Method of Optimal Reinforcement of Structures Based on Topological Derivative

The problem of o p t i d reinforcement of 2D structures is discussed in the paper. Both, structures in plane state of stress and bending Kuchhoffs plates are analyzed. The problem of minimization (maximization) of arbitmy objective functional of displacements, strains, stresses and reactions with constmint imposed on the structure cost is considered. Also other constmimts, for example geometrical constraints, can be used. In order to solve this problem, simultaneously with respect to dimensional, shape and reinforcement parameters, algorithm composed of two mutually interacted stages is proposed. In the first stage, using Lagrangian approach, optimal values of dimensional design parameters, shape design parameters, and respective Lagrange multipliers are determined in the incremental process of gradient optimization. Next, conditions of finite topology modification (ef [1]) by addition of finite fiber or rib, based on topological derivative (cf. [2]), are formulated. In the second stape. correction of parametersc characterizing stiffened structure is performed.

In the case of structures in plane state of stress, reinforcement of isotropic material by introduction of fibers is considered. The topological derivative is defined as derivative of the objective functional or constmints with respect concentration of fibers at the mint corresponding to zero concentration. In the case of disks and Kuchhoff’s dates, reinforcement by introduction of ribs is analyzed. It is assumed, that considered here structures are made of isotropic material, but after rib insertion they reveal orthotropic properties. In order to find correct location and orientation of ribs, the topological derivative with respect to introduction of infinitesimally thin rib is applied.

Here, in both cases, expressions for topological derivatives are derived and on this basis conditions of mdification are formulated. To calculate topological derivative, the adjoint method is used. In this method a new structure of the same shape and mterial constants as the primary one, but with different boundq conditions and loads, is introduced. In order to solve primary and adjoint structure, FEM is applied. When respective condition of modification is satisfied, a fiber or rib is introduced. Next, in order to correct their positions and to determine other parameters characterizing stiffened structure standard optimization is peliormed. Usually, optimal position of the stiffener only a little differs kom the initial position. Numerical examples illustrate applicability of the method.