Differentiable Lagrangian Shock Hydrodynamics with Application to Stable Shock Acceleration of Density Interfaces

TL;DR

This work addresses controlling Richtmyer-Meshkov instability (RMI) in shock-accelerated density interfaces by developing a differentiable, gradient-based optimization framework for compressible, Lagrangian hydrodynamics. It combines adjoint methods with automatic differentiation to efficiently compute $\frac{\partial O}{\partial y_0}$ for problems with $>100$ design variables, employing a time-accurate, energy-conserving discretization, checkpointing, and FE partial assembly. The authors demonstrate successful suppression of RMI while simultaneously enhancing interface acceleration, and they incorporate energy and non-negativity constraints to ensure physically feasible solutions. The approach provides a scalable pathway for optimizing complex hydrodynamic drives in fusion-relevant settings and establishes a foundation for extending to ALE remapping, multiphysics, and uncertainty quantification in future work.

Abstract

We develop a gradient based optimization approach for the equations of compressible, Lagrangian hydrodynamics and demonstrate how it can be employed to automatically uncover strategies to control hydrodynamic instabilities arising from shock acceleration of density interfaces. Strategies for controlling the Richtmyer-Meshkov instability (RMI) are of great benefit for inertial confinement fusion (ICF) where shock interactions with many small imperfections in the density interface lead to instabilities which rapidly grow over time. These instabilities lead to mixing which, in the case of laser driven ICF, quenches the runaway fusion process ruining the potential for positive energy return. We demonstrate that control of these instabilities can be achieved by optimization of initial conditions with (> 100) parameters. Optimizing over a large parameter space like this is not possible with gradient-free optimization strategies. This requires computation of the gradient of the outputs of a numerical solution to the equations of Lagrangian hydrodynamics with respect to the inputs. We show that the efficient computation of these gradients is made possible via a judicious application of (i) adjoint methods, the exact formal representation of sensitivities involving partial differential equations, and (ii) automatic differentiation (AD), the algorithmic calculation of derivatives of functions. Careful regularization of multiple operators including artificial viscosity and timestep control is required. We perform design optimization of > 100 parameter energy field driving the Richtmyer Meshkov instability showing significant suppression while simultaneously enhancing the acceleration of the interface relative to a nominal baseline case.

Figures (10)

• Figure 3: Error for the Taylor test for various perturbation magnitudes $h$. The black lines are the Taylor Error while the red are for the gradient error. The blue indicates expected scaling (quadratic) for the Taylor Error to confirm correct derivatives.
• Figure 6: Amount of additional compute time added (measured in terms of the time of the forward solve) added given a memory budget written as a percentage of the total number of states from the forward solve. The different lines are for different numbers of total states from the forward solve.
• Figure 7: Graph outline of the optimization cycle starting with an initial guess $y_0$. The solution is propagated from $y_0$ to $O$. If the objective is not sufficiently converged, then the bottom path is taken to calculate the gradient. Then, the filters can be applied then the initial guess can be updated. This process can be iterated until convergence.
• Figure 8: (a) Plots of the unoptimized (black) and optimized solutions (blue). (b) Plot of the error in the conjugate gradient solve demonstrating linear slope in the log-y scale.
• Figure 9: Domain and initial energy configuration for the RMI case study. The nodes (1, 2 ,3) indicate where the tracer particles are placed (used in Equation \ref{['eq:rmi_objective']}).