From: Timo Heister Date: Tue, 18 Jun 2024 15:04:56 +0000 (-0400) Subject: step-90: doc fixes, add funding X-Git-Tag: v9.6.0-rc1~183^2 X-Git-Url: https://gitweb.dealii.org/cgi-bin/gitweb.cgi?a=commitdiff_plain;h=83e5dcb9474b21da1199538738afc10639b776f4;p=dealii.git step-90: doc fixes, add funding --- diff --git a/examples/step-90/doc/intro.dox b/examples/step-90/doc/intro.dox index 4318c4e16e..8e5d6a7cfb 100644 --- a/examples/step-90/doc/intro.dox +++ b/examples/step-90/doc/intro.dox @@ -2,7 +2,9 @@ This program was contributed by Vladimir Yushutin and Timo Heister, Clemson University, 2023. This material is based upon work partly supported by the National -Science Foundation Award DMS- +Science Foundation Award DMS-2028346, OAC-2015848, EAR-1925575, and by the Computational +Infrastructure in Geodynamics initiative (CIG), through the NSF under Award +EAR-0949446, EAR-1550901, EAR-2149126 via the University of California -- Davis. @@ -24,7 +26,7 @@ arbitrary shape and size manifesting in the so-called ``small cut" problem and r form that restores well-conditioning of the problem. Two aspects are of our focus. First, the surface approximation is separated from the discretization of the surface PDE, -e.g. a $Q_2$ discrete level-set and a $Q_1$ solution are possible on the same bulk triangulation. +e.g., a $Q_2$ discrete level-set and a $Q_1$ solution are possible on the same bulk triangulation. Second, we make sure that the performance of TraceFEM in the parallel implementation corresponds to that of a classical fitted FEM for a two-dimensional problem. We demonstrate how to achieve both goals by using a combination of MeshWorker and NonMatching capabilities. @@ -54,7 +56,7 @@ at support points. Notice also that the mesh is graded: any cell has at most 4 n Once a desired geometry approximation $\Gamma_h$ is achieved using the iterative approach above, we can start forming the linear system using the constructed normals and quadratures. For the purposes of the tutorial we choose a non-trivial surface $\Gamma$ given by -@f{equation*} +@f{equation*}{ \frac{x^2}{4}+ y^2 + \frac{4 z^2} {(1 + 0.5 \sin(\pi x))^{2}} = 1 @f} The OY and OX views of this tamarind-shaped, exact surface $\Gamma$ are shown below along with the mesh after @@ -66,7 +68,7 @@ three iterations (the approximation $\Gamma_h$ is not shown).

Model problem

We would like to solve the simplest possible problem defined on a surface, namely the Laplace--Beltrami equation, -@f{equation*} +@f{equation*}{ -\Delta_\Gamma u + c u = f \qquad \text{in }\, \Gamma, @f} where we take $c=1$ for concreteness. We added the term $cu$ to the left-hand side so the problem becomes well-posed @@ -75,7 +77,7 @@ in the absence of any boundary; an alternative could be to take $c=0$ but impose

Manufactured exact solution

We choose the test solution and the right-hand side forcing as the restriction to $\Gamma$ of -@f{equation*} +@f{equation*}{ u(x,y,z)=xy\,,\quad f(x,y,z)=xy + 2.0\,\mathbf{n}_x \mathbf{n}_y + \kappa (y \mathbf{n}_x + x\mathbf{n}_y), @f} @@ -90,11 +92,11 @@ the cells of the background mesh with the iso surface of an approximation of the never actually create any two-dimensional meshes for the surface but only compute approximate quadrature points and surface normals. Next we distribute degrees of freedom over a thin subdomain $\Omega_h$ that completely covers $\Gamma_h$ and that consists of the intersected cells $\mathcal{T}_\Gamma^h$, -@f{equation*} +@f{equation*}{ \mathcal{T}_\Gamma^h = \{ T \in \mathcal{T}^{h} : T \cap \Gamma_h \neq \emptyset \}. @f} The finite element space where we want to find our numerical solution, $u_h$, is now -@f{equation*} +@f{equation*}{ V_h = \{ v \in C(\Omega_h) : v \in Q_p(T), \, T \in \mathcal{T}_\Gamma^h \}, @f} where $\Omega_h$ is the union of all intersected cells from $\bigcup_{T \in \mathcal{T}_\Gamma^h} \overline{T}$. @@ -106,7 +108,7 @@ we set the active_fe_index to either 0 or 1. To determine whether a cell is intersected or not, we use the class NonMatching::MeshClassifier. A natural candidate for a weak formulation involves the following (bi)linear forms -@f{align*} +@f{align*}{ a_h(u_h, v_h) = (\nabla_{\Gamma_h} u_h, \nabla_{\Gamma_h} v_h)_{\Gamma_h}+(u_h, v_h)_{\Gamma_h}\,,\qquad L_h(v_h) = (f^e,v_h)_{\Gamma_h}. @f} @@ -118,11 +120,11 @@ The latter is identified with $f^e$ on $\Gamma_h$ and it is not the normal exten However, the so-called "small-cut problem" may arise and one should introduce the stabilized version of TraceFEM: Find $u_h \in V_h$ such that -@f{equation*} +@f{equation*}{ a_h(u_h,v_h) + s_h(u_h, v_h) = L_h(v_h), \quad \forall v_h \in V_\Omega^h. @f} Here the normal-gradient stabilization $s_h$ involves the three-dimensional integration over whole (but intersected) cells and is given by -@f{equation*} +@f{equation*}{ s_h(u_h,v_h) = h^{-1}(\mathbf{n}_h\cdot\nabla u_h, \mathbf{n}_h\cdot\nabla v_h)_{\Omega_h}, @f} Note that the $h^{-1}$ scaling may be relaxed for sufficiently smooth solutions such as the manufactured one, but we @@ -130,7 +132,7 @@ choose the strong scaling to demonstrate the extreme case @cite traceFEM_review_

Discrete Level Set Function

In TraceFEM we construct the approximation $\Gamma_h$ using the interpolant $\psi_h$ of the exact level-set function on the bulk triangulation: -@f{align*} +@f{align*}{ \Gamma_h &= \{x \in \mathbb{R}^{\text{3}} : \psi_h(x) = 0 \}. @f} The exact normal vector $\mathbf{n}$ is approximated by $\mathbf{n}_h=\nabla\psi_h/\|\nabla\psi_h\|$ which, together