I need help computing the following integral:

$$ \int_{}\frac{(1+jk|\vec{r}-\vec{r}^\prime|)e^{-jk|\vec{r}-\vec{r}^\prime|}}{|\vec{r}-\vec{r}^\prime|}d\vec{r}^\prime $$

in this integral $\vec{r}$ and $\vec{r}^\prime$ are the position vectors that correspond to the observation point and source point, respectively. This integral comes from the Electric-Field Integral Equation (EFIE) kernel.

  • 5
    $\begingroup$ What is the domain you want to compute the integral for? What is the question? Do you want suggestions for a method? What have you tried so far? $\endgroup$
    – H. Rittich
    Mar 22, 2018 at 12:25
  • 1
    $\begingroup$ It seems that you are trying to solve EFIE and have troubles computing the Green's function integral. By the look of it, you have full-wave 3-D. Since you mention EFIE, you probably are discretizing only using surface elements (triangles, 2-D elements). Can you please confirm or add the required information? Including where exactly you are stuck. $\endgroup$
    – Anton Menshov
    Mar 22, 2018 at 14:28
  • $\begingroup$ in this problem integral is on a squre plan $\endgroup$
    – hamed
    Mar 23, 2018 at 0:36
  • $\begingroup$ in this integral r and r' are on the two squre plan. i discritized both of them. now I can not integral from this $\endgroup$
    – hamed
    Mar 23, 2018 at 0:40

1 Answer 1


A classic paper for evaluation of the integrals commonly present in computational electromagnetics (EM) is:

The challenge to evaluate the EFIE integrals is the required singularity treatment. If both $\mathbf{r}$ and $\mathbf{r}^\prime$ belong to the same patch in the discretized geometry, one cannot use just quadrature-based integration.

The idea is to subtract a singularity and integrate it analytically and the rest (already without a singularity), numerically. So, say you need to calculate the integral ($R=|\mathbf{r}-\mathbf{r'}|$):

$$ \int\limits_S\frac{e^{-jkR}}{R}ds'=\underbrace{\int\limits_S\frac{e^{-jkR}-1}{R}ds'}_{I_1}+\underbrace{\int\limits_S\frac{1}{R}ds'}_{I_2} $$ Here, $S$ is the source patch and $\mathbf{r}$ is fixed. Now, integral $I_1$ does not have a singularity and can be evaluated using, say, 2-D Gaussian quadrature (use barycentric coordinates and apply Gaussian quadrature over $\xi$, $\eta$ with a proper Jacobian)

Integral $I_2$ can be found analytically. The formula for a planar polygon $S$ is given in the referenced paper.

You still might want to use this technique of singularity subtraction even when observation and source patches are different, but relatively close together. Such singularity treatment will decrease the error coming from numerical integration and reduce the number of quadrature points required to achieve the accuracy for nearby interactions.

I assume, that right now you've discretized your EFIE using pulse-basis functions and you are testing your EFIE using pulses as well. That might be good enough for certain scenarios, but I will point out that the use of proper basis functions and testing procedure might be important later in your development process.

Note: there are other techniques to tackle the singularity in this Green's function, including special quadrature rules, but I personally prefer this approach.


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