Computer Aided Engineering Design

368 COMPUTER AIDED ENGINEERING DESIGN

Minimize: – VTKU

Subject to:

x

x

i iN

u

n

i

a

⎛

⎝

⎞

⎠

⎛

⎝

⎞

⎠

≤

| |

• 1 , = 1,... ,
  σ
  σ
  ε

xl≤xi≤xu (12.38)

whereσa is the allowable stress limit (10 N/mm^2 ),xi the width of the ith frame element treated as a
design variable, n is a prespecified exponent (= 3 in this case), N the total number of finite elements
andε (0.01) is a prespecified relaxation parameter having a small positive value. Out-of-plane
thickness is taken uniform as 2 mm. The notion in this topology optimization example is that if the
widths of frame elements are zero, they are non-existent. However, a very low but positive value
xl = 0.001 mm is chosen as the lower limit for the widths to prevent the global stiffness matrix from
being singular at any stage in optimization. Thus, a frame element would be considered absent from
the topology if its width assumes the lower bound. The widths of the frame elements are also bounded
from above such that they cannot exceed a value of xu = 4 mm. The stress constraints are posed so
that for xi≈xl, the effective upper limit on the stress | σi | becomes σa[ε(xu/xl)n + 1] = 10[0.01(0.4/
0.001)^3 + 1} = 6.4 × 106 which is a much larger number compared to 10, that is, stress constraints are
effectively not imposed on elements which are non-existent in the topology. However, for xi≈xu the
effective upper limit on stress | σi | is σa [ε + 1] = 1.01σawhich is very close to the allowable limit.
Eq. (12.38) is solved using the sequential quadratic programming in MATLABTM and the optimal
solution is given in Figure 12.14(b). Solid lines show the optimal connectivity while the dashed lines
depict the deformed configuration.

Exercises

1. Using any bisection method discussed in the chapter, determine the roots of

y = x^4 – 6x^3 + x^2 + 24x – 20

Note that there are a maximum of five roots of the above polynomial so that the brackets may be chosen

accordingly.

2. Using the Newton Raphson and secant methods, try and determine the roots of

y = x^5 – x^4 – 5x^3 + x^2 + 8x + 4

What are the possible difficulties one would experience with the two methods? Can these methods be

applied in case when a polynomial has multiple roots?

20 N

(a) (b)

P

Figure 12.14 Topology design example of a compliant crimper using SQP