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Shape sensitivity analysis of finite element models is useful for structural optimization and design modifications. Within numerical design optimization, semi-analytical method for sensitivity analysis is frequently applied to estimate the derivative of an objective function with respect to the design variables. Generally numerical sensitivity analysis widely suffers from severe error due to the perturbation size and find a method which is not sensitive to the perturbation size is topics under study. Complex variable methods for sensitivity analysis have some potential advantages over other methods. For first order sensitivities using the complex variable method, the implementation is straightforward, only requiring a perturbation of the finite element mesh along the imaginary axis. This paper uses a complex variable and combine it with discrete sensitivity analysis, thus present new method to obtain derivatives for linear structure. The advantage of this method are quickly, accuracy and its simple implementation. The methodologies are demonstrated using two dimensional finite element models of linear elasticity problems with known analytical solutions. Obtained sensitivity derivatives are compared to the exact solution and also finite difference solutions and show that the proposed method is effective and can predict the stable and accurate sensitivity results.

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The semi-analytical method (SAM) is an approach that computationally efficient and easy to implement. That's why this method often used for the sensitivity analysis of finite element models. However, SAM is not without defect especially in problems that rigid body motions are relatively large reveals severe inaccuracy. Such errors outcome from the pseudo load vector calculated by differentiation using the finite difference method. In the present paper, a new semi-analytical approach based on complex variables is proposed to compute the sensitivity of nonlinear finite element models. This method combines the complex variable method with the discrete sensitivity analysis to obtain the response sensitivity accurately and efficiently. The current approach maintains the computational efficiency of the semi-analytical method with higher accuracy. In addition, the current approach is insensitive to the choice of step size, a feature that simplifies its use in practical problems. The method can be used to nonlinear finite elements only requires minor modifications to existing finite element codes. In this paper, the authors demonstrate that the discrete sensitivity analysis and the complex variable method are equivalent and solve the same equation. Finally, the accuracy of the method is investigated through the various numerical examples by comparing by other methods and will show that this method is reliable and independent of step size.