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The purpose of this paper is to improve the existing differential evolution (DE) mutation operator so as to accelerate its convergence.

A new general donor form for mutation operation in DE is presented, which defines a donor as a convex combination of the triplet of individuals selected for a mutation. Three new donor schemes from that form are deduced.

The three donor schemes were empirically compared with the original DE version and three existing variants of DE by using a suite of nine well‐known test functions, and were also demonstrated by a practical application case – training a neural network to approximate aerodynamic data. The obtained numerical simulation results suggested that these modifications to the mutation operator could improve the DE's convergence performance in both the convergence rate and the convergence reliability.

Further research is still needed for adequately explaining why it was possible to simultaneously improve both the convergence rate and the convergence reliability of DE to that extent despite the well‐known “No Free Lunch” theorem. Also further research is considered necessary for outlining more distinctively the particular class of problems, where the current observations can be generalized.

More complicated engineering problems could be solved sub‐optimally, whereas their real optimal solution may never be reached subject to the current computer capability.

Though DE has demonstrated a considerably better convergence performance than the other evolutionary algorithms (EAs), its convergence rate is still far from what is hoped for by scientists. On the one hand, a higher convergence rate is always expected for any optimization method used in seeking the global optimum of a non‐linear objective function. On the other hand, since all EAs, including DE, work with a population of solutions rather than a single solution, many evaluations of candidate solutions are required in the optimization process. If evaluation of candidate solutions is too time‐consuming, the overall optimization cost may become too expensive. One often has to limit the algorithm to operate within an acceptable time, which maybe is not enough to find the global optimum (optima), but enough to obtain a sub‐optimal solution. Therefore, it is continuously necessary to investigate the new strategies to improve the current DE algorithm.

The purpose of this paper is to execute the efficiency analysis of the selected metaheuristic algorithms (MAs) based on the investigation of analytical functions and investigation optimization processes for permanent magnet motor.

A comparative performance analysis was conducted for selected MAs. Optimization calculations were performed for as follows: genetic algorithm (GA), particle swarm optimization algorithm (PSO), bat algorithm, cuckoo search algorithm (CS) and only best individual algorithm (OBI). All of the optimization algorithms were developed as computer scripts. Next, all optimization procedures were applied to search the optimal of the line-start permanent magnet synchronous by the use of the multi-objective objective function.

The research results show, that the best statistical efficiency (mean objective function and standard deviation [SD]) is obtained for PSO and CS algorithms. While the best results for several runs are obtained for PSO and GA. The type of the optimization algorithm should be selected taking into account the duration of the single optimization process. In the case of time-consuming processes, algorithms with low SD should be used.

The new proposed simple nondeterministic algorithm can be also applied for simple optimization calculations. On the basis of the presented simulation results, it is possible to determine the quality of the compared MAs.

As well known, in the finite element method, the calculation and the location of the elements of the matrix C of the coefficients requires a lot of calculation times and memory employment especially for 3D problems. Besides, once the matrix C is properly filled, the solution of the system of linear equations is computationally expensive.

The paper consists of two parts. In the first part, to quickly calculate and store only the non‐null terms of the matrix of the system, a geometrical analysis on three‐dimensional domains has been carried out. The second part of the paper deals with the solution of the system of linear equations and proposes a procedure for increasing the solution speed: the traditional method of the conjugate gradient is hybridized with an adequate genetic algorithm (Genetic Conjugate Gradient).

The proposed geometrical procedure allows us to calculate the non‐null terms and their location within the matrix C by simple recursive formulas. The results concerning the genetic conjugate gradient show that the convergence to the solution of the linear system is obtained in a much smaller number iterations and the calculation time is also significantly decreased.

The approach proposed to analyze the geometrical space has been turned out to be very useful in terms of memory saving and computational cost. The genetic conjugate gradient is an original hybrid method to solve large scale problems quicker than the traditional conjugate gradient. An application of the method has been shown for current fields generated by grounding electrodes.

Generally, iterative methods which have some instability solutions in complex structural and non-linear mechanical problems are used to compute reliability index. The purpose of this paper is to establish a non-linear conjugate gradient (NCG) optimization algorithm to overcome instability solution of the Hasofer-Lind and Rackwitz-Fiessler (HL-RF) method in first-order reliability analysis. The NCG algorithms such as the Conjugate-Descent (CD) and the Liu-Storey (LS) are used for determining the safety index. An algorithm is found based on the new line search in the reliability analysis.

In the proposed line search for calculating the safety index, search direction is computed by using the conjugate gradient approach and the HL-RF method based on the new and pervious gradient vector of the reliability function. A simple step size is presented for the line search in the proposed algorithm, which is formulated by the Wolfe conditions based on the new and previous safety index results in the reliability analysis.

From the current work, it is concluded that the proposed NCG algorithm has more efficient, robust and appropriate convergence in comparison with the HL-RF method. The proposed methods can eliminate numerical instabilities of the HL-RF iterative algorithm in highly non-linear performance function and complicated structural limit state function. The NGC optimization is applicable to reliability analysis and it is correctly converged on the reliability index. In the NCG method, the CD algorithm is slightly more efficient than the LS algorithm.

This paper usefully shows how the HL-RF algorithm and the NCG scheme are formulated in first-order reliability analysis. The proposed algorithm is validated from six numerical and structural examples taken from the literature. The HL-RF method is not converged on several non-linear mathematic and complex structural examples, while the two proposed conjugate gradient methods are appropriately converged for all examples. The CD algorithm is converged about twice faster than the LS algorithm in most of the problems. Therefore, application of the NCG method is possible in reliability analysis.

A two‐grid iterative method for 3D linear elasticity problems, discretized using quadratic tetrahedral elements is proposed. The conjugate‐gradient method is used as smoother. As compared to the conjugate‐gradient alone, it is shown, via numerical examples, that the method is much more efficient on the basis of computing time and memory allocation. The convergence property of the method is sensitive to the regularity of the problem.

The paper aims to focus on: implementation of the fast‐multipole method (FMM) to open perfect electric conductors (PEC) problems involving triangular type wire‐to‐surface junctions; investigation and analysis of the effect of wire‐to‐surface junction configuration on the conditioning of the linear systems; application of the preconditioning technique to improve the efficiency of the FMM scheme on such problems.

A complete set of formulations is proposed to evaluate the far‐field terms of the impedance matrix that represent the couplings between the wire‐to‐surface junction and standard wire and PEC surfaces. The formulations are derived in a convenient form suitable for the application of the FMM. An iterative scheme is adopted to estimate the condition number of the linear systems arising from open‐PEC problems with wire‐to‐surface junctions and to investigate the effect of wire‐to‐surface junction configuration on the conditioning of the linear systems. The Crout version of ILU (ILUC) preconditioning strategy is applied to improve the convergence rate of the iterative solver on such problems.

The solutions show that the proposed formulations have accurately evaluated the far‐field terms that represent the couplings between the wire‐to‐surface junction and standard wire and PEC surfaces. The investigation of the conditioning of open‐PEC problems with junctions shows that the effect of the wire‐to‐surface junction configuration induced to the conditioning of the linear systems is negligible. The convergence records of several open‐PEC problems involving wire‐to‐surface junctions show that the ILUC preconditioning strategy is suitable to apply to such problems, as it significantly improves the performance of the iterative solver.

The proposed FMM strategy can be applied to many practical large‐scale open‐PEC problems that involve wire‐to‐surface junctions, such as antenna arrays and electromagnetic compatibility problems, to effectively speed up the overall electromagnetic simulation progress and overcome the bottleneck associated with the dense impedance matrix of the method‐of‐moments.

The application of the FMM to open‐PEC problems that involve wire‐to‐surface junctions has yet to be reported, which has been addressed in this work. This work also investigates the conditioning of such problems and analyzes the effect of wire‐to‐surface junction configuration on the conditioning of the linear systems. In addition, the performance of the ILUC preconditioner on such problems has not been reported, which has also been included in this report.

A hierarchically preconditioned conjugate gradient (PCG) method for finite element analysis is presented. Its use is demonstrated for the difficult problem of the non‐linear analysis of 3D reinforced concrete structures. Examples highlight the dramatic savings in computer storage and more modest savings in solution times obtained using PCG especially for large problems.

To provide an analysis of transient heat conduction, which is solved using different iterative solvers for graduate and postgraduate students (researchers) which can help them develop their own research.

Three‐dimensional transient heat conduction in homogeneous materials using different time‐stepping methods such as finite difference (Θ explicit, implicit and Crank‐Nicolson) and finite element (weighted residual and least squared) methods. Iterative solvers used in the paper are conjugate gradient (CG), preconditioned gradient, least square CG, conjugate gradient squared (CGS), preconditioned CGS, bi‐conjugate gradient (BCG), preconditioned BCG, bi‐conjugate gradient stabilized (BCGSTAB), reconditioned BCGSTAB and Gaussian elimination with incomplete Cholesky factorization.

Provides information on which time‐stepping method is the most accurate, which solver is the fastest to solve a symmetric and positive system of linear matrix equations of all those considered.

Fortran 90 code given as an abstract can be very useful for graduate and postgraduate students to develop their own code.

This paper offers practical help to an individual starting his/her research in the finite element technique and numerical methods.

The purpose of this research is to propose a new choice of nonnegative parameter t in Dai–Liao conjugate gradient method.

Conjugate gradient algorithms are used to solve both constrained monotone and general systems of nonlinear equations. This is made possible by combining the conjugate gradient method with the Newton method approach via acceleration parameter in order to present a derivative-free method.

A conjugate gradient method is presented by proposing a new Dai–Liao nonnegative parameter. Furthermore the proposed method is successfully applied to handle the application in motion control of the two joint planar robotic manipulators.

The proposed algorithm is a new approach that will not either submitted or publish somewhere.