By Manohar P Kamat (ed.)
This booklet is the final word unmarried reference resource just about structural optimization. Chapters written through 35 box specialists record the cutting-edge in structural optimization which will developing one of the most promising instructions for destiny learn within the box. an important emphasis is put on engineering purposes.
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Extra resources for Structural Optimization: Status and Promise
25) shows that VC/>(H*) = 0. Also, since H(X*) is positive definite and Vg(X) is of full column rank near X*, V 2 (/>(M*) given in Eq. (26) is negative definite. These are the necessary and sufficient conditions for maximization of the dual problem defined in Eq. (27). Thus, we see that locally the original constrained minimization problem PE with respect to X is equivalent to the unconstrained local maximization of the dual problem with respect to u. Local duality result. For the problem PE, let 1) AT* be a local minimum, 2) X* be a regular point, 3) M* be the Lagrange multipliers at X* and 4) H(X*, u*) be positive definite.
ARORA Problem with Inequality Constraints The local duality for the general problem P is developed essentially along the same lines as the problem with equality constraints treated in the foregoing subsection, except that the Lagrange multipliers (dual variables) for the inequality constraints must be required to be nonnegative. Thus the Lagrangian for the problem P is defined, with M/ > 0 for / = p 4- 1, p + 2, . . , m, as L(X, ii) = f(X) + (u - g(X}} (29) The dual function for the problem P is defined (with w/ > 0 for / = /?
President. 29 Purchased from American Institute of Aeronautics and Astronautics 30 G. N. VANDERPLAATS In the most general sense, numerical optimization solves the following nonlinear, constrained problem. Find the set of design variables X that will Minimize F(X) Objective (1) Subject to gj(X) < 0 hk(X) = 0 Xf < Xi < Xj 7 j = 1 , 2, . . , m * = 1, 2, . . ,w Inequality constraints Equality constraints Side constraints (2) (3) (4) Equation (1) defines the objective function, which depends on the values of the design variables X.
Structural Optimization: Status and Promise by Manohar P Kamat (ed.)