G. A. Mohr Applied ~om~utatiooai Mechanics in BngineeriniS;, 7 Marine Avenue, St Kilda, Metboume, Australia

A~~-~~~~it~ element models of a wide range of probkns are aptimixed using the steepest descent method and merit functions appropriate to each particular problem. Here, in part I, we deal with typic4 structural problems in w&h the merit functions are base4 an the stresses caused by loading of the structure. In part II we shall deal with problems ia which the structure is loaded by a fluid flow and we alter the structure shape to improve the efficiency of the ~uid/stt~t~e interaction.

fn the first two decades since the i~tr~~c~on of the finite element method fl] little attention was directed at op~~z~~g f&&l: element mode& [2& of ~rt~G~ar note, however, bemg the pro& of GdfatEy and Gai- tagher [3j in which &fully stressed design (FSD) and the steepest descent method is used for this purpose,

In the last decade, however, interest in the optima ization of finite element models has increased [4-g], in part because of the av~~ab~lity of high speed computers f9f.

In the present paper stress criteria and the steepest descent method are applied to a simple truss probfem studied by Wang using sequeatial linear prog~rn~ ming {IO], and to the arch problem studied by M&r using optim~~ity criteria[I1] in both cases yielding ~rnpr~v~ rest&s. The techniques used can readily be extend& to frame and sheh probiems and, as will be demonstrated in part ff, to many other types of problem.

The prob~m of op~rnj~~~ the ~r~s~-~ct~o~~ areas of the members of simple truss structures provides a useful introductory example of the appli- cation of the steepest descent method to finite eiem~~t models. The procedure used here uses the following strategy:

fi) FSD is first used to obtain an ~rnp~~~ design which witI usually be close to the optimum sofution.

(2) The steepest descent method is applied to the FSD solution to obtain the optimum solution, that is the change in each of the design variables {xi 3 is given by 021

S{XJ = -iff5~/zkj) = -I,&], (XI

where R. is the step length in the direction of tke gradient vector fg) andfis the merit function which

is here the volume of material in the truss,f = G AI Li, where Ai, 15~ are the cross-~ti~na1 areas and lengths of the truss members.

The gradient vector is ikst determined by a pertur- bation 6.~~ of each design vtiabk Qn this case the member areas A,) in turu and ~~~i~t~ng the resubing change in the merit function, t?& giving

is& ~r~~b~~~o~ requiring a re-analysis of the strut- ture to determine the associated value of SA.

The o~~irnurn step Iengtb in eqn fij_ that which minimizes f, is found by a search, using gradually increased trial values of 2 until the minimum f is bracketed and then bisection or interpolation (usually Iinear or quad~~~rl2~ is used to locate the mini- mum, T&n a new gradient vector is ~a~~u~~~ed by successive ~rtur~~~~s of the cmrent vafues of the design variables and this used for the next search and the procedure is te~inated when no further re- duction in f can be obtained,

Figure I shows a sisraple empIe probkm in which there are two a~ternat~~~ foad cases, la such situations the element areas are rescakd in the FSD or gradient vector calculation using the largest stress ratio (rr/TLIM or a/CLIM, where TLIM and CLXM are the tensile and compressive stress limits) given by the afternative load cases,

fn this type of problem the foI~ow~ng rest~~~t~o~s are r~m~ende~

(1) In capsulating fgj the element areas are dou- bled, that is dn, = xi.

(2) Only one pre~~rnj~ar~ FSD step is used (fu~her Use of FSD until convergence is obtained results in member 4 vanishing in the present probEem and this is not the optimum sofution).

(3) fn all but the first steepest descent search, a first step length of IO-% is used. Then if no decrease inf is observed with this test step the search direction is

1218 G. A. Mohr

unprofitable and the current solution is taken as the estimate of the optimum. Then, using these rec- ommendations, the results in Table 1 are obtained.

The present FSD with the steepest descent pro- cedure gives a better result than that obtained by Wang [lo] using sequential (stepwise) linear program- ming (SLP). Note, however, that at the end of each LP step Wang scales all the element areas by the largest stress factor encountered in any element and this cannot yield the true optimum solution. Note also that all solutions given by the present method are valid solutions because each element is factored ac- cording to its stress level at each step.

The present procedure could easily be applied to the optimization of frame structures. In these optim- izations the sizes of the elements of the frame would follow the same process used for the foregoing truss problem, except that section limits (that is, a mini- mum allowable size for each element) must be specified, but inclusion of these in finite element programs is a trivial exercise.

Finally, in relation to optimum truss problems, the classical Hencky-Prandtl network solutions of Michell[13] are worth remembering. These are for very special cases but for the case of line (rather than point) support Rozvany and Gollub [14] report for this less restricted case solutions correspond to fields of constant strain, which give layouts consisting of a finite number of straight members and such solutions may have more practical applications.

OITIMlZATlON OF AN ARCH

Next we optimize the shape of an arch structure using the curved quadratic element of Mohr and Garner [ 153. The element has freedoms M, t, 4 at three nodes and the extensional strain, flexural curvature and transverse shear strain are given by

c = &i/ax + v/R (3)

X = ~a~f~~~l~~~ + iqjak (4)

y = adjax,- 4, (5)

where x is a curvilinear coordinate along the element and u, u are respectively the tangential and normal displacements.

Using quadratic interpolation with a dimensionless coordinate s = 2x/L (s = - I+ l), where L is the element arc length, for u, v, 4 the strain interp- olation matrix is given by

I 146 E = 30.000 TLIM = 20 I CLIM = 15

Fig. I. Truss with two alternative loadings. bers are underlined.

where

418

-I

Element num-

{j> = ((s2 - s)j2, 1 -s?, ($2 + s)/2) (7)

{f} = (2s - 1, -4s, 2s $ I}/L (8)

and a = cos c(, b = sin tl, where CL is the local slope of the element.

Using two point Gaussian quadrature the element stiffness matrix is given as

k = CMW~)B:B, + W)B;B, + P(~GA/~)B:BJ, (9)

where B is a penalty factor and the element is assumed to have constant cross-section (and hence A and I). For thick arches fl = 1 but for thin arches (as in the example studied here) p = 100t2, where t is the el- ement thickness [I 51.

In the example problem of Fig. 2, however, ,8 = I was used as the arch is very thin (span/ thickness = 160) and hence the shear strains are not taken into consideration in the optimization process.

The merit function used is the factor weight which is the material volume scaled according to the stress ratios in each element, that is we assume unit density for convenience. This is then calculated as

,f = y LwtF, (10) i=,

where L, Wand t are the element arc length, breadth and depth, respectively, and F is the largest vaiue obtained by dividing the two extreme fibre stresses by the tensile and compressive stress limits.

In shape optimization applications we use straight elements, however, as movement of the central node

Finite element optimization of structures-l 1219

Table 1. Optimization solutions for the problem of Fig. 1 using FSD with steepest descent

A, A, A, A, f Initial 1.440 2.000 1.800 2.000 45.040 FSD {I step) 0.231 0.264 0.104 0.143 4.421 Search 1 (A =0.09) 0.296 0.189 0.139 0.085 4.200 Search 2 (A = 10sm) 0.316 0.224 0.123 0.055 4.152 (A = 0.03) 0.310 0.213 0.126 0.064 4.148 Search 3 (A = lO-m) Test step increased f so stop 4.178 SLP solution

(Wana IlOll 0.364 0.261 0.121 0.084 4.837

Table 2. Optimization results for an arch with horizontally distributed load

of an element during gradient vector calculations produces a local arching effect which leads to poor results. Hence only the elevations of nodes 3, 5,7 and 9 in Fig. 2 are specified as shape function variables and 10% increments in these are used to calculate the gradient vector.

The results for the caSe of a horizontally dis- tributed load are given in Table 2. After completion of steepest descent searching the resulting shape is closely parabolic as expected, largely eliminating the bending stresses. In this thin arch, however, the remaining bending stresses still swamp the exten- sional stresses.

To obtain the final optimum shape, therefore, the arch shape is taken as parabolic; this is the design shape for elimination of bending. Scaling of this shape is then used to obtain the final optimum shape. For this, a search with a gradually increasing scale factor is used to minimize the factor weight, which is now predominantly a function of the extensional stresses.

The results are in reasonable agreement with the exact solution [16] and those of the optimality cri- terion method [Ill given in Table 2. In the latter, constant strain shell elements are elevated according to their bending stresses in order to reduce them, and scaling is used simultaneously to minimize the factor weight, calculated only from the extensional stresses. The present method is to be recommended, though, because it does not presume that bending should be eliminated.

q= 10

E = 20 x 106, v = 0.2

w = 4, t 3 0.2

TLIM = CLIM = 1000

A------ -+ Four elcmcntv @ 4 m Fig. 2. Initial shape of an arch and design shape after

application of steepest descent method.

Y3

Initial 2 Search 1 (A = 0.04) 3.06 Search 2 (2 = 0.03) 2.49 Search 3 (A = 0.012) 2.54 Search 4 (A = 0.035) 2.67 Search 5 (1 = 0.005) 254 Search 6 (A = 0.~03) 2.53 Design shape 2.8 Scaled shape (A = 2.19) 6.13 OCM technique [ 1 I] 5.6 Theoretical solution [ 161 6.06

Ys Y7 Y9 f 4 6 8 71.11 4.39 5.28 1.35 26.42 5.13 4.99 6.94 16.30 4.80 5.39 6.67 Ii.94 4.50 5.35 6.25 8.95 4.53 5.37 6.19 8.70 4.53 5.37 6.19 8.69 4.8 6.0 6.4 4.14

IO.51 13.14 14.02 2.95 9.3 11.6 13.5 -

10.40 12.99 13.86 -

An alternative procedure is to base the factor weight calculation of the present method only on the extensional stresses from the outset and again apply steepest descent. This gives an arch of similar height to that expected but, with the triangular initial shape of Fig. 2, the solution tends towards a w shape (with f = 2.73). Such a solution would involve ex- tremely high flexural stresses and is not feasible.

Finally, for the case of a surface distributed load, the steepest descent/scaling method is applied once again, giving the results in Table 3. These are reason- able but not as good as those for the horizontally distributed load case, in part because the true opti- mum shape is not now parabolic, as assumed prior to scaling. Overall, however, the effectiveness of steepest descent techniques in arch and shell problems is demonstrated, as it will be for fluid loading prob- lems in part II of this paper.

The present procedure can easily be applied to three-dimensional and shell problems, as can that of Mohr [I 11. Once again, as noted at the close of the preceding section, section limits (minimum and per- haps maximum permisssible element thicknesses) will generally need to be specified. Additional details such as these are easily added to finite element programs.

CONCLUSIONS

The steepest descent procedure used in the present work gives better results than the sequential linear programming method of Wang for trusses [IO] and the optimality criterion method of Mohr for arches 1111.

Table 3. Optimization results for an arch with surface distributed load

Y, Y5 Y7 Y9 f Initial 2 Search 1 (A = 0.04) 3.25

44.37 65.14 87.28 79.50 32.80

Search 2 (i = 0.03) 2.68 5.09 5.31 6.73 13.63 Design shape 2.8 4.8 6.0 6.4 4.14 Scaled shape(I = 1.71) 4.79 8.21 IO.26 10.94 3.53 OCM technique [l I] 4.3 7.4 9.2 IO.2 - Theory (approx) [I I] 4.27 7.32 9.15 9.76 -

1220 G. A.

Like the latter the present procedure can be gener- alized to deal with shell problems, though in these a large number of elements with thirty DOFs or more may be involved and calculation of {g) on an element by element basis may be impractical. This difficulty can be overcome by using a polynomial shape func- tion z =f(x, y) and using the relatively few co- efficients of this as the design variables.

When we also seek to optimize the thickness of the shell, another shape function t =f(.x, y) and, of course, section limits will be needed, as they will be in most structural problems.

There are, of course, many other possible appli- cations of the FEM and optimization, for example to the optimization of fluid flows coupled with the optimization of an associated structure, and simple examples of this type of problem are considered in part II.

REFERENCES

M. J. Turner, R. W. Clough, H. C. Martin and L. J. Topp, Stiffness and deflection analysis of complex struc- tures. J. Aero. Sci. 23, 8055823 (1956). F. 1. Niordson and P. Pedersen, A review of optimum structural design. Proc. 13th Int. Cong. Theoretjeal Applied Mech. University of Moscow (1972). R. A. Gellatly and R. H. Gallagher, A procedure for automated minimum weight structural design, II: Appli- cations. Aero. Quart. 17, 216224 (1966). R. J. Allwood and Y. S. Chung, An optimality criteria method applied to the design of continuous beams of

5.

6.

7.

8.

9.

10.

II.

12.

13.

14.

15.

16.

varying depth with stress, deflection and size con- straints. Comput. Struct. 20, 947-955 (1985). V. Thevendran and D. P. Thambiratnam, Minimum weight design of conical concrete water tanks. Comput. Struck. 29. 699-708 (1988). T. Kam and R. R. Chang; Optimal design of laminated composite plates with dynamic and static consider- ations. Compuf. Struct. 32, 387-396 (1989). T. Yao and-K. K. Choi, 3-D shape optimal design and automatic finite element regridding. Int. J. Numer. Meth. Engng 28, 369-381 (1989). E. A. Sadek, An optimality criterion method for dy- namic optimization of structures. int. J. Numer. Meth. Engng 28, 579-587 (1989). E. Sikiotis, V. Saouma, M. Long and W. Rogger, Finite element based optimization of structures on a Cray- MP supercomputer. Comput. Struct. 36, 901-909 (1990). C. K. Wang, Computer Methods in Advanced Structural Analysis. International Textbook Co., New York (1973). G. A. Mohr, Design of shell shape using finite elements. Comput. Siruct. 10, 745-749 (1979). M. J. Box, D. Davies and W. K. Swann, Nonlinear Optimization Techniques. ICI Monograph No. 5. Oliver and Boyd, Edinburgh (1969). A. G. M. Michell, The limits of economy in frame structures. Phil. &fag., series 6, 8, 589-593 (1904). G. I. N. Rozvany and W. Goliub, Michell layouts for various combinations of line supports-I. Int. J. Mech. Sci. 32, 1021-1029 (1990). G. A. Mohr and R. Garner, Reduced integration and penalty factors in an arch element. Int. J. Struct. 3(l), 9-1s (1983). G. A, Mohr, Finite Elements for Solids, Fluids, and Optimization. Oxford University Press, Oxford (1992).