[Paper Review] Fluctuating surface-current formulation of radiative heat transfer: theory and applications

Alejandro W. Rodríguez, M. T. Homer Reid+1 more|DSpace@MIT (Massachusetts Institute of Technology)|Apr 4, 2013

Thermal Radiation and Cooling Technologies22 citations

TL;DR

This paper introduces a fluctuating surface-current (FSC) formulation for radiative heat transfer between arbitrarily shaped bodies, leveraging surface-integral-equation (SIE) methods from classical electromagnetism. By modeling thermal fluctuations via fictitious surface currents on object boundaries, the approach enables both semi-analytical spectral solutions in high-symmetry geometries and robust numerical solutions via boundary element methods (BEM) in complex configurations, including finite slabs, cylinders, and cones.

ABSTRACT

We describe a novel fluctuating-surface current formulation of radiative heat transfer between bodies of arbitrary shape that exploits efficient and sophisticated techniques from the surface-integral-equation formulation of classical electromagnetic scattering. Unlike previous approaches to non-equilibrium fluctuations that involve scattering matrices---relating "incoming" and "outgoing" waves from each body---our approach is formulated in terms of "unknown" surface currents, laying at the surfaces of the bodies, that need not satisfy any wave equation. We show that our formulation can be applied as a spectral method to obtain fast-converging semi-analytical formulas in high-symmetry geometries using specialized spectral bases that conform to the surfaces of the bodies (e.g. Fourier series for planar bodies or spherical harmonics for spherical bodies), and can also be employed as a numerical method by exploiting the generality of surface meshes/grids to obtain results in more complicated geometries (e.g. interleaved bodies as well as bodies with sharp corners). In particular, our formalism allows direct application of the boundary-element method, a robust and powerful numerical implementation of the surface-integral formulation of classical electromagnetism, which we use to obtain results in new geometries, including the heat transfer between finite slabs, cylinders, and cones.

Motivation & Objective

• To develop a general framework for radiative heat transfer between bodies of arbitrary shape, overcoming limitations of prior scattering-matrix or time-domain approaches.
• To extend the fluctuating surface-current (FSC) formalism—previously used for equilibrium Casimir forces—to non-equilibrium radiative heat transfer between bodies at different temperatures.
• To enable both semi-analytical solutions using spectral bases (e.g., Fourier series, spherical harmonics) and fully numerical solutions via boundary element methods (BEM) for complex geometries.
• To establish a rigorous connection between non-equilibrium field correlation functions and surface-integral formulations through a novel surface-current representation.
• To demonstrate the method’s applicability to new and challenging geometries, such as finite slabs, cylinders, and cones, where analytical methods fail.

Proposed method

• Formulates radiative heat transfer in terms of fictitious surface currents on the boundaries of bodies, avoiding the need to solve wave equations for the fields.
• Uses the surface-integral-equation (SIE) framework of classical electromagnetism to relate surface currents to electromagnetic fields via Green’s function operators.
• Applies a generalized fluctuation-dissipation theorem for non-equilibrium systems, expressing field correlations as integrals over volume sources, which are then recast as surface integrals via the SIE formalism.
• Employs a complex-symmetric operator structure with a sign-flip matrix $\mathcal{S}$ to ensure reciprocity and proper symmetry in the system matrices.
• Utilizes Galerkin discretization with basis functions (e.g., RWG for BEM) to convert the continuous SIE into a linear system $M\mathbf{x} = \mathbf{b}$, where $M = W^{-1}$ is the SIE matrix.
• Enables both spectral methods (for high-symmetry bodies) and boundary element method (BEM) for arbitrary geometries, including those with sharp corners or interleaved structures.

Experimental results

Research questions

• RQ1 How can radiative heat transfer between arbitrarily shaped bodies at different temperatures be formulated without relying on scattering matrices or time-domain simulations?
• RQ2 Can the fluctuating surface-current (FSC) formalism, previously used for equilibrium Casimir forces, be generalized to non-equilibrium radiative heat transfer?
• RQ3 What is the mathematical and physical correspondence between volume-integrated field correlations in non-equilibrium systems and surface-integral representations via fictitious currents?
• RQ4 To what extent can the SIE framework and BEM be used to compute heat transfer in complex geometries such as finite slabs, cylinders, and cones?
• RQ5 How do the symmetry and definiteness properties of the SIE matrices ($M$, $L$, $\Gamma^\star$) ensure physical consistency and numerical stability in the heat transfer calculation?

Key findings

• The FSC formulation enables accurate and efficient computation of radiative heat transfer in arbitrary geometries by reducing the problem to surface unknowns, avoiding volume discretization.
• The method achieves fast-converging semi-analytical results in high-symmetry geometries (e.g., planar, spherical) by using spectral bases such as Fourier series or spherical harmonics.
• The boundary element method (BEM) implementation allows robust numerical solutions for complex configurations, including finite slabs, cylinders, and cones, which are difficult to treat analytically.
• The SIE matrices $M$ and operators $L$, $\Gamma^\star$ are shown to be negative-semidefinite and complex-symmetric under reciprocity, ensuring physical consistency and numerical stability.
• The formalism establishes a rigorous link between non-equilibrium field correlations (involving products of Green’s functions) and surface current representations, enabling a unified treatment of thermal fluctuations.
• The reciprocity relation $\Gamma^{\mathrm{T}}\star = \mathcal{S}(\Gamma\star)\mathcal{S}$ is preserved under the SIE formulation, allowing interchange of sources and fields with appropriate sign flips of magnetic components.