From 854aed80eb82f0bb560d9c8f6f163890ead36421 Mon Sep 17 00:00:00 2001 From: Wolfgang Bangerth Date: Tue, 12 Apr 2011 03:55:31 +0000 Subject: [PATCH] Add a bit of text for now. git-svn-id: https://svn.dealii.org/trunk@23577 0785d39b-7218-0410-832d-ea1e28bc413d --- deal.II/examples/step-46/doc/intro.dox | 96 ++++++++++++++++++++++++++ 1 file changed, 96 insertions(+) diff --git a/deal.II/examples/step-46/doc/intro.dox b/deal.II/examples/step-46/doc/intro.dox index 1518606e43..f29321f0a7 100644 --- a/deal.II/examples/step-46/doc/intro.dox +++ b/deal.II/examples/step-46/doc/intro.dox @@ -1,3 +1,99 @@

Introduction

+This program deals with the problem of coupling different physics in different +parts of the domain. Specifically, let us consider the following situation: + +- In a part $\Omega_f$ of $\Omega$, we have a fluid flowing that satisfies the + time independent Stokes equations (in the form that involves the strain + tensor): + @f{align*} + -2\eta\nabla \cdot \varepsilon(\mathbf v) + \nabla p &= 0, + \qquad \qquad && \text{in}\ \Omega_f\\ + -\nabla \cdot \mathbf v &= 0 && \text{in}\ \Omega_f. + @f} + Here, $\mathbf v, p$ are the fluid velocity and pressure, respectively. + We prescribe the velocity on part of the external boundary, + @f{align*} + \mathbf v = \mathbf v_0 \qquad\qquad + \text{on}\ \Gamma_{f,1} \subset \partial\Omega \cap \partial\Omega_f + @f} + while we assume free-flow conditions on the remainder of the external + boundary, + @f{align*} + (2\eta \varepsilon(\mathbf v) + p \mathbf 1) \cdot \mathbf n = 0 + \qquad\qquad + \text{on}\ \Gamma_{f,2} = \partial\Omega \cap \partial\Omega_f \backslash + \Gamma_{f,1}. + @f} +- The remainder of the domain, $\Omega_s = \Omega \backslash \Omega_f$ is + occupied by a solid whose deformation field $\mathbf u$ satisfies the + elasticity equation, + @f{align*} + -\nabla \cdot C \varepsilon(\mathbf u) = 0 \qquad\qquad + & \text{in}\ \Omega_s, + @f} + where $C$ is the rank-4 elasticity tensor (for which we will use a + particularly simple form by assuming that the solid is isotropic). + It deforms in reaction to the forces exerted by the + fluid flowing along the boundary of the solid. We assume this deformation to + be so small that it has no feedback effect on the fluid, i.e. the coupling + is only in one direction. For simplicity, we will assume that the + solid's external boundary is clamped, i.e. + @f{align*} + \mathbf u = \mathbf 0 \qquad\qquad + \text{on}\ \Gamma_{s,1} = \partial\Omega \cap \partial\Omega_s + @f} +- As a consequence of the small displacement assumption, we will pose the + following boundary conditions on the interface between the fluid and solid: + first, we have no slip boundary conditions for the fluid, + @f{align*} + \mathbf v = \mathbf 0 \qquad\qquad + \text{on}\ \Gamma_{i} = \partial\Omega_s \cap \partial\Omega_f; + @f} + secondly, the forces on the solid equal the normal strain from the fluid, + @f{align*} + (C \varepsilon(\mathbf u)) \mathbf n = + (2 \eta \varepsilon(\mathbf v) + p \mathbf 1) \mathbf n \qquad\qquad + \text{on}\ \Gamma_{i} = \partial\Omega_s \cap \partial\Omega_f. + @f} + + +This sort of coupling is of course possible by simply having two Triangulation +and two DoFHandler objects, one each for each of the two subdomains. On the +other hand, deal.II is much simpler to use if there is a single DoFHandler +object that knows about the discretization of the entire problem. + +This program is about how this can be achieved. Note that the goal is not to +present a particularly useful physical model (a realistic fluid-structure +interaction model would have to take into account the finite deformation of +the solid and the effect this has on the fluid): this is, after all, just a +tutorial program intended to demonstrate techniques, not to solve actual +problems. Furthermore, we will make the assumption that the interface between +the subdomains is aligned with cell faces. + + +

The general idea

+ +The fundamental idea to implement these sort of problems in deal.II goes as +follows: in the problem formulation, the velocity and pressure variables +$\mathbf v, p$ only live in the fluid subdomain $\Omega_f$. But let's assume +that we extend them by zero to the entire domain $\Omega$ (in the general case +this means that they will be discontinuous along $\Gamma_i$). So what is the +appropriate function space for these variables? We know that on $\Omega_f$ we +should require $\mathbf v \in H^1(\Omega_f)^d, p \in L_2(\Omega_f)$, so for +the extensions $\tilde{\mathbf v}, \tilde p$ to the whole domain the following +appears a useful set of function spaces: +@f{align*} + \tilde {\mathbf v} &\in V + = \{\tilde {\mathbf v}|_{\Omega_f} \in H^1(\Omega_f)^d, \quad + \tilde {\mathbf v}|_{\Omega_s} = 0 \} + \\ + \tilde p &\in P + = \{\tilde p|_{\Omega_f} \in L_2(\Omega_f), \quad + \tilde p|_{\Omega_s} = 0 \}. +@f} +Note that this is indeed a linear function space with obvious norm. Since no +confusion is possible in practice, we will henceforth omit the tilde again to +denote the extension of a function to the whole domain and simply refer by +$\mathbf v, p$ to both the original and the extended function. -- 2.39.5