Structural Performance Measures

As noted previously, structural topology design problems can be formulated in a number of alternative ways through utilization of assorted objective and constraint functions. Generally, the objective function measures the performance of the structure, and the constraint function limits the amount of structural material that can be used, although the roles can be reversed equally well. The significant aspects of using CSTO to design large-scale sparse structures can be demonstrated here using the linear elastic structural compliance performance measure and the critical load buckling factor.

Linear Elastic Structural Compliance

If a structure features a linear elastic response behavior, the resulting displacement field u in response to a set of applied external loads fext will be simply u = K-1 • fext where K represents the stiffness matrix of the structure. For a given set of loads, the compliance П (b) of the structure is simply

n(b) = 1 fext • u. (8)

Structural concept designs ь that are stiff with respect to the applied loads will have small compliance n(b), whereas structures that are not stiff with respect to the applied loads will have large compliance. To facilitate usage of gradient-based optimization solution techniques, it is necessary to compute the design derivatives of the compliance function. It can be shown that the design gradient of structural compliance is provided by the following expression:

Linearized Buckling Performance Measure

Linearized buckling eigenvalue analysis proceeds as follows: A prescribed force loading fxt is applied to the structure with its magnitude necessarily being less than that required to induce geometric instability in the structure. Once the resulting linear, elastostatic displacement solution u = {uj} є RN in response to

the applied loading fext is obtained (k • u = fex’), where K is the linearized stiffness matrix, then the following eigenvalue problem is solved

[K(b) + 2G(u, b)]-у = 0 (10)

In the preceding, b = {b e} є RM is again the vector of design variables; K is the tangent stiffness operator; G(u, b) is the linearized geometric stiffness matrix; Я = -(у ■ K ■ у)/(у ■ G ■ у) is an eigenvalue denoting the magnitude by which fext must be scaled to create instability in the structure, and у is a normalized eigenvector satisfying у • K • у = 1. To avoid numerical difficulties in the solution of (10) stemming from the indefinite characteristics of g, it is common (Bathe 1996) to solve a modified eigenvalue problem that deals with two positive definite matrices.

[(K + G)-у K]-v = 0 (11)

where

Л – 1 . 1 (12)

Я 1 – у

In Eq. (11), the matrix K is positive definite irrespective of the loading applied to the structure, whereas the matrix (K+G) will only be positive definite when the magnitude of the loading applied to the structural model is less than the critical magnitude that creates instability in accordance with linearized buckling theory.

The design problem is formulated to maximize the calculated minimum-buckling load factor (Я ), and accordingly the objective function fE to be minimized for this problem would simply be the reciprocal of the lowest eigenvalue Я as follows.

v •G • V

mm I — max ——

bu ^ IMI*° у • K • y

The optimization problem is thus stated to minimize the reciprocal of the first (or minimum) critical buckling load as follows

subject to the normal bound constraints on the design variables (1), the linear structural equilibrium state equation (5), and a constraint on material resources.

The design gradient of the objective function can be expressed as:

dfE _ ¥e, ¥e,5u (15)

db 5b 5u 5b

To avoid explicit computation of the term, adjoint design sensitivity analysis is employed by augmenting the objective function fE with the equilibrium state equation as follows

S = fE + u“ • r (16)

where ua is the adjoint displacement vector which functions as a matrix of Lagrange multipliers and determined by the solution of a linear adjoint problem. The design derivative of the augmented Lagrangian is then written as follows:

(17)

The last term of Eq. (17) vanishes due to satisfaction of the equilibrium constraint (r=0), and the second term can be made to vanish by selecting the adjoint displacement vector to solve the following linear adjoint equality statement

(18)

it follows that the design gradient expression for the objective

(19)

The preceding expression is valid only when the minimum eigenvalue is a simple, or non-repeated, eigenvalue. When the minimum eigenvalue is nonsimple, or repeated, the variation of the eigenvalue in design space is non-smooth, and direct usage of the expression in Eq. (19) is technically incorrect (Choi et al 1983; Seyranian et al 1994). Resolution of this issue is nontrivial, although it can be ameliorated somewhat by using small and variable move limits in the design optimization process. Despite this challenge, designs that successfully maximize the buckling stability of a structural system can nevertheless be obtained. Further details on formulation and solution of stability design problems using linearized buckling theory are provided in Rahmatalla and Swan(10).