N. V. Banichuk

Optimal Shape Design of Axisymmetric Shells for Crack Initiation and Propagation Under Cyclic Loading

ABSTRACT This paper describes a problem of axisymmetric shell optimization under fracture mechanics and geometric constraints. The problem is formulated as the weight (volume of a shell material) minimization and the meridional contour of the shell is taken as the design variable. The shell is made from quasi-brittle materials and through cracks arising are admitted. It is supposed that the shell is loaded by cyclic forces. A crack propagation process related to the stress intensity factor is described by the Paris fatigue law. The problem of finding the meridian shape (geometric design variable), of the shell having the smallest mass subject to constraint on the cyclic number for fatigue cracks, is investigated using a minimax (guaranteed) approach worked out in the theory of optimization under incomplete information. Optimal designs of the shells are found numerically with the application of genetic algorithms.

Shape Optimization of Rigid 3-D High-Speed Impactors Penetrating into Concrete Shields #

Abstract The problems of high-speed penetration of rigid pointed and truncated bodies into concrete media are considered with the application of a two-stage model introduced by Forrestal. Formulated problems of shape optimization, with constraints on the length and volume of the impactor, are investigated for the case of maximum depth of penetration (DOP). As a design variable, the number of faces of the pyramidal body is considered. The results of DOP computations are presented for different body shapes and for different problem parameters. Questions of the influence of pyramidal body truncation on the value of the optimized functional are investigated.

Probabilistic-Guaranteed Optimal Design of Membrane Shells Against Quasi-brittle Fracture

Abstract The problems described in this article concern the shape optimization of brittle or quasi-brittle (axisymmetric) elastic shells. The optimization problem takes into consideration the possibilities of cracks arising and consists in finding the shape of the shell loaded by external forces in such a way that the cost function (volume of the shell material) reaches a minimum, while satisfying some prescribed bounds on stress intensity factors. The questions discussed are characterized by incomplete information concerning crack size, crack location, and its orientation. In this context, the article presents some formulations of optimal shell design problems based on probabilistic-guaranteed approach and provides some results of analytical investigation for thin-walled shells with a through cracks.

Some Probabilistic Problems of Beam and Frame Optimization Under Longevity Constraint

Abstract The questions investigated in this article are related to an important class of problems of optimal structural design concerning the shape optimization of quasibrittle elastic bodies with cracks. The optimization problems taken into consideration consist of finding the boundary of a body loaded by cyclic quasistatical forces in such a way that the cost functional (volume of the body) reaches a minimum, while satisfying some prescribed bounds on structural longevity (i.e., on the number of cycles before destruction). These problems are characterized by incomplete information concerning initial crack size and location. In this context, the article presents some possible formulations of optimal structural problems, based on probabilistic approaches, and some examples of analytical solutions.

On the penetration of nonaxisymmetric bodies into a deformable solid medium and their shape optimization

We consider the problem of penetration of rigid pyramidal bodies (impactors) into a strained medium in the case of large speeds of penetration and estimate the depth of the impactor penetration. To this end, we use the two-stage penetration model proposed by Forrestall. We state the shape optimization problem for the penetrating body, which is based on the consideration of a set of bodies of pyramidal external shape with given fixed mass. We study both solid and hollow (shell-shaped) bodies. For the optimization functional we take the penetration depth of the penetrating body, and for the projection variable we take the number of faces of the pyramidal body. We present the results of computations of the penetration depth for different shapes of the impactor and show that, both for shells and solid impactors, the bodies of the shape of a circular cone are optimal. The problems of high-speed penetration of rigid bodies into a deformable medium are nowadays very topical problems [1] which have been studied by Russian and foreign authors [2–8].

Multiobjective Approach for Optimal Design of Layered Plates Against Penetration of Strikers

The study deals with the problem of layered plate multiobjective optimization against penetration of strikers having supersonic entry velocities. The plate is monolithic piece-wise homogeneous and composed from several plate-like layers produced from various given materials. The number of considered materials is supposed to be finite and consequently the admissible design set consists of separate discrete values. The impactor is modeled as an axisymmetric rigid body with blunted nose. The ballistic limit velocity and the mass of the plate are taken as components of the vector-functional that is optimized in Pareto sense. Best distribution of given materials is found with the help of genetic algorithm.

Multiobjective Shape Optimization of the Rigid Shell Moving into a Condensed Media

The problem of optimization of rigid bodies moving in the deformable media is considered. The shape of the axisymmetric impactors has been taken as an unknown design variable. The total resistance force, the mass of material, and the volume are taken as components of the minimized vector functional. The formulated multiobjective minimization problem for the vector functional is investigated analytically. As an example, the Pareto-optimal set of optimal shape and the Pareto front are found for the rigid thin-walled axisymmetric shells having the minimum total resistance force and the mass of the shell material.

Finding of Rigid Punch Shape and Optimal Contact Pressure Distribution

The problem of contact pressure optimization is formulated for the case of rigid punch interacted with elastic medium. Coupling of the punch penetration and action of external loads at the outside regions is taken into account. The shape of the punch is considered as an unknown design variable. The minimized integral functional characterizes the discrepancy between the actual contact pressure and the required pressure distribution. The problem is studied under condition that the total forces and moments applied to the punch and the loads acted at the outside regions are given. It is shown that the considered optimization problem can be split and transformed to two successively solved problems. Optimal shapes are found analytically for the punches having rectangular contact domains.

Optimization Problems of Contact Mechanics with Uncertainties

Abstract This article proposes a technique to optimize the shape of a rigid shell (stamp) interacted with elastic medium. As opposed to traditional optimization problems, where the loading configuration is specified, the loads are assumed uncertain in considered formulations and incorporated as unknown variables in the optimization problem statement. Guaranteed approach based on worst case scenario is applied as for formulation as for solution of the considered optimization problems with incomplete data on external forces. As a result, the optimal designs obtained are insensitive to load variations within a given admissible set.

Some problems of optimizing shell shape and thickness distribution on the basis of a genetic algorithm

We consider problems related to designing axisymmetric shells of minimal weight (mass) and the development of efficient nonlocal optimization methods. The optimization problems under study consist in simultaneous search for the optimal geometry and the shell thickness optimal distribution from the minimal weight condition under strength constraints and additional geometric constraints imposed on the thickness function, the transverse cross-section radii distribution, and the volume enclosed by the shell. Using the method of penalty functions, we reduce the above optimal design problem to a nonconvex minimization problem for the extended Lagrange functional. To find the global optimum, we apply an efficient genetic algorithm. We present the results of numerical solution of the optimal design problem for dome-like shells of revolution under the action of gravity forces. We present some data characterizing the convergence of the method developed here.