- Email: [email protected]

OPTIMIZATION USING ITERATIVE DESIGN TECHNIQUES W. R. SPILLERS and S. AL-BANNA School of Engineering and Applied Science, Columbia University, New York, N.Y. 10027, U.S.A. AMraet-A scheme is presented whereby a designer can achieve an optimal design by performing successive reanalyses (iterative design) using at each step the proper stiffness matrix. The scheme is applied to sandwich plate and frame design problems.

INTRODUCTION is concerned with iterative design techniques and their relationship to minimum weight design (optimi~tion). It is quite distinct--at least in its motiviation-from the existing and reasonably well documented work {l] concerned with the application of nonlinear programming methods to structural design. The motivation for this paper derives largely from the relative success of working designers over the more formal procedures and the fact that it is now impossible to do formally what designers do rather routinely and reasonably well. What designers, of course, do is iterative design. For example, a great deal of design today is allowable stress design in which at each step an attempt is made to satisfy an allowable stress design criterion. On the surface it would appear that such an attempt is an attempt to deal with safety since its violation ordinarily implies an unsafe structure. But in attempting to satisfy such criteria designers arrive at structures which are ‘reasonably’ optimal or at the very least buildable_ Motivated by this apparent conjunction of safety and optimization, it has been shown that attempting to satisfy an allowable stress criterion for a simple model of the truss [2] (and its continuous analog the sandwich beam [3]) leads to an optimal design. It is the intention here to extend these earlier investigations to more complex classes of structures. Specifically, a natural generalization of the truss problem is developed by approaching iterative design through Newton’s method and the Kuhn-Tucker condition. This generalization is discussed in detail for simple cases of frame and sandwich plate design. THIS PAPER

Iterative design The equations of the node method of linear structural analysis may be written in the form HF=P ~uilib~um F= KA

Hooke’s law

A =NS

member-node displacement relationship.

(1)

Briefly, F and A are the (b x 1) member force and displacement matrices, P and 6 are the (n x I) node force and displa~ment matrices, K is the (b x b) primitive sti&ess matrix, and N, the (b x n) generalized incidence matrix. It is most common to solve equation (1) as 6 = (RKI?) - ‘P 1263

(2)

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W. R. SPILLERS and S. AL-BANNA

from which F and A can be found easily. While equation (1) imply a skeletal structure, it should be noted that when continuous systems such as plates, shells, and solids are written in discrete form using finite difference or finite element techniques, their appearance is identical with equation (1). Since the matrices F, A, and S can be written as functions of the matrix K, the optimal design problem can be stated in terms of finding the primitive stiffness matrix K which corresponds to minimum weight or cost while satisfying certain associated side conditions. As a rule this is a nonlinear programming problem of considerable complexity. In the case of the truss problem mentioned above, simplification was achieved by relaxing Hooke’s law (which of course opens the question of elastic realizability). That technique will again be used here but the more complex classes of structures as a rule do not reduce to linear programming problems as does the truss; there are, however, classes of structures which reduce to homogeneous programming problems with linear constrains-the next step up in complexity from linear programming. Formally, iterative design proceeds in the following manner. At the mth step the member forces I;(“” are computed from equation (1) using the primitive stiffness matrix Pm) as J’(“‘)= j+“‘N([email protected]“N)

- ‘jJ.

(2)

On the basis of the situation at state m, a new stiffness matrix Ke”+l) is selected-ordinarily in some heuristic manner-and the forces and displacements are again computed from equation (1). When the designer is satisfied with his structure he stops iterating. Very little is actually known about the dependence of the member force matrix Ftm) upon the stiffness matrix K (N (equation 2). This equation can, however, be put into a rather interesting form using the equilibrium equation, the first of equations (1 and 2). Replacing P by RFcme2)in equation (2) gives f;(m)= @m)N(#K(NN)-

IRFCm- 1)

(3)

or F(d-.p)FF(‘“-1)

(4)

(actually equations (3-4) remain valid when F (m- ‘) is replaced by any F(O) which satisfies equilibrium.) It was pointed out in reference 4 that the matrix A(“) is idempotent (i.e. &N&0 =IA(m)) and therefore has eigenvalues of zero or one. Using the spectral norm [5], it follows from equation (4) that

which implies that the norm of F remains constant from one iteration to the next. Such a result can be regarded as necessary to iterative design since if it were not true the force matrix would either converge to zero or diverge. It was shown in [l] that iterating Kji”l+‘)=jFjm)lE/(&LJ

(6)

Optimization Using Iterative Design Techniques

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where E Young’s modulus

CT“allowable stress Li length of member i

for the truss leads to minimum weight designs. Unfortunately it is frequently not clear how to generalize equation (6) to higher order problems such as frames and plates. In fact, in some cases allowable stress design is known not to converge to minimum weight. At least part of the difficulty lies in the fact that while equation (6) has been shown to achieve the desired results it has not been well motivated from the point of view of optimization. It is the purpose of this paper to provide a generalization of equation (6); this generalization incidentally provides a clear motivation for equation (6) in terms of optimization. The simplified problem

In order to avoid some of the nonlinear programming difficulties, the optimization problem for the truss was simplified [I], by relaxing Hooke’s law, to the dual linear programming problem,

Primal problem.

Miuimize CIF& i

Dualproblem.

subject to NF=P

Maximize PC?subject to I(NS),l

where A f = a"L,/E .

Here the problem is to find F and 6 given A”, P, and N. While equation (7) can be put into one of the canonical forms of linear programming using standard changes of variables, the difficulty with the more complex problems is that in the general case, the objective function can no longer be reduced to a linear function of the independent variables. In the simple case of the frame (which does reduce to a linear programming problem), the form of the objective function is different from that of the truss and the iterative scheme, equation (6), used on the truss no longer works. In order to clarify the discussion which follows two specific problems are now formulated but it should be noted that this paper is in no way restricted to these sample problems. Sandwich plate design. Minimize IT(m)ds subject to LiU=q where

T(M) = (m,Z --m,m,+m,Z+3mxy)+,

and L

mW

J

L=(a’/ay’,

-2a2/ay2,

-2a2/axayj

(8)

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W. R. SPILLERSand

S. AL-BANNA

Frame design. Minimize

subject to RF = P

(9)

where m:, m; are themomentsat the ends of member i. In these two examples, Hooke’s law has again been relaxed allowing a formulation in terms of forces which, without repeating the discussion of earlier papers, is most easily regarded as plastic design but may also be regarded as embedding an elastic design problem into a problem in which Hooke’s law is relaxed leading to a lower bound on the original problem. In the sandwich plate problem, equation (8), it is desired to find a distribution of moments as a function of x and y which satisfies equilibrium and corresponds to a minimum volume of facing material which is written as an integral over the moments using in this case von Mises’ yield condition. (The sandwich plate problem is presented in detail in the book of Hodge [6].) In the frame problem, equation (9), it is assumed that the weight of a member is proportional to the absolute value of the maximum of its end moments which is only approximately true [7] but which, of course, implies joint loading and neglects axial load effects. While the technique to be presented in the next section has rather unexplored potential, it is clearly valid beyond these two sample problems and for that reason a more general system to which it can be applied is now presented: Primal problem. Minimize t(F) subject to RF= P. Dualproblem.

(10)

Maximize PC?subject to ]([email protected] I ](Vt,),].

The notation used here is not obvious and requires a certain amount of explanation. The primal is most straightforward and is simply the problem of minimizing a function t(F) of independent variables F subject to linear constraints. Furthermore it is assumed that the function t is convex, homogeneous of degree 1, and possesses the number derivatives required by the applications which follow. The dual problem requires finding the vector 6, which maximizes the linear function PC?and satisfies the constraints indicated. Given 6, the vector A=N6 is of the same dimension as F and it is therefore proper to write Vt(A) The symbol vt,, is used to indicate the vector Vt evaluated at some point y at which vt(y) and A are parallel and act in the same direction. In terms of this notation the inequality, k%3=~N6=ffA&‘t,I~VtF=t(F),

(11)

(see Appendix) which is typical of duality is valid. The primal problem of equation (lo), minimizing a convex, homogeneous function subject to linear constraints, is probably the next step up in complexity from linear programming. It is for the solution of this problem that an algorithm which generalizes allowable stress design is now presented. An algorithm In this section the previous notation is maintained except that the vector 6 will now be used as a Lagrangian multiplier and will not correspond to the dual variables of equation (10) since values of 6 which may not satisfy the constraints of the dual will be used.

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OptimizationUsing Iterative DesignTechniques From the primal a Lagrangian function, &t(F)[email protected]),

(12)

can now be formed. Differentiating equation (12), first with respect to 6 and then with respect to F, in the usual manner gives the Kuhn-Tucker conditions for an optimal solution F 171, iw=P vt(F)=Nd.

(13)

As an alternative to attempting to solve the primal problem directly, it is equivalent to find F, 6 which satisfy equation (13). Equations (13) form a system of nonlinear equations. One of the most common procedures for attaching such a system is Newton’s method in which the system is approximated by the first terms of its Taylor series expansion. But the fact that the function 1 is homogeneous of degree 1 implies that the Hessian matrix oft is singular (see Appendix). In order to avoid the difficulties associated with this singular matrix, the system

2tVt(F) = 2tN6

(14)

which is equivalent to equation (13) since t > 0 will be substituted for it. Let rp=P

*

Vip=2tVt,

(1%

equation (14) can then be written as iv’F=P [email protected]

(16)

Expanding equation (16) in a Taylor series and keeping only the linear terms leads at step n to the system RF(“) = P H(“-l)F(n)=2J~(n-l)N6(“) (17) where H is the Hessian of cp. In deriving equation (17) use has been made of the fact that since cp is homogeneous of degree 2, Vq(F)-VC/@‘“)+H(F”)

- (F-F")=H(F")

.F

It may be noted [8] that H is positive definite. Equation (17) describes the algorithm proposed in this paper. It is presumed that the matrix F(“-l) is known at iteration (n- 1) from which F(“’ is computed using equation (17). Since this is essentially Newton’s method it can be expected to show the convergence characteristics of Newton’s method.

W. R. SPILLERSand S. AL--BANNA

1268

Returning to the earlier discussion of iterative design, it is necessary to have some scheme whereby a new stiffness matrix can be generated given a set of forces at iteration (n- 1) from which an improved set of forces is then computed. In view of the form of equation (17), H(“-‘)/(4#“-‘) ) + p 1ays the role of the inverse primitive stiffness matrix so that the algorithm, equation (17), may be regarded as simply iterative design. This allows the designer who wishes to optimize to do so by modifying his stiffness matrix as indicated and iterating using his analysis computer programs. Finally, it should be added that the computation of the Hessian matrix H can be difficult which in turn can make Newton’s method unattractive. For the cases to which the authors have applied this algorithm the function t(F) has been separable, i.e.

In these cases it is convenient to replace equation (14) by NF=P

2t,Vt,(F,) = 2t,(N6),

(19)

which implies that the ith submatrix of Vt is multiplied by its corresponding component t,. This is done so that the system equation (19), a modified form of equation (16), remains separable which allows the Hessian to be computed explicitly in the examples which follow. Examples

Specific cases of the sandwich plate and frame problems described in an earlier section are presented in this section. For these problems the algorithm presented seems to work well and the details of the calculations are not presented. Figure 1 shows a fixed square sandwich plate loaded over one half of its surface area and the thickness contours which result after ten iterations. In this example unit values of yield stress and core height are used and the calculations follow the outline presented in the earlier sections after the problem has been reduced to a difference form. Figure 2 shows the obiective function versus the iteration number. The final weight shown has been verified using a gradient method which incidentally converged more slowly.

Schemalx

Fac/ng Thxkness

Contours

FIG. 1.

of Load

Optimization Usingkrativc DesignTechniques

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Figure 3 shows an example of the frame design problem of equation (9) and one of its optimal moment diagrams. Like the truss problem, it is always possible to solve for an optimal solution using linear programming but it is of some interest to include problems of this kind within iteraaive design techniques. That is the reason for discussing this example.

._,

27O,OUOm-/&

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W. R. SPILLERS and S. AL-BANNA

From the point of view of this paper, the frame design problem, equation (9) is, degenerate in two respects. Since the objective function t is not a function of the axial loads, it is necessary to find joint displacements-Lagrangian multipliers-which have no associated member length changes (infinite member stiffnesses) in the axial direction; that is accomplished here by using ‘large’ member areas. Furthermore, since the objective function does not possess the required number of derivatives, it isnecessary to approximate it by a function which does. The function used in numerical calculations for this paper was

When M+oo equation (20) is exact but it is a good approximation when M is a large even integer (8 has been used here in calculations). In an attempt to achieve an arbitrary starting point the initial forces were obtained using an identity matrix for the bending part of the stiffness matrix in an analysis program. The axial stiffness was maintained at 10” throughout the iterative process. (All units are in terms of inches and pounds). The convergence shown in Fig. 4 was quite good but the problem is, of course, small.

0 1296 (Opttmal

I 0

Solution of linear

Programming PrOblemI

I

,

I

I

/

I

2

3

4

5

6

Iterat/on

FIG.

4.

CONCLUDiNG REMARKS This paper has presented a motivation and an algorithm for iterative design and discussed their application to two simple design situations. One of the attractive aspects of the procedure used is that it has the appearance of analysis (using a modified stiffness matrix) which is not only familiar but also efficiently practiced. Since Hooke’s law is relaxed it is not now clear how such things as displacement constraints are to be included. But the procedure does help explain the effectiveness of existing iterative design procedures and offers certain potential for their improvement. Acknowledgement-This

work has en

supportedby the National Science Foundation.

REFERENCES [l] M. Z. COHN(Mitor), An Introduction to Structuruf Optimization, Study No. 1, Solid Mach. Div. University of Waterloo, Ontario (1%9). [i2] W. R. SPILLERS and JOHNFARRELL,On the analysis of stmctural d&n. J. Math. Analysis and Appl. 25, (2), 285-295 February (1969).

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Optimization Using Iterative Design Techniques

[3] W. R. SPILLERS,A note on the design of beams. To appear in J. uppl. Mech. [4] W. R. SPILLERSand S. AL-BANNA,Iterative design techniques. Preprint 1373, ASCE Baitimore Meeting, April (1971). 151EUOENEand KELLER ISAACSONand HERBERTBISHOP,Analysis of Numerical Methods. Wiley, New York (1966). [61P. HODGE,Plastic Analysis of Structures. McGraw-Hill, New York (1959). 171W. J. MCCUTCHEON and S. J. FENVES,A general formulation for the optimum design of frame structures.

Civil Engr. Studies, Struct. Res. Series No. 362, University of Illinois August (1970).

PI WILLARD I. ZANGWILL, Nonlinear Programming: A UnifiedApproach, p. 187. Prentice Hall, New York, (1969).

PI E. R. LORCH, Differentiable inequalities and the theory of convex bodies. Trans. A. M. Sot. 71 (2), 243-266 September (1951). (Received 8 June 1972)

APPENDIX Homogeneity and convexity Homogeneity and convexity are used elegantly in the work of Larch [8] but only simple results are required by this paper.

A scalar functionf(x) scalar a20

of a vector x is said to be homogeneous of degree r if for any f(ax) = cr’f(x) .

Euler’s theorem for homogeneous functions states that

rf(x) = 1 Xiaf/dXi ,

or

rf(x) = X * yf

from which it follows that (r- l)aj/aXj=

C Xi~‘f/dX*8X,

or

Y

(r - l)Vf=H

*X

where H is the Hessian matrix of the function.f, i.e.

Note than when r= 1 the Hessian matrix is singular since H(x) has the eigenvector x. The function of the preceding paragraph is said to be convex if

Let j’ (x) be homogeneous result [8]

of degree 1. Homogeneity and convexity come together in the f=x

which is valid for any y.

* Vf(X)TX

* Vf(y)