Computation of an Integral with a normal derivitive

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1

Sorry if this question has been asked before, but I was wondering which method I should use to compute the following computation:

[tex]$$u^s_{\infty}(\hat{r}) = \dfrac{e^{\frac{\pi i}{4}}}{\sqrt{8\pi k}} \int_{\partial\Omega}u(x)\dfrac{\partial e^{-i\omega_0\hat{r}\cdot x}}{\partial n} - \dfrac{\partial u(x)}{\partial n}e^{-i\omega_0\hat{r}\cdot x},dS(x)$$[/tex]

I have my current file attached with my attempt in it, but I get a 0 as the result. Thanks in advance for any help!

https://ngsolve.org/media/kunena/attachments/1496/Forward Pass Code.txt

Attachment: ForwardPassCode.txt

2

Hi Emilio4d46,

BoundaryFromVolumeCF(u) does not give you the normal derivative of u. It is used for discontinuous functions to evaluate them at boundaries. Thus, your integrand always cancels out to 0.

With

n = specialcf.normal(2)
us_n = BoundaryFromVolumeCF(Grad(u_s)*n)
ecomp = exp(-1j*omega_0*(cos(r)*x + sin(r)*y))
ecomp_n = CoefficientFunction((ecomp.Diff(x),ecomp.Diff(y)))*n

I get non-zero values for your code. However, you have to be careful with normal derivatives at boundaries. By replacing BoundaryFromVolumeCF(Grad(u_s)*n) with Grad(u_s)*n I get different results, as the normal derivative has a jump over the boundary. So I’m not sure if the above code does 100% what you would like to have.

Best,
Michael

3

Thank you mneunteufel, that was a big help. The discrepancy between:

BoundaryFromVolumeCF(Grad(u_s)*n)

and

Grad(u_s)*n

Is confusing not to mention the sizable discrepancy between the first order approximation of the far field and the full integral.