Second-Order Sensitivities of Inelastic Finite-Element Response by Direct Differentiation

In this paper analytical equations are developed and implemented to obtain second-order derivatives of finite-element responses with respect to input parameters. The work extends previous work on first-order response sensitivity analysis. Of particular interest in this study is the computational feasibility of obtaining second-order response sensitivities. In the past, the straightforward finite difference approach has been available, but this approach suffers from serious efficiency and accuracy concerns. In this study it is demonstrated that analytical differentiation of the response algorithm and subsequent implementation on the computer provides second-order sensitivities at a significantly reduced cost. The sensitivity results are consistent with and have the same numerical precision as the ordinary response. The computational cost advantage of the direct differentiation approach increases as the problem size increases. Several novel implementation techniques are developed in this paper to optimize the computational efficiency. The derivations and implementations are demonstrated and verified with two finite-element analysis examples.