1
$\begingroup$

The Helmholtz decomposition for a vector field B contains both volume integrals and two boundary integrals (https://en.wikipedia.org/wiki/Helmholtz_decomposition). For brevity I show just one of the boundary integrals here ($\mathbf{r}$ is a point in the volume $V$, $\delta V$ is the boundary of $V$, and $\mathbf{r'}$ is a point on the boundary $\delta V$) $$ \mathbf{B}(\mathbf{r}) = \int_{\delta V}\mathbf{B}(\mathbf{r'})\cdot\mathbf{n'}\frac{\mathbf{r}-\mathbf{r'}}{|\mathbf{r}-\mathbf{r'}|^{3}}dS' $$ where the gradient has been taken inside the integral and applied to the free space Green's function.

Clearly the integrand is singular when $\mathbf{r}=\mathbf{r'}$.

My main problem is not having the mathematical vocabulary to help my search for numerical techniques. Can someone please provide some guidance on the type of problem this is, to help with my search for numerical methods to deal with it?

Simple 3D integration techniques won't cut it close to the singularity, as the integrand won't behave like polynomials. I can't recall where I found this, but I have seen that adding a small $\delta^{2}$ (where $\delta < dx$ ($dx$ being the discretaization of the simple integration scheme I have) inside the $||$ in the denominator will remove the singularity for the discrete point $\mathbf{r}=\mathbf{r}'$. This obviosuly doesn't help with accuracy close to the singularity though.

Appreciate any help!

Improve this question
$\endgroup$

1 Answer 1

Reset to default
2
$\begingroup$

The equation in the OP does not quite make sense (why does the same field appear on both sides of the equation?), but I understand the question as inquiring how to numerically compute the Helmholtz decomposition$^\ast$ $B=-\nabla\Phi+\nabla\times A$ of a three-dimensional vector field $B$. This is worked out in Three Approaches for the Helmholtz Decomposition of a General Vector Field (2005).

$^\ast$ The integrals to evaluate are \begin{align} \Phi(\mathbf{r}) & =\frac 1 {4\pi} \int_V \frac{\nabla'\cdot\mathbf{B} (\mathbf{r}')}{|\mathbf{r}-\mathbf{r}'|} \, \mathrm{d}V' -\frac 1 {4\pi} \oint_{\delta V} \mathbf{\hat{n}}' \cdot \frac{\mathbf{B} (\mathbf{r}')}{|\mathbf{r}-\mathbf{r}'|} \, \mathrm{d}S' \\[8pt] \mathbf{A}(\mathbf{r}) & =\frac 1 {4\pi} \int_V \frac{\nabla' \times \mathbf{B}(\mathbf{r}')}{|\mathbf{r}-\mathbf{r}'|} \, \mathrm{d}V' -\frac 1 {4\pi} \oint_{\delta V} \mathbf{\hat{n}}'\times\frac{\mathbf{B} (\mathbf{r}')}{|\mathbf{r}-\mathbf{r}'|} \, \mathrm{d}S' \end{align}
Improve this answer
$\endgroup$
1
  • $\begingroup$ The equation in the OP is exactly one of the terms in the Helmholtz Decomposition so I don’t know why the responder has a problem with it, especially as they quote the HD equation in their response. The B on the RHS is the field on the boundary. Eg if the field is solenoidal and curl free then the HD is a boundary value problem.. I thank them for their suggested paper though and will take a look. $\endgroup$ Commented Nov 29, 2020 at 13:31

You must log in to answer this question.

Start asking to get answers

Find the answer to your question by asking.

Ask question

Explore related questions

See similar questions with these tags.