## ABSTRACT

Shape optimization of 3-D elasticity problems, with the BCM, is the subject of this chapter. Further details are available in Shi and Mukherjee [150].

An optimal shape design problem can be stated as a minimization problem under certain constraints. Its general form can be stated as follows:

Minimize f(b) (6.1)

Subject to gi(b) ≥ 0, i = 1, ..., Ng (6.2)

hj(b) = 0, j = 1, ..., Nh (6.3)

b () k ≤ bk ≤ b(u)k , k = 1, ..., N (6.4)

in which b = 〈b1, b2, ..., bN 〉T are the design variables, f(b) is the objective function, and gi(b) and hj(b) are the inequality and equality constraints, respectively. Equation (6.4) gives side constraints that are used to limit the search region of an optimization problem. Here, the parameters b()k and b

the lower and upper bounds, respectively, of the design variable bk. The most common mathematical programming approaches, used in gradient-

based optimization algorithms, are the successive linear programming (SLP) and successive quadratic programming (SQP) methods. In the SQP method, the optimization problem is approximated by expanding the objective function in a second order Taylor series about the current values of the design variables, while the constraints are expanded in a ﬁrst order Taylor series.