Addition Theorems for Real Vector Spherical Harmonics and Explicit Matrix Representations of the Quasi-Periodic Elastic Single Layer Potential

TL;DR

This paper introduces a novel multipole expansion method for quasi-periodic elastic potentials, deriving addition theorems for vector spherical harmonics to enable exact matrix computations and improve convergence.

Contribution

It presents new addition theorems for real vector spherical harmonics and explicit matrix representations for elastic single layer potentials in periodic arrays.

Findings

Exact matrix entries computed in closed form

Overcomes convergence issues of surface discretization

Eliminates series truncation using polylogarithm functions

Abstract

This paper develops a multipole expansion method for the quasi-periodic elastic single layer potential $\mathcal{S}_D^{\alpha,0}$ associated with the Kelvin tensor in one-dimensional periodic arrays. A key step in this approach is the derivation of translation addition theorems for the real vector spherical harmonics $V_{lm}$, $W_{lm}$, and $X_{lm}$. These addition theorems enable the exact calculation of all matrix entries of $\mathcal{S}_D^{\alpha,0}$ in closed form. By working entirely within the spherical harmonic basis, the proposed analytical method overcomes the poor convergence and mesh-dependent issues commonly caused by the direct surface discretization of weakly singular kernels. Additionally, the involved infinite sums are evaluated exactly using polylogarithm functions, which eliminates the need for series truncation. As an application, the integral equation $\mathcal{S}_D^{\alpha,0}[f]=\varphi$ is reduced to a linear system. This framework is further extended to dimer geometries consisting of two disjoint balls in each cell, where the off-diagonal matrices are explicitly formulated via the Lerch transcendent.