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Tests of the scuff-em core library

The libscuff unit of the scuff-em test suite tests several elemental calculations implemented by libscuff routines for assembling and post-processing the linear SIE system.

Table of Contents

• SIE matrix elements: Singular and non-singular 4-dimensional integrals
  • SIE matrix elements in scuff-em
    • Matrix elements
    • Edge-edge interactions
    • Panel-panel interactions
    • Desingularization
  • Reference values

$\def\vb#1{{\mathbf #1}} \def\mb#1{{\mathbb #1}} \def\mc#1{{\mathcal #1}} % \def\inp#1#2{{\big\langle #1 \big| #2 \big\rangle}} \def\Inp#1#2{{\Big\langle #1 \Big| #2 \Big\rangle}} \def\INP#1#2{{\bigg\langle #1 \bigg| #2 \bigg\rangle}} % \def\vmv#1#2#3{{\big\langle #1 \big| #2 \big | #3 \big\rangle}} \def\Vmv#1#2#3{{\Big\langle #1 \Big| #2 \Big | #3 \Big\rangle}} \def\VMV#1#2#3{{\bigg\langle #1 \bigg| #2 \bigg | #3 \bigg\rangle}} %$

SIE matrix elements: Singular and non-singular 4-dimensional integrals

This subunit tests libscuff routines for computing element of the SIE system matrix, which involve singular and nonsingular 4-dimensional integrals over pairs of triangular panels.

SIE matrix elements in scuff-em

First we review the computation of SIE matrix elements in scuff-em.

Matrix elements

In the most general case, each pair of RWG basis functions $\{\vb b_{\alpha}, \vb b_{\beta}\}$ contributes a $2\times 2$ block of entries to the system matrix:

$$\Delta \vb M(\vb b_{\alpha}, \vb b_{\beta}) =\sum_r \left(\begin{array}{cc} i\omega \mu_r \Vmv{\vb b_{\alpha}}{\mb G(k_r)}{\vb b_{\beta}} & -ik_r \Vmv{\vb b_{\alpha}}{\mb C(k_r)}{\vb b_{\beta}} \\ -ik_r \Vmv{\vb b_{\alpha}}{\mb C(k_r)}{\vb b_{\beta}} & -i\omega \epsilon_r \Vmv{\vb b_{\alpha}}{\mb G(k_r)}{\vb b_{\beta}} \\ \end{array}\right)$$

where $r$ runs over the indices of the (zero, one, or two) regions through which the basis functions interact, $\{\epsilon_r, \mu_r, k_r\}$ are the material properties and wavevector magnitude in region $r$, and $\mb G, \mb C\propto \nabla \times \mb G$ are the homogeneous dyadic Green's functions of Maxwell's equations.

If one or both basis functions lie on PEC surfaces then $\Delta\vb M$ reduces to a $2\times 1$, $1\times 2$, or $1\times 1$ matrix.

Edge-edge interactions

Matrix elements of the $\mb G$ and $\mb C$ operators are 4-dimensional integrals over the supports of the basis functions that may be written in terms of the scalar Greens' function:

$$\begin{align} \mb G_{\alpha\beta} \equiv \Vmv{\vb b_{\alpha}}{\mb G}{\vb b_{\beta}} &= \mc I_{\bullet \alpha\beta} -\frac{4}{k^2} \mc I_{\nabla \alpha\beta}, \qquad \mb C_{\alpha\beta} \equiv \Vmv{\vb b_{\alpha}}{\mb C}{\vb b_{\beta}} \mc I_{\times \alpha\beta} \\ \mc I_{\bullet\alpha\beta} &\equiv \iint \Big( \vb b_{\alpha} \cdot \vb b_{\beta}\Big) \Phi(r_{\alpha\beta}) \, d\vb x_{\alpha} d\vb x_{\beta} \\ \mc I_{\nabla\alpha\beta} &\equiv \iint \Phi(r_{\alpha\beta}) \, d\vb x_\alpha d\vb x_\beta \\ \mc I_{\times\alpha\beta} &\equiv \iint \Big[ \big(\vb b_{\alpha} \cdot \vb b_{\beta}\big)\cdot \vb r_{\alpha\beta}\Big] \Psi(r_{\alpha\beta}) \, d\vb x_{\alpha} d\vb x_{\beta} \end{align}$$

with

$$\vb r_{\alpha\beta} \equiv \vb x_{\alpha} - \vb x_{\beta}, \qquad r_{\alpha\beta} \equiv |\vb r_{\alpha\beta}|$$ $$\Phi(r) \equiv \frac{e^{ik r}}{4\pi r}, \Psi(r) \equiv (ikr-1)\frac{e^{ikr}}{4\pi r^3}.$$

Panel-panel interactions

Each edge-edge integral is a sum of 4 panel-panel integrals, e.g.

$$\mc I_{\bullet \alpha\beta} = \mc I^{++}_{\bullet\alpha\beta} -\mc I^{+-}_{\bullet\alpha\beta} -\mc I^{-+}_{\bullet\alpha\beta} +\mc I^{--}_{\bullet\alpha\beta}$$

(This is for the general case in which both basis functions are full RWG basis functions; if one or both are half-RWG functions then the number of contributing panel pairs reduces to 2 or 1.)

The panel-panel integrals are

$$\begin{align*} I^{\sigma\tau}_{\bullet\alpha\beta} &=\frac{\ell_{\alpha} \ell_{\beta}}{4A_{\alpha}^{\sigma} A_{\beta}^{\tau}} \int_{{\mc P}_{\alpha}^{\beta}} d \vb x_{\alpha} \int_{{\mc P}_{\beta}^{\tau}} d \vb x_{\beta} \big(\vb x_{\alpha} - \vb Q_{\alpha}^{\sigma}\big) \cdot \big(\vb x_{\beta} - \vb Q_{\beta}^{\tau}\big) \Phi(r_{\alpha\beta}) \\ I^{\sigma\tau}_{\nabla\alpha\beta} &=\frac{\ell_{\alpha} \ell_{\beta}}{4A_{\alpha}^{\sigma} A_{\beta}^{\tau}} \int_{{\mc P}_{\alpha}^{\beta}} d \vb x_{\alpha} \int_{{\mc P}_{\beta}^{\tau}} d \vb x_{\beta} \Phi(r_{\alpha\beta}) \\ I^{\sigma\tau}_{\times\alpha\beta} &=\frac{\ell_{\alpha} \ell_{\beta}}{4A_{\alpha}^{\sigma} A_{\beta}^{\tau}} \int_{{\mc P}_{\alpha}^{\beta}} d \vb x_{\alpha} \int_{{\mc P}_{\beta}^{\tau}} d \vb x_{\beta} \bigg[ \big(\vb x_{\alpha} - \vb Q_{\alpha}^{\sigma}\big) \times \big(\vb x_{\beta} - \vb Q_{\beta}^{\tau}\big) \cdot \vb r_{\alpha\beta} \bigg] \Psi(r_{\alpha\beta}) \end{align*}$$

Desingularization

For pairs of basis functions with 1 or more common vertices I desingularize by subtracting off the first 3 nonvanishing terms in the small-$r$ series for $\Phi(r)$ and $\Psi(r)$ and integrating these terms separately. Thus I put

$$\begin{align*} \Phi(r) \equiv C_{-1} r^{-1} + C_0 + C_1 r + C_2 r^2 + \Phi^{\text{DS}{(r) \\ \Psi(r) \equiv D_{-3} r^{-3} + D_{-1} r^{-1} + D_0 + D_1 r + \Psi^{\text{DS}}(r) \end{align*}$$

and e.g.

$$\mc{I}_{\bullet\alpha\beta} = \sum_{n=-1}^2 C_n \mc{I}^{(n)}_{\bullet \alpha \beta} +\mc{I}^{\text{DS}}_{\bullet \alpha \beta}$$

Reference values

As reference values for testing libscuff routines for evaluating RWG integrals, I consider matrix elements between RWG functions on the following surface mesh, in which the triangles are sufficiently regular that it's easy to write down and evaluate the required 4-dimensional integrals.

For example, the positive-positive-panel contribution to $\mc{I}_{\nabla 0 0}$ reads

$$\mc{I}_{\nabla 0 0}^{++} = \frac{\ell^2}{4A_0^2} \iint_{A_0\times A_0} \Phi(r_{\alpha\beta})d\vb x_{\alpha} d\vb x_{\beta}$$

Parameterize the integral according to

$$\vb x_{\alpha} = u_1 L \hat{\vb x} + u_2 L \hat{\vb y}, \quad \vb x_{\beta } = u_3 L \hat{\vb x} + u_4 L \hat{\vb y}, \quad r_{\alpha\beta} = L\sqrt{ (u_1-u_3)^3 + (u_2-u_4)^2}$$

to yield an integral over the unit 4-dimensional hypercube:

$$\mc I_{\nabla 0 0}^{++} = \ell^2 \int_{[0,1]^4} u_1 u_3 \Phi(r_{\alpha\beta}(\vb u) )d\vb u$$

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