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

Abstract

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 nT FE-models where nT 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.

Difference-based Equivalent Static Load Method with adaptive time selection and local stiffness adaption

“…For dynamic responses such as section forces, velocities, and accelerations, which are not defined in linear statics, it is necessary to find proper approximations. For velocity and acceleration, for instance, simple finite forward (Jeong et al 2010;Triller et al 2021) or central (Karev et al 2018(Karev et al , 2019 differences between adjacent load cases (corresponding to adjacent time steps) can be used.…”

“…Instead of using equidistant time steps (Triller et al 2021), we now propose how to select the value of all n T ESL-times adaptively in each cycle. The ESL-times should be placed at points in time at which nonlinearities are dominant.…”

“…The main issues result from the fact that the ESLs are always calculated based on the undeformed initial geometry. To circumvent this drawback, a difference-based approach for calculating the ESLs-the DiESL method-has been introduced recently (Triller et al 2021). It has been shown that the DiESL method enables a significant increase in approximation quality for displacements and strains from nonlinear dynamic problems while simultaneously providing faster convergence.…”

Multidisciplinary optimization of automotive mega-castings merging classical structural optimization with response-surface-based optimization enhanced by machine learning

“…The ESL-procedure has recently been extended by the introduction of the so-called DiESL-method for sizing and topology optimization problems, incorporating nonlinearities in geometry and material. It reportedly results in better approximations in the linear static response sub-problem for nonlinear dynamic problems such as crash 48 , 50 .…”

“…referring to Refs. 48 , 49 . The ESL-procedure has recently been extended by the introduction of the so-called DiESL-method for sizing and topology optimization problems, incorporating nonlinearities in geometry and material.…”