Engineering Optimization: Theory and Practice, Fourth Edition

746 Practical Aspects of Optimization

analysis for computing the values of the objective function and/or constraint functions

at any design vector. Since the objective and/or constraint functions are to be evaluated

at a large number of trial design vectors during optimization, the computation of the

derivatives of the response quantities requires substantial computational effort. It is

possible to derive approximate expressions for the response quantities. The derivatives

of static displacements, stresses, eigenvalues, eigenvectors, and transient response of

structural and mechanical systems are presented in this and the following two sections.

The equilibrium equations of a machine or structure can be expressed as

[K]Y=P (14.31)

where [K] is the stiffness matrix,Ythe displacement vector, andPthe load vector.

By differentiating Eq. (14.31) with respect to the design variablexi, we obtain

∂[K]

∂xi

Y+[K]

∂Y

∂xi

=

∂P

∂xi

(14.32)

where∂[K]/∂xidenotes the matrix formed by differentiating the elements of [K]with

respect toxi. Usually, the matrix is computed using a finite-difference scheme as

∂[K]

∂xi

≈

[K]

xi

=

[K]new−[K]

xi

(14.33)

where [K]newis the stiffness matrix evaluated at the perturbed design vec torX+Xi,

where the vectorXicontains xiin the ith location and zero everywhere else:

Xi= { 00... 0 xi 0 ... 0 }T (14.34)

In most cases the load vectorPis either independent of the design variables or a

known function of the design variables, and hence the derivatives,∂P/∂xi, can be

evaluated with no difficulty. Equations (14.32) can be solved to find the derivatives of

the displacements as

∂Y

∂xi

=[K]−^1

(

∂P

∂xi

−

∂[K]

∂xi

Y

)

(14.35)

Since [K]−^1 or its equivalent is available from the solution of Eqs. (14.31), Eqs. (14.35)

can readily be solved to find the derivatives of static displacements with respect to the

design variables.

The stresses in a machine or structure (in a particular finite element) can be deter-

mined using the relation

σ=[R]Y (14.36)

where [R] denotes the matrix that relates stresses to nodal displacements. The deriva-

tives of stresses can then be computed as

∂σ

∂xi

=[R]

∂Y

∂xi

(14.37)

wherethe matrix [R] is usually independent of the design variables and the vector

∂Y/∂xiis given by Eq. (14.35).