From 4d9e9156a92082e9649c6fc6131edf2ad22dfa6f Mon Sep 17 00:00:00 2001 From: bangerth Date: Sat, 1 Mar 2008 13:45:53 +0000 Subject: [PATCH] Add David Neckel's compressible Euler flow tutorial program. git-svn-id: https://svn.dealii.org/trunk@15823 0785d39b-7218-0410-832d-ea1e28bc413d --- deal.II/examples/step-33/Makefile | 154 + deal.II/examples/step-33/doc/intro.dox | 152 + deal.II/examples/step-33/doc/results.dox | 134 + .../examples/step-33/doc/step-33.slide.gif | Bin 0 -> 4295977 bytes .../step-33/doc/step-33.slide_adapt.gif | Bin 0 -> 14070347 bytes deal.II/examples/step-33/slide.inp | 3560 +++++++++++++++++ deal.II/examples/step-33/step-33.cc | 2114 ++++++++++ 7 files changed, 6114 insertions(+) create mode 100644 deal.II/examples/step-33/Makefile create mode 100644 deal.II/examples/step-33/doc/intro.dox create mode 100644 deal.II/examples/step-33/doc/results.dox create mode 100644 deal.II/examples/step-33/doc/step-33.slide.gif create mode 100644 deal.II/examples/step-33/doc/step-33.slide_adapt.gif create mode 100644 deal.II/examples/step-33/slide.inp create mode 100644 deal.II/examples/step-33/step-33.cc diff --git a/deal.II/examples/step-33/Makefile b/deal.II/examples/step-33/Makefile new file mode 100644 index 0000000000..2102793621 --- /dev/null +++ b/deal.II/examples/step-33/Makefile @@ -0,0 +1,154 @@ +# $Id: Makefile 14008 2006-10-17 04:05:33Z bangerth $ + + +# For the small projects Makefile, you basically need to fill in only +# four fields. +# +# The first is the name of the application. It is assumed that the +# application name is the same as the base file name of the single C++ +# file from which the application is generated. +target = $(basename $(shell echo step-*.cc)) + +# The second field determines whether you want to run your program in +# debug or optimized mode. The latter is significantly faster, but no +# run-time checking of parameters and internal states is performed, so +# you should set this value to `on' while you develop your program, +# and to `off' when running production computations. +debug-mode = on + + +# As third field, we need to give the path to the top-level deal.II +# directory. You need to adjust this to your needs. Since this path is +# probably the most often needed one in the Makefile internals, it is +# designated by a single-character variable, since that can be +# reference using $D only, i.e. without the parentheses that are +# required for most other parameters, as e.g. in $(target). +D = ../../ + + +# The last field specifies the names of data and other files that +# shall be deleted when calling `make clean'. Object and backup files, +# executables and the like are removed anyway. Here, we give a list of +# files in the various output formats that deal.II supports. +clean-up-files = *gmv *gnuplot *gpl *eps *pov + + + + +# +# +# Usually, you will not need to change something beyond this point. +# +# +# The next statement tell the `make' program where to find the +# deal.II top level directory and to include the file with the global +# settings +include $D/common/Make.global_options + + +# Since the whole project consists of only one file, we need not +# consider difficult dependencies. We only have to declare the +# libraries which we want to link to the object file, and there need +# to be two sets of libraries: one for the debug mode version of the +# application and one for the optimized mode. Here we have selected +# the versions for 2d. Note that the order in which the libraries are +# given here is important and that your applications won't link +# properly if they are given in another order. +# +# You may need to augment the lists of libraries when compiling your +# program for other dimensions, or when using third party libraries +libs.g = $(lib-deal2-2d.g) \ + $(lib-lac.g) \ + $(lib-base.g) +libs.o = $(lib-deal2-2d.o) \ + $(lib-lac.o) \ + $(lib-base.o) + + +# We now use the variable defined above which switch between debug and +# optimized mode to select the set of libraries to link with. Included +# in the list of libraries is the name of the object file which we +# will produce from the single C++ file. Note that by default we use +# the extension .g.o for object files compiled in debug mode and .o for +# object files in optimized mode (or whatever the local default on your +# system is instead of .o). +ifeq ($(debug-mode),on) + libraries = $(target).g.$(OBJEXT) $(libs.g) +else + libraries = $(target).$(OBJEXT) $(libs.o) +endif + + +# Now comes the first production rule: how to link the single object +# file produced from the single C++ file into the executable. Since +# this is the first rule in the Makefile, it is the one `make' selects +# if you call it without arguments. +$(target) : $(libraries) + @echo ============================ Linking $@ + @$(CXX) -o $@$(EXEEXT) $^ $(LIBS) $(LDFLAGS) + + +# To make running the application somewhat independent of the actual +# program name, we usually declare a rule `run' which simply runs the +# program. You can then run it by typing `make run'. This is also +# useful if you want to call the executable with arguments which do +# not change frequently. You may then want to add them to the +# following rule: +run: $(target) + @echo ============================ Running $< + @./$(target)$(EXEEXT) + + +# As a last rule to the `make' program, we define what to do when +# cleaning up a directory. This usually involves deleting object files +# and other automatically created files such as the executable itself, +# backup files, and data files. Since the latter are not usually quite +# diverse, you needed to declare them at the top of this file. +clean: + -rm -f *.$(OBJEXT) *~ Makefile.dep $(target)$(EXEEXT) $(clean-up-files) + + +# Since we have not yet stated how to make an object file from a C++ +# file, we should do so now. Since the many flags passed to the +# compiler are usually not of much interest, we suppress the actual +# command line using the `at' sign in the first column of the rules +# and write the string indicating what we do instead. +./%.g.$(OBJEXT) : + @echo ==============debug========= $( $@ \ + || (rm -f $@ ; false) + @if test -s $@ ; then : else rm $@ ; fi + + +# To make the dependencies known to `make', we finally have to include +# them: +include Makefile.dep + + diff --git a/deal.II/examples/step-33/doc/intro.dox b/deal.II/examples/step-33/doc/intro.dox new file mode 100644 index 0000000000..e720513c32 --- /dev/null +++ b/deal.II/examples/step-33/doc/intro.dox @@ -0,0 +1,152 @@ +

Introduction

+ +This program was written for fun by David Neckels (NCAR) while working +at Sandia (on the Wyoming Express bus to and from Corrales each day). +The main purpose was to better understand Euler flow. +The code solves the basic Euler equations of gas dynamics, by using a +fully implicit Newton iteration (inspired by Sandia's Aria code). The +code may be configured by an input deck to run different simulations +on different meshes, with differing boundary conditions. + +The program also uses the Trilinos linear solvers (Aztec/Amesos) and +an automatic differentiation package, Sacado. + +

Euler flow

+ +The equations for a compressible, inviscid gas (the Euler equations) are +a basic system of conservation laws, in spatial dimension $d$, +@f[ +\partial_t \mathbf{w} + \nabla \cdot \mathbf{F}(\mathbf{w}) = \mathbf{0}, +@f] +with $\mathbf{w}=(\rho,\rho v_1,\dots,\rho v_d, E)^{\top}$ for $v_i$ equal to the +flow velocity in spatial direction $i$, $\rho$ equal to the fluid density, and +$E$ the energy of the gas. The flux matrix (or system of flux functions) +are defined such that the entire system of equations are +@f{eqnarray*} + \partial_t \rho + \sum_{s=1}^d \frac{\partial(\rho v_s)}{\partial x_s} &=& 0 \\ + \partial_t (\rho v_i) + \sum_{s=1}^d \frac{\partial(\rho v_i v_s + \delta_{is} p)}{\partial x_s} &=& 0, i=1,\dots,d \\ + \partial_t E + \sum_{s=1}^d \frac{\partial((E+p)v_s)}{\partial x_s} &=& 0 \\ + p = (\gamma -1)(E-\frac{1}{2} \rho |v|^2). +@f} + +For air, $\gamma=1.4$. + +

Discretization

+ +We choose a finite element space $V_h$, and integrate our conservation law against +our (vector) test function $\mathbf{v} \in V_h$. We integrate by parts and approximate the +boundary flux with a numerical flux $\mathbf{H}$, +@f{eqnarray*} +\int_{\Omega} (\partial_t \mathbf{w}, \mathbf{v}) + (\nabla \cdot \mathbf{F}(\mathbf{w}), \mathbf{v}) & \approx & \\ +\int_{\Omega} (\partial_t \mathbf{w}, \mathbf{v}) + (\mathbf{F}(\mathbf{w}), \nabla \mathbf{v}) + h^{\eta}(\nabla \mathbf{w} , \nabla \mathbf{v}) + \int_{\partial \Omega} (\mathbf{H}(\mathbf{w}^+, \mathbf{w}^-, \mathbf{n}), \mathbf{v}^+), +@f} +where $+$ is the interior trace of a function, and $-$ represents the outer trace. +The diffusion term $h^{\eta}(\nabla \mathbf{w} , \nabla \mathbf{v})$ is introduced strictly for stability, + where $h$ is the mesh size and $\eta$ is a parameter prescribing how much diffusion to add. +Depending on the boundary condition, we prescribe the outer trace $\mathbf{w}^-$, + + +For more information, please consult Ralf Hartmann's thesis TODO:Ref. + + +Our full discretization is thus +@f{eqnarray*} +R(\mathbf{W}_{n+1}) = \\ +\int_{\Omega} (\frac{\mathbf{w}_{n+1} - \mathbf{w}_n}{\delta t}, \mathbf{v}) + \int_{\Omega} (\mathbf{F}(\tilde{\mathbf{w}}), \mathbf{v}) + h^{\eta}(\nabla \mathbf{w} , \nabla \mathbf{v}) + \int_{\partial \Omega} (\mathbf{H}(\tilde{\mathbf{w}^+}), \mathbf{w}^-((\tilde{\mathbf{w}^+})), \mathbf{n}), \mathbf{v}) & = & 0 +@f} +where $\tilde{\mathbf{w}} = \theta \mathbf{w}_{n+1} + (1-\theta) \mathbf{w}_n$ for $0 \leq \theta \leq 1$ and +$\mathbf{w}_i = \sum_k \mathbf{W}_i^k \mathbf{\phi}_k$. + +We choose the Lax-Friedrich's flux, +$\mathbf{H}(\mathbf{a},\mathbf{b},\mathbf{n}) = \frac{1}{2}(\mathbf{F}(\mathbf{a})\cdot \mathbf{n} + \mathbf{F}(\mathbf{b})\cdot \mathbf{n} + \alpha (\mathbf{a} - \mathbf{b}))$. + +We solve the nonlinear system by a Newton iteration, i.e. by iterating +@f{eqnarray*} +\partial R(\mathbf{W}^k) \delta \mathbf{W} & = & - R(\mathbf{W}^{k}) \\ +\mathbf{W}^{k+1} &=& \mathbf{W}^k + \delta \mathbf{W}, +@f} +until $|R(\mathbf{W}^k)|$ (the residual) is sufficiently small. + +

Auto-Differentiation

+ +Since computing the Jacobian $\partial R$ is a terrible beast, we use an automatic differentiation package, +Sacado, to do this. Sacado is a C++ type that supports basic arithmetic operators and functions +such as sqrt, sin, cos, pow, etc. One declares a collection Sacado type and then declares +some of this collection as degrees of freedom. These variables are used in an algorithm, and as the variables +are used, their sensitivities with respect to these degrees of freedom are continuously updated. +One can imagine that for the full Jacobian, this could be prohibitively expensive. However, we do +not use the Sacado type for the entire computation, but only element by element. The author has +used this approach side by side with a hand coded Jacobian for the Incompressible Navier-Stokes problem +and found the Sacado approach to be just as fast as using a hand coded Jacobian. Since using the +auto-differentiation requires only that one code the Residual $R(\mathbf{W})$, ensuring code correctness +and maintaining code becomes tremendously more simple. We show a very simple Sacado example: + +@code +#include + +typedef Sacado::Fad::DFad fad_double; + +main() { + + fad_double a,b,c; + + a.diff(0,2); // Set a to be dof 0, in a 2-dof system. + + b.diff(1,2); // Set b to be dof 1, in a 2-dof system. + + a = 1; b = 2; + + c = 2*a+cos(a*b); + + double *derivs = c.fastAccessDx(0); // Access derivatives + + std::cout << "dc/da = " << derivs[0] << ", dc/db=" << derivs[1] << std::endl; + +} +@endcode + +It should be noted that Sacado provides more auto-differentation capabilities than the small subset +that I use in this program. However, if one understands the example above, they will understand +how I use Sacado in this Euler flow program. + +

Trilinos solvers

+The program uses either the Aztec iterative solvers, or the Amesos direct solver, as provided by +the Trilinos package. This package is inherently designed to be used in a parallel program, however, +it may be used in serial just as easily, as is done here. The Epetra package is the basic +vector/matrix library upon which the solvers are built. This very powerful package can be used +to describe the parallel distribution of a vector, and to define sparse matrices that operate +on these vectors. Please view the commented code for more details on how these solvers are used +within the example. + +

Adaptivity

+The example uses an ad-hoc refinement indicator that shows some usefullness in shock-type problems, and +in the downhill flow example included. We refine according to the squared gradient of the density. +Hanging nodes are handled by using computing the numerical flux across cells that are of differing +refinement levels. In this way, the example combines the continuous and DG methodologies. + +Further, we enforce a maximum number of refinement levels to keep refinement under check. It is the +author's experience that adaptivity in a time dependent problem, refinement can easily lead the simulation to +a screeching halt if care is not taken. The amount of refinement is +limited in the example by letting the user specify the +maximum level of refinement that will be present anywhere in the mesh. In this way, refinement +tends not to slow the simulation to a halt. This, of course, is purely a heuristic strategy, and +if the author's advisor heard about it, the author would likely be exiled forever from the finite + element error estimation community. + +

Input Deck

+ +We use an input file deck to drive the simulation. In this way, we can alter the boundary conditions +and other important properties of the simulation without having to recompile. For more information on +the format, look at the results section, where we describe an example file in more detail. + + + + diff --git a/deal.II/examples/step-33/doc/results.dox b/deal.II/examples/step-33/doc/results.dox new file mode 100644 index 0000000000..4e4ad15b43 --- /dev/null +++ b/deal.II/examples/step-33/doc/results.dox @@ -0,0 +1,134 @@ +

Results

+ +We run the problem with the mesh slide.inp and the following input deck: +@verbatim +# Listing of Parameters +# --------------------- + +# The input grid +set mesh=slide.inp + +# Stabilization parameter +set diffusion power = 2.0 + +# Scaled value for gravity. Positive means gravity points down. +set gravity = 1.0 + +# Boundary conditions +# We may specify boundary conditions for up to MAX_BD boundaries. +# Your .inp file should have these boundaries designated. +# +subsection boundary_1 + set no penetration = true # reflective boundary condition +end + +subsection boundary_2 +# outflow boundary + #set w_3 = pressure + #set w_3 value = 1.5 - y +end + +subsection boundary_3 + set no penetration = true # reflective + #set w_3 = pressure + #set w_3 value = 1.0 +end +subsection boundary_4 + set no penetration = true #reflective +end + +# Initial Conditions +# We set the initial conditions of the conservative variables. These lines +# are passed to the expression parsing function. You should use x,y,z for +# the coordinate variables. +# +subsection initial condition + set w_0 value = 0 + set w_1 value = 0 + set w_2 value = 10*(x<-0.7)*(y> 0.3)*(y< 0.45) + (1-(x<-0.7)*(y> 0.3)*(y< 0.45))*1.0 + set w_3 value = (1.5-(1.0*1.0*y))/0.4 +end + +# Time stepping control +# +subsection time stepping + set final time = 10.0 # simulation end time + set time step = 0.02 # simulation time step +end + +subsection linear solver + set output = quiet + set method = gmres + set ilut fill = 1.5 + set ilut drop tolerance = 1e-6 + set ilut absolute tolerance = 1e-6 + set ilut relative tolerance = 1.0 +end + +# Output frequency. +# You may wish to set this > time step if you dont want output at every step +subsection output + set step = 0.01 +end + +# Refinement control +subsection refinement + set refinement = shock # none only other option + set shock value = 1.5 + set shock levels = 1 # how many levels of refinement to allow +end + +# Flux parameters +subsection flux + set stab = mesh + #set stab value = 1.0 +end +@endverbatim + +When we run the program, we get the following kind of output: +@code +T=3.14, Number of active cells: 2617 + Number of degrees of freedom: 11184 +NonLin Res: Lin Iter Lin Res +______________________________________ +* 1.684e-02 0007 1.84e-13 +* 4.414e-05 0006 3.82e-15 +* 1.750e-09 0006 1.56e-19 +* 6.831e-16 0000 1.56e-19 +max_df:124 +T=3.16, Number of active cells: 2626 + Number of degrees of freedom: 11220 +NonLin Res: Lin Iter Lin Res +______________________________________ +* 1.673e-02 0007 9.64e-14 +* 4.230e-05 0006 3.83e-15 +* 8.748e-10 0006 7.67e-20 +* 6.804e-16 0000 7.67e-20 +max_df:124 +T=3.18, Number of active cells: 2644 + Number of degrees of freedom: 11296 +NonLin Res: Lin Iter Lin Res +______________________________________ +* 1.674e-02 0007 9.29e-14 +* 3.306e-05 0007 8.07e-17 +* 4.660e-10 0006 4.05e-20 +* 6.898e-16 0000 4.05e-20 +max_df:128 +T=3.2, Number of active cells: 2647 + Number of degrees of freedom: 11312 +NonLin Res: Lin Iter Lin Res +______________________________________ +... +@endcode + +This output reports the progress of the Newton iterations and the time stepping. + +@image html step-33.slide.gif + +As we see, the heavy mass of fluid is drawn down the slope by gravity, where it collides +with the ski lodge and is flung into the air! Hopefully everyone escapes! + +@image html step-33.slide_adapt.gif + +The adaptivity follows and preceeds the flow pattern, based on this heuristic refinement scheme. + diff --git a/deal.II/examples/step-33/doc/step-33.slide.gif b/deal.II/examples/step-33/doc/step-33.slide.gif new file mode 100644 index 0000000000000000000000000000000000000000..1b35b6b886a0a69114b5ffe4b0c15f9177c022a3 GIT binary patch literal 4295977 zcmZ_UbyU>B-#7g47AzaE^pY;!NP~b%H@Kvf(h`D#f&$hq-6h>6A|XqPluI`(DJ2bp zpe~4tg}1-!y6d^mdFHPr$x#(cvA9(+&I)`QQ^(a6rx%FN2g%-+Y$?uLndpoNXQg^jPJy@!>Z zuf6q6TRUGnYY!`%yH<7q)(#=I4uQ7zAvQMm%v>#RnqKp=aJXmec*p#zubHd2rR!ZY z*8p=D3uhM&)<6h-2`$}AB&v)K`2t4s5(aOiqicf0U!fVQ0NFx7u8p2C4MsBpgAc=L zM&O7s2yH(U`VLO}7RodPVVj3SHbKZO0CfO|Zh)ct0O$k&{s4$$5b_Tl^b}xy4=|ko z%)bDxUtrE}fY>k~wg6Th1_a&%l7GMgKS0Vm5anMqR~mqTG~n**^In;14po4|2w+SB z>;D3-?m=vS(iwb$`z?UIe?sp6LzXzBk^8W`eTs!oXo9b(c9}d z{bl1T;{kWK@y_eEZXI4gVkYpk`CL^7ZQ2|(?k6PcFMZNJre+2~IuUA`maIEc&-%+m 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