Commit f1f267f9 by Praetorius, Simon

Merge branch 'docs/navier_stokes' into 'master'

documentation of navier_stokes example

See merge request !238
 Navier-Stokes equation ====================== We consider the incompressible Navier-Stokes equation in a domain $\Omega=(0,L_x)\times(0,L_y)$, math \varrho(\partial_t\mathbf{u} + (\mathbf{u}\cdot\nabla)\mathbf{u}) - \nabla\cdot\big(2\nu\mathbf{D}(\mathbf{u})\big) - \nabla p = \mathbf{f},\qquad\text{ in }\Omega \\ \nabla\cdot\mathbf{u} = 0  with velocity $\mathbf{u}$, pressure $p$, density $\varrho$, dynamic viscosity $\nu$, symmetric rate-of-strain tensor $\mathbf{D}(\mathbf{u})=\frac{1}{2}(\nabla\mathbf{u}+\nabla\mathbf{u}^\top)$, and external volume force $\mathbf{f}$, w.r.t. to boundary conditions $\mathbf{u}=\mathbf{g}$ on $\partial\Omega$. For an overview, see also [Wikipedia](https://en.wikipedia.org/wiki/Navier%E2%80%93Stokes_equations). Discretization -------------- ### Weak formulation In a weak variational formulation, we try to find $\mathbf{u}(t,\cdot)\in \mathbf{V}_\mathbf{g}:=\{\mathbf{v}\in H^1(\Omega)^d\,:\, \operatorname{tr}_{\partial\Omega}\mathbf{v} = \mathbf{g}\}$ and $p(t,\cdot)\in Q:=L_0^2(\Omega)$, such that math \int_\Omega \varrho(\partial_t\mathbf{u} + (\mathbf{u}\cdot\nabla)\mathbf{u})\cdot\mathbf{v} + \nu\nabla\mathbf{u}:\nabla\mathbf{v} + p\,\nabla\cdot\mathbf{v}\,\text{d}\mathbf{x} = \int_\Omega \mathbf{f}\cdot\mathbf{v}\,\text{d}\mathbf{x},\qquad\forall \mathbf{v}\in \mathbf{V}_0, \\ \int_\Omega q\,\nabla\cdot\mathbf{u}\,\text{d}\mathbf{x} = 0,\qquad \forall q\in Q.  for $t\in[0,T]$, given $\mathbf{u}(0,\cdot)\equiv\mathbf{u}^0$. The pair $\big(\mathbf{u}(t,\cdot), p(t,\cdot)\big)$ is thus in the product space $\mathbf{V}_\mathbf{g}\times Q$ where $\mathbf{V}_\mathbf{g}$ is also a product of $H^1$ spaces, i.e., $\mathbf{V}_\mathbf{g}=V_{g_0}\times V_{g_1}\times V_{g_2}\simeq [V_{g_i}]^d$. Later, this space will be approximated with a Taylor-Hood space of piecewise quadratic and linear functions. ### Time-discretization A linearized time-discretization with a semi-implicit backward Euler scheme then reads: For $k=0,1,2,\ldots, N$ find $\mathbf{u}^{k+1}\in\mathbf{V}_\mathbf{g},\, p\in Q$, s.t. math \int_\Omega \varrho\Big(\frac{1}{\tau}(\mathbf{u}^{k+1} - \mathbf{u}^k) + (\mathbf{u}^k\cdot\nabla)\mathbf{u}^{k+1}\Big)\cdot\mathbf{v} + \nu\nabla\mathbf{u}^{k+1}:\nabla\mathbf{v} + p\,\nabla\cdot\mathbf{v}\,\text{d}\mathbf{x} \\ = \int_\Omega \mathbf{f}(t^{k+1})\cdot\mathbf{v}\,\text{d}\mathbf{x},\qquad\forall \mathbf{v}\in \mathbf{V}_0, \\ \int_\Omega q\,\nabla\cdot\mathbf{u}^{k+1}\,\text{d}\mathbf{x} = 0,\qquad \forall q\in Q.  with $0=t^0( lagrange<2>(), flatInterleaved()), lagrange<1>(), flatLexicographic());  At this point, any Dirichlet boundary conditions are ignored in the definition of the bases. We build the Taylor-Hood basis as a composition of a product of$d\times\mathbb{V}_2$bases for the velocity components and a$\mathbb{V}_1$basis for the pressure. We have to use composite(...) to combine the different types for velocity and pressure space, while we can use power(...) for the d velocity component spaces of the same type. For efficiency, we use flat indexing strategies flatInterleaved and flatLexicographic to build a global continuous numbering of the basis functions. See the tutorial [Grids and Discrete Functions](../tutorials/grids-and-discretefunctions.md) for more details. Implementation -------------- We split the implementation into three parts, 1. the time derivative, 2. the stokes operator and 3. any external forces. For addressing the different components of the Taylor-Hood basis, we introduce the treepaths _v = Dune::Indices::_0 and _p = Dune::Indices::_1 for velocity and pressure, respectively. ### Problem framework We have an in-stationary problem consisting of a sequence of stationary equations for each timestep, thus we combine a [ProblemInstat](../reference/Problem.md#class-probleminstat) with a [ProblemStat](../reference/Problem.md#class-problemstat) . c++ using namespace AMDiS; ProblemStat prob{"stokes", grid, basis}; prob.initialize(INIT_ALL); ProblemInstat probInstat{"stokes", prob}; probInstat.initialize(INIT_UH_OLD);  For convenience, we use class template argument deduction here. ### Time derivative For simplicity of this example, we use a backward Euler time stepping and a linearization of the nonlinear advection term: c++ // define a constant fluid density double density = 1.0; Parameters::get("stokes->density", density); // a reference to 1/tau auto invTau = probInstat.invTau(); // <1/tau * u, v> auto opTime = makeOperator(tag::testvec_trialvec{}, density * invTau); prob.addMatrixOperator(opTime, _v, _v); // <1/tau * u^old, v> auto opTimeOld = makeOperator(tag::testvec{}, density * invTau * probInstat.oldSolution(_v)); prob.addVectorOperator(opTimeOld, _v); for (int i = 0; i < Grid::dimensionworld; ++i) { // <(u^old * nabla)u_i, v_i> auto opNonlin = makeOperator(tag::test_gradtrial{}, density * prob.solution(_v)); prob.addMatrixOperator(opNonlin, treepath(_v,i), treepath(_v,i)); }  ### Stokes operator The stokes part of the Navier-Stokes equation is a common pattern that emerges in several fluid equations. math \int_\Omega \nu\nabla\mathbf{u}:\nabla\mathbf{v} + p\,\nabla\cdot\mathbf{v} + q\,\nabla\cdot\mathbf{u}\,\text{d}\mathbf{x}  for$\mathbf{u},\mathbf{v}\in V$and$p,q\in Q$. c++ // define a constant fluid viscosity double viscosity = 1.0; Parameters::get("stokes->viscosity", viscosity); for (int i = 0; i < Grid::dimensionworld; ++i) { // auto opL = makeOperator(tag::gradtest_gradtrial{}, viscosity); prob.addMatrixOperator(opL, treepath(_v,i), treepath(_v,i)); } // auto opP = makeOperator(tag::divtestvec_trial{}, 1.0); prob.addMatrixOperator(opP, _v, _p); // auto opDiv = makeOperator(tag::test_divtrialvec{}, 1.0); prob.addMatrixOperator(opDiv, _p, _v);  where we have used the identity math \nabla\mathbf{u}:\nabla\mathbf{v} = \sum_i \nabla u_i\cdot\nabla v_i  ### External volume force The force may be implemented as a vector-valued function of the global coordinates or any other expression that leads to a local force vector at the quadrature points, e.g. c++ using WorldVector = typename Grid::template Codim<0>::Geometry::GlobalCoordinate; auto opForce = makeOperator(tag::testvec{}, [](WorldVector const& x) { return WorldVector{1.0, 0.0}; }); prob.addVectorOperator(opForce, _v);  Numerical example ----------------- We consider a lid-driven cavity problem in a square domain$\Omega$with$L_x=L_y=1\$ no external force, and with Dirichlet boundary conditions math \mathbf{g}(\mathbf{x}) = \begin{pmatrix}0 \\ v_1 x_1(1 - x_1)(1 - x_0)\end{pmatrix}  Thus, the boundary value is zero on top, right, and bottom boundary and parabolic on the left boundary. The boundary condition is implemented, by first setting boundary ids for parts of the grid boundary and second, setting the Dirichlet value: c++ double vel = 1.0; Parameters::get("stokes->boundary velocity", v1); // define boundary values auto g = [vel](WorldVector const& x) { return WorldVector{0.0, v1*x[1]*(1.0 - x[1])*(1.0 - x[0])}; }; // set boundary conditions for velocity prob.boundaryManager()->setBoxBoundary({1,1,1,1}); prob.addDirichletBC(1, _v, _v, g);  ### Simulation The time-stepping process with a stationary problem in each iteration can be started using an AdaptInstationary manager class: c++ // set initial conditions prob.solution(_v).interpolate(g); // start simulation AdaptInfo adaptInfo("adapt"); AdaptInstationary adapt("adapt", prob, adaptInfo, probInstat, adaptInfo); adapt.adapt();  Complete Example ---------------- The full source code of the Navier-Stokes example can be found in the repository at [examples/navier_stokes.cc](https://gitlab.mn.tu-dresden.de/amdis/amdis-core/blob/master/examples/navier_stokes.cc). Compile with bash cmake --build build --target navier_stokes.2d  and run with bash ./build-cmake/examples/navier_stokes.2d examples/init/navier_stokes.dat.2d  The flow field with density=1, viscosity=1 and boundary velocity=10 will look like \ No newline at end of file