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Shape optimisation of axisymmetric cylindrical nozzles in spherical pressure vessels subject to stress constraints
International Journal of Pressure Vessels and Piping 78 (2001) 1±9
www.elsevier.com/locate/ijpvp
Shape optimisation of axisymmetric cylindrical nozzles in spherical pressure vessels subject to stress constraints J.-S. Liu a,*, G.T. Parks b, P.J. Clarkson b a
b
School of Engineering, University of Hull, Hull HU6 7RX, UK Engineering Design Centre, Cambridge University Engineering Department, Trumpington Street, Cambridge CB2 1PZ, UK Received 4 September 2000; revised 4 November 2000; accepted 10 November 2000
Abstract In this study, optimal shapes of intersecting pressure vessels are sought using a novel topology/shape optimisation method, called Metamorphic Development (MD). An industrial benchmark design problem of ®nding the optimal pro®le of variable thickness that connects a spherical shell pressure vessel to a cylindrical nozzle is considered. Two types of intersecting structures, distinguished by ¯ush and protruding nozzles, are investigated. The optimum pro®les of minimum mass intersecting structures are found by growing and degenerating simple initial structures subject to stress constraints. The optimisation seeks to eliminate the stress peaks caused by the opening. The optimised structures are developed metamorphically in speci®ed in®nite design domains using both rectangular and triangular axisymmetric ®nite elements that are ideally suited for modelling continua with curved boundaries. It is shown that the design with a protruding nozzle would produce a better stress distribution than the design with a ¯ush nozzle. The results demonstrate the success of the method in generating suitable, practical solutions to the design problem. q 2001 Elsevier Science Ltd. All rights reserved. Keywords: Pressure vessels; Nozzles; Axisymmetric structures; Intersection structures; Stress concentration; Shape optimisation; Metamorphic development
1. Introduction The determination of optimal shapes of intersecting pressure vessels to achieve high performance and low cost is important in engineering design. To be of any practical use, a pressure vessel must contain openings through which the contents of the vessel may be accessed. These openings are regions of high stress concentration and constitute a major source of weakness, the nature of which is by no means fully understood. Cracks, which eventually lead to failure of the pressure vessel, are often initiated at these locations. The design task is to specify the amount of reinforcing material around the openings and its distribution so as to obtain strength close to that of the vessels. The intersecting vessel we shall seek to optimise is a structure connecting thin spherical and cylindrical shells, and, in particular, the practical case of a radially disposed nozzle in a spherical pressure vessel. The pressure vessel can be reinforced by a non-uniformly thickened region away from the nozzle. This well-known problem has been tackled through a variety of approaches in the past. Freiberger [1] provided an optimal tapering variation of the cylinder thick* Corresponding author. Fax: 144-1482-46-6664. E-mail address: [email protected] (J.-S. Liu).
ness for the case where the mid-surface pro®le is speci®ed. Calladine [2] proposed a plastic design approach for the reinforcement design problem by connecting the vessels with a shallow conical pad of constant thickness. Diallo and Ellyin [3] sought the optimal shape of variable thickness through a direct variational method on the basis of membrane theory and the von Mises yield condition. However, a purely membrane state of stress is an idealised situation and may not be achieved in practice, due to the considerable thickness around the opening. Ellyin [4] presented a means of seeking an optimal con®guration based on general shell theory using a direct variational formulation Ð the Rayleigh±Ritz method. In this approach, the solution is obtained by solving a system of non-linear equations and a complementary solution may be required for long nozzles. The obvious practical disadvantage of the analytical solution so obtained is that it partially blocks the nozzle. Metamorphic development (MD) is a novel optimisation method to ®nd optimal structural shapes and topologies that minimise mass and compliance subject to stress and de¯ection constraints [5,6]. The optimisation procedure starts from a minimal number of nodes and elements connecting the applied loads and support points. The structure is then developed using ®nite elements that can be of any speci®ed
0308-0161/01/$ - see front matter q 2001 Elsevier Science Ltd. All rights reserved. PII: S 0308-016 1(00)00065-X
2
J.-S. Liu et al. / International Journal of Pressure Vessels and Piping 78 (2001) 1±9
t
of pro®le changes on structural strength, detailed elastic ®nite element analysis is performed. Optimal designs are obtained for both nozzle types. The results and the MD histories are given.
t
r
r
A
A
h
h B
B
x
T Φ
T Φ
R
R
(a) With a flush nozzle
(b) With a protruding nozzle
Fig. 1. Schematic diagrams of the nozzle and spherical vessel geometries.
sizes, and a design domain may or may not be speci®ed. The optimum is sought through simultaneous growth and degeneration, i.e. by adding to and removing from the structure both nodes and elements. The MD is controlled by a growth factor, which may be positive or negative, related to current structural performance. Growth is guided to occur in areas of high strain energy or high stress. One important feature of the method is that it allows the introduction and re-introduction (if they have been removed in previous iterations) of nodes and elements, enabling the topology design space to be robustly explored. The MD method can be used to solve different kinds of topology/shape optimisation problems, such as ² minimising structural mass subject to structural response constraints, and ² minimising mean structural compliance (or mean stress/ strain energy) subject to mass/structural response constraints. In this study, the MD method is successfully used to solve the design problem mentioned above. Two typical types of intersecting structures, connecting a spherical vessel with a ¯ush nozzle or with a protruding nozzle, are considered in the optimisation. To investigate the stress distribution induced due to the applied internal pressure and the effects
Bus topology Tree topology Existing structure
Ring topology
Star topology
Growth cone (high strain energy region)
Added elements
Fig. 2. Growth cones.
2. Design problem The design task is to ®nd an optimal pro®le, variable thickness reinforcing structure at the junction of a spherical shell intersected by a radial, cylindrical nozzle. The structure must approach a minimum material volume and satisfy a minimum strength criterion. A particular loading case, of importance in most forms of pressure vessel design, will be considered, that is, constant internal pressure, p, applied on the inside surfaces of both sphere and nozzle by a weightless medium. The loading can be shape-dependent if a design surface is also a loading surface. Two basic structure types, a ¯ush nozzle with a uniform thickness pressure vessel and a protruding nozzle with the same vessel, are considered. High stress concentration occurs at the intersection between the vessels. The geometrical parameters de®ning the structures under consideration are shown in Fig. 1. In this study, the intersecting part of the pressure vessels, i.e. the pro®le between A and B (see Fig. 1), is determined using the MD optimisation procedure. The design problem can be stated as: minimise
M
1a
subject to
s max # s 0
1b
or
s intersection # s vessels
1c
where M is the structural mass, s max the maximum stress in the structure, s 0 the speci®ed limit on the maximum stress, s intersection the maximum stress in the intersection between the vessels, and s vessels the stress in the vessels themselves. 3. Optimisation methodology The structural shape optimisation problem de®ned above can be solved by a novel methodology Ð MD. The optimisation method is powered with mechanisms for both growing and degenerating a structure in order to modify its performance. Growth is guided to occur only in certain ªgrowth conesº; a growth cone being a local section of structural ªsurfaceº where high strain energy, high compliance and/or high stress occur. The growth ªgrammarsº implemented through growth cones, as shown in Fig. 2, are based on the following reasoning: ² adding new elements can create new load paths; ² adding structural material to these areas can disperse high strain energy and reduce high compliance; and
J.-S. Liu et al. / International Journal of Pressure Vessels and Piping 78 (2001) 1±9
² high stress can be decentralised by adding more elements. Conversely, elements that carry only a small load are considered to be used inef®ciently, and thus can be removed. A dynamic growth factor (DGF) is used to control growth and degeneration. The general form of the DGF is shown in Fig. 3. The values of max i1; min i1; min i2 and max i2 (the maximum and minimum numbers of elements that can be added and removed) vary throughout the optimisation. They depend on factors such as the scale of the structure (numbers of nodes and elements), the total size of structural surface, and the availability of structural symmetry in one or more directions. The values and factors that determine the growth factor may vary in each iteration. Therefore, the DGF is an adaptive function changing from one topology structure to another. The DGF is used to regulate dynamically the rates of growth and degeneration (i.e. to control the sizes of the growth cones and the parts to be removed). The DGF is related to the current structural performance, the closeness to the imposed response constraints and the calculated stress (and/or strain energy). In Fig. 3, G(Ti) represents a hybrid constraint function and is determined by comparing structural responses (such as stress or de¯ection) with speci®ed limits: G T i
m X j1
wj gj Ti
m X j1
wj uRj Ti u 2 uRpj u
2
where Rj(Ti) is jth structural response, m the number of the response constraints, Rpj the user-speci®ed target for jth response, and wj the weighting function. The values of G(Ti) at points A and B are speci®ed as: A
m X j1
wj uRpj u
3a
and B 2A=2
3b
Thus the DGF, the value of which depends through simple averaging on the values of the limits max i1; min i1 etc., is a piecewise linear function of G(Ti). 4. The optimisation process The MD optimisation process consists of the following steps: Step 1: Form an optimisation model De®ne all the kinematic boundary constraints, loads and material properties that are expected under service conditions. Specify the optimisation criteria to be used and the design constraint conditions (e.g. stresses in this application) to be imposed to optimise the structure.
3
DGF Evolution control limit
max(i)+
Dynamic growth factor
min(i)+ B min(i)−
G(Ti) A Degeneration control limit
max(i)− Fig. 3. Structural dynamic growth factor versus hybrid constraint function.
Step 2: De®ne a ®nite or in®nite design domain It is not necessary to specify a ®nite design domain. This is especially helpful if the best domain for the design is not known a priori. Nevertheless it is possible to impose restrictions on structural growth both in terms of location and direction, if appropriate. Step 3: Choose appropriate ®nite elements Finite elements appropriate to the design problem under consideration must be chosen. They can be linear or nonlinear elements with various shape functions. In this application, both triangular and quadrilateral axisymmetric elements are used. Step 4: Generate an initial structure The initial FE structure can be the simplest possible geometry connecting the loads to the supports, provided that it has a non-singular FE solution. In fact, the optimisation can start from any degree of structural development from the simplest possible structure to a heavy ground mesh. In most cases the choice of initial structure will not make much difference to the ®nal design obtained. Generally, only a rough (far from optimal) initial design is needed. However, if the user has an existing design to use as a starting point or can estimate an initial shape that is close to the optimum, this will certainly reduce the number of iterations needed to reach the optimum. Step 5: Perform a structural ®nite element analysis Elastic FEA is performed to ®nd the stress distribution over the whole structure for a strength related optimisation problem. Based on the analysis, growth cones, the parts of the structural surface that are heavily over-stressed, are located. Meanwhile, heavily under-stressed regions are also located.
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J.-S. Liu et al. / International Journal of Pressure Vessels and Piping 78 (2001) 1±9
Fig. 4. Flow chart of the MD method.
Step 6: Grow/degenerate the structure In this step, the structure is allowed to grow and degenerate volumetrically. Growth and degeneration take place simultaneously in every iteration. Material is added in overloaded locations (growth cones) and removed from underloaded locations. The growth rate is controlled by the DGF, which can be both positive and negative during the course of an optimisation. Step 7: Update the structure The FE model is now modi®ed to trace and accommodate the changes to the structure made in step 6 and to avoid possible singularities caused by them. This entails updating the structural elements, nodes etc. Steps 5±7 are repeated until a suitably optimised structure has evolved. This process will result in a homogeneous distribution of elastic energies (or stresses) in the structure, indicating that the peak stresses (and/or strain energies) are adequately reduced. However, this distribution does not identify the optimal topology for the speci®ed loads and
design constraints. The optimisation process continues until a minimum mass structure satisfying all the structural response constraints has been achieved or until design limitations prevent a further increase (or decrease) in the size of the structure. Convergence is deemed to have occurred if the performance of the structure could not be improved for a speci®ed number of consecutive iterations. In this manner, the initial design evolves into a topologically optimised design, which is a minimum weight structure satisfying all the constraints. The optimisation algorithm employs a hierarchical structure. First, objective f1(Ti) (the compliance) is minimised and objective f2(Ti) (the mass) is disregarded until the structural response constraints are satis®ed. Then f2(Ti) is minimised subject to constraints on the structural response. In the ®rst stage, a positive growth factor is used and more elements are added than removed each iteration. Conversely, in the next stage, a negative growth factor is used and more elements are removed than added each iteration. Depending on the current structural performance, these two optimisation schemes may be adopted alternately. A ¯ow chart for the optimisation procedure is given in Fig. 4. The procedure used does not require gradient-based sensitivity analysis, as the surface strain-energy/stress distribution is the sole parameter considered. The optimisation procedure can be used as a design tool in conjunction with commercial FE codes. A program has been developed to interface the optimisation program and Abaqus software [7]. 5. Results As described in Section 2, the principal objective of this study is to ®nd a minimum weight design for the intersecting part of the cylinder±sphere pressure vessel that eliminates
Design Domain
(a) Iteration 0
(b) Iteration 20
(c) Iteration 40
Fig. 5. Metamorphic development history for the ¯ush nozzle intersection.
(d) Iteration 50
J.-S. Liu et al. / International Journal of Pressure Vessels and Piping 78 (2001) 1±9
5
a
1 1 2 2 3 4
g
3
b c
5
d h 5
3 4
2
e i
3 2
f
Fig. 6. Von Mises stress distribution in the area of intersection (¯ush nozzle).
Fig. 7. Speci®c points in the optimised pressure vessel intersection structure (¯ush nozzle).
the stress peaks caused by the discontinuity due to the opening. A design requirement on the amount of reinforcement at the nozzle±sphere intersection is that the nozzle, sphere and the intersection all yield at a similar loading level and within a similar operational period. Although it is of less concern for this particular application, reducing the stress concentration factor for a component under a static load will also increase the fatigue endurance of the component, provided that the loading spectrum falls within a certain frequency band.
points are identi®ed in Fig. 7. The variations in the stresses are shown in Figs. 8 and 9. Radial stress, axial stress, hoop stress and von Mises stress are all analysed in the optimisation. A constraint that the von Mises stress should not exceed 35 MPa on the exterior surface is imposed. In this problem, the interior is not part of the design surface. Ultimately there is little that can be done to reduce excessive stress here if material cannot be added on the inside of the vessel. The optimised structure exhibits a distinct ªcornerº at point d (see Fig. 7). A larger design domain (larger value of F , see Fig. 1) gives a smoother transition but at the cost of a slight increase in mass. As can be seen in Fig. 6 the corner at d does not result in any observable stress concentration, therefore no additional material is added in the optimisation process and the corner is not ª®lled inº. Fig. 8 shows the distributions of the four stresses in the nozzle and spherical vessel and on the inner surface of the intersection. In the nozzle part (from points a to b), the stresses are essentially constant, with a nearly zero radial stress and an axial stress of about 10 MPa. The hoop stress and Mises stress, in this section, are above 16 MPa but less than 20 MPa. Starting around points b and g, both the hoop stress and Mises stress exhibit a steep increase and then a gentle rise until point h, where these two stresses reach their maximum. From point h, these two stresses start to fall gently until point i. The axial stress, however, exhibits a decrease from 10 MPa to a level slightly less than 0 MPa, and this stress level is maintained until point i. There is almost no change for the radial stress from points a to i. Between points i and e, the hoop and Mises stresses decrease signi®cantly, while the radial stress increases rapidly. From points e to f, the spherical vessel away from the intersection, the hoop and Mises stresses remain constant. The radial and axial directions are de®ned by the nozzle geometry, so within the spherical vessel the radial and axial stresses vary trigonometrically in antiphase, as expected.
5.1. Intersecting structure with a ¯ush nozzle Referring to Fig. 1(a) (a cross-sectional view of the structure), the geometric parameters are: t 50 mm; T 100 mm; r 1000 mm; R 3990 mm; h 312:4 mm; F 21:88; and p 1 MPa: The material used is mild steel. A much ®ner mesh and smaller elements are used in and near the intersecting vessel since the stress concentration in this region is high. The use of axisymmetric elements ensures that appropriate boundary conditions are imposed. The design domain is an in®nite region, with a speci®ed interior boundary (see Fig. 5(a)). The structure is permitted to grow outwards from this interior boundary, but not inwards. There is no limit on how far the structure can grow outwards. The optimisation starts from the simplest possible initial structure Ð a single layer of elements along the interior boundary of the intersection (see Fig. 5(a)). Fig. 5 shows the MD history, in the form of the meridional material area of revolution of the cylinder±sphere connection. The optimisation takes 50 iterations to converge to the optimal design. The optimised structure is shown in Fig. 5(d). The distribution of von Mises stress around the intersection is shown as a contour plot in Fig. 6. The optimisation process took just 56 min to run on a Sun Ultra1 workstation running Solaris 2.6. To give a complete picture of the various stress distributions at the surfaces of the optimised vessel, some speci®c
6
J.-S. Liu et al. / International Journal of Pressure Vessels and Piping 78 (2001) 1±9 45
von M ises stress
40
Stresses (MPa)
35
H oop stress
30
R adial stress Axial stress H oop stress von M ises stress
25 20 15 10
R adial stress
Axial stress
5 0 -5
a
b g
h
iI e
f
From points a to f (via b, g, h, i, and e) Fig. 8. Stress distribution on the interior (non-design) surface of the optimised ¯ush nozzle intersection.
Importantly, as shown in Fig. 9, the Mises stress along the exterior surface is kept beneath the speci®ed limit (35 MPa here) and has a relatively smooth distribution. The small cyclic variations are purely due to the slight geometric discontinuities in the outer surface at transitions between triangular and quadrilateral elements. Although quadrilateral and triangular can approximate a smooth surface, they cannot model it perfectly. This cyclic stress variation would, of course, be eliminated when the vessel was manufactured. The other three stresses are all have similar distributions along the surface. If the design domain is changed so that growth both inwards and outwards is allowed, then the ªoptimalº solution found by MD is one in which the opening is sealed off, as shown in Fig. 10. From an optimisation point of view this is an interesting solution, but it is hardly very practical! 5.2. Intersecting structure with a protruding nozzle To investigate the effect of nozzle type on the structural performance of the pressure vessel, an alternative intersecting structure with a protruding nozzle is optimised. To make the results directly comparable, the same geometric parameters (refer to Fig. 1), material and loads as described in 40
Section 5.1 are used. The amount by which the nozzle protrudes x 320 mm: The design domain is again an in®nite region, but, instead of having one (exterior) design surface, it has two (upper and lower) surfaces to be shaped (see Fig. 11(a)). There is no geometric restriction on growth on either design surface. For this nozzle type, the load case is shape-dependent since the lower design surface is a loading surface. Therefore, loading updating is needed in structural reanalysis to accommodate the changes in loading positions and directions. Axisymmetric boundary conditions are imposed, and a ®ner mesh and smaller elements are used in and near the intersection to detail any stress concentration in this region. The simplest possible initial layout is used as the starting design for the optimisation. This is a ¯at plate disk connecting the nozzle and the sphere (see Fig. 11(a)). The MD history is shown in Fig. 11. After 68 iterations the optimisation converged to the shape shown in Fig. 11(d). This optimisation took 84 min to complete on a Sun Ultra1 workstation running Solaris 2.6. The longer running time, compared to the ®rst example, is due in part to the greater number of iterations required for convergence and in part to the slightly larger number of elements to be analysed.
Hoop stress
35
Stresses (MPa)
30
Radial stress Axial stress Hoop stress von Mises stress
25 20
von Mises stress
15 10 5 0
Radial stress
-5
b c
d e
From points b to e (via c and d) Fig. 9. Stress distribution on the exterior surface of the optimised ¯ush nozzle intersection.
J.-S. Liu et al. / International Journal of Pressure Vessels and Piping 78 (2001) 1±9
(a) Meridional plane view
7
(b) A 3-D view Fig. 10. Optimised shape if interior boundary is not constrained
The distribution of von Mises stress around the intersection is shown as a contour plot in Fig. 12. To illustrate detailed stress distributions, some speci®c points are identi®ed in Fig. 13 and the stresses are shown in Figs. 14 and 15. The stresses analysed in the optimisation are the same as for the ¯ush nozzle, i.e. radial stress, axial stress, hoop stress and von Mises stress. For the protruding nozzle, the original design constraint speci®ed, a maximum Mises stress of 35 MPa on the design surfaces, can be satis®ed very easily. A partially developed structure, similar to that shown in Fig. 11(b), evolved by MD in only about 20 iterations satis®es this constraint. Therefore, a more stringent constraint is imposed for the protruding nozzle Ð the von Mises stress should not exceed 24 MPa on the design surfaces. Fig. 14 shows the distributions of the four different stresses in the nozzle and spherical vessel and on the inner surface of the intersection. The surface between points b and k is non-design-surface. From points b±g, the Mises stress rises, reaching a maximum at g, while the hoop stress
rises and falls. Both stresses then decrease, reaching their minimum values at point h. The axial stress decreases between points b and g, reaching its lowest value at g, and then increases until point k. All the stresses increase monotonically from point h until point k, except the radial stress, which remains constant in this region. From point k to point i, which marks the end of the interior design surface, the Mises stress is basically held at a constant value beneath its speci®ed limit (24 MPa). Between the same points, the hoop stress has a similar distribution but a slightly higher level. The same trend can be observed on the exterior design surface (from points c to d) in Fig. 15. The Mises stress is effectively held constant beneath the constraint value and the hoop stress has a similar distribution but a slightly higher level. From Fig. 14 it can be seen that all the stresses around point h are at their lowest levels. This suggests that the overprotruding part of the cylinder can be trimmed to obtain a further weight reduction. The optimised structure was
Fig. 11. Metamorphic development history for the protruding nozzle intersection.
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J.-S. Liu et al. / International Journal of Pressure Vessels and Piping 78 (2001) 1±9 Table 1 Attributes of the optimised ¯ush and protruding nozzles
3 3 4 5
Structure
Mass (kg)
Maximum Mises stress (MPa) and location
Flush nozzle Protruding nozzle Trimmed protruding nozzle
158239.5 159803.8 159605.5
39.99 (at point h, see Fig. 7) 28.28 (at point g, see Fig. 13) 28.17 (at point g, see Fig. 13)
3 4 5 4 3 2 1
modi®ed by trimming 80 mm from the bottom of the cylinder, and the resulting structure re-analysed. Fig. 16 shows the stress distribution round the intersection. Comparing Figs. 12 and 16, it can be seen that the Mises stress distributions in the untrimmed and trimmed structures are essentially the same.
3 2 1
Fig. 12. Von Mises stress distribution in the area of intersection (protruding nozzle).
5.3. Discussion From the foregoing results, it can be seen that there is no obvious stress concentration in the intersections in the optimised structures Ð a minimum material volume and a relatively uniform stress distribution have been achieved. The Mises stress on all the design surfaces is comfortably beneath the speci®ed constraint values, and the design surfaces are essentially homogenous stress surfaces. The small cyclic variations observed in Figs. 9, 14 and 15 are caused by slight surface discontinuities due to the size and shape of the elements used. It is observed that for the ¯ush nozzle the highest stresses occur at point h (an abrupt turning point) (see Figs. 6 and 8), and for the protruding nozzle at point g (see Figs. 12 and 14). These high stresses for both types of nozzle occur on non-design-surfaces and cannot be reduced signi®cantly by adding further material to the design surfaces. Table 1 compares the masses and the maximum Mises stresses within the optimised ¯ush and (trimmed) protruding nozzle intersections. Since there are two design surfaces
a
b c g d
e
h
i
k
f
Fig. 13. Speci®c points in the optimised pressure vessel intersection structure (protruding nozzle).
28
Hoop stress
Stresses (MPa)
24 20 16
von Mises stress
12
Radial stress
8
Axial stress
4 0 -4
Radial stress
Axial stress
-8
Hoop stress
von Mises stress
-12
b
g
h
k
i e
From points b to e (via g,h,k,and i) Fig. 14. Stress distribution on the interior surface of the optimised protruding nozzle intersection.
Stresses (MPa)
J.-S. Liu et al. / International Journal of Pressure Vessels and Piping 78 (2001) 1±9 28 26 24 22 20 18 16 14 12 10 8 6 4 2 0 -2
9
Hoop stress
Radial stress Hoop stress
Axial stress von Mises stress
von Mises stress
Radial stress
b
c
d
e
From points b to e (via c and d) Fig. 15. Stress distribution on the exterior surface of the optimised protruding nozzle intersection.
1 1 2 2
3 4 5
3 2 2
5
3
4
stress concentrations present is sought and a homogeneous surface stress distribution has been achieved. The method ®nds a minimum weight design for the intersection with a similar level strength as in the two pressure vessels being connected (in this case, the cylinder and the sphere). The employment of both rectangular and triangular elements makes the boundaries of the resulting optimised axisymmetric structures relatively smooth. The potential that the MD method has as a tool for investigating optimal topologies and shapes and for automatically and rapidly creating optimal, realistic, practical solutions to structural design problems has been illustrated by the examples presented.
3 Fig. 16. Von Mises stress distribution in the area of intersection (trimmed protruding nozzle).
available for the protruding nozzle (instead of one for the ¯ush nozzle), a design with a better stress distribution should be expected. Indeed, with this additional design degree of freedom, the optimisation results obtained for the protruding nozzle design are favourable from the point of view of reducing stress concentration. However, this is obtained at the cost of a heavier structure. Post-optimisation examination of the original design for the protruding nozzle enabled the nozzle to be trimmed, saving valuable weight, without compromising the structural performance. 6. Conclusions In this paper, the MD method, which is a systematic design procedure for adding material to and removing material from a structure, has been used to ®nd optimal pro®les for intersecting pressure vessels. A marked reduction in any
References [1] Freiberger W. Minimum weight design of cylindrical shells. ASME J Appl Mech 1956;23:567±80. [2] Calladine CR. On the design of reinforcement for openings and nozzles in thin spherical pressure vessels. J Mech Engng Sci 1966;8: 1±14. [3] Diallo B, Ellyin F. Optimization of connecting shell. ASCE J Engng Mech 1983;109:111±26. [4] Ellyin F. Shape optimization of intersecting pressure vessels. Proceedings of the IUTAM Symposium on Structural Optimization, 1988. p. 77±84. [5] Liu J-S, Parks GT, Clarkson PJ. Can a structure grow towards an optimum topology layout? Ð Metamorphic development: a new topology optimisation method. Proceedings of the Third World Congress of Structural and Multidisciplinary Optimisation (WCSMO-3), 1999. CDROM. [6] Liu J-S, Parks GT, Clarkson PJ. Metamorphic Development: a new topology optimisation method for continuum structures. Struct Multidisciplinary Optimization 2000;20:288±300. [7] Hibbitt, Karlsson & Sorenson, Inc. Abaqus manual, Version 5.7-5. Pawtucket RI: Hibbitt, Karlsson & Sorenson, Inc., 1998.
www.elsevier.com/locate/ijpvp
Shape optimisation of axisymmetric cylindrical nozzles in spherical pressure vessels subject to stress constraints J.-S. Liu a,*, G.T. Parks b, P.J. Clarkson b a
b
School of Engineering, University of Hull, Hull HU6 7RX, UK Engineering Design Centre, Cambridge University Engineering Department, Trumpington Street, Cambridge CB2 1PZ, UK Received 4 September 2000; revised 4 November 2000; accepted 10 November 2000
Abstract In this study, optimal shapes of intersecting pressure vessels are sought using a novel topology/shape optimisation method, called Metamorphic Development (MD). An industrial benchmark design problem of ®nding the optimal pro®le of variable thickness that connects a spherical shell pressure vessel to a cylindrical nozzle is considered. Two types of intersecting structures, distinguished by ¯ush and protruding nozzles, are investigated. The optimum pro®les of minimum mass intersecting structures are found by growing and degenerating simple initial structures subject to stress constraints. The optimisation seeks to eliminate the stress peaks caused by the opening. The optimised structures are developed metamorphically in speci®ed in®nite design domains using both rectangular and triangular axisymmetric ®nite elements that are ideally suited for modelling continua with curved boundaries. It is shown that the design with a protruding nozzle would produce a better stress distribution than the design with a ¯ush nozzle. The results demonstrate the success of the method in generating suitable, practical solutions to the design problem. q 2001 Elsevier Science Ltd. All rights reserved. Keywords: Pressure vessels; Nozzles; Axisymmetric structures; Intersection structures; Stress concentration; Shape optimisation; Metamorphic development
1. Introduction The determination of optimal shapes of intersecting pressure vessels to achieve high performance and low cost is important in engineering design. To be of any practical use, a pressure vessel must contain openings through which the contents of the vessel may be accessed. These openings are regions of high stress concentration and constitute a major source of weakness, the nature of which is by no means fully understood. Cracks, which eventually lead to failure of the pressure vessel, are often initiated at these locations. The design task is to specify the amount of reinforcing material around the openings and its distribution so as to obtain strength close to that of the vessels. The intersecting vessel we shall seek to optimise is a structure connecting thin spherical and cylindrical shells, and, in particular, the practical case of a radially disposed nozzle in a spherical pressure vessel. The pressure vessel can be reinforced by a non-uniformly thickened region away from the nozzle. This well-known problem has been tackled through a variety of approaches in the past. Freiberger [1] provided an optimal tapering variation of the cylinder thick* Corresponding author. Fax: 144-1482-46-6664. E-mail address: [email protected] (J.-S. Liu).
ness for the case where the mid-surface pro®le is speci®ed. Calladine [2] proposed a plastic design approach for the reinforcement design problem by connecting the vessels with a shallow conical pad of constant thickness. Diallo and Ellyin [3] sought the optimal shape of variable thickness through a direct variational method on the basis of membrane theory and the von Mises yield condition. However, a purely membrane state of stress is an idealised situation and may not be achieved in practice, due to the considerable thickness around the opening. Ellyin [4] presented a means of seeking an optimal con®guration based on general shell theory using a direct variational formulation Ð the Rayleigh±Ritz method. In this approach, the solution is obtained by solving a system of non-linear equations and a complementary solution may be required for long nozzles. The obvious practical disadvantage of the analytical solution so obtained is that it partially blocks the nozzle. Metamorphic development (MD) is a novel optimisation method to ®nd optimal structural shapes and topologies that minimise mass and compliance subject to stress and de¯ection constraints [5,6]. The optimisation procedure starts from a minimal number of nodes and elements connecting the applied loads and support points. The structure is then developed using ®nite elements that can be of any speci®ed
0308-0161/01/$ - see front matter q 2001 Elsevier Science Ltd. All rights reserved. PII: S 0308-016 1(00)00065-X
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t
of pro®le changes on structural strength, detailed elastic ®nite element analysis is performed. Optimal designs are obtained for both nozzle types. The results and the MD histories are given.
t
r
r
A
A
h
h B
B
x
T Φ
T Φ
R
R
(a) With a flush nozzle
(b) With a protruding nozzle
Fig. 1. Schematic diagrams of the nozzle and spherical vessel geometries.
sizes, and a design domain may or may not be speci®ed. The optimum is sought through simultaneous growth and degeneration, i.e. by adding to and removing from the structure both nodes and elements. The MD is controlled by a growth factor, which may be positive or negative, related to current structural performance. Growth is guided to occur in areas of high strain energy or high stress. One important feature of the method is that it allows the introduction and re-introduction (if they have been removed in previous iterations) of nodes and elements, enabling the topology design space to be robustly explored. The MD method can be used to solve different kinds of topology/shape optimisation problems, such as ² minimising structural mass subject to structural response constraints, and ² minimising mean structural compliance (or mean stress/ strain energy) subject to mass/structural response constraints. In this study, the MD method is successfully used to solve the design problem mentioned above. Two typical types of intersecting structures, connecting a spherical vessel with a ¯ush nozzle or with a protruding nozzle, are considered in the optimisation. To investigate the stress distribution induced due to the applied internal pressure and the effects
Bus topology Tree topology Existing structure
Ring topology
Star topology
Growth cone (high strain energy region)
Added elements
Fig. 2. Growth cones.
2. Design problem The design task is to ®nd an optimal pro®le, variable thickness reinforcing structure at the junction of a spherical shell intersected by a radial, cylindrical nozzle. The structure must approach a minimum material volume and satisfy a minimum strength criterion. A particular loading case, of importance in most forms of pressure vessel design, will be considered, that is, constant internal pressure, p, applied on the inside surfaces of both sphere and nozzle by a weightless medium. The loading can be shape-dependent if a design surface is also a loading surface. Two basic structure types, a ¯ush nozzle with a uniform thickness pressure vessel and a protruding nozzle with the same vessel, are considered. High stress concentration occurs at the intersection between the vessels. The geometrical parameters de®ning the structures under consideration are shown in Fig. 1. In this study, the intersecting part of the pressure vessels, i.e. the pro®le between A and B (see Fig. 1), is determined using the MD optimisation procedure. The design problem can be stated as: minimise
M
1a
subject to
s max # s 0
1b
or
s intersection # s vessels
1c
where M is the structural mass, s max the maximum stress in the structure, s 0 the speci®ed limit on the maximum stress, s intersection the maximum stress in the intersection between the vessels, and s vessels the stress in the vessels themselves. 3. Optimisation methodology The structural shape optimisation problem de®ned above can be solved by a novel methodology Ð MD. The optimisation method is powered with mechanisms for both growing and degenerating a structure in order to modify its performance. Growth is guided to occur only in certain ªgrowth conesº; a growth cone being a local section of structural ªsurfaceº where high strain energy, high compliance and/or high stress occur. The growth ªgrammarsº implemented through growth cones, as shown in Fig. 2, are based on the following reasoning: ² adding new elements can create new load paths; ² adding structural material to these areas can disperse high strain energy and reduce high compliance; and
J.-S. Liu et al. / International Journal of Pressure Vessels and Piping 78 (2001) 1±9
² high stress can be decentralised by adding more elements. Conversely, elements that carry only a small load are considered to be used inef®ciently, and thus can be removed. A dynamic growth factor (DGF) is used to control growth and degeneration. The general form of the DGF is shown in Fig. 3. The values of max i1; min i1; min i2 and max i2 (the maximum and minimum numbers of elements that can be added and removed) vary throughout the optimisation. They depend on factors such as the scale of the structure (numbers of nodes and elements), the total size of structural surface, and the availability of structural symmetry in one or more directions. The values and factors that determine the growth factor may vary in each iteration. Therefore, the DGF is an adaptive function changing from one topology structure to another. The DGF is used to regulate dynamically the rates of growth and degeneration (i.e. to control the sizes of the growth cones and the parts to be removed). The DGF is related to the current structural performance, the closeness to the imposed response constraints and the calculated stress (and/or strain energy). In Fig. 3, G(Ti) represents a hybrid constraint function and is determined by comparing structural responses (such as stress or de¯ection) with speci®ed limits: G T i
m X j1
wj gj Ti
m X j1
wj uRj Ti u 2 uRpj u
2
where Rj(Ti) is jth structural response, m the number of the response constraints, Rpj the user-speci®ed target for jth response, and wj the weighting function. The values of G(Ti) at points A and B are speci®ed as: A
m X j1
wj uRpj u
3a
and B 2A=2
3b
Thus the DGF, the value of which depends through simple averaging on the values of the limits max i1; min i1 etc., is a piecewise linear function of G(Ti). 4. The optimisation process The MD optimisation process consists of the following steps: Step 1: Form an optimisation model De®ne all the kinematic boundary constraints, loads and material properties that are expected under service conditions. Specify the optimisation criteria to be used and the design constraint conditions (e.g. stresses in this application) to be imposed to optimise the structure.
3
DGF Evolution control limit
max(i)+
Dynamic growth factor
min(i)+ B min(i)−
G(Ti) A Degeneration control limit
max(i)− Fig. 3. Structural dynamic growth factor versus hybrid constraint function.
Step 2: De®ne a ®nite or in®nite design domain It is not necessary to specify a ®nite design domain. This is especially helpful if the best domain for the design is not known a priori. Nevertheless it is possible to impose restrictions on structural growth both in terms of location and direction, if appropriate. Step 3: Choose appropriate ®nite elements Finite elements appropriate to the design problem under consideration must be chosen. They can be linear or nonlinear elements with various shape functions. In this application, both triangular and quadrilateral axisymmetric elements are used. Step 4: Generate an initial structure The initial FE structure can be the simplest possible geometry connecting the loads to the supports, provided that it has a non-singular FE solution. In fact, the optimisation can start from any degree of structural development from the simplest possible structure to a heavy ground mesh. In most cases the choice of initial structure will not make much difference to the ®nal design obtained. Generally, only a rough (far from optimal) initial design is needed. However, if the user has an existing design to use as a starting point or can estimate an initial shape that is close to the optimum, this will certainly reduce the number of iterations needed to reach the optimum. Step 5: Perform a structural ®nite element analysis Elastic FEA is performed to ®nd the stress distribution over the whole structure for a strength related optimisation problem. Based on the analysis, growth cones, the parts of the structural surface that are heavily over-stressed, are located. Meanwhile, heavily under-stressed regions are also located.
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J.-S. Liu et al. / International Journal of Pressure Vessels and Piping 78 (2001) 1±9
Fig. 4. Flow chart of the MD method.
Step 6: Grow/degenerate the structure In this step, the structure is allowed to grow and degenerate volumetrically. Growth and degeneration take place simultaneously in every iteration. Material is added in overloaded locations (growth cones) and removed from underloaded locations. The growth rate is controlled by the DGF, which can be both positive and negative during the course of an optimisation. Step 7: Update the structure The FE model is now modi®ed to trace and accommodate the changes to the structure made in step 6 and to avoid possible singularities caused by them. This entails updating the structural elements, nodes etc. Steps 5±7 are repeated until a suitably optimised structure has evolved. This process will result in a homogeneous distribution of elastic energies (or stresses) in the structure, indicating that the peak stresses (and/or strain energies) are adequately reduced. However, this distribution does not identify the optimal topology for the speci®ed loads and
design constraints. The optimisation process continues until a minimum mass structure satisfying all the structural response constraints has been achieved or until design limitations prevent a further increase (or decrease) in the size of the structure. Convergence is deemed to have occurred if the performance of the structure could not be improved for a speci®ed number of consecutive iterations. In this manner, the initial design evolves into a topologically optimised design, which is a minimum weight structure satisfying all the constraints. The optimisation algorithm employs a hierarchical structure. First, objective f1(Ti) (the compliance) is minimised and objective f2(Ti) (the mass) is disregarded until the structural response constraints are satis®ed. Then f2(Ti) is minimised subject to constraints on the structural response. In the ®rst stage, a positive growth factor is used and more elements are added than removed each iteration. Conversely, in the next stage, a negative growth factor is used and more elements are removed than added each iteration. Depending on the current structural performance, these two optimisation schemes may be adopted alternately. A ¯ow chart for the optimisation procedure is given in Fig. 4. The procedure used does not require gradient-based sensitivity analysis, as the surface strain-energy/stress distribution is the sole parameter considered. The optimisation procedure can be used as a design tool in conjunction with commercial FE codes. A program has been developed to interface the optimisation program and Abaqus software [7]. 5. Results As described in Section 2, the principal objective of this study is to ®nd a minimum weight design for the intersecting part of the cylinder±sphere pressure vessel that eliminates
Design Domain
(a) Iteration 0
(b) Iteration 20
(c) Iteration 40
Fig. 5. Metamorphic development history for the ¯ush nozzle intersection.
(d) Iteration 50
J.-S. Liu et al. / International Journal of Pressure Vessels and Piping 78 (2001) 1±9
5
a
1 1 2 2 3 4
g
3
b c
5
d h 5
3 4
2
e i
3 2
f
Fig. 6. Von Mises stress distribution in the area of intersection (¯ush nozzle).
Fig. 7. Speci®c points in the optimised pressure vessel intersection structure (¯ush nozzle).
the stress peaks caused by the discontinuity due to the opening. A design requirement on the amount of reinforcement at the nozzle±sphere intersection is that the nozzle, sphere and the intersection all yield at a similar loading level and within a similar operational period. Although it is of less concern for this particular application, reducing the stress concentration factor for a component under a static load will also increase the fatigue endurance of the component, provided that the loading spectrum falls within a certain frequency band.
points are identi®ed in Fig. 7. The variations in the stresses are shown in Figs. 8 and 9. Radial stress, axial stress, hoop stress and von Mises stress are all analysed in the optimisation. A constraint that the von Mises stress should not exceed 35 MPa on the exterior surface is imposed. In this problem, the interior is not part of the design surface. Ultimately there is little that can be done to reduce excessive stress here if material cannot be added on the inside of the vessel. The optimised structure exhibits a distinct ªcornerº at point d (see Fig. 7). A larger design domain (larger value of F , see Fig. 1) gives a smoother transition but at the cost of a slight increase in mass. As can be seen in Fig. 6 the corner at d does not result in any observable stress concentration, therefore no additional material is added in the optimisation process and the corner is not ª®lled inº. Fig. 8 shows the distributions of the four stresses in the nozzle and spherical vessel and on the inner surface of the intersection. In the nozzle part (from points a to b), the stresses are essentially constant, with a nearly zero radial stress and an axial stress of about 10 MPa. The hoop stress and Mises stress, in this section, are above 16 MPa but less than 20 MPa. Starting around points b and g, both the hoop stress and Mises stress exhibit a steep increase and then a gentle rise until point h, where these two stresses reach their maximum. From point h, these two stresses start to fall gently until point i. The axial stress, however, exhibits a decrease from 10 MPa to a level slightly less than 0 MPa, and this stress level is maintained until point i. There is almost no change for the radial stress from points a to i. Between points i and e, the hoop and Mises stresses decrease signi®cantly, while the radial stress increases rapidly. From points e to f, the spherical vessel away from the intersection, the hoop and Mises stresses remain constant. The radial and axial directions are de®ned by the nozzle geometry, so within the spherical vessel the radial and axial stresses vary trigonometrically in antiphase, as expected.
5.1. Intersecting structure with a ¯ush nozzle Referring to Fig. 1(a) (a cross-sectional view of the structure), the geometric parameters are: t 50 mm; T 100 mm; r 1000 mm; R 3990 mm; h 312:4 mm; F 21:88; and p 1 MPa: The material used is mild steel. A much ®ner mesh and smaller elements are used in and near the intersecting vessel since the stress concentration in this region is high. The use of axisymmetric elements ensures that appropriate boundary conditions are imposed. The design domain is an in®nite region, with a speci®ed interior boundary (see Fig. 5(a)). The structure is permitted to grow outwards from this interior boundary, but not inwards. There is no limit on how far the structure can grow outwards. The optimisation starts from the simplest possible initial structure Ð a single layer of elements along the interior boundary of the intersection (see Fig. 5(a)). Fig. 5 shows the MD history, in the form of the meridional material area of revolution of the cylinder±sphere connection. The optimisation takes 50 iterations to converge to the optimal design. The optimised structure is shown in Fig. 5(d). The distribution of von Mises stress around the intersection is shown as a contour plot in Fig. 6. The optimisation process took just 56 min to run on a Sun Ultra1 workstation running Solaris 2.6. To give a complete picture of the various stress distributions at the surfaces of the optimised vessel, some speci®c
6
J.-S. Liu et al. / International Journal of Pressure Vessels and Piping 78 (2001) 1±9 45
von M ises stress
40
Stresses (MPa)
35
H oop stress
30
R adial stress Axial stress H oop stress von M ises stress
25 20 15 10
R adial stress
Axial stress
5 0 -5
a
b g
h
iI e
f
From points a to f (via b, g, h, i, and e) Fig. 8. Stress distribution on the interior (non-design) surface of the optimised ¯ush nozzle intersection.
Importantly, as shown in Fig. 9, the Mises stress along the exterior surface is kept beneath the speci®ed limit (35 MPa here) and has a relatively smooth distribution. The small cyclic variations are purely due to the slight geometric discontinuities in the outer surface at transitions between triangular and quadrilateral elements. Although quadrilateral and triangular can approximate a smooth surface, they cannot model it perfectly. This cyclic stress variation would, of course, be eliminated when the vessel was manufactured. The other three stresses are all have similar distributions along the surface. If the design domain is changed so that growth both inwards and outwards is allowed, then the ªoptimalº solution found by MD is one in which the opening is sealed off, as shown in Fig. 10. From an optimisation point of view this is an interesting solution, but it is hardly very practical! 5.2. Intersecting structure with a protruding nozzle To investigate the effect of nozzle type on the structural performance of the pressure vessel, an alternative intersecting structure with a protruding nozzle is optimised. To make the results directly comparable, the same geometric parameters (refer to Fig. 1), material and loads as described in 40
Section 5.1 are used. The amount by which the nozzle protrudes x 320 mm: The design domain is again an in®nite region, but, instead of having one (exterior) design surface, it has two (upper and lower) surfaces to be shaped (see Fig. 11(a)). There is no geometric restriction on growth on either design surface. For this nozzle type, the load case is shape-dependent since the lower design surface is a loading surface. Therefore, loading updating is needed in structural reanalysis to accommodate the changes in loading positions and directions. Axisymmetric boundary conditions are imposed, and a ®ner mesh and smaller elements are used in and near the intersection to detail any stress concentration in this region. The simplest possible initial layout is used as the starting design for the optimisation. This is a ¯at plate disk connecting the nozzle and the sphere (see Fig. 11(a)). The MD history is shown in Fig. 11. After 68 iterations the optimisation converged to the shape shown in Fig. 11(d). This optimisation took 84 min to complete on a Sun Ultra1 workstation running Solaris 2.6. The longer running time, compared to the ®rst example, is due in part to the greater number of iterations required for convergence and in part to the slightly larger number of elements to be analysed.
Hoop stress
35
Stresses (MPa)
30
Radial stress Axial stress Hoop stress von Mises stress
25 20
von Mises stress
15 10 5 0
Radial stress
-5
b c
d e
From points b to e (via c and d) Fig. 9. Stress distribution on the exterior surface of the optimised ¯ush nozzle intersection.
J.-S. Liu et al. / International Journal of Pressure Vessels and Piping 78 (2001) 1±9
(a) Meridional plane view
7
(b) A 3-D view Fig. 10. Optimised shape if interior boundary is not constrained
The distribution of von Mises stress around the intersection is shown as a contour plot in Fig. 12. To illustrate detailed stress distributions, some speci®c points are identi®ed in Fig. 13 and the stresses are shown in Figs. 14 and 15. The stresses analysed in the optimisation are the same as for the ¯ush nozzle, i.e. radial stress, axial stress, hoop stress and von Mises stress. For the protruding nozzle, the original design constraint speci®ed, a maximum Mises stress of 35 MPa on the design surfaces, can be satis®ed very easily. A partially developed structure, similar to that shown in Fig. 11(b), evolved by MD in only about 20 iterations satis®es this constraint. Therefore, a more stringent constraint is imposed for the protruding nozzle Ð the von Mises stress should not exceed 24 MPa on the design surfaces. Fig. 14 shows the distributions of the four different stresses in the nozzle and spherical vessel and on the inner surface of the intersection. The surface between points b and k is non-design-surface. From points b±g, the Mises stress rises, reaching a maximum at g, while the hoop stress
rises and falls. Both stresses then decrease, reaching their minimum values at point h. The axial stress decreases between points b and g, reaching its lowest value at g, and then increases until point k. All the stresses increase monotonically from point h until point k, except the radial stress, which remains constant in this region. From point k to point i, which marks the end of the interior design surface, the Mises stress is basically held at a constant value beneath its speci®ed limit (24 MPa). Between the same points, the hoop stress has a similar distribution but a slightly higher level. The same trend can be observed on the exterior design surface (from points c to d) in Fig. 15. The Mises stress is effectively held constant beneath the constraint value and the hoop stress has a similar distribution but a slightly higher level. From Fig. 14 it can be seen that all the stresses around point h are at their lowest levels. This suggests that the overprotruding part of the cylinder can be trimmed to obtain a further weight reduction. The optimised structure was
Fig. 11. Metamorphic development history for the protruding nozzle intersection.
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J.-S. Liu et al. / International Journal of Pressure Vessels and Piping 78 (2001) 1±9 Table 1 Attributes of the optimised ¯ush and protruding nozzles
3 3 4 5
Structure
Mass (kg)
Maximum Mises stress (MPa) and location
Flush nozzle Protruding nozzle Trimmed protruding nozzle
158239.5 159803.8 159605.5
39.99 (at point h, see Fig. 7) 28.28 (at point g, see Fig. 13) 28.17 (at point g, see Fig. 13)
3 4 5 4 3 2 1
modi®ed by trimming 80 mm from the bottom of the cylinder, and the resulting structure re-analysed. Fig. 16 shows the stress distribution round the intersection. Comparing Figs. 12 and 16, it can be seen that the Mises stress distributions in the untrimmed and trimmed structures are essentially the same.
3 2 1
Fig. 12. Von Mises stress distribution in the area of intersection (protruding nozzle).
5.3. Discussion From the foregoing results, it can be seen that there is no obvious stress concentration in the intersections in the optimised structures Ð a minimum material volume and a relatively uniform stress distribution have been achieved. The Mises stress on all the design surfaces is comfortably beneath the speci®ed constraint values, and the design surfaces are essentially homogenous stress surfaces. The small cyclic variations observed in Figs. 9, 14 and 15 are caused by slight surface discontinuities due to the size and shape of the elements used. It is observed that for the ¯ush nozzle the highest stresses occur at point h (an abrupt turning point) (see Figs. 6 and 8), and for the protruding nozzle at point g (see Figs. 12 and 14). These high stresses for both types of nozzle occur on non-design-surfaces and cannot be reduced signi®cantly by adding further material to the design surfaces. Table 1 compares the masses and the maximum Mises stresses within the optimised ¯ush and (trimmed) protruding nozzle intersections. Since there are two design surfaces
a
b c g d
e
h
i
k
f
Fig. 13. Speci®c points in the optimised pressure vessel intersection structure (protruding nozzle).
28
Hoop stress
Stresses (MPa)
24 20 16
von Mises stress
12
Radial stress
8
Axial stress
4 0 -4
Radial stress
Axial stress
-8
Hoop stress
von Mises stress
-12
b
g
h
k
i e
From points b to e (via g,h,k,and i) Fig. 14. Stress distribution on the interior surface of the optimised protruding nozzle intersection.
Stresses (MPa)
J.-S. Liu et al. / International Journal of Pressure Vessels and Piping 78 (2001) 1±9 28 26 24 22 20 18 16 14 12 10 8 6 4 2 0 -2
9
Hoop stress
Radial stress Hoop stress
Axial stress von Mises stress
von Mises stress
Radial stress
b
c
d
e
From points b to e (via c and d) Fig. 15. Stress distribution on the exterior surface of the optimised protruding nozzle intersection.
1 1 2 2
3 4 5
3 2 2
5
3
4
stress concentrations present is sought and a homogeneous surface stress distribution has been achieved. The method ®nds a minimum weight design for the intersection with a similar level strength as in the two pressure vessels being connected (in this case, the cylinder and the sphere). The employment of both rectangular and triangular elements makes the boundaries of the resulting optimised axisymmetric structures relatively smooth. The potential that the MD method has as a tool for investigating optimal topologies and shapes and for automatically and rapidly creating optimal, realistic, practical solutions to structural design problems has been illustrated by the examples presented.
3 Fig. 16. Von Mises stress distribution in the area of intersection (trimmed protruding nozzle).
available for the protruding nozzle (instead of one for the ¯ush nozzle), a design with a better stress distribution should be expected. Indeed, with this additional design degree of freedom, the optimisation results obtained for the protruding nozzle design are favourable from the point of view of reducing stress concentration. However, this is obtained at the cost of a heavier structure. Post-optimisation examination of the original design for the protruding nozzle enabled the nozzle to be trimmed, saving valuable weight, without compromising the structural performance. 6. Conclusions In this paper, the MD method, which is a systematic design procedure for adding material to and removing material from a structure, has been used to ®nd optimal pro®les for intersecting pressure vessels. A marked reduction in any
References [1] Freiberger W. Minimum weight design of cylindrical shells. ASME J Appl Mech 1956;23:567±80. [2] Calladine CR. On the design of reinforcement for openings and nozzles in thin spherical pressure vessels. J Mech Engng Sci 1966;8: 1±14. [3] Diallo B, Ellyin F. Optimization of connecting shell. ASCE J Engng Mech 1983;109:111±26. [4] Ellyin F. Shape optimization of intersecting pressure vessels. Proceedings of the IUTAM Symposium on Structural Optimization, 1988. p. 77±84. [5] Liu J-S, Parks GT, Clarkson PJ. Can a structure grow towards an optimum topology layout? Ð Metamorphic development: a new topology optimisation method. Proceedings of the Third World Congress of Structural and Multidisciplinary Optimisation (WCSMO-3), 1999. CDROM. [6] Liu J-S, Parks GT, Clarkson PJ. Metamorphic Development: a new topology optimisation method for continuum structures. Struct Multidisciplinary Optimization 2000;20:288±300. [7] Hibbitt, Karlsson & Sorenson, Inc. Abaqus manual, Version 5.7-5. Pawtucket RI: Hibbitt, Karlsson & Sorenson, Inc., 1998.