From 941791c7517dc145834b52227f558cb8915b1bf9 Mon Sep 17 00:00:00 2001 From: Giuseppe Orlando Date: Sun, 1 May 2022 20:24:59 +0200 Subject: [PATCH] Solver for incompressible NS based on Discontinuous Galerkin TR-BDF2 implemented with MatrixFree --- NavierStokes_TRBDF2_DG/CMakeLists.txt | 54 + NavierStokes_TRBDF2_DG/README.md | 104 + NavierStokes_TRBDF2_DG/doc/adaptive_mesh.png | Bin 0 -> 39096 bytes NavierStokes_TRBDF2_DG/doc/author | 1 + NavierStokes_TRBDF2_DG/doc/builds-on | 3 + NavierStokes_TRBDF2_DG/doc/dependencies | 2 + NavierStokes_TRBDF2_DG/doc/drag.png | Bin 0 -> 67566 bytes NavierStokes_TRBDF2_DG/doc/entry-name | 1 + NavierStokes_TRBDF2_DG/doc/lift.png | Bin 0 -> 41456 bytes NavierStokes_TRBDF2_DG/doc/tooltip | 1 + .../doc/velocity_magnitude.png | Bin 0 -> 92506 bytes NavierStokes_TRBDF2_DG/equation_data.h | 83 + .../navier_stokes_TRBDF2_DG.cc | 3045 +++++++++++++++++ NavierStokes_TRBDF2_DG/parameter-file.prm | 39 + NavierStokes_TRBDF2_DG/runtime_parameters.h | 182 + 15 files changed, 3515 insertions(+) create mode 100644 NavierStokes_TRBDF2_DG/CMakeLists.txt create mode 100644 NavierStokes_TRBDF2_DG/README.md create mode 100644 NavierStokes_TRBDF2_DG/doc/adaptive_mesh.png create mode 100644 NavierStokes_TRBDF2_DG/doc/author create mode 100644 NavierStokes_TRBDF2_DG/doc/builds-on create mode 100644 NavierStokes_TRBDF2_DG/doc/dependencies create mode 100644 NavierStokes_TRBDF2_DG/doc/drag.png create mode 100644 NavierStokes_TRBDF2_DG/doc/entry-name create mode 100644 NavierStokes_TRBDF2_DG/doc/lift.png create mode 100644 NavierStokes_TRBDF2_DG/doc/tooltip create mode 100644 NavierStokes_TRBDF2_DG/doc/velocity_magnitude.png create mode 100644 NavierStokes_TRBDF2_DG/equation_data.h create mode 100644 NavierStokes_TRBDF2_DG/navier_stokes_TRBDF2_DG.cc create mode 100644 NavierStokes_TRBDF2_DG/parameter-file.prm create mode 100644 NavierStokes_TRBDF2_DG/runtime_parameters.h diff --git a/NavierStokes_TRBDF2_DG/CMakeLists.txt b/NavierStokes_TRBDF2_DG/CMakeLists.txt new file mode 100644 index 0000000..fcb74dd --- /dev/null +++ b/NavierStokes_TRBDF2_DG/CMakeLists.txt @@ -0,0 +1,54 @@ +## +# CMake script for the NavierStokes_TRBDF2_DG program: +## + +# Set the name of the project and target: +SET(TARGET "NS_TRBDF2_DG") + +# Declare all source files the target consists of. Here, this is only +# the one .cc file, but as you expand your project you may wish +# to add other source files as well. If your project becomes much larger, +# you may want to either replace the following statement by something like +# FILE(GLOB_RECURSE TARGET_SRC "source/*.cc") +# FILE(GLOB_RECURSE TARGET_INC "include/*.h") +# SET(TARGET_SRC ${TARGET_SRC} ${TARGET_INC}) +# or switch altogether to the large project CMakeLists.txt file discussed +# in the "CMake in user projects" page accessible from the "User info" +# page of the documentation. +SET(TARGET_SRC + navier_stokes_TRBDF2_DG.cc + ) + +# Usually, you will not need to modify anything beyond this point... + +CMAKE_MINIMUM_REQUIRED(VERSION 2.8.12) + +FIND_PACKAGE(deal.II 9.2.0 + HINTS ${deal.II_DIR} ${DEAL_II_DIR} ../ ../../ $ENV{DEAL_II_DIR} + ) +IF(NOT ${deal.II_FOUND}) + MESSAGE(FATAL_ERROR "\n" + "*** Could not locate a (sufficiently recent) version of deal.II. ***\n\n" + "You may want to either pass a flag -DDEAL_II_DIR=/path/to/deal.II to cmake\n" + "or set an environment variable \"DEAL_II_DIR\" that contains this path." + ) +ENDIF() + +# +# Are all dependencies fulfilled? +# +IF(NOT DEAL_II_WITH_MPI OR NOT DEAL_II_WITH_P4EST) # keep in one line + MESSAGE(FATAL_ERROR " +Error! This tutorial requires a deal.II library that was configured with the following options: + DEAL_II_WITH_MPI = ON + DEAL_II_WITH_P4EST = ON +However, the deal.II library found at ${DEAL_II_PATH} was configured with these options + DEAL_II_WITH_MPI = ${DEAL_II_WITH_MPI} + DEAL_II_WITH_P4EST = ${DEAL_II_WITH_P4EST} +which conflict with the requirements." + ) +ENDIF() + +DEAL_II_INITIALIZE_CACHED_VARIABLES() +PROJECT(${TARGET}) +DEAL_II_INVOKE_AUTOPILOT() diff --git a/NavierStokes_TRBDF2_DG/README.md b/NavierStokes_TRBDF2_DG/README.md new file mode 100644 index 0000000..e490b57 --- /dev/null +++ b/NavierStokes_TRBDF2_DG/README.md @@ -0,0 +1,104 @@ +# TRBDF2-DG projection solver for the incompressible Navier-Stokes equations + +## Compiling and Running +To generate a makefile for this code using CMake, type the following command into the terminal from the main directory: + + cmake . -DDEAL_II_DIR=/path/to/deal.II + +To compile the code in release mode use: + + make release + +This command will create the executable, NS_TRBDF2_DG. + +To run the code on N processors type the following command into the terminal from the main directory, + + mpirun -np N ./NS_TRBDF2_DG + +The output of the code will be in .vtu format and be written to disk in parallel. The results can be viewed using ParaView. A parameter file called parameter-file.prm has to be present in +the same folder of the executable, following the same structure employed in step-35. Two extra fields are present: saving_directory with the name of the folder where the results should be saved (which has therefore to be created before launching the program) and refinement_iterations that specifies how often the remeshing procedure has to be performed. + + +### The Navier-Stokes equations and the time discretization strategy ### + +In this section, we briefly describe the problem and the approach employed. A detailed explanation of the numerical scheme is reported in [1]. We consider the classical unsteady incompressible Navier-Stokes equations, written in non-dimensional form as: + +$$ +\begin{align*} +\frac{\partial \mathbf{u}}{\partial t} + \nabla\cdot\left(\mathbf{u} \otimes\mathbf{u}\right) + \nabla p &= \frac{1}{Re}\Delta\mathbf{u} + \mathbf{f} \\ +\nabla\cdot\mathbf{u} &= 0, +\end{align*} +$$ + +where $Re$ denotes the Reynolds number. In the case of projection methods, difficulties arise in choosing the boundary conditions to be imposed for the Poisson equation which is to be solved at each time step to compute the pressure. An alternative that allows to avoid or reduce some of these problems is the so-called artificial compressibility formulation. In this formulation, the incompressibility constraint is relaxed and a time evolution equation for the pressure is introduced, which is characterized by an artificial sound speed $ c, $ so as to obtain: + +$$ +\begin{align*} +\frac{\partial\mathbf{u}}{\partial t} + \nabla\cdot\left(\mathbf{u}\otimes\mathbf{u}\right) + \nabla p &= \frac{1}{Re}\Delta\mathbf{u} + \mathbf{f} \\ +\frac{1}{c^2}\frac{\partial p}{\partial t} + \nabla\cdot\mathbf{u} &= 0. +\end{align*} +$$ + +For the sake of simplicity, we shall only consider $\mathbf{f} = \mathbf{0}$. The numerical scheme is an extension of the projection method introduced in [2] based on the TR-BDF2 method. For a generic time-dependent problem $\bm{u}^{'} = \mathcal{N}(\bm{u})$, the TR-BDF2 method can be described in terms of two stages as follows: + +$$ +\begin{align*} +\frac{\bm{u}^{n+\gamma} - \bm{u}^{n}}{\gamma\Delta t} &= \frac{1}{2}\mathcal{N}\left(\bm{u}^{n+\gamma}\right) + \frac{1}{2}\mathcal{N}\left(\bm{u}^{n}\right) \\ +\frac{\bm{u}^{n+1} - \bm{u}^{n + \gamma}}{\left(1 - \gamma\right)\Delta t} &= \frac{1}{2 - \gamma}\mathcal{N}\left(\bm{u}^{n+1}\right) + \frac{1 - \gamma}{2\left(2 - \gamma\right)}\mathcal{N}\left(\bm{u}^{n+\gamma}\right) + \frac{1 - \gamma}{2\left(2 - \gamma\right)}\mathcal{N}\left(\bm{u}^{n}\right). +\end{align*} +$$ + +Following then the projection approach described in [2], the momentum predictor equation for the first stage reads: + +$$ +\begin{eqnarray} +&&\frac{\mathbf{u}^{n+\gamma,\*} - \mathbf{u}^{n}}{\gamma\Delta t} - \frac{1}{2Re}\Delta\mathbf{u}^{n+\gamma,\*} + \frac{1}{2}\nabla\cdot\left(\mathbf{u}^{n+\gamma,\*}\otimes\mathbf{u}^{n+\frac{\gamma}{2}}\right) = \nonumber \\ +&&\frac{1}{2Re}\Delta\mathbf{u}^{n} - \frac{1}{2}\nabla\cdot\left(\mathbf{u}^{n}\otimes\mathbf{u}^{n+\frac{\gamma}{2}}\right) - \nabla p^n \nonumber \\ +&&\mathbf{u}^{n+\gamma,\*}\rvert_{\partial\Omega} = \mathbf{u}_D^{n+\gamma}. \nonumber +\end{eqnarray} +$$ + +Notice that, in order to avoid solving a nonlinear system at each time step, an approximation is introduced in the nonlinear momentum advection term, so that $\mathbf{u}^{n + \frac{\gamma}{2}}$ is defined by extrapolation as + +$$ +\mathbf{u}^{n + \frac{\gamma}{2}} = \left(1 + \frac{\gamma}{2\left(1-\gamma\right)}\right)\mathbf{u}^{n} - \frac{\gamma}{2\left(1-\gamma\right)}\mathbf{u}^{n-1}. +$$ + +For what concerns the pressure, we introduce the intermediate update +$\mathbf{u}^{n+\gamma,\*\*} = \mathbf{u}^{n+\gamma,\*} + \gamma\Delta t\nabla p^{n}$, and we solve the following Helmholtz equation + +$$ +\frac{1}{c^2}\frac{p^{n+\gamma}}{\gamma^2\Delta t^2} -\Delta p^{n+\gamma} = +- \frac{1}{\gamma\Delta t} \nabla\cdot\mathbf{u}^{n+\gamma,\*\*} + \frac{1}{c^2}\frac{p^{n }}{\gamma^2\Delta t^2} +$$ + +and, finally, we set $\mathbf{u}^{n+\gamma} = \mathbf{u}^{n+\gamma,\*\*} - \gamma\Delta t\nabla p^{n+\gamma}$. +The second stage of the TR-BDF2 scheme is performed in a similar manner (see [1] for all the details). + +### Some implementation details ### + +A matrix-free approach was employed like for step-37 or step-50. Another feature of the library which it is possible to employ during the numerical simulations is the mesh adaptation capability. On each element $K$ we define the quantity + +$$ +\eta_K = \text{diam}(K)^2\left\|\nabla \times \mathbf{u}\right\|^2_K +$$ + +that acts as local refinement indicator. The preconditioned conjugate gradient method implemented in the function SolverCG was employed to solve the Helmholtz equations, whereas, for the momentum equations, the GMRES solver +implemented in the function SolverGMRES was used. +A Jacobi preconditioner is used for the two momentum predictors, whereas a Geometric Multigrid preconditioner is employed for the Helmholtz equations (see step-37). + +#### Test case #### + +We test the code with a classical benchmark case, namely the flow past a cylinder in 2D at $Re = 100$ (see [1] for all the details). The image shows the contour plot of the velocity magnitude at $ t = T_{f} = 400 $. The evolution of the lift and drag coefficients from $ t = 385 $ to $ t = T_{f} $ are also reported and the expected periodic behaviour is retrieved. + +![contour](./doc/velocity_magnitude.png) + +![drag](./doc/drag.png) ![lift](./doc/lift.png) +![adaptive grid](./doc/adaptive_mesh.png) + +## References ## + +[1] G. Orlando, A. Della Rocca, P.F. Barbante, L. Bonaventura, and +N. Parolini. An efficient and accurate implicit DG solver for the incompressible Navier-Stokes equations. International Journal for Numerical Methods in Fluids, 2022. DOI: 10.1002/FLD.5098 + +[2] A. Della Rocca. Large-Eddy Simulations of Turbulent Reacting Flows with Industrial Applications. PhD thesis. 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