Numerical methods for evaluating the sensitivity of element modal strain energy

Various methods for the calculation of the sensitivity of element modal strain energy (MSE) are surveyed and classified as the finite difference method, the indirect method and the direct algebraic method. Also, based on the variation principle, a method is presented to accurately calculate the sensitivity of element MSE for undamped systems with distinct eigenvalues. The method computes the sensitivity by constructing a Lagrange function. Once the Lagrange multipliers are evaluated, the sensitivity can be determined directly. Note the Lagrange multipliers are independent of the number of design variables. More importantly, the proposed method is robust since the linear system is independent of the derivatives of system matrices and some weight constants are introduced in the coefficient matrices of the linear system to reduce the condition number. An operational flop count to compare the relative computational cost of the indirect method, Yan and Renʹs method and the proposed method is evaluated as a function of system matrix size, the number of mode shape of interest, the number of design variables and the number of individual element stiffness matrices. The storage capacity and robustness are also considered. General guidelines are established to choose the computational method for a given problem. Finally, three examples are used to show the effectiveness of the results. It is shown that the proposed method is more robust than the indirect method and the direct algebraic method.