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The purpose of this paper is to elaborate the effective method of adaptation of the external penalty function to the genetic algorithm.

In the case of solving the optimization tasks with constraints using the external penalty function, the penalty term has a larger value than the primary objective function. The sigmoidal transformation is introduced to solve this problem. A new method of determining the value of the penalty coefficient in subsequent iterations associated with the changing penalty has been proposed. The proposed approach has been applied to the optimization of an electromagnetic linear actuator, and the mathematical model of the devices contains equations of the magnetic field, by taking into account the nonlinearity of ferromagnetic material.

The proposed new approach of the penalty function method consists in the reduction of the external penalty function in successive penalty iterations instead of its increase as it is in the classical method. In addition, the method of normalization of constraints during the formulation of optimization problem has a significant impact on the obtained results of optimization calculations.

The proposed approach can be applied to solve constrained optimization tasks in designing of electromagnetic devices.

The purpose of this paper is to elaborate algorithm and software for the optimization of the actuator–capacitor system taking the dynamics parameters into account. The system is applied for driving the valve of plasma gun. Two optimization strategies are proposed and pondered. The penalty function approach has been expanded in detail.

The field-circuit mathematical model of the dynamics operation consists of the strongly coupled equations of the transient electromagnetic fields and the equations of the electric circuit. The numerical implementation is based on finite element method and step-by-step Cranck–Nicholson procedure. The genetic algorithm has been used in the optimization procedure. The sigmoidal transformation has been applied to adjust the classical external penalty function method to the genetic algorithm.

The modification consists in adaptation of the penalty function to the genetic algorithm. In the proposed approach, operations involving successive iterations of increasing penalty function and operations containing genetic iterations are intertwined with each other. The differences between these two procedures are getting blurred. The proposed approach is very effective. It is possible to achieve optimal solution even more than ten times faster than using the classical method.

The proposed approach can be successfully applied to designing and optimization of different electromagnetic devices, including functional constraints.

The purpose of this paper is to improve and to extend the use of original rank‐niche evolution strategy (RNES) algorithm to solve constrained and unconstrained multiobjective optimization problems.

A new mutation step size is developed for evolution strategy. A mixed ranking procedure is used to improve the quality of the fitness function. A self‐adaptive sharing radius is developed to save computational time. Four constraint‐treating methods are developed to solve constrained optimization problems. Two of them do not use penalty function approach.

The improved RNES algorithm finds better quality Pareto‐optimal solutions more efficiently than the previous version. For most test problems, the solutions obtained by improved RNES are better than, or at least can be compared with, results from other papers.

The application of any evolutionary algorithm to real structural optimization problems would face a problem of spending huge computational time. Some approximate analysis method needs to be incorporated with RNES to solve practical problems.

This paper provides an easier approach to find Pareto‐optimal solutions using an evolutionary algorithm. The algorithm can be used to solve both unconstrained and constrained problems.

Most studies of power‐law fluids are carried out using stress‐based system of Navier‐Stokes equations; and least‐squares finite element models for vorticity‐based equations of power‐law fluids have not been explored yet. Also, there has been no study of the weak‐form Galerkin formulation using the reduced integration penalty method (RIP) for power‐law fluids. Based on these observations, the purpose of this paper is to fulfill the two‐fold objective of formulating the least‐squares finite element model for power‐law fluids, and the weak‐form RIP Galerkin model of power‐law fluids, and compare it with the least‐squares finite element model.

For least‐squares finite element model, the original governing partial differential equations are transformed into an equivalent first‐order system by introducing additional independent variables, and then formulating the least‐squares model based on the lower‐order system. For RIP Galerkin model, the penalty function method is used to reformulate the original problem as a variational problem subjected to a constraint that is satisfied in a least‐squares (i.e. approximate) sense. The advantage of the constrained problem is that the pressure variable does not appear in the formulation.

The non‐Newtonian fluids require higher‐order polynomial approximation functions and higher‐order Gaussian quadrature compared to Newtonian fluids. There is some tangible effect of linearization before and after minimization on the accuracy of the solution, which is more pronounced for lower power‐law indices compared to higher power‐law indices. The case of linearization before minimization converges at a faster rate compared to the case of linearization after minimization. There is slight locking that causes the matrices to be ill‐conditioned especially for lower values of power‐law indices. Also, the results obtained with RIP penalty model are equally good at higher values of penalty parameters.

Vorticity‐based least‐squares finite element models are developed for power‐law fluids and effects of linearizations are explored. Also, the weak‐form RIP Galerkin model is developed.

The paper proposed a fast procedure for solving the reliability-based robust design optimization (RBRDO) by modifying the RBRDO formulation and transforming it into a series of RBRDO subproblems. Then for each subproblem, the objective function, constraint function and reliability index are approximated using Taylor series expansion, and their approximate forms depend on the deterministic design vector rather than the random vector and the uncertain estimation in the inner loop of RBRDO can be avoided. In this way, it can greatly reduce the evaluation number of performance function. Lastly, the trust region method is used to manage the above sequential RBRDO subproblems for convergence.

As is known, RBRDO is nested optimization, where the outer loop updates the design vector and the inner loop estimate the uncertainties. When solving the RBRDO, a large evaluation number of performance functions are needed. Aiming at this issue, the paper proposed a fast integrated procedure for solving the RBRDO by reducing the evaluation number for the performance functions. First, it transforms the original RBRDO problem into a series of RBRDO subproblems. In each subproblem, the objective function, constraint function and reliability index caused are approximated using simple explicit functions that solely depend on the deterministic design vector rather than the random vector. In this way, the need for extensive sampling simulation in the inner loop is greatly reduced. As a result, the evaluation number for performance functions is significantly reduced, leading to a substantial reduction in computation cost. The trust region method is then employed to handle the sequential RBRDO subproblems, ensuring convergence to the optimal solutions. Finally, the engineering test and the application are presented to illustrate the effectiveness and efficiency of the proposed methods.

The paper proposes a fast procedure of solving the RBRDO can greatly reduce the evaluation number of performance function within the RBRDO and the computation cost can be saved greatly, which makes it suitable for engineering applications.

The standard deviation of the original objective function of the RBRDO is replaced by the mean and the reliability index of the original objective function, which are further approximated by using Taylor series expansion and their approximate forms depend on the deterministic design vector rather than the random vector. Moreover, the constraint functions are also approximated by using Taylor series expansion. In this way, the uncertainty estimation of the performance functions (i.e. the mean of the objective function, the constraint functions) and the reliability index of the objective function are avoided within the inner loop of the RBRDO.

The purpose of this paper is to present the application of an adaptive bacterial foraging (BF) algorithm for the design optimization of an energy efficient induction motor.

The induction motor design problem is formulated as a mixed integer nonlinear optimization problem. A set of nine independent variables is selected, and to make the machine feasible and practically acceptable, six constraints are imposed on the design. Two different objective functions are considered, namely, the annual active material cost, and the sum of the annual active material cost, annual cost of the active power loss of the motor and annual energy cost required to supply such power loss. A new adaptive BF algorithm is used for solving the optimization problem. A generic penalty function method, which does not require any penalty coefficient, is employed for constraint handling.

The adaptive BF algorithm is validated for two sample motors and benchmarked with the genetic algorithm, particle swarm optimization, simple BF algorithm, and conventional design methods. The results show that the proposed algorithm outperforms the other methods in both the solution quality and convergence rate. The annual cost of the induction motor is remarkably reduced when designed on the basis of minimizing its annual total cost, instead of minimizing its material cost only.

To the best of the knowledge, none of the existing work has applied the BF algorithms for electrical machine design problems. Therefore, the solution to this problem constitutes the main contribution of the paper. According to the huge number of induction motors operating all over the world, the BF techniques used in their design, on minimum annual cost basis, will lead to a tremendous saving in global energy consumption.

Contact problems are among the most difficult ones in mechanics. Due to its practical importance, the problem has been receiving extensive research work over the years. The finite element method has been widely used to solve contact problems with various grades of complexity. Great progress has been made on both theoretical studies and engineering applications. This paper reviews some of the main developments in contact theories and finite element solution techniques for static contact problems. Classical and variational formulations of the problem are first given and then finite element solution techniques are reviewed. Available constraint methods, friction laws and contact searching algorithms are also briefly described. At the end of the paper, a bibliography is included, listing about seven hundred papers which are related to static contact problems and have been published in various journals and conference proceedings from 1976.

The purpose of this paper is the presentation of a technique to be integrated in a numerical airfoil optimization scheme, for the exact satisfaction of a strict equality cross-sectional area constraint.

An airfoil optimization framework is presented, based on Area-Preserving Free-Form Deformation (AP FFD) technique. A parallel metamodel-assisted differential evolution (DE) algorithm is used as an optimizer. In each generation of the DE algorithm, before the evaluation of the fitness function, AP FFD is applied to each candidate solution, via coupling a classic B-Spline-based FFD with an area correction step. The area correction step is achieved by solving a sub problem, which consists of computing and applying the minimum possible offset to each one of the free-to-move control points of the FFD lattice, subject to the area preservation constraint.

The proposed methodology is able to obtain better values of the objective function, compared to both a classic penalty function approach and a generic framework for handling constraints, which suggests the separation of constraints and objectives (separation-sub-swarm), without any loss of the convergence capabilities of the DE algorithm, while it also guarantees an exact area preservation. Due to the linearity of the area constraint in each axis, the extraction of an inexpensive closed-form solution to the sub problem is possible by using the method of Lagrange multipliers.

AP FFD can be easily incorporated into any 2D shape optimization/design process, as it is a time-saving and easy-to-implement repair algorithm, independent from the nature of the problem at hand.

The proposed methodology proved to be an efficient tool in facing airfoil design problems, enhancing the rigidity of the optimal airfoil by preserving its cross-sectional area to a predefined value.

This paper presents a new finite element (FE) approach to modeling edge effects in beam sandwich structures. The approach is based on a mixed 2D and beam formulation conjuncted by means of a penalty function method. Several results from analysis of sandwich beams and frames bending with different boundary conditions and laminate properties are solved in order to demonstrate the accuracy of the algorithms and software developed. The influence of the penalty factor on the spectral condition number of the stiffness matrix and on the residual norm of the solution is also investigated for different isotropic and sandwich structures. The FE analysis of complex sandwich beam joint is subsequently presented. Results of this analysis show the advantages of the developed approach for large problems.

The purpose of this paper is to propose a numerical approaching analysis method combining the sequential unconstrained minimization technique and finite element method to identify the loading condition and geometry of smart structures accurately.

A new load identification model is built and the finite element approaching method is proposed by the combination of finite element method and optimization technique.

The approaching algorithm has good convergence and fast approximation speed; the accuracy can meet the engineering requirements. The approaching model is simple, and the precision is controllable and it can be used to solve the load identification problem of the smart material structure.

In view of the cited papers, the information sensed by the smart structure is limited, discrete and contains certain errors. How to derive the cause from the limited, error-containing discrete information is an important problem that needs to be solved by the self-diagnosis function. A load identification model based on structural displacement response is established and a numerical approximation method is proposed by combining the finite element method with the optimization technique; the load magnitude and position of the structure are identified according to the displacement measurement values of the internal finite point in the structure under the load condition.