 RMS Di =n 10 i1
provides an independent performance index against which distorted re¯ector antennas can be compared. The deviations Di were calculated in directions normal to the de®ned pro®le. Thus, the RMS deviations from the design pro®le of the composite re¯ector form one of the objective functions to be minimised in the design optimisation. Lightweight space structures are desirable from the point of view of reducing launch and orbit transfer costs and reducing the torque and power required to slew and point on-orbit. It is of practical interest to design a re¯ector structure of reduced surface errors but of less structural mass by ®nding an optimal stiness distribution. Therefore, structural mass is taken as one of the objectives which is to be minimised. An antenna structure must be sti enough to avoid undesirable interaction with close-loop control system and to withstand the various disturbing forces without suering unacceptable distortions. Structural fundamental frequencies at deployed and stowed conditions are also included for optimisation. Therefore, the design optimisation problem was required to minimise the structural mass, re¯ector surface RMS error, maximum displacement and maximum stress of the structure, and to increase structural frequencies at stowed and deployed shapes to their given
best level (the target) values. All these topics are of great importance to the structural designer. The re¯ector surface panel and ribs were constructed with carbon ®bre sheets and aluminium honeycomb sandwiches. The local use of stiening ribs improves the quality of the structure by avoiding the use of heavier face sheets over the entire structure. This structure has 133 nodes, 198 three-side and four-side, irregular composite laminate element, and 774 degrees of freedom. The optimisation studies were used to determine possible new ®bre/matrix combinations as ply candidates and as a new rib stiening system. The design variables used for the optimisation represent the shapes of the ribs, individual ply thicknesses of the honeycomb face sheets, ply orientations (®bre angles), and honeycomb-core thicknesses, in both panels and ribs. These design variables provided the designer with more control to ®ne tune the structure and these sizing variables and geometry variables will be considered simultaneously in the optimisation. 6. Optimisation results In the optimisation, four dierent loading cases are considered simultaneously. These are, extremely cold temperature, extremely hot temperature, a temperature gradient distribution from 0°C to )180°C in the structure, and the stowed re¯ector in 8=11:2=30 Gs accelerations in the x=y=z directions in the launch case. The original design had a fundamental frequency of 4.35 Hz in the deployed shape and 24.3 Hz in stowed shape with an overall mass of 18.6 kg. Other performances of the original design can be found in Table 2. For this antenna structure, the structural frequency requires to be increased to the values of over 8 Hz for deployed shape and over 28 Hz for stowed shape in the launch case, and other performances (i.e. re¯ector surface RMS error, structural mass, maximum displacement and maximum stress) at all loading cases need to be minimised. This optimisation problem can be solved quickly because evaluation for the RMS distortion is trivial compared to evaluating antenna radiation performance. After the optimisation iteration, a re¯ector structure which is much stronger, stier, lighter and more accurate
Table 2 Performances of original structure at four dierent loading cases RMS D (mm) Maximum stress rmax (MN/m2 ) Structural mass (kg) Structural frequency (Hz) Maximum displacement dmax (mm)
Case 1 ()180°C)
Case 2 (+115°C)
Case 3 (thermal gradient)
Case 4 (launch case)
2.28 136.0 18.6 4.35 4.88
1.05 62.7 18.6 4.35 2.25
1.98 136.0 18.6 4.35 4.27
1.43 33.0 18.6 24.3 4.75
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than the original design was obtained. By performing the optimisation, the OPI is greatly enhanced which is increased from the original score of 9.77 to the optimised score of 83.27 (Fig. 4). Figs. 5±8 illustrate that all the structural performances at all the loading cases have been successfully improved. The convergence history of the optimisation shows that the current procedure converges to a much better design than the original one based on the optimisation criteria. The structural performances at all the loading cases before and after optimisation are shown in Tables 1 and 3, respectively. The signi®cant improvement in the design, following the optimisation, can be observed as follow:
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Fig. 7. The convergence history of maximum stresses at different loading cases.
Fig. 8. The convergence history of structural mass and fundamental frequencies. Fig. 4. The convergence history of overall objectives.
Fig. 5. The convergence history of re¯ector surface accuracies (RMSs) at dierent loading cases.
Fig. 6. The convergence history of maximum displacements at dierent loading cases.
1. Under loading cases 1 ()180°C), 2 (115°C) and 3 (thermal gradient): A. The RMS errors are reduced from the original 2.28, 1.05 and 1.98 mm to the optimised 0.042, 0.019 and 0.285 mm, respectively. B. The maximum stresses are reduced from the original 136.0, 62.7 and 136.0 MPa to the optimised 130.0, 59.7 and 130.0 MPa, respectively. C. The maximum displacements are reduced from the original 4.88, 2.25 and 4.27 mm to the optimised 0.457, 0.211 and 1.13 mm, respectively. 2. Under loading case 4 (launch case): A. The RMS error is reduced from the original 1.43 mm to the optimised 0.857 mm. B. The maximum stress is reduced from the original 33.0 MPa to the optimised 19.8 MPa. C. The maximum displacement is reduced from the original 4.75 mm to the optimised 2.99 mm. 3. In addition to the above A. The ®nal design has a total structural mass of 12.7 kg, a reduction of 32% from the original mass of 18.6 kg. B. The structural fundamental frequency is increased from the original 4.35/24.3 Hz to the optimised 9.07/29.1 Hz for deployed/stowed shape. It can be seen that it has been possible to obtain a stress-safe design with two-thirds of the original mass and a substantially improved RMS. Figs. 9±12 provide a more complete comparison between the characteristics
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Table 3 Performances of optimised structure at four dierent loading cases RMS D (mm) Maximum stress rmax (MN/m2 ) Structural mass (kg) Structural frequency (Hz) Maximum displacement dmax (mm)
Case 1 ()180°C)
Case 2 (+115°C)
Case 3 (thermal gradient)
Case 4 (launch case)
0.042 130.0 12.7 9.07 0.457
0.019 59.7 12.7 9.07 0.211
0.285 130.0 12.7 9.07 1.13
0.857 19.8 12.7 29.1 2.99
Fig. 9. The comparison of re¯ector surface accuracies (RMSs) between the original and optimised structures.
Fig. 10. The comparison of the maximum displacements between the original and optimised structures.
Fig. 11. The comparison of the maximum stresses between the original and optimised structures.
Fig. 12. The comparison of structural masses and frequencies between the original and optimised structures.
J.-S. Liu, L. Hollaway / Computers and Structures 78 (2000) 637±647
of the re¯ector before and after optimisation at all the loading cases considered. The best level values and the given acceptable limits on each performance parameter at each loading case are also shown in these ®gures. Figs. 13±16 illustrate the convergence histories for the mean values, SDs, PPIs and CPIs for all the performance parameters and all the loading cases considered in the optimisation. The values of means, SDs, PPIs and CPIs for the original and optimised system are listed in Tables 4 and 5. It can be seen that the structural performance parameters, at all the loading cases in the optimised structure, have much higher mean values and PPIs and much lower standard deviations than those in the original design. Also, at each loading case, the antenna will behave in such a way that all the performances have increased reliability and decreased the possibility of performing unsatisfactorily.
For the optimised antenna structure, Tables 4 and 5 show that the worst-case temperature pro®le corresponds to a condition of temperature gradients across the re¯ector surface and not to a situation of absolute temperature excursion from the ambient fabrication temperature. The determination of ®bre orientation angles is an important subject in the optimum design of composite materials. The fact that the design variables for the ®bre angles in both the composite surface panels and composite stiening ribs did not alter in the optimisation procedure, veri®ed that the 45° ®bre angles for these components are the best choice. The iteration history of the design variables is shown in Figs. 17 and 18. The optimised re¯ector geometry is shown in Fig. 19. The change in geometric shapes of the stiening ribs in the original design and the optimised structure can be
Fig. 15. The convergence history of the PPIs.
Fig. 13. The convergence history of the means of performances in the PPM.
Fig. 14. The convergence history of the standard deviations of performances in the PPM.
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Fig. 16. The convergence history of the working/loading cases performance indices.
Table 4 System parameter pro®le analysis of the original antenna structure Mean RMS error Maximum stress Structural mass Structural frequency Maximum displacement
Standard deviation
PPI
Original
Optimised
Original
Optimised
Original
Optimised
3.96 7.11 3.40 1.99 3.42
9.78 7.51 9.30 10.00 9.48
1.08 2.57 0.00 0.00 1.00
0.31 2.49 0.00 0.00 0.48
3.62 6.19 3.40 1.10 3.06
9.77 6.68 9.30 10.00 9.46
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Table 5 Loading case pro®le analysis of the original antenna structure Mean Case Case Case Case
1 2 3 4
()180°C) (+115°C) (thermal gradient) (launch case)
Standard deviation
CPI
Original
Optimised
Original
Optimised
Original
Optimised
2.64 4.57 3.08 5.62
8.76 9.78 8.45 9.86
1.26 2.75 1.21 2.30
1.89 0.27 1.77 0.28
1.93 2.52 2.20 4.93
8.17 9.78 7.93 9.85
Fig. 17. The convergence history of design variables (thicknesses of layers and honeycomb cores).
Fig. 18. The convergence history of structural geometric design variables.
clearly observed by comparing the shapes shown in Figs. 2 and 19. The modal analysis for the optimised antenna structure shows that the lowest structural natural frequencies for both the deployed and stowed cases are 9.07 and 29.1 Hz, respectively. These frequencies are increased from their original values of 4.35 and 24.3 Hz, and satis®ed the given target values which should be over 8 and 28 Hz, respectively. The optimised structure produces signi®cantly higher modal frequencies than the original structure, due to the relatively sti nature of the optimised structure.
Fig. 19. The optimised re¯ector geometry.
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7. Conclusions
References
The optimisation technique is shown to have an intrinsic applicability to multi-objective optimisation. Although like any non-linear programming problem, the method cannot guarantee the ®nding of a global optimum because of the Pareto feature of some problems, it makes it feasible to optimise complicated structures and systems considering many loading conditions. One of its primary bene®ts in the application is the elimination of the potentially expensive separate optimisations for each objective. For solutions of structural problems, the optimisation method has been coupled with the A B A Q U S ®nite element code which is used to determine system response variables as functions of design variables. The technique has been demonstrated on various structural systems built of various types of ®nite elements and dierent materials, contributing to a wide range of mechanical properties and cost to the design objectives. The method may not necessarily make every performance at every loading case reach its speci®ed aspiration value, especially in the case of con¯ict objectives and many loading cases simultaneously considered in an optimisation. Using currently available composite materials, with their actual physico-mechanical properties, it is possible to optimise the reinforcement of the bearing layers of the structures. These materials oer the structural designers a wide range of new degrees of freedom to think in terms of simultaneous optimisation of structural con®guration and structural material. In this optimisation model developed, design variables can incorporate the tailorable variables of composite materials such as ply angles, laminate thicknesses, ®bre volume ratios, and shape parameters. The application of the method on this composite re¯ector structure appeared very encouraging.
[1] Charnes A, Cooper W. Management models and the industrial applications of linear programming. New York: Wiley; 1961. [2] Ijiri Y. Management goals and accounting for control. Chicago, IL: Rand McNally; 1965. [3] Lee SM. Goal programming for decision analysis. Philadelphia, PA: Auerback; 1972. [4] Ignizio JP. Generalized goal programming: an overview. Comput Oper Res 1983;10:277±89. [5] Rao SS, Sundararaju K, Prakash BG, Balakrishna C. Fuzzy goal programming approach for structural optimization. AIAA J 1992;30:1425±32. [6] El-Sayed MEM, Jang TS. Structural optimization using unconstrained nonlinear goal programming algorithm. Comput Struct 1994;52:723±7. [7] Liu J-S, Hollaway L. Multidisciplinary optimisation of antenna structure-electromagnetic systems. Proceedings of the Second World Congress of Structural and Multidisciplinary Optimisation. Zakopane, Poland. 26±30 May 1997. p. 473±8. [8] Farmer JT, Wahls DM, Wright RL, Tahernia AL. Thermal distortion analysis of a antenna-support truss in geocynchronous orbit. J Spacecraft Rockets 1992;29: 386±93. [9] Florio FA, Joslo AT. Thermo/structural analysis of a large ¯exible paraboloid antenna. J Spacecraft 1968;5: 1417±24. [10] PrudÕhon C, Gautier C, Flechais A. Development of the large N-star re¯ectors. Int Conf Antennas. JINA-94. Nice, France. November 1994. [11] Liu J-S, Thompson G. The multi-factor design evaluation of antenna structures by parameter pro®le analysis. J Engng Manufac. Proc Instn Mech Engnrs 1996;210(B5): 449±56. [12] Thompson G, Goeminne J. The design review of engineering systems including maintainability and reliability using parameter pro®les. Int Conf Engng Design. The Hague, Netherlands. August 1993.
www.elsevier.com/locate/compstruc
Design optimisation of composite panel structures with stiening ribs under multiple loading cases Jing-Sheng Liu a, Len Hollaway b,* b
a Department of Engineering, Cambridge University, Cambridge CB2 1PZ, UK Department of Civil Engineering, Composite Structures Research Unit, University of Surrey, Guildford, Surrey GU2 5XH, UK
Received 31 July 1998; accepted 8 December 1999
Abstract An optimisation procedure for composite panel structures with stiening ribs is presented in this paper. The procedure employs standard FE structural analysis, structural system pro®le analysis, and multi-factor optimisation techniques to predict an optimum structural design. A composite antenna re¯ector structure in space environment is optimised. Laminate, sandwich con®guration and rib shape in the structure are optimised in this application. The optimisation improves antenna performance under multiple loading cases and minimises structural mass simultaneously. Ó 2000 Elsevier Science Ltd. All rights reserved. Keywords: Structural con®gurations; Optimisation; Re¯ector; Advanced polymer composites; Panel; Stiening rib
1. Introduction Sandwich systems, which are manufactured by cocuring or secondary bonding, are the most used structural solution for high accuracy lightweight panel structures. A honeycomb sandwich panel is a layered construction typically formed by bonding two thin face sheets to a thick core. This sandwich construction provides a very lightweight structural con®guration for many load conditions. The face sheets take the membrane and bending loads while the core resists the shear loads. On the other hand, carbon-®bre-reinforced composite materials provide an extremely small coecient of thermal expansion and one of the lightest weight material systems due to their excellent speci®c stiness. Therefore, a sandwich panel employing ®bre-reinforced composite material faces brings together the excellence of both the structural con®guration and materials system.
*
Corresponding author. Tel.: +44-1483-879280; fax: +441483-450984. E-mail address: [email protected] (L. Hollaway).
In this work, a composite space antenna structure with parabolic re¯ector has been optimised. The panels of the re¯ector are stiened with ribs. Both the panels and the ribs are constructed from sandwich con®guration utilising aluminium honeycomb cores; the face sheets of both the panels and the ribs are fabricated from laminates. The individual plies are ``composites'' of carbon ®bre ®laments encapsulated unidirectionally in a matrix material giving them orthogonal material properties. An optimum design for the structure and con®guration was sought. In the optimisation, the re¯ector surface accuracy (typically, the RMS value of the displacements from the ideal re¯ector surface), structural strength, stiness, natural frequencies, and structural mass were optimised simultaneously, while considering the thermal deformations due to the temperature changes in space environment and a launch case. The possibility of achieving a design that eciently optimises multiple performances, coupled with the dif®culty in selecting the values of a large set of design variables, makes structural optimisation an important tool for the design of laminated composite structures. The proposed method takes advantage of the concept of the goal programming (GP) method which minimises
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the deviations between the achievement of the goals and their aspiration levels. Linear goal programming (LGP) was initiated by Charnes and Cooper [1] and the LGP algorithms were further developed by Ijiri [2], Lee [3] and many others. The extension of GP to non-linear optimisation problems has been considered by Ignizio [4], Rao et al. [5], El-Sayed and Jang [6]. However, the method proposed here diers from general GP methods mainly in that · It treats complex engineering design problems, which may have multiple objectives and multiple loading cases, in a systematic way by employing a parameter pro®le analysis. · It introduces an assessment system which brings scores and indexes into the range of 0±10 for all the non-commensurable performances (i.e. have dierent units) and loading cases. · It makes no dierence for performance constraints and objective functions in the formulation. A multi-objective optimisation computer program, M O S T (multifactor optimisation of structures technique), has been developed to accommodate and implement the optimisation method. The M O S T system is a synergistic combination of ®nite element static/dynamic analysis, system assessment, and optimisation techniques, together with some pre-phase programs of design optimisation, input/output, model update, and interfacing programs. It utilises A B A Q U S and the theory of laminated plate to build a comprehensive analysis/design capability for structural composites. The optimisation system control program is written in U N I X shell scripts. It can control optimisation ¯ow and execute application programs in U N I X environment so that all the functions can be integrated in a system. It can also simulate the electromagnetic analysis of distorted antenna systems (a speci®cally useful feature in optimising antenna systems) [7]. The program can account for various environmental conditions and loading cases. These features make M O S T a powerful, cost eective and reliable tool to optimise complex structural systems.
2. The analyses of a composite re¯ector panel structure The antenna is an oset system. The re¯ector has a projected aperture size of 3:6 2:6 m2 and a highly accurate surface. The nominal surface is a section of a paraboloid having a focal length of 1.8 m and oset by 0.4 m from the paraboloid axis (Fig. 1). The re¯ector dish is fabricated from a graphite composite honeycomb sandwich panel structure stiened by a ribbed backing structure which is formed from a lattice of beams, also of honeycomb sandwich construction. Fig. 2 shows the backside of the re¯ector. All these ribs are assembled and are oset bonded towards the rear of the re¯ector.
Fig. 1. An oset re¯ector.
Fig. 2. The original re¯ector geometry.
The ribs can vary in height over the structure. In the original structure, all the ribs have a height of 0.1 m. The dish sandwich panel is a 0.01 m thick aluminium alloy honeycomb core covered with graphite ®bre reinforced epoxy (GFRE) face sheets. The sheets on both sides of the core are constructed with 0.0001 m thick GFRE layers in a [0/90/45/)45] lay-up. Usually, the front surface is coated with a metalised material to provide the required high radio frequency (RF) re¯ectivity and to minimise the temperature excursions and the resulting distortions of the panels. The rib sandwich panel is a 0.02 m thick aluminium alloy honeycomb core covered with face sheets in a [0/90/45/)45] lay-up with 0.0001 m GFRE layers. The overall mass of the re¯ector is 18.6 kg. All the thermal and dynamic loads are carried by both the surface sandwich shell and backing structure. In its deployed position, the re¯ector has a four-point interface with the spacecraft, consisting of two antenna deployment mechanisms and two release assemblies mounted on the spacecraft. The backing structure provides suitable attachment points for the hold-down and release mechanisms. In the stowed state, there are two more interface points of the re¯ector with the spacecraft.
J.-S. Liu, L. Hollaway / Computers and Structures 78 (2000) 637±647
These two points are located in the upper middle area of the re¯ector front surface. The structure is analysed to determine the distortions of the parabolic re¯ector surface, subject to solar heating in a synchronous orbit. The in¯uence of thermal strains on surface accuracy is complex and dependent to a great extent on the detailed design. As the temperatures change throughout the orbit, the structural elements expand or contract depending on their thermal expansion properties and the change in element temperature relative to its undeformed temperature (22°C for this analysis). The properties of surface shell elements and rib plate elements were taken from the physical structure of the sandwiches. The lay-ups of these elements are assumed to be constant across the whole re¯ector. A simpli®cation of the analysis was achieved by neglecting the anisotropy of the honeycomb cores. The following composite physical properties are used in the analysis: Face sheets: E1 289 GPa, E2 6:1 GPa, G12 G13 G23 4:21 GPa, ls 0:29, qs 1750 kg/m3 , a11 ÿ1:15 10ÿ6 =°C, a22 a33 36:2 10ÿ6 =°C. Honeycomb core: E1 200 MPa, E2 200 MPa, G12 G13 G23 140 MPa, lc 0:3, qc 32 kg/ m3 , a 22 10ÿ6 =°C. Three thermal patterns were investigated, these were uniform cooling, uniform heating and a transition temperature ®eld (Fig. 3) resulting from re¯ector shadow-
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Table 1 Some static analysis results of a space re¯ector structure
RMS (mm) rmax (MN/m2 ) dmax (mm)
Case 1 ()180°C)
Case 2 (+115°C)
Case 3 (thermal gradient)
Case 4 (launch case)
2.28 136.0 4.88
1.05 62.7 2.25
1.98 136.0 4.27
1.43 33.0 4.75
ing. Temperature ranges were estimated from previously published work [8,9], and brief transients were not considered in this study. Overall variations of ÿ180°C to 115°C were used to estimate structural distortions due to thermal loads. The re¯ector assembly has been analysed in the deployed and stowed con®gurations with thermal and launch loads. The launch loads are simulated by giving accelerations in the x=y=z directions [10]. Four different worst space-loading cases are selected to give maximum and minimum absolute temperatures, maximum thermal gradient, and launch accelerations. These cases are Case 1: Extremely low temperature (ÿ180°C). Case 2: Extremely high temperature (115°C). Case 3: A temperature gradient distribution from 0°C to ÿ180°C in the structure. Case 4: Stowed re¯ector in 8/11.2/30 Gs accelerations in the x=y=z directions in a launch case. The results of the analyses are presented as displacement and Von Mises stress for the selected loading conditions. The re¯ector surface RMS deviations, maximum stresses, and maximum displacements of the structure at the selected four worse loading cases are listed in Table 1, where the RMS values are the deviations of deformed re¯ector surface measured with normal deviation. Ordinarily, the stiness of a spacecraft structure is expressed in terms of its natural frequency. Normal mode analyses are performed entirely on the models. The lowest vibration frequencies of the structure in deployed and stowed con®gurations are 4.35 and 24.3 Hz, respectively.
3. Design assessment ± a systematic method
Fig. 3. Re¯ector isotherms for a sun angle.
A systematic method for evaluating engineering design is used. Using the concept of parameter pro®les [11,12], the procedure reviews, in a non-dimensional manner, the pro®le of the performance of structural system with respect to dierent loading cases. This method evaluates a structural design by considering many individual performance parameters at a spectrum of working/loading cases simultaneously in addition to considering mass and cost.
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3.1. Performance data matrix The basis of the analysis method is a matrix of data which describes system performance under dierent working/loading cases of the structure, (see the PDM given later). The matrix, called performance data matrix (PDM), is a schematic representation of a collection of data. The matrix lists every item of the loading cases considered and also every performance parameter relevant to the individual loading cases. The matrix is de®ned by the set of performance parameters Pi included in the analysis at loading cases Cj considered. Thus, the data point dij is the performance of the structure with respect to performance Pi at case Cj .
P1 P .. 2 . Pm
C1
C2
...
Cn
d11 ..d21 . dm1
d12 ..d22 . dm2
... ...
d1n ..d2n . dmn
...
Here, Pi the ith parameter describing the system performance, Cj the jth loading case of the structure, dij the data point.
C1
C2
...
Cn
D11 ..D21 . Dm1
D12 ..D22 . Dm2
... ... . . . ...
D1n ..D2n . Dmn
i 1; 2; . . . ; m;
n X 1 1 : Ui D ij j1
2
1
where dij is the actual value of the performance taken from the PDM shown earlier; lij and bij are the lower
3
When the ith parameter is very vulnerable, i.e., some data points Dij are close to 0, Ui and (PPI)i will be close to 0. Similarly, CPIj Vj m;
j 1; 2; . . . ; n;
4
where m 1 X 1 : Vj D ij i1
The calculation of the data point Dij for only one acceptable limit (e.g. lower limit) is, in principle, as follows: dij ÿ lij 10; bij ÿ lij
The information obtained from the parameter pro®le matrix makes it possible to evaluate the structural system. For each row and column, the mean and standard deviation (SD) for each parameter and loading case are calculated. The SD is a measure of the degree of the dispersion of the data around the mean. A well-designed system should have a low SD and a high mean which is close to 10. The existence of high SDs signi®es that the system will be likely to have signi®cant problematic areas. Therefore, a high SD for a row indicates a variable performance at dierent loading cases in the system for a particular parameter. A high SD for a column indicates that the system, at that loading case, will have signi®cant problematic performance. It is possible to analyse the system at a more advanced level. A parameter performance index (PPI) and a case performance index (CPI) can be de®ned:
where
The character of a structural system is assessed by a review of the pro®le of the performance parameters at dierent loading cases, and with respect to the proximity of actual performance to the acceptable limit and the best level value of the performance. An evaluation matrix called parameter pro®le matrix (PPM) is used in the assessment method. The data point Dij in the PPM (see the PPM given later) is a non-dimensional number in the range 0±10 which is determined by the closeness of the actual performance dij to the acceptable limit and the best level value of the performance.
Dij
3.3. Performance assessment by the analysis of the matrix
PPIi Ui n;
3.2. Parameter pro®le matrix
P1 P .. 2 . Pm
limit and best level value, respectively. Expression (1) is valid for lij < dij < bij ; for dij > bij , Dij 10, and for dij < lij , Dij 0. The data point Dij for the cases of acceptable upper limit and double acceptable limits can be calculated in a similar way.
5
When the structure system is vulnerable at a particular jth loading case, then Vj and (CPI)j will be close to 0. The mean values, SDs, PPIs and CPIs give an overall performance rating for each system performance and each loading case, respectively. The indices are calculated by summing up the inverse of the data points to avoid the eect of any particularly low scores being hidden by high scores in other respects which is possible when only the mean is calculated. For ease of analysis, the performance indices are brought into the range 0±10, no matter how many data points are used in each calculation. This enables dierent parameters and loading cases to be compared in order to gain an overall per-
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spective of the character of the system. The system may be reviewed by using this information as follows: 1. A comparison of PPIs will indicate whether the system performs better with respect to some performances than others. 2. A comparison of CPIs will show whether the system performs signi®cantly better at some loading cases than others. By comparing the indices CPIs, the weakest loading case can be identi®ed. Once this is given, it is possible to determine which performance parameter has the most in¯uence on that weak behaviour. If this is given for one item only, then it is sucient to search the particular column for the lowest performance data point. This will identify the parameter whose performance needs to be improved.
4. The optimisation problem The results in performance data matrix represent a variety of performance parameters of a complicated structural system at various loading conditions. Optimisation of such a structural system signi®es the improvement of these performance parameters under the loading cases considered and the side constraints of the design variables. The stated non-linear programming problem has a large number of objectives, variables and constraints. In real problems, these objectives are usually antagonist functions. In mathematic±analytic terms, optimising some of the functions may degrade others. For this reason and owing to the complexity of the structure and of the material, it may not be possible to ®nd a simple closed-form relationship that includes these functions and the design variables. In this work, a goal system is established by transforming every performance parameter into a set of goal functions in the range of 0±10 with respect to loading cases. For every performance parameter, the target has the same numerical value and is speci®ed as 10. The values of goal functions re¯ect closeness to the predetermined targets. Hence, the original optimisation problem is converted to the problem of minimising the deviations between all these goal functions and their pseudotargets. This is equivalent to minimising the distance between the performance Pi and its given best value Pi ; i 1; . . . ; m; i.e. min kPi ÿ Pi k;
i 1; . . . ; m:
6
From matrix pro®le analysis, it has been known that the PPI is a measure of the vulnerability of each performance parameter and the CPI is a measure of the vulnerability of each loading case. Therefore, the integration of PPI and CPI should be a measure of the
641
vulnerability of the particular parameter/loading-case combination. High vulnerability results in low indices and high superiority results in high indices. A single multiplication therefore seems most appropriate: Sij PPIi CPIj ; i 1; 2; . . . ; m; j 1; 2; . . . ; n:
7
Each performance parameter and loading case may be weighted according to its importance, and the data points are calculated as Sij W pi PPIi W cj CPIj ; i 1; 2; . . . ; m; j 1; 2; . . . ; n;
8
where Wpi and Wcj are weighting factors in the range of 0±1 re¯ecting the preference for dierent parameters and dierent loading cases, respectively. This design synthesis concept provides a framework for formulating the quanti®able portion of a system design on which advanced optimisation techniques can be brought to bear. The optimisation objective function should be an overall measurement of design quality of a structural system. An overall performance index (OPI), which is a qualitative score, can be established for the system considering all the performances and all the loading cases. Mathematically, this is expressed as (for the weighted case) OPI
m X n 100 X Sij : m n i1 j1
9
The OPI is in the range of 0±100. This function comprises all the m parameters considered under dierent total n loading conditions. Optimisation techniques make it possible to force the performances to approach their best level values. The nearer the performances are to their acceptable limits, the more severe will be Ôthe punishmentsÕ. The optimisation problem stated above is complicated by the fact that the objective does not always have continuous ®rst and second derivatives for some engineering problems. Consequently, the problem has been tackled by means of numerical processes. However, the numerical calculation of the gradient and Hessian matrix may be costly if not impossible. Therefore, the optimisation problem is solved by maximising the objective function using the eective zero-order method which utilises conjugate search directions. An eective polynomial interpolation unidimensional search technique is also used in the algorithm. The procedure consists of the steps of analysis of a prior established starting design, sensitivity analysis, and the development of preferential values of the design variables. The analysis step may be mathematically ÔexactÕ, but non-linearities in the systemÕs response with respect to changes in the design variables make the development step approximate.
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These two steps are repeated iteratively to achieve the ®nal design. 5. The optimisation of a composite re¯ector structure Minimising structural distortions caused by temperature changes encountered during orbital ¯ight are the most critical aspect in designing structural and material systems for space antenna structures. Surface pro®le accuracy is of prime importance for performance of antenna re¯ectors for space communication satellites. Deviations of the re¯ector shape from the ideal shape could cause a change in position to the focal point, reduction in the peak antenna gain, and an increase in side-lobe level. The surface RMS error !1=2 n X 2 RMS Di =n 10 i1
provides an independent performance index against which distorted re¯ector antennas can be compared. The deviations Di were calculated in directions normal to the de®ned pro®le. Thus, the RMS deviations from the design pro®le of the composite re¯ector form one of the objective functions to be minimised in the design optimisation. Lightweight space structures are desirable from the point of view of reducing launch and orbit transfer costs and reducing the torque and power required to slew and point on-orbit. It is of practical interest to design a re¯ector structure of reduced surface errors but of less structural mass by ®nding an optimal stiness distribution. Therefore, structural mass is taken as one of the objectives which is to be minimised. An antenna structure must be sti enough to avoid undesirable interaction with close-loop control system and to withstand the various disturbing forces without suering unacceptable distortions. Structural fundamental frequencies at deployed and stowed conditions are also included for optimisation. Therefore, the design optimisation problem was required to minimise the structural mass, re¯ector surface RMS error, maximum displacement and maximum stress of the structure, and to increase structural frequencies at stowed and deployed shapes to their given
best level (the target) values. All these topics are of great importance to the structural designer. The re¯ector surface panel and ribs were constructed with carbon ®bre sheets and aluminium honeycomb sandwiches. The local use of stiening ribs improves the quality of the structure by avoiding the use of heavier face sheets over the entire structure. This structure has 133 nodes, 198 three-side and four-side, irregular composite laminate element, and 774 degrees of freedom. The optimisation studies were used to determine possible new ®bre/matrix combinations as ply candidates and as a new rib stiening system. The design variables used for the optimisation represent the shapes of the ribs, individual ply thicknesses of the honeycomb face sheets, ply orientations (®bre angles), and honeycomb-core thicknesses, in both panels and ribs. These design variables provided the designer with more control to ®ne tune the structure and these sizing variables and geometry variables will be considered simultaneously in the optimisation. 6. Optimisation results In the optimisation, four dierent loading cases are considered simultaneously. These are, extremely cold temperature, extremely hot temperature, a temperature gradient distribution from 0°C to )180°C in the structure, and the stowed re¯ector in 8=11:2=30 Gs accelerations in the x=y=z directions in the launch case. The original design had a fundamental frequency of 4.35 Hz in the deployed shape and 24.3 Hz in stowed shape with an overall mass of 18.6 kg. Other performances of the original design can be found in Table 2. For this antenna structure, the structural frequency requires to be increased to the values of over 8 Hz for deployed shape and over 28 Hz for stowed shape in the launch case, and other performances (i.e. re¯ector surface RMS error, structural mass, maximum displacement and maximum stress) at all loading cases need to be minimised. This optimisation problem can be solved quickly because evaluation for the RMS distortion is trivial compared to evaluating antenna radiation performance. After the optimisation iteration, a re¯ector structure which is much stronger, stier, lighter and more accurate
Table 2 Performances of original structure at four dierent loading cases RMS D (mm) Maximum stress rmax (MN/m2 ) Structural mass (kg) Structural frequency (Hz) Maximum displacement dmax (mm)
Case 1 ()180°C)
Case 2 (+115°C)
Case 3 (thermal gradient)
Case 4 (launch case)
2.28 136.0 18.6 4.35 4.88
1.05 62.7 18.6 4.35 2.25
1.98 136.0 18.6 4.35 4.27
1.43 33.0 18.6 24.3 4.75
J.-S. Liu, L. Hollaway / Computers and Structures 78 (2000) 637±647
than the original design was obtained. By performing the optimisation, the OPI is greatly enhanced which is increased from the original score of 9.77 to the optimised score of 83.27 (Fig. 4). Figs. 5±8 illustrate that all the structural performances at all the loading cases have been successfully improved. The convergence history of the optimisation shows that the current procedure converges to a much better design than the original one based on the optimisation criteria. The structural performances at all the loading cases before and after optimisation are shown in Tables 1 and 3, respectively. The signi®cant improvement in the design, following the optimisation, can be observed as follow:
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Fig. 7. The convergence history of maximum stresses at different loading cases.
Fig. 8. The convergence history of structural mass and fundamental frequencies. Fig. 4. The convergence history of overall objectives.
Fig. 5. The convergence history of re¯ector surface accuracies (RMSs) at dierent loading cases.
Fig. 6. The convergence history of maximum displacements at dierent loading cases.
1. Under loading cases 1 ()180°C), 2 (115°C) and 3 (thermal gradient): A. The RMS errors are reduced from the original 2.28, 1.05 and 1.98 mm to the optimised 0.042, 0.019 and 0.285 mm, respectively. B. The maximum stresses are reduced from the original 136.0, 62.7 and 136.0 MPa to the optimised 130.0, 59.7 and 130.0 MPa, respectively. C. The maximum displacements are reduced from the original 4.88, 2.25 and 4.27 mm to the optimised 0.457, 0.211 and 1.13 mm, respectively. 2. Under loading case 4 (launch case): A. The RMS error is reduced from the original 1.43 mm to the optimised 0.857 mm. B. The maximum stress is reduced from the original 33.0 MPa to the optimised 19.8 MPa. C. The maximum displacement is reduced from the original 4.75 mm to the optimised 2.99 mm. 3. In addition to the above A. The ®nal design has a total structural mass of 12.7 kg, a reduction of 32% from the original mass of 18.6 kg. B. The structural fundamental frequency is increased from the original 4.35/24.3 Hz to the optimised 9.07/29.1 Hz for deployed/stowed shape. It can be seen that it has been possible to obtain a stress-safe design with two-thirds of the original mass and a substantially improved RMS. Figs. 9±12 provide a more complete comparison between the characteristics
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J.-S. Liu, L. Hollaway / Computers and Structures 78 (2000) 637±647
Table 3 Performances of optimised structure at four dierent loading cases RMS D (mm) Maximum stress rmax (MN/m2 ) Structural mass (kg) Structural frequency (Hz) Maximum displacement dmax (mm)
Case 1 ()180°C)
Case 2 (+115°C)
Case 3 (thermal gradient)
Case 4 (launch case)
0.042 130.0 12.7 9.07 0.457
0.019 59.7 12.7 9.07 0.211
0.285 130.0 12.7 9.07 1.13
0.857 19.8 12.7 29.1 2.99
Fig. 9. The comparison of re¯ector surface accuracies (RMSs) between the original and optimised structures.
Fig. 10. The comparison of the maximum displacements between the original and optimised structures.
Fig. 11. The comparison of the maximum stresses between the original and optimised structures.
Fig. 12. The comparison of structural masses and frequencies between the original and optimised structures.
J.-S. Liu, L. Hollaway / Computers and Structures 78 (2000) 637±647
of the re¯ector before and after optimisation at all the loading cases considered. The best level values and the given acceptable limits on each performance parameter at each loading case are also shown in these ®gures. Figs. 13±16 illustrate the convergence histories for the mean values, SDs, PPIs and CPIs for all the performance parameters and all the loading cases considered in the optimisation. The values of means, SDs, PPIs and CPIs for the original and optimised system are listed in Tables 4 and 5. It can be seen that the structural performance parameters, at all the loading cases in the optimised structure, have much higher mean values and PPIs and much lower standard deviations than those in the original design. Also, at each loading case, the antenna will behave in such a way that all the performances have increased reliability and decreased the possibility of performing unsatisfactorily.
For the optimised antenna structure, Tables 4 and 5 show that the worst-case temperature pro®le corresponds to a condition of temperature gradients across the re¯ector surface and not to a situation of absolute temperature excursion from the ambient fabrication temperature. The determination of ®bre orientation angles is an important subject in the optimum design of composite materials. The fact that the design variables for the ®bre angles in both the composite surface panels and composite stiening ribs did not alter in the optimisation procedure, veri®ed that the 45° ®bre angles for these components are the best choice. The iteration history of the design variables is shown in Figs. 17 and 18. The optimised re¯ector geometry is shown in Fig. 19. The change in geometric shapes of the stiening ribs in the original design and the optimised structure can be
Fig. 15. The convergence history of the PPIs.
Fig. 13. The convergence history of the means of performances in the PPM.
Fig. 14. The convergence history of the standard deviations of performances in the PPM.
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Fig. 16. The convergence history of the working/loading cases performance indices.
Table 4 System parameter pro®le analysis of the original antenna structure Mean RMS error Maximum stress Structural mass Structural frequency Maximum displacement
Standard deviation
PPI
Original
Optimised
Original
Optimised
Original
Optimised
3.96 7.11 3.40 1.99 3.42
9.78 7.51 9.30 10.00 9.48
1.08 2.57 0.00 0.00 1.00
0.31 2.49 0.00 0.00 0.48
3.62 6.19 3.40 1.10 3.06
9.77 6.68 9.30 10.00 9.46
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Table 5 Loading case pro®le analysis of the original antenna structure Mean Case Case Case Case
1 2 3 4
()180°C) (+115°C) (thermal gradient) (launch case)
Standard deviation
CPI
Original
Optimised
Original
Optimised
Original
Optimised
2.64 4.57 3.08 5.62
8.76 9.78 8.45 9.86
1.26 2.75 1.21 2.30
1.89 0.27 1.77 0.28
1.93 2.52 2.20 4.93
8.17 9.78 7.93 9.85
Fig. 17. The convergence history of design variables (thicknesses of layers and honeycomb cores).
Fig. 18. The convergence history of structural geometric design variables.
clearly observed by comparing the shapes shown in Figs. 2 and 19. The modal analysis for the optimised antenna structure shows that the lowest structural natural frequencies for both the deployed and stowed cases are 9.07 and 29.1 Hz, respectively. These frequencies are increased from their original values of 4.35 and 24.3 Hz, and satis®ed the given target values which should be over 8 and 28 Hz, respectively. The optimised structure produces signi®cantly higher modal frequencies than the original structure, due to the relatively sti nature of the optimised structure.
Fig. 19. The optimised re¯ector geometry.
J.-S. Liu, L. Hollaway / Computers and Structures 78 (2000) 637±647
647
7. Conclusions
References
The optimisation technique is shown to have an intrinsic applicability to multi-objective optimisation. Although like any non-linear programming problem, the method cannot guarantee the ®nding of a global optimum because of the Pareto feature of some problems, it makes it feasible to optimise complicated structures and systems considering many loading conditions. One of its primary bene®ts in the application is the elimination of the potentially expensive separate optimisations for each objective. For solutions of structural problems, the optimisation method has been coupled with the A B A Q U S ®nite element code which is used to determine system response variables as functions of design variables. The technique has been demonstrated on various structural systems built of various types of ®nite elements and dierent materials, contributing to a wide range of mechanical properties and cost to the design objectives. The method may not necessarily make every performance at every loading case reach its speci®ed aspiration value, especially in the case of con¯ict objectives and many loading cases simultaneously considered in an optimisation. Using currently available composite materials, with their actual physico-mechanical properties, it is possible to optimise the reinforcement of the bearing layers of the structures. These materials oer the structural designers a wide range of new degrees of freedom to think in terms of simultaneous optimisation of structural con®guration and structural material. In this optimisation model developed, design variables can incorporate the tailorable variables of composite materials such as ply angles, laminate thicknesses, ®bre volume ratios, and shape parameters. The application of the method on this composite re¯ector structure appeared very encouraging.
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