مطالعه بر روی تجزیه و تحلیل حساسیت طراحی بر اساس متغیر مختلط در مسئله مقادیر ویژه

In the sensitivity techniques, the adjoint variable method is quite popular because it reduces computation time and save computer resources. Commonly, the adjoint variable method employs exact analytical differentiation with respect to the design variables. But, it can be cumbersome to precisely differentiate every given type of finite element. For improving this trouble, the numerical differentiation scheme can replace this exact manner of differentiation. Even though the numerical differentiation has some advantages, it suffers from severe inaccuracy due to the perturbation size dilemma. This paper employs a complex variable which is not much influenced by the perturbation size. Then, the adjoint variable method combined with complex variables is applied to obtain the shape and size sensitivity for structural optimization. Numerical examples demonstrate that the proposed method can predict stable sensitivity results and that its accuracy is remarkably superior to traditional sensitivity evaluation methods.

There are several methods for calculating the sensitivity of structural response with respect to design parameters, including the direct sensitivity method (DSM) and the adjoint variable method (AVM). Generally, the DSM, which is based on numerical derivatives, cannot avoid numerical error due to perturbation size. There have been various approaches to improving this sensitivity problem over the last few decades. The global difference method(GDM) is one such approach, and is a simple means of calculating the sensitivity by comparison to the analytical method. It is also independent of the type of finite elements involved in the calculation. Moreover, if a reasonable perturbation size is chosen, the GDM provides a reliable solution in geometrically nonlinear problems. However, this method may not be efficient since it requires a large amount of computer resources and its result is severely dependent upon the perturbation sizes of the design variables. The traditional semi-analytic method (TSAM) has been proposed as a compromise between accuracy and efficiency since the TSAM is expected to be as easy to implement and as accurate as the analytical methods are [1], [2], [3], [4], [5], [6] and [7]. Moreover, it does not require excessive computer resources because it calculates the sensitivity at the element-level. However, the numerical derivative calculated by TSAM generates truncation errors and round-off errors that depend on the magnitude of the perturbation size. For overcoming the inaccuracies arising from perturbation size, the rigid body mode separation scheme has been proposed by Van Keulen, De Boer, Parente and Vaz [8], [9] and [10]. This method eliminates the severe errors caused by the influence of the rigid body modes in the TSAM through the separation and the exact differentiation of the rigid body modes. Still, this method does not assure the rapid convergence of the sensitivity for large perturbation values. To improve the sensitivity accuracy in the range of the large perturbation sizes, the matrix inverse has been introduced by evaluating the higher order terms with a Von Neumann series [11]. Recently, Cho and Kim have proposed an iterative scheme that is combined with the mode decomposition technique and series expansion to alleviate the truncation error of large perturbation ranges as well as the round-off error of small perturbation sizes [12]. By comparison to these DSMs, the AVM is remarkably advantageous in cases where the number of design variables is greater than that of the constraints, such as in cases of displacement or stress. In practical design situations, it is usually necessary to consider several load cases. In a multiple load-case simulation the adjoint method becomes more attractive. Therefore, the AVM has been widely used in the calculation of sensitivity of the static problem [13], [14] and [15], and it has been applied to evaluate the eigenvalue sensitivity in distinct eigenvalue problems [16], [17] and [18]. However, in the AVM, it is cumbersome to obtain the corresponding analytical design sensitivity for every type of finite element since it is common to use the exact derivatives. This defect can be avoided only if the use of exact derivatives can be replaced with a reliable alternate numerical differentiation. Cho and Kim have proposed a sensitivity computation that is combined with the iterative scheme and numerical derivatives in the AVM [19]. The iterative scheme of this method is based on a combination with the mode decomposition technique to alleviate the truncation error as well as the round-off error. However, these sensitivity methods based on numerical derivatives cannot completely overcome the perturbation troubles. This paper employs complex variables that have been mainly applied for the CFD problem for the purposes of numerical differentiation. The complex variable method (CVM) has a simple formula that presents the accurate derivatives and that assures robust and stable sensitivity results because it does not involve the subtractive cancellation error, as well as working accurately and efficiently regardless of the perturbation size. In the beginning, Lyness and Moler employed the complex variable for the approximation of nth derivatives in complex planes [20] and introduced the CVM for the estimation of the nth derivatives of analytic functions [21]. After that, Suqire and Trapp employed this method for obtaining the first derivative accurately and easily [22]. Martins et al. implemented automated Fortran/C++ code by using the CVM and demonstrated the advantages of the CVM in computing sensitivity [23] and [24]. For an application problem, the CVM has been employed for CFD sensitivity and multidisciplinary design optimization of aero-structural models [25] and [26]. This method has been employed for nonlinear finite element codes and extended to efficient pseudo spectral simulation codes that use the fast Fourier transform (FFT) [27]. Recently, it has been used to solve the iterative sensitivity equation in steady-state discrete sensitivity analyses [28]. As mentioned above, the CVM has been mainly employed for estimating the sensitivity of computational fluid dynamics or obtaining the higher order nth derivatives in the real function. In the present study, a combination of the AVM and the CVM (ACVM) is proposed for calculating the eigenvalue sensitivities related to size and shape design variables. The key features of ACVM are its simplicity and reliability in the calculation of the structural sensitivity based on the finite element code. The ACVM can be easily implemented in a commercial FEM program and does not depend on elements without perturbation trouble. Through a few numerical examples, the accuracy of the ACVM is compared with that of a traditional sensitivity scheme and it is demonstrated that its reliability is independent of perturbation values.

This paper provided a robust design sensitivity method by combining the adjoint variable method and the complex variable method (ACVM) in the eigenvalue problem. Adjoint variable method (AVM) is efficient and save computation time compared to other sensitivity schemes because it calculates the sensitivity values only in position that analyzer is willing to obtained. Moreover, once the adjoint variable is obtained, it can be successively or repeatedly used for the calculation of the sensitivity regardless of the design variable. But, it is common that AVM employs the exact manner for differentiation of the stiffness matrix and mass matrix. It can be cumbersome to precisely differentiate by exact manner for every given type of finite element. For improving the efficiency of AVM, the exact derivatives can be replaced with the numerical difference scheme. But, the traditional sensitivity schemes such as TSAM or GDM are still severely dependent on the perturbation values even though there have been various studies to avoid this defect. Thus, by employing the advantages of AVM and the complex variable which is not much influenced on the perturbation size, this paper presented newly the numerical sensitivity calculation methodology based on AVM. Through a few numerical examples, this study has demonstrated the superiority of the ACVM to traditional sensitivity evaluation methods. If only small perturbation sizes are selected, then the truncation error can be ignored, the ACVM provides a stable solution to/for the structural sensitivity of the eigenvalue problem. Additionally, the error value does not have a significant effect on the sensitivity field in the overall perturbation size, as was shown in the numerical examples. In conclusion, the sensitivity errors produced through traditional sensitivity evaluation methods could be significantly reduced through application of the ACVM. The numerical sensitivity in the framework of the adjoint variable method can be improved by considering complex variables. Compared to traditional sensitivity schemes, the method of calculating sensitivities presented in this paper offers high efficiency and assures good performances in all ranges of the perturbation value.