By P. Brousse (eds.)

ISBN-10: 3211813764

ISBN-13: 9783211813768

ISBN-10: 3709127386

ISBN-13: 9783709127384

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**Additional resources for Structural Optimization**

**Example text**

2 Relations Between Solutions of Problems (49) and (56) (cf. 3 Elongation Rate Vector of Non Null Bars Solutions of Primal Problem (49) when the Velocity Vector of the Joints is Solution of Dual Problem (56) (cf. (39)) In each solution of dual problem (56), the elongation rate of { each bar of any solution of (49) with a non null cross section area is equal to ± Qk fa, wherd\ is the initial length of this bar. 4 A Virtual Fail of Solutions of Primal Problem (49) If the strictly positive constant Ct.

This is not the case for the following theorem. E42 Let us 8uppose that f is convex, g is concave, and F has an interior point. If the problem P has a solution then the problem D has also a solution y and 0 (y) = f(x).

There exists an open ball centred at x. having (3 for radius, contained inS. Then Taylor theorem gives General Methods ... 37 =I f"(~)(h,h) f(x +h)- f(x) + e(h)ll h 11 2 with limit e (h) = 0 when h tends to zero. By coercivity of f"(x) it follows f (x + h) - f (~) ;;;:. I (a + 2e (h)) 11 h 11 2 Now, there exists 6, 0 < 6 < 13 such that II h II a + 2 e (h) > 0. Thus the function f does have a strict local minimum at x. < 6 implies 11-9 Convexity Case Let f be a numerical differentiable convex (strictly convex) function on an _9pen convex set C in a normed space, and let x be a point of C.

### Structural Optimization by P. Brousse (eds.)

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