Adjoint-based deviational Monte Carlo methods for phonon transport calculations

In the field of linear transport, adjoint formulations exploit linearity to derive powerful reciprocity relations between a variety of quantities of interest. In this paper, we develop an adjoint formulation of the linearized Boltzmann transport equation for phonon transport. We use this formulation for accelerating deviational Monte Carlo simulations of complex, multiscale problems. Benefits include significant computational savings via direct variance reduction, or by enabling formulations which allow more efficient use of computational resources, such as formulations which provide high resolution in a particular phase-space dimension (e.g., spectral). We show that the proposed adjoint-based methods are particularly well suited to problems involving a wide range of length scales (e.g., nanometers to hundreds of microns) and lead to computational methods that can calculate quantities of interest with a cost that is independent of the system characteristic length scale, thus removing the traditional stiffness of kinetic descriptions. Applications to problems of current interest, such as simulation of transient thermoreflectance experiments or spectrally resolved calculation of the effective thermal conductivity of nanostructured materials, are presented and discussed in detail.

Figure 1

Schematic of a transient thermoreflectance experiment. Point O denotes the center of the heating pulse, also taken to be the origin of the Cartesian $(x,y,z)$ set of axes. The system is assumed infinite in the $z>0$ direction and in the $x\text{−}y$ plane.

Figure 2

Temperature as a function of time in a transient thermoreflectance experiment. Results shown for the pulse center, $6\mu \text{m}$ away from the pulse center, and $12\mu \text{m}$ away from the pulse center.

Figure 3

Ratio between the standard deviation in the temperature measurement, ${\sigma }_T$, and the temperature, for three cylindrical detectors with height 10, 5, and 1 nm.

Figure 4

Sketch of the nanoporous structure studied in Sec. 4b. The dashed square represents the boundary of the computational domain, along which periodic boundary conditions are applied (see Refs. [5, 8]).

Figure 5

Frequency-resolved differential contribution to the thermal conductivity (measured heat flux per unit temperature gradient) from the longitudinal acoustic (LA) modes in the problem defined in Fig. 4. Result is normalized by the corresponding frequency-resolved differential contribution to the bulk thermal conductivity, and calculated using the adjoint method. Results shown for square pores of side 25 and 50 nm; the spacing between the pores is 2 $\mu \mathrm{m}$ in both cases. These calculations used $28000$ particles per frequency cell, for a total of 1399 frequency cells (for a total of approximately 40 million particles).

Figure 6

Frequency-resolved differential contribution to the thermal conductivity (measured heat flux per unit temperature gradient) from the longitudinal acoustic (LA) modes in the problem defined in Fig. 4. Result is normalized by the corresponding frequency-resolved differential contribution to the bulk thermal conductivity, and calculated using the forward method. Results shown for square pores of side 25 and 50 nm; the spacing between the pores is 2 $\mu \mathrm{m}$ in both cases. These calculations used a total of 40 million particles.

Figure 7

Contour plot of the solution (63) of Laplace's equation in a thin film with sinusoidal Dirichlet boundary conditions.

Figure 8

Temperature profile along the segment $AB$ in Fig. 7. Comparison between the solution to the BTE calculated using the adjoint method of Sec. 3a and the ACA method of Sec. 6a. The analytical solution of Laplace's equation is also shown.

Figure 9

Standard deviation ${\sigma }_{q_{x_2}^{\prime \prime }}$ of the particle contributions to the estimate of the heat flux at point $A$ of Fig. 7 in the $x_2$ direction, in the single free path model. In the ACA method, the standard deviation is proportional to $\langle \text{Kn}\rangle$. The latter outperforms the adjoint method with fixed control for $\langle \text{Kn}\rangle \le 0.2$.

Figure 10

(a) Standard deviation ${\sigma }_{q_{x_2}^{\prime \prime }}$ of the particle contributions to the estimate of the heat flux at point $B$ of Fig. 7 in the $x_2$ direction, in the variable free path model. (b) Standard deviation ${\sigma }_{q_{x_2}^{\prime \prime }}$ of the particle contributions to the estimate of the heat flux at point $A$ of Fig. 7 in the $x_2$ direction, in the variable free path model.

Figure 11

Evolution of the contributions of particles to the final estimate, when the adjoint method is used with the control (B1) along with a temperature field which is the solution of Laplace's equation for $\langle \text{Kn}\rangle =0.01$ (left) and $\langle \text{Kn}\rangle =0.001$ (right). This calculation was performed using the single free path model.

Figure 12

Evolution of the contributions of particles to the final estimate, when the adjoint method is used with the control (B1) along with a temperature field which is not a solution of Laplace's equation, for $\langle \text{Kn}\rangle =0.01$ (left) and $\langle \text{Kn}\rangle =0.001$ (right). This calculation was performed using the single free path model.