From: Simon Sticko Date: Sat, 22 Oct 2022 08:39:45 +0000 (+0200) Subject: Fix documentation misprint, modulus should be norm. X-Git-Tag: v9.5.0-rc1~350^2 X-Git-Url: https://gitweb.dealii.org/cgi-bin/gitweb.cgi?a=commitdiff_plain;h=455f7c878a997afcbb0c56ab11420d5b1298b838;p=dealii.git Fix documentation misprint, modulus should be norm. --- diff --git a/include/deal.II/non_matching/immersed_surface_quadrature.h b/include/deal.II/non_matching/immersed_surface_quadrature.h index bc59656502..953ebfac01 100644 --- a/include/deal.II/non_matching/immersed_surface_quadrature.h +++ b/include/deal.II/non_matching/immersed_surface_quadrature.h @@ -78,23 +78,29 @@ namespace NonMatching * * When dim = spacedim - 1, this class represents a (spacedim-2)-dimensional * integral. That is, if spacedim = 3 we have a line integral immersed in a - * face. Here, the transformation between the face, $F$, and reference face, - * $\hat{F}$, reads + * face. Let $\hat{r}(t)$, $t \in [0,T]$ be an arc-length parameterizations of + * $\hat{F}\cap \hat{S}$, i.e., the part of the surface that intersects the + * face in reference space. This means that $\bar{r}(t) = F_K(\hat{r}(t))$ is + * a parameterization of $S\cap F$. The transformation of the line integral + * now reads * @f[ - * \int_{S\cap F} f dr = - * \int_{S\cap F} f |d\bar{r}| = - * \int_{\hat{S}\cap\hat{F}} f \circ F_{K} | J d\hat{r}| - * \approx \sum_{q} f \left(F_{K}(\hat{x}_{q}) \right) |J_q \hat{t}_q| w_q, + * \int_{S\cap F} f dr + * = \int_{0}^T f(\bar{r}(t)) \left \|\frac{d\bar{r}}{dt} \right \| dt + * = \int_{0}^T f(F_K(\hat{r}(t))) \left \| J \frac{d\hat{r}}{dt} \right \| dt + * \approx \sum_{q} f \left(F_{K}(\hat{x}_{q}) \right) \|J(\hat{x}_q) + * \hat{t}_q \| w_q, * @f] - * where $\hat{t}_q = \hat{n}_q \times \hat{n}_F$ is the tangent to the curve - * at $\hat{x}_q$ and $\hat{n}_F$ is the face normal. It would be possible to - * compute the tangent by only knowing the normal to the curve in the face - * plane (i.e. the dim-dimensional normal). However, when these quadratures - * are used, the weak form typically involves the so-called conormal, which - * can not be computed without knowing the surface normal in + * where $\hat{t}_q = \frac{d\hat{r}}{dt}(x_q) $ is the tangent to the curve + * at $\hat{x}_q$. This tangent can also be computed as + * $t_q = \hat{n}_q \times \hat{n}_F / \| \hat{n}_q \times \hat{n}_F \|$ where + * $\hat{n}_F$ is the face normal. It would be possible to compute the tangent + * by only knowing the normal to the curve in the face plane (i.e. the + * dim-dimensional normal). However, when these quadratures are used, the weak + * form typically involves the so-called conormal, which can not be computed + * without knowing the surface normal in * $\mathbb{R}^{\text{spacedim}}$. The conormal is the unit vector parallel to - * the projection of the face normal into the surface plane. This is - * essentially the same thing as the normalized + * the projection of the face normal into the surface plane. This is the same + * as the normalized * @ref GlossBoundaryForm "boundary form". */ template