Skip to main content
U.S. flag

An official website of the United States government

Dot gov

Official websites use .gov
A .gov website belongs to an official government organization in the United States.


Secure .gov websites use HTTPS
A lock ( ) or https:// means you've safely connected to the .gov website. Share sensitive information only on official, secure websites.

Semi-discrete Green's function for solution of anisotropic thermal/electrostatic Boussinesq and Mindlin problems: Application to two-dimensional materials



Vinod K. Tewary, Edward J. Garboczi


The Green's function (GF) for the steady state Laplace/Poisson equation is derived for an anisotropic finite two-dimensional (2D) composite material by solving a combined Boussinesq- Mindlin problem. The source term for the GF is a delta-function located somewhere in the bulk of the solid (Mindlin problem). The boundary condition for the GF is prescribed in terms of a delta function at a single point on the outer boundary (Boussinesq problem). The calculated GF is, therefore, quite general and can be used to find the full solution of the Laplace/Poisson equation for an arbitrary but integrable distribution of sources and boundary values in a composite material. The piecewise continuous composite, consisting of a host material containing one or more inclusions, is assumed to be infinite and periodic in one direction (X- direction) but of finite arbitrary size in the perpendicular direction in the plane (Z- direction). A semi-discrete semi-analytic model is developed for the calculation of the GF subject to the boundary conditions described above. The primary equation is solved by using a partial Fourier transform technique. Our model is semi-discrete in the sense that the solution space is discretized only in the periodic X-direction but not in the Z-direction. This allows the space to be filled up exactly in terms of the discrete elements even if the shape of the inclusions are geometrically irregular. Further, our model is semi-analytic in the sense that the Fourier integration in the Z direction is done exactly, so that a 2D problem has only a 1D discretization. An approximate analytical estimate shows that the numerical convergence of our semi-discrete model is at least an order of magnitude faster than conventional fully discretized models (e.g., finite-element or boundary-element models). As an application of the method, GF numerical results are presented at all points on the plane of 2D phosphorene containing a metallic inclusion.

Physical Review Materials


Boussinesq problem, Green's function, Laplace equation, Mindlin problem, phosphorene, Poisson equation, thermal electrostatic modeling, two-dimensional materials.
Created October 3, 2019, Updated January 27, 2021