optimization in problems of elasticity with unknown

OPTIMIZATION IN PROBLEMS OF ELASTICITY WITH UNKNOWN BOUNDARIES H.

V. Banichuk, V. G. Bel'skii, and V. V. Kobelev

jzv. AN SSSR. Mekhanika Tverdogo Tela, Vol. 19, No. 3, pp. 46-52, 1984 UDC 539.3

In this papei" we will investigate problems of shape optimization for elastic solids. fYie optimization criteria are provided by integral and local functionals. A new procedure is proposed for deriving the necessary optimality conditions for the problem with unknown boundaries under arbitrary boundary conditions. The derivation makes essential use of a formula that expresses the variation of a functional in terms of the variations of the boundaries and of the state functions. An algorithm for numerical determination of opti­ cal shapes of two-dimensional elastic solids is proposed; it is based on the necessary optimality conditions and on analysis of sensitivity. The algorithm is iterative, and involves successive solution of direct problems of determining the stress-strain state by the finite-element method and of constructing improving variations of the shape of the region occupied by the elastic solid. Optimal shapes of plane structural elements are determined numerically for various cases of loading. 1. STATEMENT OF OPTIMIZATION PROBLEM Consider a deformable elastic solid that occupies region ft with boundary T. On part of the boundary V displacements u. = U. are specified, while on part Ta the loads (T = - F + TV) are specified: a..n. = T., where n. are the components of the unit normal vec-or to surface r (n.n. = 1). Latin subscripts assume the values 1, 2 and 3; summation is performed over indexes that repeat twice. Assuming the conditions of statics and of smallness of the strains to be satisfied, we can write the basic equations as follows: Ot^i+q^O,

Oy—V»ilWu(Ujk,J+U|,»)

(1.1)

where A..,7 are the elastic moduli; a., are the components of the stress tensor; u^ are the displacements; and q. are the volume forces acting on the solid; a subscript after a ziomma denotes differentiation with respect to the corresponding coordinate. The stress-strain state of the solid (or structural element) is estimated by the functional JF(aihuk)dQ (1.2) where P is a specified function of the variables u, , a... The optimization problem invoives the determination of the varied part of r that minimizes the volume of the elastic solid: / """J"" 1 ' 5 ^

'

(1.3)

0

and such that the constraints on the stress-strain state I :£ I Q are satisfied, where I Q is a specified constant. Variations of the boundary that do not alter the connectedness (or topological class of region ft are permitted in (1.3); the functions u\ and T. are defined for each permis© 1984 by AHerton Press, Inc.

sible configuration of the region on the boundary r. Let us offer some clarification. The local constraints on strength and stiffness can be satisfied by the particular s p e - '''°3 cification of P. These constraints can be taken into account by using the relations be-i'| tween the norms in space of continuous functions and the norms in space of functions tha't-l are integrable to the p-th degree. Thus, for sufficiently large p , the strength constralfflf maxo£(Ofl)