Engineering Analysis with Boundary Elements 110 (2020) 56–68 Contents lists available at ScienceDirect Engineering Analysis with Boundary Elements journal homepage: www.elsevier.com/locate/enganabound Semi-discrete Green’s function for solution of anisotropic thermal/electrostatic Boussinesq and Mindlin problems: Application to two-dimensional material systems ✩ V.K. Tewary ∗ , E.J. Garboczi Applied Chemicals and Materials Division, NIST, Boulder, CO 80305, USA a r t i c l e i n f o Keywords: Two-dimensional material systems Electrostatic/thermal response Green’s function Laplace and Poisson equations Boussinesq and Mindlin problems Anisotropic phosphorene a b s t r a c t Green’s function (GF) for steady state Laplace/Poisson equation is derived for an anisotropic, finite two- dimensional (2D) composite material by solving a combined Boussinesq–Mindlin problem. A semi-discrete model of the material is developed in which only one Cartesian coordinate axis is discretized, while the other is treated as a continuous variable. The Fourier integral for the continuous coordinate is obtained analytically. Thus, a 2D problem needs only a 1D discretization. An approximate analytical estimate shows that the numerical con- vergence of our model is at least an order of magnitude better than fully discretized models. Numerical results are reported for the GF for a phosphorene composite containing an array of metallic inclusions. The GF is use- ful as a starting solution in boundary element calculations. It can be used for deriving the full solution of the Laplace/Poisson equation for an arbitrary distribution of sources and boundary values, used for modeling heat flow and electrostatic potential distribution in a 2D composite. These material systems are of strong topical in- terest because of their potential application in revolutionary new solid-state devices for energy conversion and quantum computing. This paper is another step towards developing GF based characterization techniques for modern 2D materials. 1. Introduction Modern two-dimensional (2D) materials are being considered as pri- mary materials for thermoelectric and quantum computing devices and many other revolutionary applications [1]. Currently there is a spe- cial interest in phosphorene because of its unusual physical proper- ties and possible application in solid state devices [2–7]. Its thermal conductivity is highly anisotropic, which makes it especially interest- ing, as well as relatively more challenging, for mathematical model- ing. For example, the thermal conductivity of black phosphorene has been estimated [8] to be 110 W/m-K and 36 W/m-K along its arm- chair and zigzag directions at room temperatures. Modeling and char- acterization of these materials are, therefore, subjects of strong topical interest. An earlier paper [9] suggested that the Green’s function (GF) is not just a mathematical artifact but is a physical quantity that can be mea- sured using a point probe technique such as SPM (Scanning Probe Mi- croscopy). Mathematically, GFs are known to be useful tools for solving the Laplace/Poisson equation. If they can be measured, they can provide a direct and reliable characterization technique that will better enable ✩ Contribution of the National Institute of Standards and Technology, an agency of the US Federal Govt. Not subject to copyright in the USA ∗ Corresponding author. E-mail address: vinod.tewary@nist.gov (V.K. Tewary). the industrial applications of modern 2D solids. However, to deconvolve the measurements and to extract physically meaningful data, it is neces- sary to have highly efficient and robust computational tools for calculat- ing the GFs and for identifying the discriminants in the GF that provide the characteristic features of the materials. This paper is a step in that direction, with special reference to phosphorene and its composites. Formally, the GF [10,11] gives the response of a system to a point probe, which is exactly what is measured by SPM type techniques. At macroscales, the 2D GF for an infinite solid in the continuum model can be written in terms of logarithmic functions. For finite solids, the GF must be calculated by specifying appropriate boundary conditions. Such problems can be broadly classified as the classic Boussinesq or Mindlin problems [11–13]. In the Boussinesq problem, Dirichlet bound- ary conditions are prescribed at the boundary: the value of the unknown function is forced to unity at a single point on the boundary and zero at all other points on the boundary. In the Mindlin problem the point probe, referred to as the source, is buried in the bulk. The Mindlin and the Boussinesq GFs can then give the solution of problems of physi- cal interest for an arbitrary distribution of sources subject to prescribed boundary conditions using linear superposition. For brevity, we will https://doi.org/10.1016/j.enganabound.2019.10.004 Received 8 May 2019; Received in revised form 26 September 2019; Accepted 10 October 2019 Available online 26 October 2019 0955-7997/Published by Elsevier Ltd.