Design sensitivity and Hessian matrix of generalized eigenproblems

Highlight
• A generalized eigenproblem is formed and its normalizations are presented and discussed.
• A unified consideration of the computation of the sensitivity and Hessian matrix is possible.
• The developed methods obtain eigenderivatives directly without using the left eigenvectors.
• The computational effort of obtaining the Hessian matrix is reduced remarkably.
• Two transcendental eigenproblems are considered to show the validity of the methods.

A generalized eigenproblem is formed and its normalizations are presented and discussed. Then a unified consideration of the computation of the sensitivity and Hessian matrix is studied for both the self-adjoint and non-self-adjoint cases. In the self-adjoint case, a direct algebraic method is presented to determine the eigensolution derivatives simultaneously by solving a linear system with a symmetric coefficient matrix. In the non-self-adjoint case, an algebraic method is presented to determine the eigensolution derivatives directly and simultaneously without having to use the left eigenvectors. In this sense, the method has advantages in computational cost and storage capacity. It is shown that the second order derivatives of eigensolutions can also be obtained by solving a linear system and the computational effort of obtaining Hessian matrix is reduced remarkably since only the recalculation of the right-hand vector of the linear system is required. The presented methods are accurate, compact, numerically stable and easy to implement. Finally, two transcendental eigenproblem examples are used to demonstrate the validity of the presented methods. The first example is considered as an example of the case of non-self-adjoint systems, which can result from feedback control systems. The other example is used to illustrate the case of self-adjoint systems by considering the three bar truss structure which is a viscoelastic composite structure and consists of two aluminum truss components and one viscoelastic truss. In addition, the capacity of predicting the changes of eigenvalues and eigenvectors with respect to the changes of design parameters is studied.