Direct method for second-order sensitivity analysis of modal strain energy

This paper proposed a Lagrange-based method for calculation of the second-order sensitivity of modal strain energy (MSE) values. According to scheme of the direct method, a Lagrange functional of the element MSE, which added the eigenproblem and the normalization augmented by Lagrange multipliers as constraints, is constructed firstly. After that, the Lagrange multipliers can be decided by setting variations of the Lagrange functional with respect to state variables to zeros. And then, the sensitivity of MSE can be yielded by derivative of the Lagrange functional easily. The accuracy of the new method and the affection of normalization criterion are verified by the numerical instances of a simply supported beam and a truss structure. A portal frame is employed to testify the prediction ability of both the first-order and the second-order sensitivities of MSE values. As the numerical example of the portal frame shows, the computation complexity of the new method can be reduced significantly in comparison of indirect methods at similar accuracy. (C) 2019 Published by Elsevier Ltd.