This work presents a framework to solve DAE systems. The framework obtains the time-domain response of the DAE system using the BDF scheme, and the gradient of functions of interest with respect to design variables using the adjoint method.

The novelty of the framework is the use of CSDL that enables the user to write high-level python code and automates the computation of the required derivatives. Users need only provide a CSDL model of the DAE and the FoI instead of deriving the gradient themselves or providing complicated function handles. Thus, the framework saves time for users who are prototyping DAE formulations and solving complex engineering problems.

Validation

The framework is validated against three benchmark problems: spring-mass-damper-system, pendulum system, and RLC circuit; each with different functions of interest and design variables. It is observed that the difference between the gradients from the adjoint method and the finite difference method is negligible.

While this work demonstrates the framework on the benchmark problems, the framework is applicable to any general DAE system. For example, aircraft wings can be represented as nonlinear beam elements. This leads to a DAE system with 18 states per node in the beam. It has been shown that such a system can efficiently size structural components of the aircraft wing subjected to dynamic loads specified by 14-CFR regulations.

A. Spring-Mass-Damper System

The percentage difference between the results from adjoint method and finite difference method is 0.0207%.

Mass displacement as a function of time

Angular displacement and angular velocity of the pendulum as a function of time

Current in the RLC circuit as a function of time