Design optimization of dynamic flexible multibody systems using the discrete adjoint variable method

TL;DR

This work extends the discrete adjoint variable method (DAVM) to design optimization of dynamic flexible multibody systems by deriving adjoint equations directly from discretized motion equations, yielding a linear algebraic system rather than differential-algebraic equations. Integrated with a geometric variational integrator, the approach computes exact gradients of the discrete objective $\phi$ with respect to design variables $\mathbf{a}$, including both geometrical and non-geometrical parameters, without requiring state-derivative calculations. By employing rotation-free ANCF for beams and natural coordinates for rigid bodies, the method maintains a constant mass matrix and simplifies constraint handling, enabling efficient sensitivity analysis and gradient-based optimization. Three numerical examples—sensitivity analysis of a flexible pendulum, optimization of a rigid-spring-beam assembly, and optimization of an automotive double-wishbone suspension—demonstrate accuracy compared with prior results and substantial computational savings (e.g., DAVM > FD by factors up to ~40), illustrating the practical impact for high-dimensional design and generative design workflows.

Abstract

The design space of dynamic multibody systems (MBSs), particularly those with flexible components, is considerably large. Consequently, having a means to efficiently explore this space and find the optimum solution within a feasible timeframe is crucial. It is well known that for problems with several design variables, sensitivity analysis using the adjoint variable method extensively reduces the computational costs. This paper presents the novel extension of the discrete adjoint variable method to the design optimization of dynamic flexible MBSs. The extension involves deriving the adjoint equations directly from the discrete, rather than the continuous, equations of motion. This results in a system of algebraic equations that is computationally less demanding to solve compared to the system of differential algebraic equations produced by the continuous adjoint variable method. To describe the proposed method, it is integrated with a numerical time-stepping algorithm based on geometric variational integrators. The developed technique is then applied to the optimization of MBSs composed of springs, dampers, beams and rigid bodies, considering both geometrical (e.g., positions of joints) and non-geometrical (e.g., mechanical properties of components) design variables. To validate the developed methods and show their applicability, three numerical examples are provided.

Figures (17)

• Figure 1: Two-noded Euler-Bernoulli beam element
• Figure 2: Rigid body
• Figure 3: A rigid-spring-beam assembly
• Figure 4: (a) Pendulum at $t=0$ (b) Deflected pendulum at time instance $t$
• Figure 5: Difference between the sensitivity values with respect to $h$, $\rho$ and $E$ fromPi2012 and the proposed DAVM