Sensitivity analysis and optimization of vibration modes in continuum systems

The sensitivity analysis of objective functions including the eigenmodes of continuum systems governed by scalar Helmholtz equations is carried out in continuum form. In addition, based on the sensitivity, the mode shapes are specified through numerical optimization. Using the continuum sensitivity and adjoint equation, the physical nature of them can be analyzed, which helps to explain the nature of the target optimization problem. Moreover, the continuum sensitivity and adjoint equation contribute to the quick numerical implementation of sensitivity analysis using software that can solve an arbitrary partial differential equation directly. A scalar Helmholtz equation in 1D or 2D domain is considered. The sensitivity analysis is performed for the general objective function formulated as a function of the eigenmode in continuum form. A minimization problem using the least squared error (i.e., difference) between the eigenvector and target mode shape is set as a sample objective function for both the first and second eigenmodes. The sensitivity and the adjoint equation are derived for this objective function. 1D and 2D numerical sensitivity analysis and optimization examples are studied to illustrate the validity of the derived sensitivity.