In this paper, an efficient GV1-CG is developed to optimizing the modified conjugate gradient algorithm by using a new conjugate property. This is to to increase the speed of the convergence and retain the characteristic mass convergence using the conjugate property. This used property is proposed to public functions as it is not necessary to be a quadratic and convex function. The descent sharp property and comprehensive convergence for the proposed improved algorithm have been proved. Numerical results on some test function indicate that the new CG-method outperforms many of the similar methods in this field.

The conjugate gradient method plays a very important role in several fields, to solve problems of large sizes. To improve the efficiency of this method, a lot of work has been done; in this paper, we propose a new modification of PRP method to solve a large scale unconstrained optimization problems in relation with strong Wolf Powell Line Search property, when the latter was used under some conditions, a global convergence result was proved. In comparison with other known methods the efficiency of this method proved that it is better in the number of iterations and in time on $90$ proposed problems by use of Matlab.

In this paper, the conjugate gradient (CG) method is employed for identifying the parameters of crack in a functionally graded beam from natural frequency measurement. The crack is modeled as a massless rotational spring with sectional flexibility. Using the Euler-Bernoulli beam theory on two separate beams and applying the compatibility requirements of the crack, the characteristic equation can be obtained as a function of natural frequency and location and depth of crack. In direct problem, the natural frequency is computed using analytical analysis. Moreover, the location and depth of crack are determined by measuring the three natural frequencies of beam in inverse problem. In this study, the CG method is utilized in inverse problem to determine the location and depth of crack. The obtained results show the efficiency of CG algorithm in terms of accuracy and the convergence speed.

Different methods for analyzing nonlinear systems have been proposed. Most of them are not fast enough and also fail to deal with particular set of equations like nonlinear analysis of structures. In this paper, the preconditioned biconjugate gradient (PBCG) is combined with the Newton-Raphson method to reduce the convergence time of each iteration in nonlinear solution process. This algorithm replaces the procedure of inversing the tangent stiffness matrix with a new iterative method to solve the linearized system of equations. Also a new method is introduced to reduce computational time by applying preconditioning to the improved biconjugate gradient method. The proposed method is useful to analyze complex behavior structures, including unloading, snap-through buckling, and inelastic post-buckling analyses. In this paper a new preconditioner is proposed which is suitable for nonlinear analysis of structures. Results show that using the new methodology maintains the accuracy in an acceptable level while reduces computational cost and time, significantly.