Design with Several Eigenvalue Constraints by Finite Elements and Linear Programming

Abstract A finite element discretization, combined with a powerful numerical eigenvalue procedure, has proved to be a unified approach to eigenvalue analysis of elastic solids. Treating the sensitivity analysis as an integrated part of this approach, one obtains gradients of the eigenvalues without any new eigenvalue analysis. This forms the necessary information for an optimal redesign which is formulated as a linear programming problem. By a sequence of optimal redesigns, one then obtains a solution to the problem of optimal design or a solution to an inverse eigenvalue problem. Taking as an example the vibration of Timoshenko beams, we focus on the gradient functions, on the dependence of slenderness, and on the inherent problem of local optima.