Convergence analysis of the thermal discrete dipole approximation

Abstract

The thermal discrete dipole approximation (T-DDA) is a numerical approach for modeling nearfield radiative heat transfer in complex three-dimensional geometries. In this work, the convergence of the T-DDA is investigated by comparison against the exact results for two spheres separated by a vacuum gap. The error associated with the T-DDA is reported for various sphere sizes, refractive indices and vacuum gap thicknesses. The results reveal that for a fixed number of subvolumes, the accuracy of the T-DDA degrades as the refractive index and the sphere diameter to gap ratio increase. A converging trend is observed as the number of subvolumes increases. The large computational requirements associated with increasing the number of subvolumes, and the shape error induced by large sphere diameter to gap ratios, are mitigated by using a nonuniform discretization scheme. Nonuniform discretization is shown to significantly accelerate the convergence of the T-DDA, and is thus recommended for near-field thermal radiation simulations. Errors less than 5% are obtained in 74% of the cases studied by using up to 82712 subvolumes. Additionally, the convergence analysis demonstrates that the T- † Corresponding authors. Tel.: +1 801 581 5721, Fax: +1 801 585 9825 E-mail addresses: mfrancoeur@mech.utah.edu (M. Francoeur), sheila.edalatpour@utah.edu (S. Edalatpour) 2 DDA is very accurate when dealing with surface polariton resonant modes dominating radiative heat transfer in the near field.

Near-field thermal electromagnetic transport: An overview

“…Note that the convergence and accuracy of the T-DDA have been discussed in Ref. [44]. The total dipole moment p i of sub-volume i is the sum of an induced dipole moment p i ind and a fluctuating dipole moment p i fl .…”

“…In Eq. (23) (22) such that there is no need to compute the instantaneous total dipole moment, which is a quantity that is not experimentally observable [44].…”

“…By applying the discretization scheme described above, the volume integral equation for the electric field (Eq. (17)) can be written in terms of dipole moments [44]: (20) where k 0 is the magnitude of the wavevector in vacuum, while …”

“…(20), E inc is the incident field column vector, while P is the column vector of unknown total dipole moments [44]. The system of equations (22) is stochastic, and its direct solution provides the instantaneous total dipole moment in each sub-volume.…”

“…In terms of modeling, closed-form solutions of near-field thermal electromagnetic transport have been derived for special geometries such as one-dimensional layered media [28,29], two large spheres [30][31][32], a sphere and a surface [33,34] and an arbitrary number of nanoparticles modeled as electric point dipoles [35,36] using the method of dyadic Green's function. Moreover, a number of numerical methods have been adapted to near-field thermal radiation: the finite-difference time-domain method [37][38][39], the finite-difference frequencydomain method [40], the boundary element method [41], the method of moments [42] and the discrete dipole approximation, which has been referred to as the thermal discrete dipole approximation (T-DDA) [43,44]. Additional information about near-field thermal radiation modeling and its potential engineering applications can be found in the numerous review papers published during the past decade [28,[45][46][47][48][49][50][51][52].…”