Modern additive technologies make it possible to create structures of variable thickness and of any shape. Thus, designers face problems of optimal design of a new type, and these are problems of topological optimization. Such problems are to determine the optimal form of the structure or the optimal distribution of material over the structure. As a rule, the criterion of optimality is the mass of the structure. However, the structure must retain its bearing capacity under a certain load. The symmetric two-shear adhesive joint of the main plate with two overlays of the same shape on both sides is the object of study in this article. The main goal of this study was to determine the optimal form of overlays with variable thicknesses under certain restrictions. The main restriction is the strength of the structure. Furthermore, additional restrictions are imposed on the minimum and maximum thickness of the overlay. Therefore, the solution to the problem is presented in the form of a set of the following tasks: building a mathematical model of the adhesive joint, building a numerical solution to the primal problem using the finite difference method, and building a genetic optimization algorithm. In the presented article, to improve the convergence of the genetic algorithm is proposed to use an island model that consists of several populations. The main feature of the proposed model of the genetic algorithm lies in the fact that on one of the "islands" mutations occur more frequently and with higher dispersion than on the other two "islands". On the one hand, this decision ensures a high rate of evolutionary selection, and on the other hand, the stability of the results is achieved. Several modeling problems are solved in this article. The main results of this research include the following: nonlinear dependence of the overlay length on the applied load was determined; restrictions on the minimum thickness of the overlay, which cause the appearance of a certain “plateau” at the edge of the overlay, the thickness of which is equal to the minimum allowable were defined.

A genetic algorithm for solving the problem of optimal beam material distribution along length at a given restriction on maximum sagging value is suggested. A review of literature sources is conducted and it was shown that existing solutions cover partial cases only in which the position of the point with maximum sagging was defined previously. In the paper presented I-section beam with constant proportions is considered, i.e., beam width, caps, and web thickness are proportional to beam height in the current cross-section. A statically determined beam is being considered. The load applied to a beam can be arbitrary, including cases of non-symmetrical loads and differently oriented ones. The position of point(s) at which beam sagging is maximum are unknown at the beginning of optimization and are found in the process solution. The problem is solved in the linear definition. Beam mass was assumed to be an optimization criterion. The method of finite differences is used for beam sagging finding, i.e., for the solution of the differential equation of the bending beam with a variable cross-section. Discretization allows transforming the problem of design into the problem of beam height determination at a system of reference points. At this stage, found values of beam height must satisfy restrictions on reference point displacements. The suggested technique allows controlling beam displacement quite flexibly because restrictions on point displacement are considered separately and do not depend on each other. The suggested objective function is the linear superposition of beam mass and the possible penalty in case of beam maximum sagging over exceeding predefined values. The application of a genetic algorithm allows getting sets of beam thicknesses those, which guaranty reaching the minimum of the objective function. The model problem is solved. It is shown that the suggested algorithm allows effectively solves problems of optimal design of beams with restrictions on the maximum sagging value. The suggested approach can be developed for strength restrictions, statically undetermined structures, etc.

Adhesive lap joints are widely used in modern structures. Known analytical mathematical models of the stress state of lap joints describe the joints of straight beams or cylindrical coaxial pipes. It is assumed that the stress state of these structures depends on only one coordinate. The study of the stress state of plates with defects, which are reinforced by patches, in most cases requires the use of at least two-dimensional mathematical models. In this work, it is shown that the axial symmetry of the plate, the cut-out, the patch, and the applied load makes it possible to reduce the problem to a one-dimensional problem in the polar coordinate system. An adhesive lap joint with circular symmetry is considered for the first time. The stress-strain state of the structure is described in an analytical form. Comparison of the results of calculating the stress state of the joint with the results of finite element modelling showed good adequacy of the proposed mathematical model.