optimal structural remodeling of multi-objective systems

Compurers & Slrucrures Vol. 18, No. 4, PP. 61%628, 1984 Printed in Great Britain.

004s7949/84 s3.00 + .oo 01984 Pergamon Press Ltd.

OPTIMAL STRUCTURAL REMODELING OF MULTI-OBJECTIVE SYSTEMS

B. F'RASAD Engineering and Research Staff, Research, Ford Motor Company, Dearborn, MI 48121, U.S.A.

and

J. F. EMERK~N PACE Systems, Inc., Los Gatos, California, U.S.A.

(Received 19 November 1983; received for publication 19 April 1983)

Abstract-The paper describes a statistical formulation for the structure modification problems, where two or more performance measuring functions (e.g. stiffness, strength, frequency, modeshape, weight, etc.) can be specified at several points on the structure. “Optimal Remodeling” refers to the best possible form of modification which would satisfy the prescribed limits imposed on these performance measures. The design variables may be any parameter associated with a generic finite element model such as sectional, material, geometric, or shape properties. The formulation is based on statistical system identification methods and penalty approaches that move as many performance measures as possible into an acceptable region. As an illustration of these concepts, examples have been chosen from two types of problems: (a) shape optimization to minimize stress concentrations and (b) fully stressed design (FSD).

1. INTRODUCHON

Mathematical techniques have been tried in a number of different forms such as optimality criteria [ l] recur- sive methods, state space method, mathematical programming[2], variational approach, etc. to produce an optimal or best solution to a design problem. Basically, most of the techniuques in one form or the other, consider a merit function, which is to be optimized (minimized or maximized) subject to a given set of constraints. The weight and cost have been the most commonly used merit functions in structural design as evidenced by the large number of papers published (Ref. [l], e.g. gives over 200 references.) Several other merit functions have also been used depending upon the specific design situations and the overall objectives. In a number of cases, stresses have been used as merit functions for fully stressed designs and stress concentration problems[3-51. The amplitude of the dynamic response (modeshapes) was used for reducing vibration and for improving vehicle ride[6]. In Refs. [7-91, a set of system frequency goals was used to obtain desired system performance characteristics. These references are not complete but merely indicative.

It is frequently difficult to identify one quantifiable function that satisfactorily describes the quality of a stipulated design. A single basis for the measure of the performance characteristics, moreover, may not be an adequate and dependable criteria for the design to perform well in multi-operating environments. Often, two or more independent functionals (e.g. stiffness, strength, frequency, modeshape amplitude, weight, etc.)--specified at several sampling points are necessary. They are further restricted by a pair of upper and lower limits on these functionals to get a desired degree of satisfaction.

Unfortunately, in the literature, there exists no convenient method which could handle this type of multi-objective system easily. Most of the work on multi-objective systems is limited to the use of system identification techniques [8-91. But these techniques as such are only designed to attain prescribed target values rather than to guide the design to lie within an allowable band (region of acceptability). There exists also very little work where this technique was shown to be successfully used for both static and dynamic identifications simultaneously. Olhoff and Taylor [l l] have used variational principles; however, their technique is only limited to weight identification before and after modification. Others,[6,12, 131 who have considered more than one function have tried least square approaches or some weighted residual forms. The latter forms are not really different than problems with a single objective system since all the functions have been effectively merged into an independent functional, which is to be finally optimized.

A general class of structural remodeling problems is considered here where limits on several performance variables can be specified independently. Optimal remodeling refers to the best form of modification which would satisfy these performance variables. A technique is also proposed to attach weighting factors to the performance functions so that if the process is halted as a result of competing requirements, a favorable course could be deter- mmed. The development pertains to all types of problems where the measures of the performance variables consist of one or more of the following functionals: stiffness, strength, frequency, modeshape amplitude and weight. The design variables may be selected to be almost any parameter associated with a generaic finite element model[l4] such as sectional,

619

620 B. PIUSAD and J. F. EMERSON

material, geometric or shape properties. Additional “side limits” can also be imposed directly on the design variables to avoid undesirable solutions from an engineering standpoint.

2. STATEMENT OF THE OPTIMAL

REMODELING PROBLEM

The problem is to find a solution for the design parameters veD, which would drive most of the following functionals into their acceptable ranges:

MY’” I t+(D) I UT; i = 1,. . ,p

o~“la,(D)lo~; i=p+l,...,q

1y<I,(D)<Izr; i=q+l,...,r (1)

w~<wi,(D)<w~; i=s+l,...,m

where u, Q, 1 and rj represent the deflection, stress, eigenvalue (frequency) and eigenvector (modeshape) amplitude. These functions are named here as “performance variables”. They are assumed to be continuous functions of the design variable v. The sub- scripts are: j is the nodal point for deflection or modeshape amplitude; I is an element group consis- ting of one or more elements of a given type; k is the number of eigenvalue/eigenvector of interest; e is the type of the elements, e.g. plate, membrane, beam, etc.; and m is the total number of entries for the functionals.

The design variables are also constrained to lie in a design space D(v) such that:

v,,, I v I u,,,.

These are called “side constraints” because they are imposed directly on the variables. Two types of such constraints can be included: first, constraints imposed to avoid the possibility of highly distorted geometry or shape such as negative areas, manufacturing toler- ance, etc. Second, constraints imposed by design requirements such as needed to avoid undesirable numerical solutions from a practical design standpoint.

2.1 Scale invariant problem In order to avoid the possible difficulty in solving

a poorly scaled remodeling system, the problem posed in eqn (l), can be transformed to find a solution of the design parameter which would drive most of the functional values into a unitized space such that:

where the normalized functonal f; can be any of the

performance variables defined as:

With this transformation, every functional freedom in eqn (1) is normalized and is limited to an equal range, e.g.f;(D)E[O, 11. An acceptable design from an optimal remodeling standpoint is then one which satisfies these limits (eqns 3-7) as closely as possible.

3. DERIVATION OF SYSTEM REMODELING

EQUATIONS

The following derivation is based on statistical identification methods[8,9] which consider each of the performance variables as independent functions of the design variables. The formulation maintains uniformity of the response functions through a common finite element representation of the model. The work presented here is to propose a general method which will identify the properties of this FE structural model capable of providing a set of performance measuring goals. The method uses values of the structural properties originally assigned to the model as the starting point. The method then modifies original property values to drive the performance measures to lie within the target bounds. Accuracy of the initial estimates for the structural parameters and the designer’s confidence on the bounds imposed for the functionals are incorporated into the procedure. Figure 1 shows a macro- schematic of the optimal remodeling sequences.

3.1 Basic equations The solutions for static deformation u, strain E and

stresses Q are found using the finite element equations:

Ku =p (8)

C.7 = 66 (9) and

.s=Bu (10)

where K is the stiffness matrix, u is the displacement vector, P is the external force vector, D, B are the elasticity matrix and strain matrix, respectively. Com- bining eqns (9) and (10)

u = dBu = Su (11)

where S is a stress matrix. The eigenvalues and eigenvectors of a multi-degree

Optimal structural remodeling of multi-obj~tive systems 621

l Initial Structural Parometar Values, vg

l Possible Variance On vQ

# Lower And Upper Bounds

8 Specified Limits On Deflection. Stress, Frequency Mode Shape And Weight

, Description Of Sompling Points For Each Function01

, Variance On The Bounds Imposed For The Functionols

Design Sensitivity Proc.

~

T Matrix

t Optimal Remodeling Proc.

, New Values Of Structural Porometer I 8 Uncertainty In New Structural

Parometer Values I

Fig. 1. Diagram showing the information flow sequences in the optimal remodeling procedure.

of freedom system is obtained by solving the set of dynamic (characteristic) equations.

[W&j = W+#,J (12)

where M is the mass matrix, & = kth eigenvalue, Wk = (2,) ‘/* = kth natural frequency and #k = eigenvector (modeshape) of kth mode. The mass and stiffness matrices are functions of the structural parameters of the system, and, therefore, deflections, stresses, eigenvalues are also implicit functions of these same parameters.

3.2 Taylor’s appro~~~a~io~ For a small perturbation about the initial design

point, the functional relationship between the response/weight (performance variables) and the structural parameters (design variables), u can be expressed using Taylor’s series expansion as:

or IF) = &J + Vl{~ - %j (14)

where each of the vectors {C), fli>, {AT and f@ represents the magnitude of the response functions evaluated at the co~esponding sampling points on the structure [see also eqns (3)-(7)]. {C) represents the weight corresponding to a group of elements. The total number of imposed limits on performance

variables is m. The set Z = { 1, . . . , m ) of performance indices is partitioned into five disjunct subsets I,={1 ,..., pj, Z,=jp+l,..., qf, z,= q+l, . . . ( r),Z,={r+l,..., sf,Z,={s+t ,..., m \ con- taining stiffness, strength, frequency, modal amplitude and volume~wei~t functional, respectively. Any one or more of the subsets, I,, . . , , 15, may be empty. vO is the initial estimate of the structural parameters and u represents their new values. T is considered here to denote the sensitivity matrix-arranged in the same order as that of performance variable set I. It consists of partial derivatives of the appropriate response functions with respect to design vector v.

(15)

mxn

Equation (13) represents a linear approximation of the functionals at the initial point which can be expressed more compactly as:

{AF) = [T](Auf. (16)

3.3 Sensitivity matrix T A large proportion of the computational time is

usually spent in computing the derivatives of the

622 B. PRAS~~D and J. F. EMERSON

response functions. Efficient schemes of computing the derivatives have, therefore, been utilized in most optimization works. In order to be consistent, authors have put together a fairly general capability of design sensitivity[l5] which runs on EISI/EAL[l6]. Ref. [15] contains in detail descriptions about the procedure and its theoretical development. The procedure is capable of predicting sensitivity of design (variations in specified performance variables such as stiffness. strength, frequency and modeshapes) to changes in design parameters. The design variables can be chosen from a combination of the following parameters: sectional, materials, geometrical and shape properties, forming a part of generic finite element model. The procedure is employed here to extract the appropriate response functions at the sampling points to form the sensitivity matrix T as required in eqn (15).

3.4 Basic statistical model The developnent of a linear model (eqn (16) is

based on the assumption that a small perturbation of a structural element property causes small per- turbations in the response of the system, and the relationships are piecewise linear. Iterative use of eqns (IS)+ 16) would be convergent if there exists a solution (at least for a single combination of o) that satisfies all the bounds imposed on the chosen functionals. Equation (16) could be used directly to find such an estimator for v. In physical problems, however, if the bounds are not properly specified, some the the functionals may block further motion of the others towards an acceptable region. To account for this eqn (16) can be modified as:

AF = fT]{Ao) + (t: ). (17)

A vector {(t ) has been added to eqn (16) to account for the permissible variation in the region of acceptability. The vector (6) can thus be assumed to have, for smoothness, a zero mean and a covariance [X:,,]. The statistical model of the response functionals can now be formulated based on eqn (1’7). Consider the PI variables t’,, v2, ojr v,, . . . , u, to be a set of random variabies. Thus, the m functionals F of these variables will themselves to a set of random variables. If we denote the jth random variable’s expected value:

E(v,) = vO, or E(v) = v, (18

and the covariance matrix of u, which must be specified by the designer as:

t&J= EIfo - s&J - %rl (19)

then, eqn (17) can be interpreted as a Taylor’s series expansion of the functionals F about the mean values of the design variables v,. Note that the covariance matrix Z;,, indicates the possible variation that is allowed to occur in the predicted estimates of v. Some of the relevant relationships between mean and covariance of F and u are derived in the Appendix.

3.5 Inverse statistical formuiat~~n In this section, the objective is to determine the

solution for the inverse problem-that is, to find a

best estimate of the structural parameter set v such that the performance measures lie within the specified bounds and variance X,,. We are, therefore, seeking a solution for v in the form:

{Au*} = [R]{AFj (20)

where R is a system remodeling matrix to be determined. Since the eqn (20) represents an inverse form of eqn (17), Au* should lie within the statistical variations of Au. The matrix R can thus be found such that the variance of the difference between the true value of {Au} and the estimated value {AU*} given by eqn (ZO), i.e. the new functional Q:

IQ1 = EfjAu * -Abvj{Av*-Aujl] (21)

is minimize for all possible choices of R. This is equivalent to setting:

S[Q] = 0 for V6[R]

eqn (21) can be rewritten using eqn (20) as:

[Ql = EK[Rl(AF) - ~Au~)(~~l~A~~ - (A+‘1 (22)

and taking the expected value, Q becomes:

Q = [RIPF,JIR~~ - [RltW - &vlr[RIT + L,. WI

The variation of Q (SQ = 0) can now be obtained as:

0 = WW[&,l[Rl’ - LoI) f ([RILI - &rl’)[~NT. (24)

Thus, for all possible variations of [SR], eqn (24) implies:

WI = Lf&~l- ’ (25)

substituting [R] into the eqn (20), the solution for the best estimate of the structural parameter can be written as:

3.6 Remodeling equations Equation (26) gives a relationship that can be used

to determine the change in the structural parameter that a model has to undergo to cause a desired AF change. &] and [C,] represent the covariance and cross-covariance matrix with respect to F and (F and v), respectively. In the Appendix, fed and fed matrices have been detived in terms of user-supplied covariance matrices for v and L ([C,,] and IA,d) and computed senstitivity matrix [T] at the beginning of each iteration. Two cases have been considered:

(a) when & is defined to be a covariance matrix of AFi, i.e.

& = ~[~A~~~A~~~. (27)

The eqn (20) forms a remodeling equation for the purpose of determining AU.

Optimal structural remodeling of multi-objective systems 623

{Au) = [Rl{AF} (28)

where the remodeling matrix R can be formed based on eqn (25) and expressions (A. 10) and (A. 11) given in the Appendix:

R = LIWl’UIPATIT+ PcJ-’ (29)

(b) when Z, is looked as covariance of the percentage change in AF, that is:

CW = ~[{AF,/F,i){AF,I~~~}Tl~ (30)

In order to keep the meaning of {AF} the same as in eqn (28) the remodeling equations can be modified as:

{Aa } = PWl{AF}. (31)

As shown in the Appendix for case (a), R can be similarly derived in this case as:

R = [c,,~[NT~~([NTI[c,,I[NTIT + rw) - 1 (32)

where r-’ 1

(33)

N is a (m x m) diagonal matrix. It may be noted that if N is used as a unit matrix, the eqns (31H32) revert back to those obtained for case (a). Thus, once during an iteration, the Au* change is evaluated in either remodeling situations (a) or (b), the new design parameter can be determined using:

{u*} = {ua} + (Au*} (34)

when (a} is replaced by {a*}, a new estimate for the covariance of {u *} can be computed as:

X.,t.=E[(Au*-Au}{Au*-Au}T]

= &I - [R IL1

= Ll - UW~IP,oI. (35)

4. IMPLEMENTATION

Figure 2 describes the operations in revising the structural property estimates. If the relationship between the performance mreasuring functions and the parameters of interest were truly linear, the method would converge to the best estimate in a single step. However, this is often not the case. For determinate structures only, the stress and deflections are proportional to the inverse of the areas for rod elements and thickness for membrane-plate and shear-panel elements. For indeterminate structures, such relationships between these types of functions and design parameters are rather complex (problem dependent) and cannot be expressed explicitly. Con- sequently, a solution must be obtained by repeated iterations of the remodeling equations. At the end of each iteration, the convergence must be evaluated and if not satisfactory, the steps (Fig. 2) must be repeated each time using {Au*}, Ez, from the last revision along with the original {F,}, Z,, and &,.

4.1 Determination of AF vector In order to complete the definition of the re-

modeling formulation, we must determine how the AF vector is formed. Figure 3 shows a plot for the region of acceptability of a typical functional F,. The curves for F,i and F,, represent the lower and upper bounds that a functional F, has to satisfy. They are shown here to vary with u but in most cases, they are often specified as constant. The dotted line shows the median of the acceptable region F,,,i, that could also be independently specified if required.

F1, I Fm, I F”i

F,,,i, if not available, is taken as:

(36)

(37)

Given these boundaries of acceptability, the computed functional, at any iteration step would either lie within the region or would fall outside. The vector AF

f--J stop

I I _ 1 Estimated Structure I I Finite Elanmnl Analysis Revise The Estamote

I Model Properties I Compute Response Of The Struciurol

I 1 _ -) Functtons - Model PropertIes - Model PropertIes

A” = A”* xv” = z”“* Eqs I3218 (35)

{A:), &v*

Measures Et Sampling

L--- “‘st-“-era~lp”_ _ _ -1 Subsequent lterotion

Fig. 2. The operations diagram for the optimal remodeling procedure.

624 B. PRASAD and J. F. EMERSON

L I

5 MEDIAN F,,

DESIGN VARIABLE V - - , ” - DEFLECTION

I c - STRESS

FI CAN z A - FREQUENCY

4 - MODE SHAPE

- WEIGHT

Fig. 3. Plot showing the reion of acceptability of a typical functional F,.

can thus be formed as:

{AF} = 0: if F,i < Fi 5 F,,, (38)

a, (F, - FA if F, is outside (39)

The second part represents a penalty for the performance limit violation. Note that the distance is measured from the median so that the design would not asymptotically approach the boundaries Fui or F/,. xi is defined as weighing function for the functional which may be chosen by the designer. It provides a flexibility of assigning different relative importance to the functionals during a design iteration. The AF vector is formed at the beginning of each iteration. To start with, all a,‘s are set equal to 1.0 and later relaxed based on the conditions discussed below. After few iterations, ultimately either all performance limits will be satisfied or the problem will become so constrained that it is not possible to improve the design further.

If the first condition is met, i.e. limits are all satisfied, the desirable design is achieved. However, if one intends to refine it further, this could be done by tightening the upper and lower bounds on these functionals near to their original values and con- tinuing to iterate further. The procedure could be repeated several times until the design is satisfactorily obtained. If the second condition occurs where the limits will not permit further improvements, it is important to determine which limits are blocking further motion. Blocking is generally caused by the active performance measuring sets that are close to the boundaries. It is easy to identify those based on the following checks:

or

In a typical design problem, a significant number of performance variables are directed towards goal improvement tasks rather than being a part of the physical limits. If this is so, the bounds on these functionals, which are not physically dependent, can be relaxed to possibly cause some more improvement. This can be done similar to the one described earlier. In the present case, however, the following steps have

been found more convenient. The values of a, associated with those measures which are critical (or near critical) are decreased, making them little less important (effective) as compared to others. This allows for some further improvement to take place. This is easier than changing the bounds on the required functionals and can be controlled automatically based on the results of a previous iteration.

4.2 Convergence criteria The incorporation of the suggested change Au for

updating the initial design vector is governed by the following criteria:

(a) If the move limit is specified, no element of the v vector is updated more than what is prescribed by the corresponding move limits.

(b) If the linear scaling is used, all elements of the parameter change vector Au are scaled in the same proportion such that no elements are out of bounds.

(c) If the selective scaling option is used, only those elements of Au vector that are out of bounds are set to the prescribed limits; elements not out of bounds remain unchanged.

When one or more of the above options are specified, the v vector is updated at each iteration and design proceeds unless one of the following convergence criteria is finally met

(i) The suggested parameter change Au, is less than one percentage of their original value.

(ii) The reduction in the critical performance index value during the last iteration is smaller than one percent of its mean value.

5. PROBLEM AREAS AND OTHER CONSIDERATIONS

5.1 Potential areas of applications The procedure described represents a comprehen-

sive development and could be applied to a number of different design problems. The most noteworthy of these are listed in the following:

(a) Fine tuning a model to match the predicted analytical results with that obtained from the tests.

(b) Designing the shape of components to minimize the stress concentration effects.

(c) Designing a class of structures where fully stressed design criteria is required for optimality.

(d) Multi-point stiffness remodeling without significant weight differential.

(e) Identifying the properties of the model to pre- dict a desired stress contour, frequency pattern, deflection or modal profiles.

In these classes of problems, the minimum weight criteria is not imposed as an explicit design requirement. During the implementation, the different design variables such as sectional sizes, material, geometrical or shape properties are all treated as members of a single array-a common design vector without regard to their types.

5.2 Stress concentration problem If CC,, i = 1, m are used as weighting function for

the effective stresses in each group, the stress concentration problem can be written in the form:

min(max[X, 6,(c)])

van i

(42)

where the components di (v), i = 1, m represent normalized (see eqn 3) effective stresses in groups of elements where the stress concentrations are domi- nant. This class of problem has also been referred in literature as multi-criteria optimization[l7]. They turn the original problem (eqn 42), into a sequence of scalar optimization problems which may be solved by standard non-linear programming techniques [ 171. In order to apply the present theory to solve this type of problem, the original problem can be reformulated as:

(45)

Two design cases are considered to illustrate the possible uses of the procedure.

Cure (a). For this case, it is assumed that the desired stress levels of the members can be specified. The remodeling problem is proposed as to find the parameters A,, A,, and H to achieve a given set of target stresses (10,000 kN/m2 in each bar). In terms of eqn (38), the problem can be formulated as:

Find minimum stress o* such that Fi = oi - 10,000 (46)

{Au} = { cc_(a_ _O~‘;” ~i~~~s~ut ofrange I I rmr

for all rri, i = 1, m and where the structure satisfies the remodeling eqns (29) or (32). omi represents the median stress value (see Fig. 3).

1.0 I H 5 3.0

A,, A, > 0. (47)

5.3 Side constraints and design variable linking

Case (b). Here, it is assumed that the target values are not known a priori, the intent is to minimize the maximum stress. To solve this problem, a different design option is used; the F, in eqn (46) is updated based on

The side constraints (such as the ones imposed in eqn 2) are usually considered to be quite practical, although other forms, such as requiring a joint to lie on a plane or within a sphere, could easily be included (through generic modeling[l4]). Any of the parameters or coordinate variables may be linked to any other variable through a functional relationship:

where

F, = o, - Q,,,~

cl + a2 Qnl, 2 .

f(% x,) = c (43)

in which c is a constant and f represents a chosen continuous function. This linking equation can be of any desired form to suit a practical usefulness, e.g. a linear form may enable a designer to impose geometric symmetry about the coordinate axis or else some required clearance. The functional relationship between variables can readily be incorporated through generic modeling techniques. This further allows to include linking of shape-related parameters so that the generated shapes are consistent and conform (provide continuity of slope, curvature, etc.) with the rest of the structure.

To apply the procedure, a sample configuration with a = 4 m, b = 1 .O m and a problem with two independent design variables (A, = 2A,) were selected. The design was started from an initial value of A, = 4.0 m and H = 3.0 m in each case. The results converged to H,,, = 2.0 m and Alopt = 4.472 x 10-4m2 in 5 iterations for case (a). In case (b), the design also converged to H,,, = 2.0 m in 5 iterations, but there were many solutions for A,.

The problem is so simple that the exact solutions for the two cases can be obtained using eqns (44H48). This provides for case (a):

6. EXAMPLES and

3 112 H,,, = ab ~

[ 1 a2 - 4b2

Two examples[l8] have been used to show the effectiveness and usefulness of the optima1 remodeling procedure to structural designs. The em- phasis is on the application of the remodeling procedure to problems of different types rather than on the presentation of general design data; hence, only sample configurations are considered.

I’* x 10_4 (50)

6.1 Example 1. Unsymmetrical 2-bar truss This example (Fig. 4) was purposely chosen to

verify the results of the optima1 remodeling procedure during development phase. The performance limits were prescribed on the stresses. The design variables were the cross-sectional areas of the bars, A,, A, and the height of the truss, H. The stresses Q, and or in the bars are non-linear functions of these variables and can be expressed as:

PbJm II ,\

and for case (b). the solution can be obtained by setting 0, = u2. It results into the same value for H,,, as given in eqn (49), which becomes independent of A,. The optima1 remodeling results were thus accurate in both cases.

CJ’ = H(a + b)A,

,&y-Jz& I

31 ____-- ;

4 P= IOkN (44)

Fig. 4. Example l-2 bar truss.


626 B. PRASAD and J. F. EMERSON

Fig. 5. Example 2-Connecting rod eye-end.

6.2 Example 2. Connecting rod eye-end The second example[5] is chosen to iliustrate the

use of the technique for reducing stress concentration effects. It is desired to obtain an optimal profile of a connecting rod eye-end in the region of its connection with the eye (see Fig. 5) so that the stress concentration effects are minimized. The area of interest is defined from Point-A to Point-E. The connecting rod eye was loaded with a distributed load proportional to the cosine of the angle and parallel to the axes of the rod so as to produce a uniform tensile stress distribution (18,OOOpsi) in the shaft portion of the connecting rod. The end-most plane of the shaft was fully restrained so as to react out the loads introduced at the eye. For the purpose of this example, only the inplane membrane stresses were considered. An ac- tual connecting rod problem is highly complex due to the nature of the bearing loads and stiffened cross section. A simplified application was chosen so as not to over-complicate this example with excess detail.

The profile or curve defined from Point-A to Point-D was assumed to be expressed as a fifth-degree polynomial and a straight line segment between Point-D and Point-E. The slopes were defined as continuous at both ends of the curve. Point-A was defined with a fixed set of position coordinates. The general form of the curve is as follows:

n =a,+a,S +az<2+a353+a454+as55 (51)

where a,, a,, aI, a3, a4 and a, are to be evaluated from the design parameters. Three geometric design parameters (Xl, Y2, Y3) were chosen to get the required number of equations to solve for a’s in eqn (51). These were:

Xl-Horizontal (x-dir) distance from Point-D to Point-E:

YZ--Vertical (y-dir) location of Point B; Y3-Vertical (y-dir) location of Point C; Two equations can be obtained from the continuity

of the slopes at the ends and an additional two from the requirement that the slopes at the two ends of the curve be horizontal. The remaining two equations are obtained by evaluating the curve at two intermediate design points B and C. This led us to uniquely determine the unknowns (a’s) in terms of the desired design parameters.

The finite element model (see Fig. 5) was developed from 148 linear membrane elements and 193 nodal points. An additional 21 rod elements were used in

the outer fiber for accurate stress/strain monitoring. The loading and finite element model are symmetric about the connecting rod axes so a half model was utilized and symmetric boundary conditions were imposed along the plane, y = 0.

As a measure of the performance, fifteen outer fiber stresses along the transition area (A-E) were chosen to minotor the design remodeling process. The objective was to reduce the maximum outer fiber stress intensity factor (SIF) to an acceptable level (SIF = 1.6) when SIF is defined as

SIF = %!!! (52) CO

where omax is the maximum and go is the nominal stress. For the initial starting design, the intensity factor (SIF) was 2.36. The strategy was to use fifteen of the outer fiber stresses as performance indicators. The procedure is considered converged and an acceptable design is achieved if all the stress factors were reduced to lie within the prescribed SIF value (aJo I 1.6) and the parameters were to lie within the side limits. The side constraints were imposed to protect the occurrence of non-acceptable design during remodeling. The limits imposed were:

0.2 I X1 < 0.3

4.5 5 Y2 < 5.80 (53) 2.8 < Y3 I 5.0.

The iteration started fray an initial design value of Xl = 0.2, Y2 = 4.5 and Y3 = 3.0. Move limits of 10% on the design parameters were used to stabilize the solution process. The parameters were independently controlled by the individual side limits or move limits (refer to Section 4.2~) depending upon which were critical at that time. A linear penalty function approach (see Section 4.1) with cci = 1 was imposed on the violations of the stress indicators. A feasible solution was achieved at the end of seven iterations. The final stress intensity factor was (SIF = 1.59). The stress profiles for a few of the iterations are given in Fig. 6, and the corresponding geometric shapes are given in Fig. 7.

25 I I I I I SYMBOL ITERATION -____

20 40 60 60

X-DIR POSITION

Fig. 6. Iteration history for example 2.


PROFILE ITERATION 13. ___- -- I

- 2 - 4

- 7 14.

fiw , 15.

Fig. 7. Geometrical shapes for some key iterations in 16. example 2.

7. CONCLUSIONS

A statistical formulation for optimal remodeling of I’. structures to satisfy a set of prescribed measuring goals has been presented. It uses the original property of the finite element model as the starting point,

18,

linearly perturbs them in a piecewise fashion and finally arrives at a set of values that drives the performance measures within or close to the bounds. The required remodeling process preserves the consis-

R. A. Potter and K. D. Willmerts, Optimum design of a vehicle suspension system. ASME Paper No. 73-DET-46 (1973). H. T. Kulkarni, B. Prasad and J. F. Emerson, Generic modeling for complex component design, Proe. of4th fnt. Co@. on Vehicle Structural Me&. SAE Paper No. 811320 (Nov. 1981). B. Prasad and J. F. Emerson, A general capability of design sensitivity for finite element systems. Prof. AIAA/ASME/ASCE/AHS 23rd Structures, Structural Dynamics and Materials Conference, New Orleans, Lou- isiana, May 10-12 (1982). W. D. Whetstone. EISI-EAL: Engineering analvsis lan- guage, Proc. of ihe 2nd Specialty ?onf. on”Com&ng in Civil Ertgng. ASCE, Baltimore, Maryland, 9-13 June (1980). J. Koski, Multi-criterion optimization in structural design, Proc. of Int. Symp on Optimum Structural Design, Tucson, Arizona, pp. 10.29-10.36 (1981). B. Prasad and J. F. Emerson, Optimal structural remodeling. Computers in Engineering, 1983 (ASME) Vol. 3, pp. 99-108 (1983).

APPENDIX Statistical re~at~o~hips

tency of the model. The examples, while restricted in total scope, provide mainly an understanding of this

In eqn (I?), a Taylor’s series approximation of the

process and demonstrate its basic concept. Wider response function is obtained as:

applications may be desirable to provide some additional insight. (AF) = [Tl+ } + @} (Al)

1.

2.

3.

4.

5.

6.

7.

8.

9.

IO.

II.

12.

REFERENCES V. B. Venkayya, Structural optimization: a review and some recommendatjons. Int. J. Num. Meth. Engng 13, 203-228 (1978). B. Prasad and R. T. Haftka, Organization of PARS-A structural resizing system. Adances in Computer Tech-

nology, (Edited by Ali Seireg), Vol. 2, pp. 261-273 (1980). Also in Advances in Engineering Sofiware, 4(l), 9-19 (1982). S. S. Bhavikatti and C. V. Ramakrishnan, Optimum shape design of rotating disks. Computers and Struc- tures, 11, 397-401 (1980). C. V. Ramakrishnan and A. Francavilla. Structural shape optimization using penalty functions, J. Sgruc- tural Mech. 3(4), 403-432 (1975). A. Francavitla, C. V. Ramakrishnan and 0. C. Zi- enkiewicz, Optimization of shape to minimize stress concentration. J. Strain Anal. 10, 63-70 (1975). S. R. Johnson and J. W. Subhedar, Computer optimization of engine mounting systems, SAE Paper No. 790974, 3rd Int. Con/I Vehicle Structural Mech. SW,. 19-26 (1979). ”

_ .

J. E. Taylor, Scaling of a discrete structural model to match measured model frequencies. AZAA J. 15(11), 1647-1649 (1977). J. D. Collins, G. C. Hart, T. K. Hasselman and B. Kennedy, Statistical identification of structures. AIAA J. 12(2), 185-190 (1974). G. C. Hart and J. D. Collins, The treatment of random- ness in finite element modeling. Trans. SAE 81(4) 2408-2416 (1970). D. L. Bartel and R. W. Marks, The optimum design of mechanical systems with competing de&n objectiv&. J. Engng for Industry, ASME Trans., Series B, !M( I ), 17 1 f 19741. x-- >-

N. Olhoff and J. E. Taylor, On optimal structural remodeling. J. Optimization Theory and Applications f7(4), 571-581 (1979).

I: AsAn - - E,. WI

Mean and covuriance of AF J. A. Bennett, Automated design of multi-objective systems, ASME Paper 75-DET-111, Presented at the

Since {Au 1 is statistically distributed (e.g. a multi-variate

Design Engineering Technical Conference, Washington, normal distribution), the response A.F through the linear

D.C., 17-19 Sept. (1975). transformation would also be statistically distributed. It can be shown, easily through eqns (AlHA3), that mean of AF

with 6 representing the statistical variation in the region of acceptability with mean:

and specified covariance [Z,,]. Prior to the derivation of remodeling equation, it is assumed that the unknown structural parameter {u} is also statistically distributed with mean equal to its

(A3)

starting value v, (at the beginning of iteration) and a specified covariance = [X,1.

Mean and covariance of Av By definition, the mean of Au is given by:

PA,, = E@a}l 644)

and the covariance of Au as:

PA,> = E[{v} - {V,}] = E[(v}] - E[{Q}] = 0. (A7)

Also since v, is a mean of v, 2, is given by eqn (19). This implies:

628 B. F'RASAD and J. F. EMERSON

is The covariance of {AF] and {Au], which will be needed in the derivation of remodeling equation can be established as:

~‘af = E[{AF]] = 0 (A9 & = E[{AF}{A~Y-l

and covariance is: = ~Wl{A~~l + if ),{A~}‘]

= 1 Wwl LF = El{AF){AF}T

(All) where E[{L}{Av}~] = 0 b ecause

= WPI{A~~) + {c )U--l{~~ } + {c I,‘1 the vectors (t) and {Au} are

statistically independent.

optimal structural remodeling of multi-objective systems

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