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Adjoint DSMC Method for Spatially Inhomogeneous Boltzmann Equation with General Boundary Conditions

Russel Caflisch, Yunan Yang

Abstract

This manuscript derives adjoint equations for the numerical solution of the spatially inhomogeneous Boltzmann equation using Direct Simulation Monte Carlo (DSMC). The formulation accounts for spatial transport and a range of boundary conditions, including periodic boundaries, specular reflection, thermal reflection, and prescribed inflow. Numerical experiments are presented to validate the resulting adjoint system. These adjoint formulations are intended for use in gradient-based optimization, sensitivity analysis, and design problems involving rarefied gas dynamics.

Adjoint DSMC Method for Spatially Inhomogeneous Boltzmann Equation with General Boundary Conditions

Abstract

This manuscript derives adjoint equations for the numerical solution of the spatially inhomogeneous Boltzmann equation using Direct Simulation Monte Carlo (DSMC). The formulation accounts for spatial transport and a range of boundary conditions, including periodic boundaries, specular reflection, thermal reflection, and prescribed inflow. Numerical experiments are presented to validate the resulting adjoint system. These adjoint formulations are intended for use in gradient-based optimization, sensitivity analysis, and design problems involving rarefied gas dynamics.
Paper Structure (31 sections, 92 equations, 3 figures, 1 algorithm)

This paper contains 31 sections, 92 equations, 3 figures, 1 algorithm.

Table of Contents

  1. Introduction
  2. Background
  3. Boltzmann Equation
  4. The DSMC Method
  5. The Adjoint DSMC Method
  6. Adjoint DSMC Method With Periodic and Specular Reflecting Boundary Conditions
  7. Periodic Boundary Condition
  8. Reflecting Boundary Condition
  9. Adjoint DSMC Method With Thermal Boundary Conditions
  10. The Lagrangian
  11. Adjoint equation derivation
  12. Recovering forward DSMC
  13. Gradient for $m_x$ and $m_v$
  14. Final-time condition for the adjoint system
  15. Adjoint equations
  16. ...and 16 more sections

Figures (3)

  • Figure 1: Thermal-thermal BCs (\ref{['subsec:test1']}): (a) and (b) show the relative error between the adjoint gradient and the finite-difference gradient for $T_L$ and $T_R$, respectively, as the number of particles $N$ increases; (c) and (d) show the standard deviation of the adjoint gradient for $T_L$ and $T_R$, respectively. The standard deviation decays according to the Monte Carlo rate $\mathcal{O}(\frac{1}{\sqrt{N}})$. The three columns correspond to different spatial bin sizes: from left to right, $\Delta x = 0.05$, $0.1$, and $0.2$, respectively.
  • Figure 2: Thermal and specular reflections BCs (\ref{['subsec:test2']}): (a) and (b) show the relative error between the adjoint gradient and the finite-difference gradient for $T_0$ and $T_L$, respectively, as the number of particles $N$ increases; (c) and (d) show the standard deviation of the adjoint gradient for $T_0$ and $T_L$, respectively. The standard deviation decays with a rate of $\mathcal{O}(\frac{1}{\sqrt{N}})$.
  • Figure 3: Inflow BCs (\ref{['subsec:test3']}): (a) and (b) show the absolute error between the adjoint gradient and the finite-difference gradient for $T_L$ and $T_R$, respectively, as $N$ increases; (c) and (d) show the standard deviation of the adjoint gradient for $T_L$ and $T_R$, respectively. The standard deviation decays according to the Monte Carlo rate $\mathcal{O}(\frac{1}{\sqrt{N}})$.

Theorems & Definitions (1)

  • Remark 1: Practical implementation