The difference-based equivalent static load method: an improvement of the ESL method’s nonlinear approximation quality

Nonlinear dynamic structural optimization is a real challenge, in particular for problems that require the use of explicit solvers, e.g., crash. Here, the number of design variables is typically very limited. A way to overcome this drawback is to use linear auxiliary load cases which are derived from nonlinear dynamic analysis results in order to enable the application of linear static response optimization. The equivalent static load method (ESLM) provides a well-defined procedure to create such linear auxiliary load cases. The main idea here is that after the selection of a number of representative time steps, a set of equivalent static loads (ESLs) is computed for each time step such that the resulting displacement field in the linear static analysis is identical to the respective field in the nonlinear dynamic analysis. Each set of ESLs defines an auxiliary load case, which is used in the linear static response optimization. The crucial point is that the finite element (FE)-model for each auxiliary load case describes the undeformed initial geometry. This can lead to insufficient approximation quality in the linear static system for highly nonlinear problems. To overcome this drawback, a difference-based extension of the ESL method called DiESL has been developed for nonlinear dynamic response optimization problems. Here, the FE-model for each auxiliary load case describes the deformed nonlinear geometry at the respective time, and the corresponding ESLs create only the displacement field leading to the deformed state of the subsequent ESL time step. Consequently, responses in each linear auxiliary load case (corresponding to a time step) are computed as the accumulated sum of the previous linear auxiliary load cases. Furthermore, the linear static response optimization problem consists not only of one but of n T FE-models where n T is the number of selected time steps. Such a multi-model optimization (MMO) can be solved with commercial FE solvers. It turns out that the DiESL approach leads to a significant improvement of the nonlinear approximation quality and faster convergence to the optimum when compared to standard ESLM. This will be demonstrated and discussed based on selected test examples.